Physics-informed reduced-order modelling with equivariant spectral submanifolds

# Physics-informed reduced-order modelling with equivariant spectral submanifolds
Source: [https://arxiv.org/html/2608.04239](https://arxiv.org/html/2608.04239)
Georg Maierhofer Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK gam37@cam\.ac\.uk

###### Abstract

Spectral submanifold \(SSM\) reduction has emerged as a mathematically principled route to reliable nonlinear reduced\-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition \(DMD\)\. The computation of SSMs, however, remains computationally expensive, particularly for high\-dimensional systems\. In this work, we introduce equivariant spectral submanifold \(eSSM\) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full\-order model into the reduction process\. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions\. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness\. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science\.

#### Keywords:

equivariance, spectral submanifolds, model reduction, nonlinear dynamics, data\-driven modelling

#### MSC codes:

37D10, 37C81, 37M21

## 1Introduction

Reduced order models are of paramount importance in many scientific disciplines facilitating the simulation of the main features of complex dynamics in a computationally efficient manner\[[6](https://arxiv.org/html/2608.04239#bib.bib56)\]\. There are a range of techniques that can be applied for this task, including linear methods such as dynamic mode decomposition \(DMD\)\[[51](https://arxiv.org/html/2608.04239#bib.bib47),[54](https://arxiv.org/html/2608.04239#bib.bib48),[35](https://arxiv.org/html/2608.04239#bib.bib53)\], as well as nonlinear methods such as the Koopman operator framework\[[9](https://arxiv.org/html/2608.04239#bib.bib14),[18](https://arxiv.org/html/2608.04239#bib.bib15)\]and spectral submanifold \(SSM\) reduction\[[28](https://arxiv.org/html/2608.04239#bib.bib41),[27](https://arxiv.org/html/2608.04239#bib.bib45)\]\. Such reduced\-order methods are particularly useful in the context of data\-driven modelling, where they can provide a principled and interpretable framework for modelling and predicting complex dynamics often outperforming other types of data\-driven models in terms of predictive accuracy and generalisation\[[47](https://arxiv.org/html/2608.04239#bib.bib32),[55](https://arxiv.org/html/2608.04239#bib.bib13),[57](https://arxiv.org/html/2608.04239#bib.bib31)\]\.

In particular, SSMs have recently emerged as a powerful tool for model reduction of nonlinear dynamical systems\. SSMs are invariant manifolds that serve as the smoothest nonlinear continuation of spectral subspaces of the linearised system around a fixed point\. As such, SSMs provide a natural and rigorous framework for reduced\-order modelling that incorporates the nonlinear aspect of the dynamics without having to resort to linearisation or choose ad\-hoc observables\. The existence of a unique SSM was first proved by Haller and Ponsioen\[[28](https://arxiv.org/html/2608.04239#bib.bib41)\]using the parametrisation method of Cabré et al\.\[[12](https://arxiv.org/html/2608.04239#bib.bib38),[13](https://arxiv.org/html/2608.04239#bib.bib39),[14](https://arxiv.org/html/2608.04239#bib.bib40)\]in the case of stable spectral subspaces and later extended to mixed internal stability types in\[[27](https://arxiv.org/html/2608.04239#bib.bib45)\]\. Since then, there have been several successful developments of capable and versatile toolboxes for the efficient computation of SSMs and their associated reduced dynamics\. In particular, SSMTool\[[32](https://arxiv.org/html/2608.04239#bib.bib44),[33](https://arxiv.org/html/2608.04239#bib.bib42)\]is a powerful Matlab toolbox for the computation of SSMs in mechanical systems, while the data\-driven SSMLearn reduction method introduced by\[[15](https://arxiv.org/html/2608.04239#bib.bib1),[3](https://arxiv.org/html/2608.04239#bib.bib37)\]allows the discovery of SSM reduced models directly from data\. However, several challenges remain including the cost of the optimisation problems associated with finding parametrisations of the spectral submanifold and the reduced dynamics which grows with reduced order dimension and the polynomial degree in the reduction algorithm, as well as the incorporation of physical symmetries in the reduced models to faithfully reflect underlying physical principles\.

The incorporation of such physical symmetries has been a long\-studied problem in the context of numerical methods\[[43](https://arxiv.org/html/2608.04239#bib.bib36),[25](https://arxiv.org/html/2608.04239#bib.bib35),[42](https://arxiv.org/html/2608.04239#bib.bib12),[8](https://arxiv.org/html/2608.04239#bib.bib34),[22](https://arxiv.org/html/2608.04239#bib.bib33),[20](https://arxiv.org/html/2608.04239#bib.bib11)\]and has also seen increasing attention in the context of data\-driven modelling with the idea of creating more physically consistent and robust models learned from data\. In particular, in the context of DMD recent work by\[[5](https://arxiv.org/html/2608.04239#bib.bib55)\]has introduced a physics\-informed DMD \(piDMD\) method that incorporates symmetries into the DMD framework\. In symbolic regression\[[10](https://arxiv.org/html/2608.04239#bib.bib54),[48](https://arxiv.org/html/2608.04239#bib.bib46)\], symmetries and conservation laws are treated in\[[56](https://arxiv.org/html/2608.04239#bib.bib58)\]and, in the context of Koopman operator learning, this has been explored in recent work by\[[49](https://arxiv.org/html/2608.04239#bib.bib17),[29](https://arxiv.org/html/2608.04239#bib.bib16)\]\.

Notably, structure\-preserving reduced\-order modelling remains largely unexplored in the context of SSMs\. A first step in this direction is the recent work of\[[34](https://arxiv.org/html/2608.04239#bib.bib52)\], who exploited the shift\-reflect symmetry of pipe flow by restricting the dynamics to a symmetry\-invariant subspace, in which the edge state becomes amenable to SSM\-based reduction\. In that approach, however, the symmetry serves only to constrain the ambient dynamics prior to reduction: the SSM parametrisation and the reduced dynamics are still computed without reference to the group action, so the reduction step itself does not benefit computationally from the symmetry\.

In this work we fill this gap by showing how linear symmetries of the full\-order model can be incorporated directly into the SSM reduction process\. We prove that SSMs of equivariant systems are themselves equivariant submanifolds, and that suitably chosen charts and the associated reduced dynamics inherit induced actions of the symmetry group\. Building on these results, we develop an equivariant SSM reduction algorithm \(eSSM\) that constrains both the manifold parametrisation and the reduced dynamics to symmetry\-adapted coefficient spaces\. This yields significantly faster computations as well as higher model robustness, since the learned model cannot drift away from the symmetric dynamics of the full system\.

The remainder of this manuscript is structured as follows\. We begin in §[2](https://arxiv.org/html/2608.04239#S2)by reviewing the basic theory of SSMs before developing their equivariance properties\. In §[3](https://arxiv.org/html/2608.04239#S3)we then discuss the computation of SSMs and the associated reduced dynamics as well as associated equivariance properties, before introducing our novel eSSM reduction algorithm in §[4](https://arxiv.org/html/2608.04239#S4)\. We demonstrate the advantages of this approach on several benchmark problems in §[5](https://arxiv.org/html/2608.04239#S5), including a test problem from the Common Task Framework for Science\. Finally, concluding remarks are provided in §[6](https://arxiv.org/html/2608.04239#S6)\.

The code associated with this manuscript is publicly available as an installable Python packageeSSMat[https://github\.com/GeorgAUT/eSSM](https://github.com/GeorgAUT/eSSM)\.

## 2Spectral submanifolds and equivariance

In this work we consider the following general form of nonlinear dynamical systems around a given fixed point𝐱=𝟎\\mathbf\{x\}=\\mathbf\{0\}:

𝐱˙=𝐀𝐱\+𝐟​\(𝐱\),𝐱∈ℝn,\\displaystyle\\dot\{\\mathbf\{x\}\}=\\mathbf\{A\}\\mathbf\{x\}\+\\mathbf\{f\}\(\\mathbf\{x\}\),\\quad\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\},\(1\)where𝐀\\mathbf\{A\}is a linear operator \(the linearisation of the system at𝐱=𝟎\\mathbf\{x\}=\\mathbf\{0\}\) and𝐟=𝒪​\(‖𝐱‖2\)\\mathbf\{f\}=\\mathcal\{O\}\(\\\|\\mathbf\{x\}\\\|^\{2\}\)is a smooth nonlinear function𝐟∈C∞​\(ℝn,ℝn\)\\mathbf\{f\}\\in C^\{\\infty\}\(\\mathbb\{R\}^\{n\},\\mathbb\{R\}^\{n\}\)\. We assume that its fixed point at𝐱=𝟎\\mathbf\{x\}=\\mathbf\{0\}is hyperbolic, i\.e\. the spectrumSpect⁡\(𝐀\)\\operatorname\{Spect\}\(\\mathbf\{A\}\)does not intersect the imaginary axis,0∉Re​\[Spect⁡\(𝐀\)\]0\\notin\\mathrm\{Re\}\\left\[\\operatorname\{Spect\}\(\\mathbf\{A\}\)\\right\]\.

Spectral submanifolds are a nonlinear extension of the concept of spectral subspaces of the linearised system\. Let us denote the eigenvalues of𝐀\\mathbf\{A\}byλj=αj\+i​ωj,j=1,…,n,\\lambda\_\{j\}=\\alpha\_\{j\}\+i\\omega\_\{j\},j=1,\\ldots,n,ordered by

Re​λ1≤Re​λ2≤…≤Re​λn\.\\displaystyle\\mathrm\{Re\}\\lambda\_\{1\}\\leq\\mathrm\{Re\}\\lambda\_\{2\}\\leq\\ldots\\leq\\mathrm\{Re\}\\lambda\_\{n\}\.Since𝐀\\mathbf\{A\}is a real matrix, its eigenvalues are either real or come in complex conjugate pairs, and we denote the corresponding eigenvectors by𝐯j\\mathbf\{v\}\_\{j\}\.

###### Definition 2\.1\.

The real modal eigenspacesEjE\_\{j\}are defined as the real span of the eigenvectors associated withλj\\lambda\_\{j\}\. In particular, ifλj\\lambda\_\{j\}is a real eigenvalue, thenEj=spanℝ⁡\{𝐯j\}E\_\{j\}=\\operatorname\{span\}\_\{\\mathbb\{R\}\}\\\{\\mathbf\{v\}\_\{j\}\\\}, while ifλj\\lambda\_\{j\}is a complex eigenvalue, thenEj=spanℝ⁡\{𝐯j,𝐯¯j\}E\_\{j\}=\\operatorname\{span\}\_\{\\mathbb\{R\}\}\\\{\\mathbf\{v\}\_\{j\},\\overline\{\\mathbf\{v\}\}\_\{j\}\\\}\.

The modal eigenspacesEjE\_\{j\}are invariant under the linear flow generated by𝐀\\mathbf\{A\}\. A direct sum of modal eigenspaces,E=⨁j∈JEjE=\\bigoplus\_\{j\\in J\}E\_\{j\}, is called a spectral subspace and is also invariant under the flow generated by𝐀\\mathbf\{A\}\. For a given spectral subspaceEE, we can ask whether there exists an invariant manifold of the full nonlinear system \([1](https://arxiv.org/html/2608.04239#S2.E1)\) that is tangent toEEat the fixed point𝐱=𝟎\\mathbf\{x\}=\\mathbf\{0\}and that captures the nonlinear dynamics associated with the modes inEE\. Based purely on this tangency requirement there are infinitely many such invariant manifolds, however,\[[27](https://arxiv.org/html/2608.04239#bib.bib45),[28](https://arxiv.org/html/2608.04239#bib.bib41)\]proved that under certain non\-resonance conditions, there exists a unique invariant manifold𝒲​\(E\)\\mathcal\{W\}\(E\)that is as smooth as the full system and that is tangent toEEat the fixed point\. This “smoothest” invariant manifold is called the spectral submanifold \(SSM\) associated with the spectral subspaceEE\. The existence of this SSM is guaranteed under certain non\-resonance conditions on the eigenvalues of𝐀\\mathbf\{A\}\. These resonance conditions are common practice in the relevant literature and generally hold for generic parameter configurations of typical dissipative systems, however, they may fail for simple toy examples with non\-generic parameter choices\.

###### Definition 2\.2\(Global non\-resonance condition\)\.

We say that𝐀\\mathbf\{A\}satisfies the global non\-resonance condition if, with the exception of possible1:11\{:\}1resonances created by repeated eigenvalues, for anyj∈\{1,…,n\}j\\in\\\{1,\\dots,n\\\}and any𝐦=\(m1,…,mn\)∈ℕn\\mathbf\{m\}=\(m\_\{1\},\\dots,m\_\{n\}\)\\in\\mathbb\{N\}^\{n\}with\|𝐦\|:=∑k=1nmk≥2\|\\mathbf\{m\}\|:=\\sum\_\{k=1\}^\{n\}m\_\{k\}\\geq 2we have

λj≠∑k=1nmk​λk\.\\displaystyle\\lambda\_\{j\}\\;\\neq\\;\\sum\_\{k=1\}^\{n\}m\_\{k\}\\,\\lambda\_\{k\}\.

Under this global non\-resonance condition, we then have the following existence result proved by\[[27](https://arxiv.org/html/2608.04239#bib.bib45)\]adapted to our setting\.

###### Theorem 2\.3\(Theorem 1 in\[[27](https://arxiv.org/html/2608.04239#bib.bib45)\]\)\.

Assume that𝐀\\mathbf\{A\}is semi\-simple and satisfies the global non\-resonance condition of Definition[2\.2](https://arxiv.org/html/2608.04239#S2.Thmtheorem2)and letEEbe a spectral subspace of𝐀\\mathbf\{A\}\. Then there is a unique invariant manifold𝒲​\(E\)\\mathcal\{W\}\(E\)of classC∞C^\{\\infty\}and dimensiondim⁡E\\operatorname\{dim\}Ethat is tangent toEEat𝐱=𝟎\\mathbf\{x\}=\\mathbf\{0\}\. Moreover,𝒲​\(E\)\\mathcal\{W\}\(E\)admits, near the origin, a local representation as a graph overEEwith integer\-powered Taylor expansion\.

### 2\.1Linear symmetries and equivariance

For completeness let us recall the definition of equivariance for a dynamical system\. We will, throughout this work, focus on linear symmetries that fix the origin, i\.e\.S​𝟎=𝟎S\\mathbf\{0\}=\\mathbf\{0\}for allS∈𝒢S\\in\\mathcal\{G\}\.

###### Definition 2\.5\.

The dynamical system \([1](https://arxiv.org/html/2608.04239#S2.E1)\) is said to be equivariant with respect to a linear symmetry group𝒢⊂G​L​\(n,ℝ\)\\mathcal\{G\}\\subset GL\(n,\\mathbb\{R\}\)acting onℝn\\mathbb\{R\}^\{n\}if for allS∈𝒢S\\in\\mathcal\{G\}and𝐱∈ℝn\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\}we have

S​\(A​𝐱\+𝐟​\(𝐱\)\)=A​S​𝐱\+𝐟​\(S​𝐱\)\.\\displaystyle S\\left\(A\\mathbf\{x\}\+\\mathbf\{f\}\(\\mathbf\{x\}\)\\right\)=AS\\mathbf\{x\}\+\\mathbf\{f\}\(S\\mathbf\{x\}\)\.\(2\)

Such linear symmetries naturally arise in many applications, for example as a result of spatial symmetries in the underlying domain \(e\.g\. evolution of waves on periodic or spherical domains\) and intrinsic symmetries of the dynamical equations themselves, such as permutation symmetries of coupled identical units\. The following is a simple guiding example of such a system with a linear symmetry that we will use as a running example throughout the paper, further examples are provided in §[5](https://arxiv.org/html/2608.04239#S5)\.

###### Example 2\.6\.

Our first example is a damped oscillator chain withℓ\\ellmasses and an additional nonlinear spring attached to the leftmost mass, cf\. Figure[1](https://arxiv.org/html/2608.04239#S2.F1), similarly to the basic setup provided in\[[4](https://arxiv.org/html/2608.04239#bib.bib59), §4\.1\]\. The equations of motion are

𝐌​𝐪¨\+𝐂​𝐪˙\+𝐊𝐪\+𝐟nl​\(𝐪,𝐪˙\)=𝟎,\\displaystyle\\mathbf\{M\}\\ddot\{\\mathbf\{q\}\}\+\\mathbf\{C\}\\dot\{\\mathbf\{q\}\}\+\\mathbf\{K\}\\mathbf\{q\}\+\\mathbf\{f\}\_\{\\mathrm\{nl\}\}\(\\mathbf\{q\},\\dot\{\\mathbf\{q\}\}\)=\\mathbf\{0\},where𝐪=\(q1,…,qℓ\)⊤\\mathbf\{q\}=\(q\_\{1\},\\ldots,q\_\{\\ell\}\)^\{\\top\}is the vector of displacements from equilibrium\. The mass matrix is diagonal,

𝐌=diag⁡\(m1,m,…,m\),m1=1\.5,m=1,\\displaystyle\\mathbf\{M\}=\\operatorname\{diag\}\(m\_\{1\},m,\\ldots,m\),\\qquad m\_\{1\}=1\.5,\\ m=1,and the stiffness matrix is the standard tridiagonal coupling matrix with a free right end,

𝐊=k​\(2−1−12−1⋱⋱⋱−12−1−11\),k=1\.\\displaystyle\\mathbf\{K\}=k\\begin\{pmatrix\}2&\-1&&&\\\\ \-1&2&\-1&&\\\\ &\\ddots&\\ddots&\\ddots&\\\\ &&\-1&2&\-1\\\\ &&&\-1&1\\end\{pmatrix\},\\qquad k=1\.Damping is of Rayleigh type,𝐂=α​𝐌\+β​𝐊\\mathbf\{C\}=\\alpha\\mathbf\{M\}\+\\beta\\mathbf\{K\}, withα=2⋅10−3\\alpha=2\\cdot 10^\{\-3\}andβ=5⋅10−3\\beta=5\\cdot 10^\{\-3\}\. The nonlinear spring acts only on the leftmost mass and depends on both its displacement and velocity:

𝐟nl​\(𝐪,𝐪˙\)=m1​\(3​q13\+7​q14​q˙1\+5​q˙13\)​𝐞1,\\displaystyle\\mathbf\{f\}\_\{\\mathrm\{nl\}\}\(\\mathbf\{q\},\\dot\{\\mathbf\{q\}\}\)=m\_\{1\}\\bigl\(3\\,q\_\{1\}^\{3\}\+7\\,q\_\{1\}^\{4\}\\,\\dot\{q\}\_\{1\}\+5\\,\\dot\{q\}\_\{1\}^\{\\,3\}\\bigr\)\\,\\mathbf\{e\}\_\{1\},where𝐞1\\mathbf\{e\}\_\{1\}is the first standard basis vector\. Introducing the state𝐱=\(𝐪⊤,𝐪˙⊤\)⊤∈ℝ2​ℓ\\mathbf\{x\}=\(\\mathbf\{q\}^\{\\top\},\\dot\{\\mathbf\{q\}\}^\{\\top\}\)^\{\\top\}\\in\\mathbb\{R\}^\{2\\ell\}, the system can be recast in first\-order form as

𝐱˙=𝐀𝐱\+𝐅​\(𝐱\),𝐀=\(𝟎𝐈−𝐌−1​𝐊−𝐌−1​𝐂\),𝐅​\(𝐱\)=\(𝟎−𝐌−1​𝐟nl​\(𝐪,𝐪˙\)\)\.\\displaystyle\\dot\{\\mathbf\{x\}\}=\\mathbf\{A\}\\mathbf\{x\}\+\\mathbf\{F\}\(\\mathbf\{x\}\),\\qquad\\mathbf\{A\}=\\begin\{pmatrix\}\\mathbf\{0\}&\\mathbf\{I\}\\\\ \-\\mathbf\{M\}^\{\-1\}\\mathbf\{K\}&\-\\mathbf\{M\}^\{\-1\}\\mathbf\{C\}\\end\{pmatrix\},\\qquad\\mathbf\{F\}\(\\mathbf\{x\}\)=\\begin\{pmatrix\}\\mathbf\{0\}\\\\ \-\\mathbf\{M\}^\{\-1\}\\mathbf\{f\}\_\{\\mathrm\{nl\}\}\(\\mathbf\{q\},\\dot\{\\mathbf\{q\}\}\)\\end\{pmatrix\}\.
We note that the system is equivariant with respect to the spatial symmetry group𝒢=⟨−𝐈2​ℓ⟩=\{𝐈2​ℓ,−𝐈2​ℓ\}≅C2\\mathcal\{G\}=\\langle\-\\mathbf\{I\}\_\{2\\ell\}\\rangle=\\\{\\mathbf\{I\}\_\{2\\ell\},\-\\mathbf\{I\}\_\{2\\ell\}\\\}\\cong C\_\{2\}\(whereC2C\_\{2\}denotes the cyclic group of order 2\), which acts on the phase spaceℝ2​ℓ\\mathbb\{R\}^\{2\\ell\}by negation\. In other words, the system is invariant under the transformation𝐱↦−𝐱\\mathbf\{x\}\\mapsto\-\\mathbf\{x\}, which corresponds to simultaneously reversing the direction of all displacements and velocities and can easily be verified by noting that𝐅​\(𝐱\)\\mathbf\{F\}\(\\mathbf\{x\}\)contains only odd terms\.

![Refer to caption](https://arxiv.org/html/2608.04239v1/x1.png)Figure 1:Schematic of the chain of oscillators, with nonlinear force at left end\.

### 2\.2Equivariance of SSMs

The uniqueness of the SSM𝒲​\(E\)\\mathcal\{W\}\(E\)in Theorem[2\.3](https://arxiv.org/html/2608.04239#S2.Thmtheorem3)is especially useful for the treatment of equivariant systems\. In particular, we can immediately deduce the following result about the equivariance of SSMs\.

###### Theorem 2\.7\.

Suppose that the system \([1](https://arxiv.org/html/2608.04239#S2.E1)\) is equivariant with respect to a linear symmetry group𝒢\\mathcal\{G\}and letEEbe a spectral subspace of𝐀\\mathbf\{A\}\. Then the following statements hold:

1. \(i\)the spectral subspaceEEis invariant under the action of𝒢\\mathcal\{G\}, i\.e\. for allS∈𝒢S\\in\\mathcal\{G\}we haveS​E=ESE=E;
2. \(ii\)the unique smoothest SSM𝒲​\(E\)\\mathcal\{W\}\(E\)associated withEEis also invariant under the action of𝒢\\mathcal\{G\}, i\.e\. for allS∈𝒢S\\in\\mathcal\{G\}we haveS​𝒲​\(E\)=𝒲​\(E\)S\\mathcal\{W\}\(E\)=\\mathcal\{W\}\(E\)\.

###### Proof\.

LetS∈𝒢S\\in\\mathcal\{G\}be a linear symmetry of the system\. Let us differentiate the equivariance condition \([2](https://arxiv.org/html/2608.04239#S2.E2)\) with respect to𝐱\\mathbf\{x\}at𝐱=𝟎\\mathbf\{x\}=\\mathbf\{0\}to obtain \(since𝐟​\(𝟎\)=𝟎\\mathbf\{f\}\(\\mathbf\{0\}\)=\\mathbf\{0\}andD​𝐟​\(𝟎\)=𝟎D\\mathbf\{f\}\(\\mathbf\{0\}\)=\\mathbf\{0\}\):

S​𝐀=𝐀​S\.\\displaystyle S\\mathbf\{A\}=\\mathbf\{A\}S\.Subtracting from \([2](https://arxiv.org/html/2608.04239#S2.E2)\) implies that for any𝐱∈ℝn\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\}we also have

S​𝐟​\(𝐱\)=𝐟​\(S​𝐱\)\.\\displaystyle S\\mathbf\{f\}\(\\mathbf\{x\}\)=\\mathbf\{f\}\(S\\mathbf\{x\}\)\.This implies thatSScommutes with the linear flow generated by𝐀\\mathbf\{A\}, which in turn implies \(together with invertibility ofSS\) thatS​E=ESE=Efor any spectral subspaceEEof𝐀\\mathbf\{A\}\.

It remains to show that the unique smoothest SSM𝒲​\(E\)\\mathcal\{W\}\(E\)is also invariant under the action ofSS\. Letφt:ℝn→ℝn\\varphi\_\{t\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}^\{n\}be the flow map of the system \([1](https://arxiv.org/html/2608.04239#S2.E1)\)\. Since the system is equivariant with respect to𝒢\\mathcal\{G\}, we have that for allS∈𝒢S\\in\\mathcal\{G\}and𝐱∈ℝn\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\},

dd​t​S​φt​\(𝐱\)=S​\(𝐀𝐱\+𝐟​\(𝐱\)\)=𝐀​S​𝐱\+𝐟​\(S​𝐱\)=dd​t​φt​\(S​𝐱\)\.\\displaystyle\\frac\{d\}\{dt\}S\\varphi\_\{t\}\(\\mathbf\{x\}\)=S\(\\mathbf\{A\}\\mathbf\{x\}\+\\mathbf\{f\}\(\\mathbf\{x\}\)\)=\\mathbf\{A\}S\\mathbf\{x\}\+\\mathbf\{f\}\(S\\mathbf\{x\}\)=\\frac\{d\}\{dt\}\\varphi\_\{t\}\(S\\mathbf\{x\}\)\.Thus, by smoothness of \([1](https://arxiv.org/html/2608.04239#S2.E1)\) and uniqueness of solutions, the flow\-map isSS\-equivariant, i\.e\.S​φt​\(𝐱\)=φt​\(S​𝐱\)S\\varphi\_\{t\}\(\\mathbf\{x\}\)=\\varphi\_\{t\}\(S\\mathbf\{x\}\)\. Let us now considerS​\(𝒲​\(E\)\)S\(\\mathcal\{W\}\(E\)\)\. SinceSSis linear and invertible, this is𝒞∞\\mathcal\{C\}^\{\\infty\}\-manifold with the following properties:

1. \(i\)𝟎=S​𝟎∈S​𝒲​\(E\)\\mathbf\{0\}=S\\mathbf\{0\}\\in S\\mathcal\{W\}\(E\), since𝟎∈𝒲​\(E\)\\mathbf\{0\}\\in\\mathcal\{W\}\(E\);
2. \(ii\)T𝟎​S​𝒲​\(E\)=S​T𝟎​𝒲​\(E\)=S​E=ET\_\{\\mathbf\{0\}\}S\\mathcal\{W\}\(E\)=S\\,T\_\{\\mathbf\{0\}\}\\mathcal\{W\}\(E\)=SE=E, sinceT𝟎​𝒲​\(E\)=ET\_\{\\mathbf\{0\}\}\\mathcal\{W\}\(E\)=Eby Theorem[2\.3](https://arxiv.org/html/2608.04239#S2.Thmtheorem3)andS​E=ESE=Eby the first part of this proposition;
3. \(iii\)for any𝐲=S​𝐱∈S​𝒲​\(E\)\\mathbf\{y\}=S\\mathbf\{x\}\\in S\\mathcal\{W\}\(E\)with𝐱∈𝒲​\(E\)\\mathbf\{x\}\\in\\mathcal\{W\}\(E\), φt​\(𝐲\)=φt​\(S​𝐱\)=S​φt​\(𝐱\)∈S​𝒲​\(E\),\\displaystyle\\varphi\_\{t\}\(\\mathbf\{y\}\)=\\varphi\_\{t\}\(S\\mathbf\{x\}\)=S\\varphi\_\{t\}\(\\mathbf\{x\}\)\\in S\\mathcal\{W\}\(E\),thusS​𝒲​\(E\)S\\mathcal\{W\}\(E\)is invariant under the flowφt\\varphi\_\{t\}\.

By uniqueness in Theorem[2\.3](https://arxiv.org/html/2608.04239#S2.Thmtheorem3)we thus must haveS​𝒲​\(E\)=𝒲​\(E\)S\\mathcal\{W\}\(E\)=\\mathcal\{W\}\(E\), which completes the proof\. ∎

## 3Computing spectral submanifolds and reduced dynamics on SSMs

The central pillar of SSM\-based reduced order modelling, as first introduced by\[[32](https://arxiv.org/html/2608.04239#bib.bib44)\], is to find a parametrisation of the SSM𝒲​\(E\)\\mathcal\{W\}\(E\)as a graph over the spectral subspaceEEand to then obtain the reduced dynamics on𝒲​\(E\)\\mathcal\{W\}\(E\)by restriction\. The standard construction of SSM\-based reduced order models\[[32](https://arxiv.org/html/2608.04239#bib.bib44)\]takes the parametrisation of the SSM along the orthogonal complementE⟂E^\{\\perp\}\. We will see that in the equivariant setting it is advantageous to allow the construction of this parametrisation along a more general complement ofEEthat is not necessarily orthogonal\. Thus, let us begin by fixing a complementEcE^\{c\}ofEE, i\.e\. a subspace such thatℝn=E⊕Ec\\mathbb\{R\}^\{n\}=E\\oplus E^\{c\}\. Note this choice ofEcE^\{c\}is not unique, and we will discuss two natural choices forEcE^\{c\}in the equivariant setting in §[3\.1](https://arxiv.org/html/2608.04239#S3.SS1)before fixing the choiceEcE^\{c\}to the most efficient choice for the rest of the manuscript\. A similar “oblique projection” construction was considered in the context of strongly non\-normal𝐀\\mathbf\{A\}in\[[7](https://arxiv.org/html/2608.04239#bib.bib57)\], however the use of this in the equivariant setting has not previously been explored\. Throughout this section we will focus on compact symmetry groups𝒢\\mathcal\{G\}\.

Letd=dim​\(𝒲​\(E\)\)=dim​\(E\)d=\\mathrm\{dim\}\(\\mathcal\{W\}\(E\)\)=\\mathrm\{dim\}\(E\)\. We begin by parametrisingEEandEcE^\{c\}with a suitable basis𝐔1=\(𝐮1​…​𝐮d\)∈ℝn×d\\mathbf\{U\}\_\{1\}=\\left\(\\mathbf\{u\}\_\{1\}\\ldots\\mathbf\{u\}\_\{d\}\\right\)\\in\\mathbb\{R\}^\{n\\times d\}and𝐔2=\(𝐮d\+1​…​𝐮n\)∈ℝn×\(n−d\)\\mathbf\{U\}\_\{2\}=\\left\(\\mathbf\{u\}\_\{d\+1\}\\ldots\\mathbf\{u\}\_\{n\}\\right\)\\in\\mathbb\{R\}^\{n\\times\(n\-d\)\}respectively, such that\[𝐔1∣𝐔2\]∈ℝn×n\[\\mathbf\{U\}\_\{1\}\\mid\\mathbf\{U\}\_\{2\}\]\\in\\mathbb\{R\}^\{n\\times n\}is invertible\. The inverse then defines the dual basis𝐕1∈ℝn×d,𝐕2∈ℝn×\(n−d\)\\mathbf\{V\}\_\{1\}\\in\\mathbb\{R\}^\{n\\times d\},\\mathbf\{V\}\_\{2\}\\in\\mathbb\{R\}^\{n\\times\(n\-d\)\}given by

\(𝐕1⊤𝐕2⊤\)=\[𝐔1∣𝐔2\]−1,\\displaystyle\\begin\{pmatrix\}\\mathbf\{V\}\_\{1\}^\{\\top\}\\\\ \\mathbf\{V\}\_\{2\}^\{\\top\}\\end\{pmatrix\}=\[\\mathbf\{U\}\_\{1\}\\mid\\mathbf\{U\}\_\{2\}\]^\{\-1\},which satisfies𝐕1⊤​𝐔1=𝐈d\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{U\}\_\{1\}=\\mathbf\{I\}\_\{d\},𝐕2⊤​𝐔2=𝐈n−d\\mathbf\{V\}\_\{2\}^\{\\top\}\\mathbf\{U\}\_\{2\}=\\mathbf\{I\}\_\{n\-d\},𝐕1⊤​𝐔2=0\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{U\}\_\{2\}=0and𝐕2⊤​𝐔1=0\\mathbf\{V\}\_\{2\}^\{\\top\}\\mathbf\{U\}\_\{1\}=0\. This means that the projectionπE\\pi\_\{E\}ontoEEalongEcE^\{c\}is given byπE=𝐔1​𝐕1⊤\\pi\_\{E\}=\\mathbf\{U\}\_\{1\}\\mathbf\{V\}\_\{1\}^\{\\top\}, while the projectionπEc\\pi\_\{E^\{c\}\}ontoEcE^\{c\}alongEEis given byπEc=𝐔2​𝐕2⊤\\pi\_\{E^\{c\}\}=\\mathbf\{U\}\_\{2\}\\mathbf\{V\}\_\{2\}^\{\\top\}\.

The reduced coordinates𝜼\\bm\{\\eta\}onEEcan then be obtained from a given point𝐲∈𝒲​\(E\)\\mathbf\{y\}\\in\\mathcal\{W\}\(E\)by projection alongEcE^\{c\}\(cf\. Figure[2](https://arxiv.org/html/2608.04239#S3.F2)\), i\.e\.𝜼=𝐕1⊤​𝐲\\bm\{\\eta\}=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}\. By Theorem[2\.3](https://arxiv.org/html/2608.04239#S2.Thmtheorem3)we can then, in a neighbourhood of𝟎\\mathbf\{0\}, find a parametrisation of𝒲​\(E\)\\mathcal\{W\}\(E\)as a graph overEEwith integer\-powered Taylor expansion, i\.e\. \(for a fixed choice ofEcE^\{c\}\) there exists a unique function𝐡:E→ℝn\\mathbf\{h\}:E\\rightarrow\\mathbb\{R\}^\{n\}such that𝐡​\(𝟎\)=𝟎\\mathbf\{h\}\(\\mathbf\{0\}\)=\\mathbf\{0\},𝐕1⊤​𝐡​\(𝜼\)=𝟎\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{h\}\(\\bm\{\\eta\}\)=\\mathbf\{0\}for all𝜼∈ℝd\\bm\{\\eta\}\\in\\mathbb\{R\}^\{d\}, andD𝜼​𝐡​\(𝟎\)=𝟎D\_\{\\bm\{\\eta\}\}\\mathbf\{h\}\(\\mathbf\{0\}\)=\\mathbf\{0\}such that

𝐲=𝐔1​𝜼\+𝐡​\(𝜼\)\.\\displaystyle\\mathbf\{y\}=\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{h\}\(\\bm\{\\eta\}\)\.\(3\)The reduced dynamics onEEare then given by

𝜼˙=𝐫​\(𝜼\)=𝐕1⊤​\(𝐀𝐔1​𝜼\+𝐀𝐡​\(𝜼\)\+𝐟​\(𝐔1​𝜼\+𝐡​\(𝜼\)\)\)\.\\displaystyle\\dot\{\\bm\{\\eta\}\}=\\mathbf\{r\}\(\\bm\{\\eta\}\)=\\mathbf\{V\}\_\{1\}^\{\\top\}\\left\(\\mathbf\{A\}\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{A\}\\mathbf\{h\}\(\\bm\{\\eta\}\)\+\\mathbf\{f\}\(\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{h\}\(\\bm\{\\eta\}\)\)\\right\)\.\(4\)
![Refer to caption](https://arxiv.org/html/2608.04239v1/x2.png)Figure 2:Projection ontoEEalongEcE^\{c\}\.In principle, if𝐟\\mathbf\{f\}is analytic, then𝐡\\mathbf\{h\}and𝐫\\mathbf\{r\}can be written as a convergent series expansion in𝜼\\bm\{\\eta\}\(cf\.\[[27](https://arxiv.org/html/2608.04239#bib.bib45)\]\)\. However, generically, the expansion of the vector field𝐫\\mathbf\{r\}contains coefficients that are messy and partly redundant artefacts of the particular choice of representation of the system \([1](https://arxiv.org/html/2608.04239#S2.E1)\)\. Thus\[[32](https://arxiv.org/html/2608.04239#bib.bib44),[15](https://arxiv.org/html/2608.04239#bib.bib1)\]advocate for the representation of the reduced dynamics in an extended Poincaré normal form style \(cf\.\[[46](https://arxiv.org/html/2608.04239#bib.bib7),[1](https://arxiv.org/html/2608.04239#bib.bib8),[24](https://arxiv.org/html/2608.04239#bib.bib9)\]\) to promote model sparsity and help develop efficient and robust algorithms for computing the reduced dynamics from data\. For this we seek a nonlinear change of coordinates𝜼=𝐭​\(𝐳\)\\bm\{\\eta\}=\\mathbf\{t\}\(\\mathbf\{z\}\)such that the transformed vector field𝐳˙=𝐧​\(𝐳\)\\dot\{\\mathbf\{z\}\}=\\mathbf\{n\}\(\\mathbf\{z\}\)has a diagonal linear part and as few nonlinear terms in its Taylor expansion as possible\. The “extended” part of the normal form refers to the fact that we do not seek to remove all non\-resonant terms from \([4](https://arxiv.org/html/2608.04239#S3.E4)\)\. Following\[[15](https://arxiv.org/html/2608.04239#bib.bib1)\]we do not remove the non\-resonant terms that would lead to small denominators in the Taylor expansion of𝐭\\mathbf\{t\}\. This means effectively that near\-resonant terms are assigned to the reduced dynamics and all other terms to the change of coordinates\. Specifically, we let𝐁:=𝐕1⊤​𝐀𝐔1∈ℝd×d\\mathbf\{B\}:=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{A\}\\mathbf\{U\}\_\{1\}\\in\\mathbb\{R\}^\{d\\times d\}, i\.e\.𝐁\\mathbf\{B\}is the linearisation of the reduced dynamics inEEaround𝟎\\mathbf\{0\}, and we seek a change of coordinates𝐭:ℂd→ℂd\\mathbf\{t\}:\\mathbb\{C\}^\{d\}\\rightarrow\\mathbb\{C\}^\{d\}such that the transformed vector field𝐧:ℂd→ℂd,\\mathbf\{n\}:\\mathbb\{C\}^\{d\}\\rightarrow\\mathbb\{C\}^\{d\},given by

𝐳˙=𝐧​\(𝐳\),𝜼=𝐭​\(𝐳\),\\displaystyle\\dot\{\\mathbf\{z\}\}=\\mathbf\{n\}\(\\mathbf\{z\}\),\\quad\\bm\{\\eta\}=\\mathbf\{t\}\(\\mathbf\{z\}\),\(5\)has a diagonal linear part and as few nonlinear terms in its Taylor expansion as possible\. Since the reduced coordinates𝜼\\bm\{\\eta\}are real, we require that the change of coordinates𝐭\\mathbf\{t\}maps the invariant subspace\{𝐳:zk=z¯k′​for conjugate pairs​\(k,k′\)\}\\\{\\mathbf\{z\}:z\_\{k\}=\\bar\{z\}\_\{k^\{\\prime\}\}\\text\{ for conjugate pairs \}\(k,k^\{\\prime\}\)\\\}intoℝd\\mathbb\{R\}^\{d\}and that𝐧\\mathbf\{n\}is real on this invariant subspace\. We diagonalise

𝐁=𝐖​𝚲​𝐖−1,𝚲=diag⁡\(λ1,…,λd\),\\displaystyle\\mathbf\{B\}=\\mathbf\{W\}\\bm\{\\Lambda\}\\mathbf\{W\}^\{\-1\},\\quad\\bm\{\\Lambda\}=\\operatorname\{diag\}\(\\lambda\_\{1\},\\dots,\\lambda\_\{d\}\),\(6\)where the eigenvalues and the columns of𝐖∈ℂd×d\\mathbf\{W\}\\in\\mathbb\{C\}^\{d\\times d\}come in complex conjugate pairs \(and the eigenvalues correspond to the eigenvalues of𝐀\\mathbf\{A\}associated with the spectral subspaceEE\)\. Our new coordinates and normal form are then sought as expansions

𝐭​\(𝐳\)=𝐖𝐳\+∑\|𝐦\|≥2𝐭𝐦​𝐳𝐦,𝐧​\(𝐳\)=𝚲​𝐳\+∑\|𝐦\|≥2𝐧𝐦​𝐳𝐦,\\displaystyle\\mathbf\{t\}\(\\mathbf\{z\}\)=\\mathbf\{W\}\\mathbf\{z\}\+\\sum\_\{\|\\mathbf\{m\}\|\\geq 2\}\\mathbf\{t\}\_\{\\mathbf\{m\}\}\\mathbf\{z\}^\{\\mathbf\{m\}\},\\qquad\\mathbf\{n\}\(\\mathbf\{z\}\)=\\bm\{\\Lambda\}\\mathbf\{z\}\+\\sum\_\{\|\\mathbf\{m\}\|\\geq 2\}\\mathbf\{n\}\_\{\\mathbf\{m\}\}\\mathbf\{z\}^\{\\mathbf\{m\}\},subject to the change of coordinates condition between the reduced dynamics and the normal form

D​𝐭​\(𝐳\)​𝐧​\(𝐳\)=𝐫​\(𝐭​\(𝐳\)\)\.\\displaystyle D\\mathbf\{t\}\(\\mathbf\{z\}\)\\,\\mathbf\{n\}\(\\mathbf\{z\}\)=\\mathbf\{r\}\(\\mathbf\{t\}\(\\mathbf\{z\}\)\)\.\(7\)Solving \([7](https://arxiv.org/html/2608.04239#S3.E7)\) order by order in𝐳\\mathbf\{z\}yields the homological equations

\(𝐦⋅𝝀−λj\)​t~j,𝐦\+nj,𝐦=gj,𝐦,\\displaystyle\\left\(\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}\\right\)\\widetilde\{t\}\_\{j,\\mathbf\{m\}\}\+n\_\{j,\\mathbf\{m\}\}=g\_\{j,\\mathbf\{m\}\},\(8\)where𝐭~:=𝐖−1​𝐭\\widetilde\{\\mathbf\{t\}\}:=\\mathbf\{W\}^\{\-1\}\\mathbf\{t\}andgj,𝐦g\_\{j,\\mathbf\{m\}\}collects terms arising from \([4](https://arxiv.org/html/2608.04239#S3.E4)\) and from lower\-order coefficients\. At each order, \([8](https://arxiv.org/html/2608.04239#S3.E8)\) is a single scalar equation in the two unknownst~j,𝐦\\widetilde\{t\}\_\{j,\\mathbf\{m\}\}andnj,𝐦n\_\{j,\\mathbf\{m\}\}, so every monomial may be assigned to exactly one of the two maps: removing it from the reduced dynamics \(nj,𝐦=0n\_\{j,\\mathbf\{m\}\}=0,t~j,𝐦=gj,𝐦/\(𝐦⋅𝝀−λj\)\\widetilde\{t\}\_\{j,\\mathbf\{m\}\}=g\_\{j,\\mathbf\{m\}\}/\(\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}\)\) is possible whenever𝐦⋅𝝀≠λj\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\\neq\\lambda\_\{j\}, but produces a small denominator whenever the system is near an inner resonance\. The extended normal form formulation in\[[15](https://arxiv.org/html/2608.04239#bib.bib1)\]therefore retains these terms in the reduced dynamics by fixing a thresholdδ\>0\\delta\>0and defining the near\-resonant index set

ℐδ:=\{\(j,𝐦\):\|Im​\(𝐦⋅𝝀−λj\)\|≤δ\}\.\\displaystyle\\mathcal\{I\}\_\{\\delta\}:=\\left\\\{\(j,\\mathbf\{m\}\):\|\\mathrm\{Im\}\(\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}\)\|\\leq\\delta\\right\\\}\.We then set

\(t~j,𝐦,nj,𝐦\)=\{\(gj,𝐦/\(𝐦⋅𝝀−λj\),0\),\(j,𝐦\)∉ℐδ,\(0,gj,𝐦\),\(j,𝐦\)∈ℐδ\.\\displaystyle\(\\widetilde\{t\}\_\{j,\\mathbf\{m\}\},n\_\{j,\\mathbf\{m\}\}\)=\\begin\{cases\}\(g\_\{j,\\mathbf\{m\}\}/\(\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}\),0\),&\(j,\\mathbf\{m\}\)\\notin\\mathcal\{I\}\_\{\\delta\},\\\\ \(0,g\_\{j,\\mathbf\{m\}\}\),&\(j,\\mathbf\{m\}\)\\in\\mathcal\{I\}\_\{\\delta\}\.\\end\{cases\}\(9\)With this convention, the coefficients of𝐭\\mathbf\{t\}and𝐧\\mathbf\{n\}are uniquely determined order by order\. The motivation for considering only the imaginary parts of the eigenvalues is to make sure that the normal form depends continuously on the damping \(i\.e\. the real parts of the eigenvalues\), reducing to the classical normal form of the limiting conservative system asRe⁡𝝀→𝟎\\operatorname\{Re\}\\bm\{\\lambda\}\\to\\mathbf\{0\}, and follows the convention established in\[[15](https://arxiv.org/html/2608.04239#bib.bib1)\]\.

We will discuss how this extended normal form can be computed from data in §[4](https://arxiv.org/html/2608.04239#S4)and how equivariance can be preserved in §[3\.1](https://arxiv.org/html/2608.04239#S3.SS1)\.

### 3\.1Equivariant SSM reduction

In order to preserve equivariance in the reduced model, it is crucial to choose the complementEcE^\{c\}in a way that is compatible with the symmetry group𝒢\\mathcal\{G\}\. In particular, we need to ensure thatEcE^\{c\}is also invariant under the action of𝒢\\mathcal\{G\}, i\.e\. for allS∈𝒢S\\in\\mathcal\{G\}we haveS​Ec=EcSE^\{c\}=E^\{c\}\. There are two natural ways to achieve this\.

1. \(I\)Spectral complement:We pickEcE^\{c\}to be the direct sum of the remaining spectral subspaces of𝐀\\mathbf\{A\}, i\.e\.Ec=⨁j∉JEjE^\{c\}=\\bigoplus\_\{j\\notin J\}E\_\{j\}, whereE=⨁j∈JEjE=\\bigoplus\_\{j\\in J\}E\_\{j\}\. Invariance ofEcE^\{c\}under the action of𝒢\\mathcal\{G\}is then guaranteed by Theorem[2\.7](https://arxiv.org/html/2608.04239#S2.Thmtheorem7)\. However, for largenn, it is expensive to compute the spectral complementEcE^\{c\}since it requires computing all eigenvalues and eigenvectors of𝐀\\mathbf\{A\}, which is not feasible for large\-scale systems\. Thus, in practice, we prefer to use the second option below, which is computationally cheaper and still guarantees equivariance of the reduced model\.
2. \(II\)Orthogonal complement with respect to the equivariant inner product:For a compact group𝒢\\mathcal\{G\}, an alternative choice is to defineEcE^\{c\}as the orthogonal complement ofEEwith respect to theequivariant inner product⟨⋅,⋅⟩𝒢\\langle\\cdot,\\cdot\\rangle\_\{\\mathcal\{G\}\}as defined in Definition[3\.2](https://arxiv.org/html/2608.04239#S3.Thmtheorem2)\. EveryS∈𝒢S\\in\\mathcal\{G\}acts as an isometry of⟨⋅,⋅⟩𝒢\\langle\\cdot,\\cdot\\rangle\_\{\\mathcal\{G\}\}, so the orthogonal complement of the𝒢\\mathcal\{G\}\-invariant subspaceEEis again𝒢\\mathcal\{G\}\-invariant\.

###### Definition 3\.2\.

Let𝒢\\mathcal\{G\}be a compact linear symmetry group acting onℝn\\mathbb\{R\}^\{n\}, equipped with its normalised Haar measureμ\\mu\(so thatμ​\(𝒢\)=1\\mu\(\\mathcal\{G\}\)=1\)\. The equivariant inner product⟨⋅,⋅⟩𝒢\\langle\\cdot,\\cdot\\rangle\_\{\\mathcal\{G\}\}is defined as

⟨𝐯,𝐰⟩𝒢:=∫𝒢⟨S​𝐯,S​𝐰⟩​dμ​\(S\),\\displaystyle\\langle\\mathbf\{v\},\\mathbf\{w\}\\rangle\_\{\\mathcal\{G\}\}:=\\int\_\{\\mathcal\{G\}\}\\langle S\\mathbf\{v\},S\\mathbf\{w\}\\rangle\\,\\differential\\mu\(S\),where⟨⋅,⋅⟩\\langle\\cdot,\\cdot\\rangleis the standard inner product onℝn\\mathbb\{R\}^\{n\}\. For a finite group, the Haar measure is the normalised counting measure and this reduces to the average⟨𝐯,𝐰⟩𝒢=1\|𝒢\|​∑S∈𝒢⟨S​𝐯,S​𝐰⟩\\langle\\mathbf\{v\},\\mathbf\{w\}\\rangle\_\{\\mathcal\{G\}\}=\\frac\{1\}\{\|\\mathcal\{G\}\|\}\\sum\_\{S\\in\\mathcal\{G\}\}\\langle S\\mathbf\{v\},S\\mathbf\{w\}\\rangle\.

With any𝒢\\mathcal\{G\}\-invariant choice ofEcE^\{c\}, it turns out that the reduced dynamics on the SSM𝒲​\(E\)\\mathcal\{W\}\(E\)also inherits an equivariance property from the full system, which can be used to significantly simplify the computations of the parametrisation map𝐡\\mathbf\{h\}and the reduced vector field𝐫\\mathbf\{r\}as well as to improve the model fidelity of the reduced model\. For this we introduce the standard restrictionS\|E:=𝐕1⊤​S​𝐔1S\|\_\{E\}:=\\mathbf\{V\}\_\{1\}^\{\\top\}S\\mathbf\{U\}\_\{1\}forS∈𝒢S\\in\\mathcal\{G\}and let:

𝒢\|E:=\{S\|E,S∈𝒢\}\.\\displaystyle\\mathcal\{G\}\|\_\{E\}:=\\\{S\|\_\{E\},S\\in\\mathcal\{G\}\\\}\.
###### Lemma 3\.3\.

The restriction𝒢\|E\\mathcal\{G\}\|\_\{E\}is a well\-defined linear symmetry group acting onℝd≅E\\mathbb\{R\}^\{d\}\\cong E,𝒢\|E⊂G​L​\(d,ℝ\)\\mathcal\{G\}\|\_\{E\}\\subset GL\(d,\\mathbb\{R\}\)\.

###### Proof\.

SinceEEis invariant under the action of𝒢\\mathcal\{G\}\(Theorem[2\.7](https://arxiv.org/html/2608.04239#S2.Thmtheorem7)\), and𝐔1​𝐕1⊤\\mathbf\{U\}\_\{1\}\\mathbf\{V\}\_\{1\}^\{\\top\}is the projection ontoEE\(cf\. Figure[2](https://arxiv.org/html/2608.04239#S3.F2)\) we have, for anyS1,S2∈𝒢S\_\{1\},S\_\{2\}\\in\\mathcal\{G\},

S1\|E​S2\|E=𝐕1⊤​S1​𝐔1​𝐕1⊤​S2​𝐔1=𝐕1⊤​S1​S2​𝐔1=\(S1​S2\)\|E,\\displaystyle S\_\{1\}\|\_\{E\}S\_\{2\}\|\_\{E\}=\\mathbf\{V\}\_\{1\}^\{\\top\}S\_\{1\}\\mathbf\{U\}\_\{1\}\\mathbf\{V\}\_\{1\}^\{\\top\}S\_\{2\}\\mathbf\{U\}\_\{1\}=\\mathbf\{V\}\_\{1\}^\{\\top\}S\_\{1\}S\_\{2\}\\mathbf\{U\}\_\{1\}=\(S\_\{1\}S\_\{2\}\)\|\_\{E\},where the second equality follows from the fact that the columns ofS2​𝐔1S\_\{2\}\\mathbf\{U\}\_\{1\}are inS2​E=ES\_\{2\}E=E, where𝐔1​𝐕1⊤=πE\\mathbf\{U\}\_\{1\}\\mathbf\{V\}\_\{1\}^\{\\top\}=\\pi\_\{E\}acts as the identity\. Thus the restriction is closed under group multiplication and hence a group homomorphism\. The result follows\. ∎

We can now show that the parametrisation map𝐡\\mathbf\{h\}and the reduced vector field𝐫\\mathbf\{r\}are also equivariant with respect to this restricted group action\.

###### Proposition 3\.4\.

If \([1](https://arxiv.org/html/2608.04239#S2.E1)\) is equivariant with respect to a compact linear symmetry group𝒢\\mathcal\{G\}andEEis a spectral subspace of𝐀\\mathbf\{A\}, then, in a neighbourhood of the origin𝟎\\mathbf\{0\}, the parametrisation map𝐡:E→ℝn\\mathbf\{h\}:E\\rightarrow\\mathbb\{R\}^\{n\}and the reduced vector field𝐫:E→E\\mathbf\{r\}:E\\rightarrow Eare equivariant with respect to the restricted group action𝒢\|E\\mathcal\{G\}\|\_\{E\}, in the sense that

𝐡​\(S\|E​𝜼\)=S​𝐡​\(𝜼\),𝐫​\(S\|E​𝜼\)=S\|E​𝐫​\(𝜼\),\\displaystyle\\mathbf\{h\}\(S\|\_\{E\}\\bm\{\\eta\}\)=S\\mathbf\{h\}\(\\bm\{\\eta\}\),\\qquad\\mathbf\{r\}\(S\|\_\{E\}\\bm\{\\eta\}\)=S\|\_\{E\}\\mathbf\{r\}\(\\bm\{\\eta\}\),for allS∈𝒢S\\in\\mathcal\{G\}and𝛈∈E\\bm\{\\eta\}\\in E\.

###### Proof\.

LetS∈𝒢S\\in\\mathcal\{G\}be a linear symmetry of the system, and denote byΩ⊂ℝn\\Omega\\subset\\mathbb\{R\}^\{n\}the neighbourhood of𝟎\\mathbf\{0\}in which the parametrisation \([3](https://arxiv.org/html/2608.04239#S3.E3)\) is valid\. Note, in the following we will assume thatS​𝐲∈ΩS\\mathbf\{y\}\\in\\Omegafor𝐲∈𝒲​\(E\)∩Ω\\mathbf\{y\}\\in\\mathcal\{W\}\(E\)\\cap\\Omega\. This can be achieved by shrinkingΩ\\Omegato⋂S∈𝒢S​Ω\\bigcap\_\{S\\in\\mathcal\{G\}\}S\\Omega, which is non\-empty by compactness of𝒢\\mathcal\{G\}\. Then, by virtue ofEEandEcE^\{c\}being𝒢\\mathcal\{G\}\-invariant, we have for any𝐲∈𝒲​\(E\)∩Ω\\mathbf\{y\}\\in\\mathcal\{W\}\(E\)\\cap\\Omega,

S\|E​𝐕1⊤​𝐲=𝐕1⊤​S​𝐔1​𝐕1⊤​𝐲\\displaystyle S\|\_\{E\}\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}=\\mathbf\{V\}\_\{1\}^\{\\top\}S\\mathbf\{U\}\_\{1\}\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}=𝐕1⊤​S​𝐔1​𝐕1⊤​\(𝐔1​𝜼\+𝐡​\(𝜼\)\)\\displaystyle=\\mathbf\{V\}\_\{1\}^\{\\top\}S\\mathbf\{U\}\_\{1\}\\mathbf\{V\}\_\{1\}^\{\\top\}\(\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{h\}\(\\bm\{\\eta\}\)\)=𝐕1⊤​S​𝐔1​𝜼=𝐕1⊤​S​\(𝐔1​𝜼\+𝐡​\(𝜼\)\)=𝐕1⊤​S​𝐲,\\displaystyle=\\mathbf\{V\}\_\{1\}^\{\\top\}S\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}=\\mathbf\{V\}\_\{1\}^\{\\top\}S\(\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{h\}\(\\bm\{\\eta\}\)\)=\\mathbf\{V\}\_\{1\}^\{\\top\}S\\mathbf\{y\},where we have used the fact thatS​Im​\(𝐡\)⊂S​Ec=Ec⊂ker​\(𝐕1⊤\)S\\mathrm\{Im\}\(\\mathbf\{h\}\)\\subset SE^\{c\}=E^\{c\}\\subset\\mathrm\{ker\}\(\\mathbf\{V\}\_\{1\}^\{\\top\}\)and wrote𝜼=𝐕1⊤​𝐲\\bm\{\\eta\}=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}for the last equality\. ThusS\|E​𝐕1⊤​𝐲S\|\_\{E\}\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}is the unique reduced coordinate ofS​𝐲S\\mathbf\{y\}\. Lifting this back to𝒲​\(E\)\\mathcal\{W\}\(E\)\(using the fact that𝒲​\(E\)\\mathcal\{W\}\(E\)isSS\-invariant by Theorem[2\.7](https://arxiv.org/html/2608.04239#S2.Thmtheorem7)\) we thus have

S​𝐔1​𝜼\+S​𝐡​\(𝜼\)=S​𝐲=𝐔1​S\|E​𝜼\+𝐡​\(S\|E​𝜼\)\.\\displaystyle S\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+S\\mathbf\{h\}\(\\bm\{\\eta\}\)=S\\mathbf\{y\}=\\mathbf\{U\}\_\{1\}S\|\_\{E\}\\bm\{\\eta\}\+\\mathbf\{h\}\(S\|\_\{E\}\\bm\{\\eta\}\)\.Applying the dual bases𝐕1⊤\\mathbf\{V\}\_\{1\}^\{\\top\}and𝐕2⊤\\mathbf\{V\}\_\{2\}^\{\\top\}respectively to this equation \(i\.e\. the projection onto coordinates ofE,EcE,E^\{c\}\) we have \(noting that𝐕1⊤​𝐔2=0\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{U\}\_\{2\}=0,𝐕2⊤​𝐔1=0\\mathbf\{V\}\_\{2\}^\{\\top\}\\mathbf\{U\}\_\{1\}=0and thatIm​\(𝐔1\)=E,Im​\(𝐡\)⊂Ec\\mathrm\{Im\}\(\\mathbf\{U\}\_\{1\}\)=E,\\mathrm\{Im\}\(\\mathbf\{h\}\)\\subset E^\{c\}withE,EcE,E^\{c\}invariant under the action ofSS\):

S​𝐔1​𝜼\\displaystyle S\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}=πE​\(S​𝐔1​𝜼\+S​𝐡​\(𝜼\)\)=𝐔1​𝐕1⊤​\(𝐔1​S\|E​𝜼\+𝐡​\(S\|E​𝜼\)\)=𝐔1​S\|E​𝜼,\\displaystyle=\\pi\_\{E\}\(S\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+S\\mathbf\{h\}\(\\bm\{\\eta\}\)\)=\\mathbf\{U\}\_\{1\}\\mathbf\{V\}\_\{1\}^\{\\top\}\\left\(\\mathbf\{U\}\_\{1\}S\|\_\{E\}\\bm\{\\eta\}\+\\mathbf\{h\}\(S\|\_\{E\}\\bm\{\\eta\}\)\\right\)=\\mathbf\{U\}\_\{1\}S\|\_\{E\}\\bm\{\\eta\},S​𝐡​\(𝜼\)\\displaystyle S\\mathbf\{h\}\(\\bm\{\\eta\}\)=πEc​\(S​𝐔1​𝜼\+S​𝐡​\(𝜼\)\)=𝐔2​𝐕2⊤​\(𝐔1​S\|E​𝜼\+𝐡​\(S\|E​𝜼\)\)=𝐡​\(S\|E​𝜼\)\.\\displaystyle=\\pi\_\{E^\{c\}\}\\left\(S\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+S\\mathbf\{h\}\(\\bm\{\\eta\}\)\\right\)=\\mathbf\{U\}\_\{2\}\\mathbf\{V\}\_\{2\}^\{\\top\}\\left\(\\mathbf\{U\}\_\{1\}S\|\_\{E\}\\bm\{\\eta\}\+\\mathbf\{h\}\(S\|\_\{E\}\\bm\{\\eta\}\)\\right\)=\\mathbf\{h\}\(S\|\_\{E\}\\bm\{\\eta\}\)\.Finally, using this, we have

𝐫​\(S\|E​𝜼\)\\displaystyle\\mathbf\{r\}\(S\|\_\{E\}\\bm\{\\eta\}\)=𝐕1⊤​\(𝐀𝐔1​S\|E​𝜼\+𝐀𝐡​\(S\|E​𝜼\)\+𝐟​\(𝐔1​S\|E​𝜼\+𝐡​\(S\|E​𝜼\)\)\)\\displaystyle=\\mathbf\{V\}\_\{1\}^\{\\top\}\\left\(\\mathbf\{A\}\\mathbf\{U\}\_\{1\}S\|\_\{E\}\\bm\{\\eta\}\+\\mathbf\{A\}\\mathbf\{h\}\(S\|\_\{E\}\\bm\{\\eta\}\)\+\\mathbf\{f\}\(\\mathbf\{U\}\_\{1\}S\|\_\{E\}\\bm\{\\eta\}\+\\mathbf\{h\}\(S\|\_\{E\}\\bm\{\\eta\}\)\)\\right\)=𝐕1⊤​S​\(𝐀𝐔1​𝜼\+𝐀𝐡​\(𝜼\)\+𝐟​\(𝐔1​𝜼\+𝐡​\(𝜼\)\)\)\\displaystyle=\\mathbf\{V\}\_\{1\}^\{\\top\}S\\left\(\\mathbf\{A\}\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{A\}\\mathbf\{h\}\(\\bm\{\\eta\}\)\+\\mathbf\{f\}\(\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{h\}\(\\bm\{\\eta\}\)\)\\right\)=S\|E​𝐫​\(𝜼\)\.\\displaystyle=S\|\_\{E\}\\mathbf\{r\}\(\\bm\{\\eta\}\)\.∎

Having established the properties of the reduced dynamics on the SSM𝒲​\(E\)\\mathcal\{W\}\(E\), we turn to the canonical construction of an equivariant normal form for the reduced dynamics, cf\. \([5](https://arxiv.org/html/2608.04239#S3.E5)\)\. It turns out that the change of variables𝐟\\mathbf\{f\}and𝐧\\mathbf\{n\}, uniquely constructed as per \([9](https://arxiv.org/html/2608.04239#S3.E9)\), are also equivariant with respect to appropriate restricted group action of𝒢\\mathcal\{G\}\. We will discuss the implications of this equivariance for the data\-driven computation of the extended normal form in §[4](https://arxiv.org/html/2608.04239#S4)\.

###### Proposition 3\.5\.

Suppose \([1](https://arxiv.org/html/2608.04239#S2.E1)\) is equivariant with respect to the compact linear symmetry group𝒢\\mathcal\{G\}\. Let𝐭,𝐧\\mathbf\{t\},\\mathbf\{n\}be constructed termwise from \([9](https://arxiv.org/html/2608.04239#S3.E9)\) and, forS∈𝒢S\\in\\mathcal\{G\}, letS^:=𝐖−1​\(S\|E\)​𝐖\\widehat\{S\}:=\\mathbf\{W\}^\{\-1\}\(S\|\_\{E\}\)\\mathbf\{W\}be the restricted action in the diagonalising coordinates𝐖\\mathbf\{W\}defined in \([6](https://arxiv.org/html/2608.04239#S3.E6)\)\. Then𝐭^:=𝐖−1​𝐭\\hat\{\\mathbf\{t\}\}:=\\mathbf\{W\}^\{\-1\}\\mathbf\{t\}and𝐧\\mathbf\{n\}are both equivariant with respect to the restricted group action of𝒢\\mathcal\{G\}, i\.e\.

𝐭^​\(S^​𝐳\)=S^​𝐭^​\(𝐳\),𝐧​\(S^​𝐳\)=S^​𝐧​\(𝐳\),∀S∈𝒢,𝐳∈ℂd\.\\displaystyle\\hat\{\\mathbf\{t\}\}\(\\widehat\{S\}\\mathbf\{z\}\)=\\widehat\{S\}\\hat\{\\mathbf\{t\}\}\(\\mathbf\{z\}\),\\qquad\\mathbf\{n\}\(\\widehat\{S\}\\mathbf\{z\}\)=\\widehat\{S\}\\mathbf\{n\}\(\\mathbf\{z\}\),\\qquad\\forall S\\in\\mathcal\{G\},\\ \\mathbf\{z\}\\in\\mathbb\{C\}^\{d\}\.

###### Proof\.

We will exploit uniqueness of the construction of𝐭,𝐧\\mathbf\{t\},\\mathbf\{n\}as described in §[3](https://arxiv.org/html/2608.04239#S3)\. LetS∈𝒢S\\in\\mathcal\{G\}and define

𝐭′​\(𝐳\):=S\|E−1​𝐭​\(S^​𝐳\),𝐧′​\(𝐳\):=S^−1​𝐧​\(S^​𝐳\)\.\\displaystyle\\mathbf\{t\}^\{\\prime\}\(\\mathbf\{z\}\):=S\|\_\{E\}^\{\-1\}\\mathbf\{t\}\(\\widehat\{S\}\\mathbf\{z\}\),\\qquad\\mathbf\{n\}^\{\\prime\}\(\\mathbf\{z\}\):=\\widehat\{S\}^\{\-1\}\\mathbf\{n\}\(\\widehat\{S\}\\mathbf\{z\}\)\.We will show that\(𝐭′,𝐧′\)\(\\mathbf\{t\}^\{\\prime\},\\mathbf\{n\}^\{\\prime\}\)satisfies the same conjugacy equation \([7](https://arxiv.org/html/2608.04239#S3.E7)\) and has the same linear parts and near\-resonant support as\(𝐭,𝐧\)\(\\mathbf\{t\},\\mathbf\{n\}\)\. By uniqueness of the solution to the term\-wise conditions \([8](https://arxiv.org/html/2608.04239#S3.E8)\), we then conclude𝐭′=𝐭\\mathbf\{t\}^\{\\prime\}=\\mathbf\{t\}and𝐧′=𝐧\\mathbf\{n\}^\{\\prime\}=\\mathbf\{n\}, which gives the desired equivariance properties\.

Differentiating𝐭′\\mathbf\{t\}^\{\\prime\}and using the conjugacy equation \([7](https://arxiv.org/html/2608.04239#S3.E7)\) atS^​𝐳\\widehat\{S\}\\mathbf\{z\}together with linearity of the group action and the equivariance of𝐫\\mathbf\{r\},

D​𝐭′​\(𝐳\)​𝐧′​\(𝐳\)\\displaystyle D\\mathbf\{t\}^\{\\prime\}\(\\mathbf\{z\}\)\\,\\mathbf\{n\}^\{\\prime\}\(\\mathbf\{z\}\)=\(S\|E\)−1​D​𝐭​\(S^​𝐳\)​S^​S^−1​𝐧​\(S^​𝐳\)=\(S\|E\)−1​D​𝐭​\(S^​𝐳\)​𝐧​\(S^​𝐳\)\\displaystyle=\(S\|\_\{E\}\)^\{\-1\}\\,D\\mathbf\{t\}\(\\widehat\{S\}\\mathbf\{z\}\)\\widehat\{S\}\\widehat\{S\}^\{\-1\}\\,\\mathbf\{n\}\(\\widehat\{S\}\\mathbf\{z\}\)=\(S\|\_\{E\}\)^\{\-1\}\\,D\\mathbf\{t\}\(\\widehat\{S\}\\mathbf\{z\}\)\\,\\mathbf\{n\}\(\\widehat\{S\}\\mathbf\{z\}\)=\(S\|E\)−1​𝐫​\(𝐭​\(S^​𝐳\)\)=\(S\|E\)−1​𝐫​\(S\|E​𝐭′​\(𝐳\)\)=𝐫​\(𝐭′​\(𝐳\)\),\\displaystyle=\(S\|\_\{E\}\)^\{\-1\}\\,\\mathbf\{r\}\\big\(\\mathbf\{t\}\(\\widehat\{S\}\\mathbf\{z\}\)\\big\)=\(S\|\_\{E\}\)^\{\-1\}\\,\\mathbf\{r\}\\big\(S\|\_\{E\}\\,\\mathbf\{t\}^\{\\prime\}\(\\mathbf\{z\}\)\\big\)=\\mathbf\{r\}\\big\(\\mathbf\{t\}^\{\\prime\}\(\\mathbf\{z\}\)\\big\),so\(𝐭′,𝐧′\)\(\\mathbf\{t\}^\{\\prime\},\\mathbf\{n\}^\{\\prime\}\)satisfies the same conjugacy equation \([7](https://arxiv.org/html/2608.04239#S3.E7)\) as\(𝐭,𝐧\)\(\\mathbf\{t\},\\mathbf\{n\}\)\. Next we show that the linear parts of𝐭′\\mathbf\{t\}^\{\\prime\}and𝐧′\\mathbf\{n\}^\{\\prime\}coincide with those of𝐭\\mathbf\{t\}and𝐧\\mathbf\{n\}\. The linear part of𝐭′\\mathbf\{t\}^\{\\prime\}is given by

D​𝐭′​\(𝟎\)=\(S\|E\)−1​D​𝐭​\(𝟎\)​S^=\(S\|E\)−1​𝐖​S^=\(S\|E\)−1​𝐖​\(𝐖−1​\(S\|E\)​𝐖\)=𝐖=D​𝐭​\(𝟎\),\\displaystyle D\\mathbf\{t\}^\{\\prime\}\(\\mathbf\{0\}\)=\(S\|\_\{E\}\)^\{\-1\}D\\mathbf\{t\}\(\\mathbf\{0\}\)\\widehat\{S\}=\(S\|\_\{E\}\)^\{\-1\}\\mathbf\{W\}\\widehat\{S\}=\(S\|\_\{E\}\)^\{\-1\}\\mathbf\{W\}\(\\mathbf\{W\}^\{\-1\}\(S\|\_\{E\}\)\\mathbf\{W\}\)=\\mathbf\{W\}=D\\mathbf\{t\}\(\\mathbf\{0\}\),and the linear part of𝐧′\\mathbf\{n\}^\{\\prime\}is given by

D​𝐧′​\(𝟎\)=S^−1​D​𝐧​\(𝟎\)​S^=S^−1​𝚲​S^=𝚲=D​𝐧​\(𝟎\),\\displaystyle D\\mathbf\{n\}^\{\\prime\}\(\\mathbf\{0\}\)=\\widehat\{S\}^\{\-1\}D\\mathbf\{n\}\(\\mathbf\{0\}\)\\widehat\{S\}=\\widehat\{S\}^\{\-1\}\\bm\{\\Lambda\}\\widehat\{S\}=\\bm\{\\Lambda\}=D\\mathbf\{n\}\(\\mathbf\{0\}\),where we have used that𝐁=𝐕1⊤​𝐀𝐔1\\mathbf\{B\}=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{A\}\\mathbf\{U\}\_\{1\}commutes withS\|ES\|\_\{E\}for allS∈𝒢S\\in\\mathcal\{G\}, so that𝚲\\bm\{\\Lambda\}commutes withS^\\widehat\{S\}\. It remains to show that the assignment of nonlinear terms to𝐭′\\mathbf\{t\}^\{\\prime\}and𝐧′\\mathbf\{n\}^\{\\prime\}, i\.e\. their near\-resonant support, is the same as that for𝐭\\mathbf\{t\}and𝐧\\mathbf\{n\}given in \([9](https://arxiv.org/html/2608.04239#S3.E9)\)\. In principle the linear substitution𝐳↦S^​𝐳\\mathbf\{z\}\\mapsto\\widehat\{S\}\\mathbf\{z\}preserves the degree of a monomial, but a single monomial𝐳𝐦\\mathbf\{z\}^\{\\mathbf\{m\}\}may be mapped to a linear combination of monomials of the same degree under this map\.

Let us consider a single monomial𝐞j​𝐳𝐦\\mathbf\{e\}\_\{j\}\\mathbf\{z\}^\{\\mathbf\{m\}\}in the Taylor expansion of𝐧\\mathbf\{n\}, contributingz1m1​⋯​zdmdz\_\{1\}^\{m\_\{1\}\}\\cdots z\_\{d\}^\{m\_\{d\}\}to thejj\-th component of𝐧\\mathbf\{n\}\. The assignment of this monomial to𝐧\\mathbf\{n\}or𝐭\\mathbf\{t\}is determined by the size of\|Im​\(𝐦⋅𝝀−λj\)\|\|\\mathrm\{Im\}\(\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}\)\|relative to the thresholdδ\\delta\. Under the action ofS^\\widehat\{S\}, this monomial is mapped to

S^−1​𝐞j​\(S^​𝐳\)𝐦=\(∑i=1d\(S^−1\)i​j​𝐞i\)​∏l=1d\(∑k=1dS^l​k​zk\)ml,\\displaystyle\\widehat\{S\}^\{\-1\}\\mathbf\{e\}\_\{j\}\(\\widehat\{S\}\\mathbf\{z\}\)^\{\\mathbf\{m\}\}=\\Big\(\\sum\_\{i=1\}^\{d\}\(\\widehat\{S\}^\{\-1\}\)\_\{ij\}\\,\\mathbf\{e\}\_\{i\}\\Big\)\\prod\_\{l=1\}^\{d\}\\Big\(\\sum\_\{k=1\}^\{d\}\\widehat\{S\}\_\{lk\}\\,z\_\{k\}\\Big\)^\{m\_\{l\}\},which is a linear combination of degree\-\|𝐦\|\|\\mathbf\{m\}\|monomial vector fields𝐞i​𝐳𝐦′\\mathbf\{e\}\_\{i\}\\mathbf\{z\}^\{\\mathbf\{m\}^\{\\prime\}\}with\|𝐦′\|=\|𝐦\|\|\\mathbf\{m\}^\{\\prime\}\|=\|\\mathbf\{m\}\|\. Now,S^\\widehat\{S\}commutes with the diagonal matrix𝚲\\bm\{\\Lambda\}, i\.e\.S^l​k​\(λk−λl\)=\(S^​𝚲−𝚲​S^\)l​k=0\\widehat\{S\}\_\{lk\}\(\\lambda\_\{k\}\-\\lambda\_\{l\}\)=\(\\widehat\{S\}\\bm\{\\Lambda\}\-\\bm\{\\Lambda\}\\widehat\{S\}\)\_\{lk\}=0, soS^l​k=0\\widehat\{S\}\_\{lk\}=0wheneverλl≠λk\\lambda\_\{l\}\\neq\\lambda\_\{k\}, and likewise forS^−1\\widehat\{S\}^\{\-1\}\. In other wordsS^\\widehat\{S\}couples only coordinates belonging to the same eigenvalue\. This implies the following

1. \(i\)S^−1​𝐞j\\widehat\{S\}^\{\-1\}\\mathbf\{e\}\_\{j\}involves only the basis vectors𝐞i\\mathbf\{e\}\_\{i\}withλi=λj\\lambda\_\{i\}=\\lambda\_\{j\};
2. \(ii\)each factor\(S^​𝐳\)l=∑k:λk=λlS^l​k​zk\(\\widehat\{S\}\\mathbf\{z\}\)\_\{l\}=\\sum\_\{k:\\lambda\_\{k\}=\\lambda\_\{l\}\}\\widehat\{S\}\_\{lk\}z\_\{k\}involves only variableszkz\_\{k\}withλk=λl\\lambda\_\{k\}=\\lambda\_\{l\}, so every monomial𝐳𝐦′\\mathbf\{z\}^\{\\mathbf\{m\}^\{\\prime\}\}produced by expanding the product carries the same eigenvalue weight,𝐦′⋅𝝀=∑l=1dml​λl=𝐦⋅𝝀\\mathbf\{m\}^\{\\prime\}\\cdot\\bm\{\\lambda\}=\\sum\_\{l=1\}^\{d\}m\_\{l\}\\lambda\_\{l\}=\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\.

Hence every term𝐞i​𝐳𝐦′\\mathbf\{e\}\_\{i\}\\mathbf\{z\}^\{\\mathbf\{m\}^\{\\prime\}\}appearing inS^−1​𝐞j​\(S^​𝐳\)𝐦\\widehat\{S\}^\{\-1\}\\mathbf\{e\}\_\{j\}\(\\widehat\{S\}\\mathbf\{z\}\)^\{\\mathbf\{m\}\}has the same value for the resonance condition as the original term𝐞j​𝐳𝐦\\mathbf\{e\}\_\{j\}\\mathbf\{z\}^\{\\mathbf\{m\}\}:

𝐦′⋅𝝀−λi=𝐦⋅𝝀−λj,\\displaystyle\\mathbf\{m\}^\{\\prime\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{i\}=\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\},and therefore, for any contribution𝐞i​𝐳𝐦′\\mathbf\{e\}\_\{i\}\\mathbf\{z\}^\{\\mathbf\{m\}^\{\\prime\}\}to𝐧′\\mathbf\{n\}^\{\\prime\}or𝐭′\\mathbf\{t\}^\{\\prime\}, we have\(i,𝐦′\)∈ℐδ\(i,\\mathbf\{m\}^\{\\prime\}\)\\in\\mathcal\{I\}\_\{\\delta\}if and only if\(j,𝐦\)∈ℐδ\(j,\\mathbf\{m\}\)\\in\\mathcal\{I\}\_\{\\delta\}\. This means that the support of the Taylor expansion of𝐧′\\mathbf\{n\}^\{\\prime\}is exactly the same as that of𝐧\\mathbf\{n\}, and likewise for𝐭^′\\widehat\{\\mathbf\{t\}\}^\{\\prime\}and𝐭^\\widehat\{\\mathbf\{t\}\}\. Since the support of the Taylor expansions of𝐧\\mathbf\{n\}and𝐭\\mathbf\{t\}are disjoint, this implies the term\-wise coefficients which are obtained as solutions of \([8](https://arxiv.org/html/2608.04239#S3.E8)\) are exactly the same as those of𝐭\\mathbf\{t\}and𝐧\\mathbf\{n\}, so that𝐭′=𝐭\\mathbf\{t\}^\{\\prime\}=\\mathbf\{t\}and𝐧′=𝐧\\mathbf\{n\}^\{\\prime\}=\\mathbf\{n\}\. The result follows\. ∎

### 3\.2Taylor expansions of equivariant functions

We noted in the previous section that the central ingredient in the construction of equivariant SSM\-based reduced order models are functions \(𝐡,𝐫,𝐭,𝐧\\mathbf\{h\},\\mathbf\{r\},\\mathbf\{t\},\\mathbf\{n\}\) with integer power Taylor expansions which are equivariant with respect to a linear symmetry group\. In this section we examine the characterisation of the permissible terms in such an equivariant expansion as well as the reduction in degrees of freedom that can be achieved by incorporating the symmetry\. To begin with, we introduce the following notation:

###### Definition 3\.7\.

For a given integerk∈ℕk\\in\\mathbb\{N\}, we denote byϕk\(m\)​\(𝐱\)\\bm\{\\phi\}\_\{k\}^\{\(m\)\}\(\\mathbf\{x\}\)the vector of all monomials of degreekkin the variables𝐱=\(x1,…,xm\)\\mathbf\{x\}=\(x\_\{1\},\\dots,x\_\{m\}\), i\.e\.

ϕk\(m\)​\(𝐱\)=\(𝐱𝐣\)\|𝐣\|=k=\(x1k,x1k−1​x2,…,xmk\)⊤\.\\displaystyle\\bm\{\\phi\}\_\{k\}^\{\(m\)\}\(\\mathbf\{x\}\)=\(\\mathbf\{x\}^\{\\mathbf\{j\}\}\)\_\{\|\\mathbf\{j\}\|=k\}=\(x\_\{1\}^\{k\},x\_\{1\}^\{k\-1\}x\_\{2\},\\dots,x\_\{m\}^\{k\}\)^\{\\top\}\.

###### Theorem 3\.8\(Taylor expansion of equivariant functions\)\.

Suppose𝐠:ℝm→ℝn\\mathbf\{g\}:\\mathbb\{R\}^\{m\}\\rightarrow\\mathbb\{R\}^\{n\}is𝒞∞\\mathcal\{C\}^\{\\infty\}and has a convergent Taylor series with integer powers around𝟎\\mathbf\{0\},

𝐠​\(𝐱\)=∑k≥0𝐂k​ϕk\(m\)​\(𝐱\),\\displaystyle\\mathbf\{g\}\(\\mathbf\{x\}\)=\\sum\_\{k\\geq 0\}\\mathbf\{C\}\_\{k\}\\bm\{\\phi\}\_\{k\}^\{\(m\)\}\(\\mathbf\{x\}\),where𝐂k∈ℝn×Nk\\mathbf\{C\}\_\{k\}\\in\\mathbb\{R\}^\{n\\times N\_\{k\}\}collects the coefficients of the monomials of degreekkandNk=\(k\+m−1k\)N\_\{k\}=\\binom\{k\+m\-1\}\{k\}is the number of such monomials\. Suppose further that𝐠\\mathbf\{g\}is equivariant with respect to the compact symmetry groupℋ⊂GL​\(n,ℝ\)\\mathcal\{H\}\\subset\\mathrm\{GL\}\(n,\\mathbb\{R\}\), with Haar measureμ\\muand representationσ\\sigmainℝm\\mathbb\{R\}^\{m\}, i\.e\.

𝐠​\(σ​\(S\)​𝐱\)=S​𝐠​\(𝐱\),∀S∈ℋ,𝐱∈ℝm\.\\displaystyle\\mathbf\{g\}\(\\sigma\(S\)\\mathbf\{x\}\)=S\\mathbf\{g\}\(\\mathbf\{x\}\),\\qquad\\forall S\\in\\mathcal\{H\},\\ \\mathbf\{x\}\\in\\mathbb\{R\}^\{m\}\.\(10\)Then the Taylor coefficients𝐂k\\mathbf\{C\}\_\{k\}satisfy the following three properties:

1. \(i\)Linear constraints on the coefficients\. The equivariance condition \([10](https://arxiv.org/html/2608.04239#S3.E10)\) imposes linear constraints on the Taylor coefficients𝐂k\\mathbf\{C\}\_\{k\}, which can be expressed as S​𝐂k=𝐂k​𝐃k​\(S\),∀S∈ℋ,k≥0,\\displaystyle S\\mathbf\{C\}\_\{k\}=\\mathbf\{C\}\_\{k\}\\mathbf\{D\}\_\{k\}\(S\),\\qquad\\forall S\\in\\mathcal\{H\},\\ k\\geq 0,\(11\)where𝐃k​\(S\)\\mathbf\{D\}\_\{k\}\(S\)is the representation ofℋ\\mathcal\{H\}on the space of homogeneous polynomials of degreekkinℝm\\mathbb\{R\}^\{m\}as defined in \([14](https://arxiv.org/html/2608.04239#S3.E14)\)\.
2. \(ii\)The admissible coefficients in the Taylor expansion of𝐠\\mathbf\{g\}are given precisely by 𝐂k∈⋂S∈ℋker⁡\(𝐌k​\[S\]\),\\displaystyle\\mathbf\{C\}\_\{k\}\\in\\bigcap\_\{S\\in\\mathcal\{H\}\}\\operatorname\{ker\}\(\\mathbf\{M\}\_\{k\}\[S\]\),where𝐌k​\[S\]\\mathbf\{M\}\_\{k\}\[S\]is the linear operator defined by 𝐌k​\[S\]​\(𝐂\):=S​𝐂k−𝐂k​𝐃k​\(S\)\.\\displaystyle\\mathbf\{M\}\_\{k\}\[S\]\(\\mathbf\{C\}\):=S\\mathbf\{C\}\_\{k\}\-\\mathbf\{C\}\_\{k\}\\mathbf\{D\}\_\{k\}\(S\)\.
3. \(iii\)Projection onto the admissible space\. Let us define the group\-averaging operatorℛk:ℝn×Nk→ℝn×Nk\\mathcal\{R\}\_\{k\}:\\mathbb\{R\}^\{n\\times N\_\{k\}\}\\rightarrow\\mathbb\{R\}^\{n\\times N\_\{k\}\}, ℛk​\(𝐂\):=∫ℋS​𝐂𝐃k​\(S\)−1​dμ​\(S\),\\displaystyle\\mathcal\{R\}\_\{k\}\(\\mathbf\{C\}\):=\\int\_\{\\mathcal\{H\}\}S\\mathbf\{C\}\\mathbf\{D\}\_\{k\}\(S\)^\{\-1\}\\mathrm\{d\}\\mu\(S\),\(12\)which for finite groupsℋ\\mathcal\{H\}simplifies toℛk=1\|ℋ\|​∑S∈ℋS​𝐂𝐃k​\(S\)−1\\mathcal\{R\}\_\{k\}=\\frac\{1\}\{\|\\mathcal\{H\}\|\}\\sum\_\{S\\in\\mathcal\{H\}\}S\\mathbf\{C\}\\mathbf\{D\}\_\{k\}\(S\)^\{\-1\}\. Thenℛk\\mathcal\{R\}\_\{k\}is a linear projection \(ℛk∘ℛk=ℛk\\mathcal\{R\}\_\{k\}\\circ\\mathcal\{R\}\_\{k\}=\\mathcal\{R\}\_\{k\}\), whose image is exactly the set of admissible coefficients im​ℛk=⋂S∈ℋker⁡\(𝐌k​\[S\]\)\.\\displaystyle\\mathrm\{im\}\\mathcal\{R\}\_\{k\}=\\bigcap\_\{S\\in\\mathcal\{H\}\}\\operatorname\{ker\}\(\\mathbf\{M\}\_\{k\}\[S\]\)\.

###### Proof of Theorem[3\.8](https://arxiv.org/html/2608.04239#S3.Thmtheorem8)\.

Let us begin by proving property \(i\)\.By linearity of the action ofℋ\\mathcal\{H\}, the equivariance condition \([10](https://arxiv.org/html/2608.04239#S3.E10)\) must hold at any order in the Taylor series meaning we have, for everyk≥0k\\geq 0,

𝐂k​ϕk\(m\)​\(σ​\(S\)​𝐱\)=S​𝐂k​ϕk\(m\)​\(𝐱\)\.\\displaystyle\\mathbf\{C\}\_\{k\}\\bm\{\\phi\}^\{\(m\)\}\_\{k\}\(\\sigma\(S\)\\mathbf\{x\}\)=S\\mathbf\{C\}\_\{k\}\\bm\{\\phi\}\_\{k\}^\{\(m\)\}\(\\mathbf\{x\}\)\.Let us denote by𝐃k​\(S\)\\mathbf\{D\}\_\{k\}\(S\)the representation of the action ofℋ\\mathcal\{H\}on the space of homogeneous polynomials of degreekkinℝm\\mathbb\{R\}^\{m\}, i\.e\.

𝐃k​\(S\)​ϕk\(m\)​\(𝐱\)=ϕk\(m\)​\(σ​\(S\)​𝐱\)\.\\displaystyle\\mathbf\{D\}\_\{k\}\(S\)\\bm\{\\phi\}\_\{k\}^\{\(m\)\}\(\\mathbf\{x\}\)=\\bm\{\\phi\}\_\{k\}^\{\(m\)\}\(\\sigma\(S\)\\mathbf\{x\}\)\.\(14\)Then the above equation can be rewritten as

𝐂k​𝐃k​\(S\)​ϕk\(m\)​\(𝐱\)=S​𝐂k​ϕk\(m\)​\(𝐱\),\\displaystyle\\mathbf\{C\}\_\{k\}\\mathbf\{D\}\_\{k\}\(S\)\\bm\{\\phi\}\_\{k\}^\{\(m\)\}\(\\mathbf\{x\}\)=S\\mathbf\{C\}\_\{k\}\\bm\{\\phi\}\_\{k\}^\{\(m\)\}\(\\mathbf\{x\}\),which must hold for all𝐱∈ℝm\\mathbf\{x\}\\in\\mathbb\{R\}^\{m\}, i\.e\. we must have

S​𝐂k=𝐂k​𝐃k​\(S\),∀S∈ℋ,k≥0,\\displaystyle S\\mathbf\{C\}\_\{k\}=\\mathbf\{C\}\_\{k\}\\mathbf\{D\}\_\{k\}\(S\),\\qquad\\forall S\\in\\mathcal\{H\},\\ k\\geq 0,which is exactly the statement of property \(i\)\.

Property \(ii\) then immediately follows, since the admissible coefficients are precisely those that satisfy the linear constraints in property \(i\)\.

Finally, we prove property \(iii\)\.Consider the linear mapρk​\(S\):ℝn×Nk→ℝn×Nk\\rho\_\{k\}\(S\):\\mathbb\{R\}^\{n\\times N\_\{k\}\}\\to\\mathbb\{R\}^\{n\\times N\_\{k\}\}on the space of Taylor coefficients of degreekk, given byρk​\(S\)​𝐂:=S​𝐂​𝐃k​\(S\)−1\\rho\_\{k\}\(S\)\\mathbf\{C\}:=S\\,\\mathbf\{C\}\\,\\mathbf\{D\}\_\{k\}\(S\)^\{\-1\}\. Since𝐃k\\mathbf\{D\}\_\{k\}is a representation ofℋ\\mathcal\{H\}we have, for anyS,T∈ℋS,T\\in\\mathcal\{H\},𝐃k​\(S\)​𝐃k​\(T\)=𝐃k​\(S​T\)\\mathbf\{D\}\_\{k\}\(S\)\\mathbf\{D\}\_\{k\}\(T\)=\\mathbf\{D\}\_\{k\}\(ST\), and thus, for any𝐂∈𝐑n×Nk\\mathbf\{C\}\\in\\mathbf\{R\}^\{n\\times N\_\{k\}\},

ρk​\(S\)​ρk​\(T\)​𝐂=S​T​𝐂​𝐃k​\(T\)−1​𝐃k​\(S\)−1=\(S​T\)​𝐂​𝐃k​\(S​T\)−1=ρk​\(S​T\)​𝐂,\\displaystyle\\rho\_\{k\}\(S\)\\rho\_\{k\}\(T\)\\mathbf\{C\}=S\\,T\\,\\mathbf\{C\}\\,\\mathbf\{D\}\_\{k\}\(T\)^\{\-1\}\\mathbf\{D\}\_\{k\}\(S\)^\{\-1\}=\(ST\)\\,\\mathbf\{C\}\\,\\mathbf\{D\}\_\{k\}\(ST\)^\{\-1\}=\\rho\_\{k\}\(ST\)\\mathbf\{C\},soρk\\rho\_\{k\}is a representation ofℋ\\mathcal\{H\}on the coefficient space\. By property \(i\) a coefficient𝐂k\\mathbf\{C\}\_\{k\}is admissible if and only ifρk​\(S\)​𝐂k=𝐂k\\rho\_\{k\}\(S\)\\mathbf\{C\}\_\{k\}=\\mathbf\{C\}\_\{k\}for allS∈ℋS\\in\\mathcal\{H\}, i\.e\. if and only if it is a fixed point for allρk​\(S\)\\rho\_\{k\}\(S\), i\.e\.𝐂k∈Fix⁡\(ρk\)\\mathbf\{C\}\_\{k\}\\in\\operatorname\{Fix\}\(\\rho\_\{k\}\)\. Let us now considerℛk​\(𝐂\)\\mathcal\{R\}\_\{k\}\(\\mathbf\{C\}\)\. We have, for anyT∈ℋT\\in\\mathcal\{H\}

ρk​\(T\)​ℛk​\(𝐂\)\\displaystyle\\rho\_\{k\}\(T\)\\mathcal\{R\}\_\{k\}\(\\mathbf\{C\}\)=T​∫ℋS​𝐂𝐃k​\(S\)−1​dμ​\(S\)​𝐃k​\(T\)−1\\displaystyle=T\\int\_\{\\mathcal\{H\}\}S\\mathbf\{C\}\\mathbf\{D\}\_\{k\}\(S\)^\{\-1\}\\mathrm\{d\}\\mu\(S\)\\mathbf\{D\}\_\{k\}\(T\)^\{\-1\}=∫ℋ\(T​S\)​𝐂𝐃k​\(T​S\)−1​dμ​\(S\)=ℛk​\(𝐂\),\\displaystyle=\\int\_\{\\mathcal\{H\}\}\(TS\)\\mathbf\{C\}\\mathbf\{D\}\_\{k\}\(TS\)^\{\-1\}\\mathrm\{d\}\\mu\(S\)=\\mathcal\{R\}\_\{k\}\(\\mathbf\{C\}\),where in the final equality we reindexedS↦S​TS\\mapsto STusing the group invariance property of the Haar measure\. Thusℛk​\(𝐂\)\\mathcal\{R\}\_\{k\}\(\\mathbf\{C\}\)is fixed byρk​\(T\)\\rho\_\{k\}\(T\)for allT∈ℋT\\in\\mathcal\{H\}, and henceim⁡ℛk⊆Fix⁡\(ρk\)\\operatorname\{im\}\\mathcal\{R\}\_\{k\}\\subseteq\\operatorname\{Fix\}\(\\rho\_\{k\}\)\. Conversely, if𝐂∈Fix⁡\(ρk\)\\mathbf\{C\}\\in\\operatorname\{Fix\}\(\\rho\_\{k\}\)thenρk​\(S\)​𝐂=𝐂\\rho\_\{k\}\(S\)\\mathbf\{C\}=\\mathbf\{C\}for everySSand

ℛk​\(𝐂\)=∫ℋρk​\(S\)​𝐂​dμ​\(S\)=∫ℋ𝐂​dμ​\(S\)=𝐂,\\displaystyle\\mathcal\{R\}\_\{k\}\(\\mathbf\{C\}\)=\\int\_\{\\mathcal\{H\}\}\\rho\_\{k\}\(S\)\\mathbf\{C\}\\mathrm\{d\}\\mu\(S\)=\\int\_\{\\mathcal\{H\}\}\\mathbf\{C\}\\mathrm\{d\}\\mu\(S\)=\\mathbf\{C\},thusℛk\\mathcal\{R\}\_\{k\}fixesFix⁡\(ρk\)\\operatorname\{Fix\}\(\\rho\_\{k\}\)pointwise, i\.e\.Fix⁡\(ρk\)⊆im⁡ℛk\\operatorname\{Fix\}\(\\rho\_\{k\}\)\\subseteq\\operatorname\{im\}\\mathcal\{R\}\_\{k\}andℛk2=ℛk\\mathcal\{R\}\_\{k\}^\{2\}=\\mathcal\{R\}\_\{k\}\. Thusℛk\\mathcal\{R\}\_\{k\}is a projection withim⁡ℛk=Fix⁡\(ρk\)=⋂Sker⁡𝐌k​\[S\]\\operatorname\{im\}\\mathcal\{R\}\_\{k\}=\\operatorname\{Fix\}\(\\rho\_\{k\}\)=\\bigcap\_\{S\}\\operatorname\{ker\}\\mathbf\{M\}\_\{k\}\[S\], and𝐂k\\mathbf\{C\}\_\{k\}is admissible iff𝐂k=ℛk​\(𝐂k\)\\mathbf\{C\}\_\{k\}=\\mathcal\{R\}\_\{k\}\(\\mathbf\{C\}\_\{k\}\)\. ∎

Theorem[3\.8](https://arxiv.org/html/2608.04239#S3.Thmtheorem8)can be used to count the number of independent coefficients in the Taylor expansion of an equivariant function:

###### Corollary 3\.11\.

Letℋ\\mathcal\{H\}be a finite group, then the number of admissible coefficients in the Taylor expansion of an equivariant function at degreekk,

pk:=dim\(⋂S∈ℋker⁡𝐌k​\[S\]\),\\displaystyle p\_\{k\}:=\\dim\\\!\\Big\(\\bigcap\_\{S\\in\\mathcal\{H\}\}\\operatorname\{ker\}\\mathbf\{M\}\_\{k\}\[S\]\\Big\),can be characterised by:

pk=tr⁡ℛk=1\|ℋ\|​∑S∈ℋtr⁡\(S\)​tr⁡\(𝐃k​\(S\)−1\),\\displaystyle p\_\{k\}\\;=\\;\\operatorname\{tr\}\\mathcal\{R\}\_\{k\}\\;=\\;\\frac\{1\}\{\|\\mathcal\{H\}\|\}\\sum\_\{S\\in\\mathcal\{H\}\}\\operatorname\{tr\}\(S\)\\,\\operatorname\{tr\}\\\!\\big\(\\mathbf\{D\}\_\{k\}\(S\)^\{\-1\}\\big\),whereℛk\\mathcal\{R\}\_\{k\}is the operator defined in \([12](https://arxiv.org/html/2608.04239#S3.E12)\)\.

###### Proof\.

We note thatℛk\\mathcal\{R\}\_\{k\}is a projection onto⋂Sker⁡𝐌k​\[S\]\\bigcap\_\{S\}\\operatorname\{ker\}\\mathbf\{M\}\_\{k\}\[S\]by Theorem[3\.8](https://arxiv.org/html/2608.04239#S3.Thmtheorem8)\(iii\), sorank⁡ℛk=dim\(im⁡ℛk\)=pk\\operatorname\{rank\}\\mathcal\{R\}\_\{k\}=\\dim\(\\operatorname\{im\}\\mathcal\{R\}\_\{k\}\)=p\_\{k\}and, being idempotent,rank⁡ℛk=tr⁡ℛk\\operatorname\{rank\}\\mathcal\{R\}\_\{k\}=\\operatorname\{tr\}\\mathcal\{R\}\_\{k\}\. The final expression forpkp\_\{k\}follows from the fact thattr⁡\(A⊗B\)=tr⁡\(A\)​tr⁡\(B\)\\operatorname\{tr\}\(A\\otimes B\)=\\operatorname\{tr\}\(A\)\\operatorname\{tr\}\(B\)and that, in vectorised form,vec⁡\(ρk​\(S\)​𝐂\)=\(𝐃k​\(S\)−⊤⊗S\)​vec⁡\(𝐂\)\\operatorname\{vec\}\(\\rho\_\{k\}\(S\)\\mathbf\{C\}\)=\(\\mathbf\{D\}\_\{k\}\(S\)^\{\-\\top\}\\otimes S\)\\operatorname\{vec\}\(\\mathbf\{C\}\), for anyS∈ℋS\\in\\mathcal\{H\}\. ∎

###### Example 3\.12\(Effect of equivariance on the damped oscillator chain\)\.

We continue the set\-up of Example[2\.6](https://arxiv.org/html/2608.04239#S2.Thmtheorem6)which is equivariant under the symmetry group𝒢=\{𝐈,−𝐈\}\\mathcal\{G\}=\\\{\\mathbf\{I\},\-\\mathbf\{I\}\\\}\. LettingEEbe a spectral subspace with associated SSM𝒲​\(E\)\\mathcal\{W\}\(E\), the chart\-equivariant parametrisation𝐡\\mathbf\{h\}\(Prop\.[3\.4](https://arxiv.org/html/2608.04239#S3.Thmtheorem4)\) must satisfy𝐡​\(−𝐳\)=−𝐡​\(𝐳\)\\mathbf\{h\}\(\-\\mathbf\{z\}\)=\-\\mathbf\{h\}\(\\mathbf\{z\}\)\. Moreover, we see that

𝐃k​\(−I\)=\(−1\)k​𝐈Nk,\\displaystyle\\mathbf\{D\}\_\{k\}\(\-I\)=\(\-1\)^\{k\}\\mathbf\{I\}\_\{N\_\{k\}\},thus

ℛk​\(𝐂\)=12​\(𝐂\+\(−1\)k\+1​𝐂\)=\{𝐂,k​even,0,k​odd\.\\displaystyle\\mathcal\{R\}\_\{k\}\(\\mathbf\{C\}\)=\\frac\{1\}\{2\}\\big\(\\mathbf\{C\}\+\(\-1\)^\{k\+1\}\\mathbf\{C\}\\big\)=\\begin\{cases\}\\mathbf\{C\},&k\\text\{ even\},\\\\ 0,&k\\text\{ odd\}\.\\end\{cases\}This means that the equivariant Taylor expansion of𝐡\\mathbf\{h\}contains only odd\-degree monomials, and thus the number of degrees of freedom in the Taylor expansion is significantly reduced\. Note that this “pruning” of monomials is a special feature of the sign symmetry𝒢=⟨−𝐈⟩\\mathcal\{G\}=\\langle\-\\mathbf\{I\}\\rangle; in general, the kernel of𝐌k\\mathbf\{M\}\_\{k\}need not be a coordinate subspace, and the admissible coefficients are not obtained by simply discarding certain monomials from the expansion\. Instead, we find a reparametrisationvec⁡\(𝐖k\)=𝐁k​𝐰k\\operatorname\{vec\}\(\\mathbf\{W\}\_\{k\}\)=\\mathbf\{B\}\_\{k\}\\mathbf\{w\}\_\{k\}to reduce the dimensionality of the fitting problem, as described in further detail in §[4\.2](https://arxiv.org/html/2608.04239#S4.SS2)\.

## 4Data\-driven SSM reduction with equivariance

We now show how we can use the above theory to develop a data\-driven SSM reduction method that preserves equivariance\. The key idea is to incorporate the restrictions obtained in §[3\.2](https://arxiv.org/html/2608.04239#S3.SS2)into the optimisation problem for computing the reduced dynamics from data, which leads to a smaller parameter space and thus a more efficient and robust algorithm\.

For the data\-driven eSSM reduction we assume that we have access to trajectories sampled uniformly in time,𝐲k\(j\)∈ℝn\\mathbf\{y\}\_\{k\}^\{\(j\)\}\\in\\mathbb\{R\}^\{n\},k=1,…,Njk=1,\\dots,N\_\{j\},j=1,…,Jj=1,\\dots,J, generated by the underlying system \([1](https://arxiv.org/html/2608.04239#S2.E1)\),

𝐲k\(j\)=𝐱​\(tk;𝐱0\(j\)\),tk=\(k−1\)​Δ​t,k=1,…,Nj,\\displaystyle\\mathbf\{y\}\_\{k\}^\{\(j\)\}=\\mathbf\{x\}\(t\_\{k\};\\mathbf\{x\}\_\{0\}^\{\(j\)\}\),\\qquad t\_\{k\}=\(k\-1\)\\Delta t,\\quad k=1,\\dots,N\_\{j\},\(15\)where𝐱​\(t;𝐱0\)\\mathbf\{x\}\(t;\\mathbf\{x\}\_\{0\}\)is the solution of \([1](https://arxiv.org/html/2608.04239#S2.E1)\) with initial condition𝐱0\\mathbf\{x\}\_\{0\}at timett\. For notational simplicity, we will restrict the presentation of the algorithm to the case of a single trajectory of data, i\.e\.J=1J=1\. Information from multiple trajectories can be incorporated into the algorithm by stacking the data matrices:

𝐘=\[𝐘\(1\),…,𝐘\(J\)\]∈ℝn×∑j=1JNj,where​𝐘\(j\)=\[𝐲1\(j\),…,𝐲Nj\(j\)\]∈ℝn×Nj,\\displaystyle\\mathbf\{Y\}=\[\\mathbf\{Y\}^\{\(1\)\},\\dots,\\mathbf\{Y\}^\{\(J\)\}\]\\in\\mathbb\{R\}^\{n\\times\\sum\_\{j=1\}^\{J\}N\_\{j\}\},\\quad\\text\{where \}\\mathbf\{Y\}^\{\(j\)\}=\[\\mathbf\{y\}\_\{1\}^\{\(j\)\},\\dots,\\mathbf\{y\}\_\{N\_\{j\}\}^\{\(j\)\}\]\\in\\mathbb\{R\}^\{n\\times N\_\{j\}\},taking care to apply any finite difference approximations of time derivatives per trajectory separately\. In addition, we assume that we have knowledge of thefinitesymmetry group𝒢\\mathcal\{G\}of the system\.

### 4\.1Delay embedding and equivariance

In applications when observations of the full state𝐱∈ℝn\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\}are not available, it is common practice in SSM reduction and similar data\-driven methods to rely on Takens’ embedding theorem\[[53](https://arxiv.org/html/2608.04239#bib.bib49),[19](https://arxiv.org/html/2608.04239#bib.bib51),[50](https://arxiv.org/html/2608.04239#bib.bib50)\]to reconstruct a suitable state space using delay embedding of lower\-dimensional observations\[[15](https://arxiv.org/html/2608.04239#bib.bib1),[3](https://arxiv.org/html/2608.04239#bib.bib37)\]\. In particular, for lower\-dimensional observations𝐇​\(𝐱\)∈ℝm\\mathbf\{H\}\(\\mathbf\{x\}\)\\in\\mathbb\{R\}^\{m\},m<nm<n, a delay embedding is typically constructed as

𝚽​\(𝐱\)=\(𝐇​\(𝐱\),𝐇​\(φτ​\(𝐱\)\),…,𝐇​\(φ\(p−1\)​τ​\(𝐱\)\)\)∈ℝm​p,\\displaystyle\\bm\{\\Phi\}\(\\mathbf\{x\}\)=\\big\(\\mathbf\{H\}\(\\mathbf\{x\}\),\\,\\mathbf\{H\}\(\\varphi\_\{\\tau\}\(\\mathbf\{x\}\)\),\\,\\dots,\\,\\mathbf\{H\}\(\\varphi\_\{\(p\-1\)\\tau\}\(\\mathbf\{x\}\)\)\\big\)\\in\\mathbb\{R\}^\{mp\},\(16\)whereφt\\varphi\_\{t\}denotes the flow map of \([1](https://arxiv.org/html/2608.04239#S2.E1)\),τ\>0\\tau\>0is a delay \(in practice an integer multiple of the sampling intervalΔ​t\\Delta t\), andp∈ℕp\\in\\mathbb\{N\}is the number of delays\. By Takens’ embedding theorem\[[53](https://arxiv.org/html/2608.04239#bib.bib49)\]and its extensions\[[50](https://arxiv.org/html/2608.04239#bib.bib50)\], for generic observables andm​p≥2​dim𝒲​\(E\)\+1mp\\geq 2\\dim\\mathcal\{W\}\(E\)\+1the map𝚽\\bm\{\\Phi\}restricts to an embedding of𝒲​\(E\)\\mathcal\{W\}\(E\), so that the eSSM reduction can be performed on the delay\-embedded data\.

We note that, if the observable𝐇\\mathbf\{H\}is equivariant with respect to the symmetry group𝒢\\mathcal\{G\}, then the delay embedding𝚽\\bm\{\\Phi\}is also equivariant with respect to a lifted representation of𝒢\\mathcal\{G\}on the delay\-embedded space\. This means that the equivariance properties of the original system are preserved under delay embedding, allowing for the application of equivariant SSM reduction techniques even when only lower\-dimensional observations are available\. This statement is made precise in the following result\.

###### Proposition 4\.2\.

Let \([1](https://arxiv.org/html/2608.04239#S2.E1)\) be equivariant with respect to the linear symmetry group𝒢⊂G​L​\(n,ℝ\)\\mathcal\{G\}\\subset GL\(n,\\mathbb\{R\}\), and suppose the observable𝐇:ℝn→ℝm\\mathbf\{H\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}^\{m\}is𝒢\\mathcal\{G\}\-covariant, i\.e\. there is a linear representationσ:𝒢→G​L​\(m,ℝ\)\\sigma:\\mathcal\{G\}\\to GL\(m,\\mathbb\{R\}\)such that𝐇​\(S​𝐱\)=σ​\(S\)​𝐇​\(𝐱\)\\mathbf\{H\}\(S\\mathbf\{x\}\)=\\sigma\(S\)\\mathbf\{H\}\(\\mathbf\{x\}\)for allS∈𝒢S\\in\\mathcal\{G\},𝐱∈ℝn\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\}\. Then

𝚽​\(S​𝐱\)=\(𝐈p⊗σ​\(S\)\)​𝚽​\(𝐱\),S∈𝒢,𝐱∈ℝn,\\displaystyle\\bm\{\\Phi\}\(S\\mathbf\{x\}\)=\\big\(\\mathbf\{I\}\_\{p\}\\otimes\\sigma\(S\)\\big\)\\,\\bm\{\\Phi\}\(\\mathbf\{x\}\),\\qquad S\\in\\mathcal\{G\},\\ \\mathbf\{x\}\\in\\mathbb\{R\}^\{n\},\(17\)i\.e\.𝒢\\mathcal\{G\}acts on the delay\-embedded spaceℝm​p\\mathbb\{R\}^\{mp\}by the block\-diagonal representationσ~=𝐈p⊗σ\\widetilde\{\\sigma\}=\\mathbf\{I\}\_\{p\}\\otimes\\sigma, and the dynamics induced on𝚽​\(ℝn\)\\bm\{\\Phi\}\(\\mathbb\{R\}^\{n\}\)are equivariant with respect toσ~​\(𝒢\)\\widetilde\{\\sigma\}\(\\mathcal\{G\}\)\.

###### Proof\.

Since \([1](https://arxiv.org/html/2608.04239#S2.E1)\) is𝒢\\mathcal\{G\}\-equivariant, so is the flow mapφt\\varphi\_\{t\}\(see proof of Theorem[2\.7](https://arxiv.org/html/2608.04239#S2.Thmtheorem7)\)\. Thus, each block of \([16](https://arxiv.org/html/2608.04239#S4.E16)\) satisfies

𝐇​\(φk​τ​\(S​𝐱\)\)=𝐇​\(S​φk​τ​\(𝐱\)\)=σ​\(S\)​𝐇​\(φk​τ​\(𝐱\)\),k=0,…,p−1,\\displaystyle\\mathbf\{H\}\\big\(\\varphi\_\{k\\tau\}\(S\\mathbf\{x\}\)\\big\)=\\mathbf\{H\}\\big\(S\\varphi\_\{k\\tau\}\(\\mathbf\{x\}\)\\big\)=\\sigma\(S\)\\,\\mathbf\{H\}\\big\(\\varphi\_\{k\\tau\}\(\\mathbf\{x\}\)\\big\),\\qquad k=0,\\dots,p\-1,which completes the proof of \([17](https://arxiv.org/html/2608.04239#S4.E17)\)\. In particular, if𝐱​\(t\)\\mathbf\{x\}\(t\)solves \([1](https://arxiv.org/html/2608.04239#S2.E1)\), then so doesS​𝐱​\(t\)S\\mathbf\{x\}\(t\), and by \([17](https://arxiv.org/html/2608.04239#S4.E17)\) the corresponding embedded trajectories are𝚽​\(𝐱​\(t\)\)\\bm\{\\Phi\}\(\\mathbf\{x\}\(t\)\)andσ~​\(S\)​𝚽​\(𝐱​\(t\)\)\\widetilde\{\\sigma\}\(S\)\\bm\{\\Phi\}\(\\mathbf\{x\}\(t\)\)\. Henceσ~​\(S\)\\widetilde\{\\sigma\}\(S\)maps𝚽​\(ℝn\)\\bm\{\\Phi\}\(\\mathbb\{R\}^\{n\}\)into itself and sends embedded trajectories to embedded trajectories, i\.e\. the dynamics induced on𝚽​\(ℝn\)\\bm\{\\Phi\}\(\\mathbb\{R\}^\{n\}\)are equivariant with respect toσ~​\(S\)\\widetilde\{\\sigma\}\(S\)for everyS∈𝒢S\\in\\mathcal\{G\}\. ∎

### 4\.2Algorithmic details of the eSSM reduction method

We formulate the equivariant SSM reduction \(eSSM\) algorithm as an extension of the SSMLearn method introduced in\[[15](https://arxiv.org/html/2608.04239#bib.bib1)\]\. Our algorithm consists of two main steps: \(i\) identification and parametrisation of the spectral submanifold𝒲​\(E\)\\mathcal\{W\}\(E\); and \(ii\) computation of the reduced dynamics on the SSM in extended normal form style\. Each of these steps is described in detail below, along with the interaction of the symmetry group𝒢\\mathcal\{G\}with each step\. Our trajectory data𝐘=\[𝐲1,…,𝐲N\]∈ℝn×N\\mathbf\{Y\}=\[\\mathbf\{y\}\_\{1\},\\dots,\\mathbf\{y\}\_\{N\}\]\\in\\mathbb\{R\}^\{n\\times N\}is assumed to be generated from the underlying system \([1](https://arxiv.org/html/2608.04239#S2.E1)\) as in \([15](https://arxiv.org/html/2608.04239#S4.E15)\), and we are only able to access the data𝐘\\mathbf\{Y\}and the symmetry group𝒢\\mathcal\{G\}, but not the underlying system \([1](https://arxiv.org/html/2608.04239#S2.E1)\) itself\.

#### Step \(i\): Identification and parametrisation of the spectral submanifold𝒲​\(E\)\\mathcal\{W\}\(E\)from data

Our ultimate goal in Step \(i\) will be to identify equivariant𝐔1,𝐕1\\mathbf\{U\}\_\{1\},\\mathbf\{V\}\_\{1\}and𝐡\\mathbf\{h\}from data such that the following equivariant least\-squares objective is minimised:

∑i=1N‖𝐲i−𝐔1​𝜼i−𝐡​\(𝜼i\)‖𝒢2,where​𝜼i=𝐕1⊤​𝐲i,\\displaystyle\\sum\_\{i=1\}^\{N\}\\big\\\|\\,\{\\mathbf\{y\}\}\_\{i\}\-\{\\mathbf\{U\}\}\_\{1\}\\bm\{\\eta\}\_\{i\}\-\\mathbf\{h\}\(\\bm\{\\eta\}\_\{i\}\)\\,\\big\\\|\_\{\\mathcal\{G\}\}^\{2\},\\,\\,\\text\{where\}\\,\\,\\bm\{\\eta\}\_\{i\}=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}\_\{i\},\(18\)together with the graph parametrisation

𝐲=𝐔1​𝜼\+𝐡​\(𝜼\),𝐡​\(𝜼\)=∑k=2M𝐖k​ϕk​\(𝜼\),𝐕1⊤​𝐖k=0,k=2,…,M\.\\displaystyle\\mathbf\{y\}=\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{h\}\(\\bm\{\\eta\}\),\\qquad\\mathbf\{h\}\(\\bm\{\\eta\}\)=\\sum\_\{k=2\}^\{M\}\\mathbf\{W\}\_\{k\}\\,\\bm\{\\phi\}\_\{k\}\(\\bm\{\\eta\}\),\\quad\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{W\}\_\{k\}=0,\\quad k=2,\\dots,M\.\(19\)To begin with, we note that the equivariant inner product⟨⋅,⋅⟩𝒢\\langle\\cdot,\\cdot\\rangle\_\{\\mathcal\{G\}\}can be efficiently computed using the Gram matrix

𝐏𝒢=1\|𝒢\|​∑S∈𝒢S⊤​S\.\\displaystyle\\mathbf\{P\}\_\{\\mathcal\{G\}\}=\\frac\{1\}\{\|\\mathcal\{G\}\|\}\\sum\_\{S\\in\\mathcal\{G\}\}S^\{\\top\}S\.In particular, if we let𝐏𝒢=𝐋𝐋⊤\\mathbf\{P\}\_\{\\mathcal\{G\}\}=\\mathbf\{L\}\\mathbf\{L\}^\{\\top\}be the Cholesky factorisation of𝐏𝒢\\mathbf\{P\}\_\{\\mathcal\{G\}\}, then we can compute the equivariant inner product as⟨𝐯,𝐰⟩𝒢=⟨𝐋⊤​𝐯,𝐋⊤​𝐰⟩\\langle\\mathbf\{v\},\\mathbf\{w\}\\rangle\_\{\\mathcal\{G\}\}=\\langle\\mathbf\{L\}^\{\\top\}\\mathbf\{v\},\\mathbf\{L\}^\{\\top\}\\mathbf\{w\}\\rangle\. We refer to the coordinates𝐲~i=𝐋⊤​𝐲i\\widetilde\{\\mathbf\{y\}\}\_\{i\}=\\mathbf\{L\}^\{\\top\}\\mathbf\{y\}\_\{i\}as the whitened coordinates, and we denote the corresponding whitened data matrix as𝐘~=\[𝐲~1,…,𝐲~N\]\\widetilde\{\\mathbf\{Y\}\}=\[\\widetilde\{\\mathbf\{y\}\}\_\{1\},\\dots,\\widetilde\{\\mathbf\{y\}\}\_\{N\}\]\. In these coordinates, the least\-squares objective \([18](https://arxiv.org/html/2608.04239#S4.E18)\) can be rewritten as

∑i=1N‖𝐲~i−𝐔~1​𝜼i−𝐡~​\(𝜼i\)‖2,where​𝜼i=𝐕~1⊤​𝐲~i,\\displaystyle\\sum\_\{i=1\}^\{N\}\\big\\\|\\,\\widetilde\{\\mathbf\{y\}\}\_\{i\}\-\\widetilde\{\\mathbf\{U\}\}\_\{1\}\\bm\{\\eta\}\_\{i\}\-\\widetilde\{\\mathbf\{h\}\}\(\\bm\{\\eta\}\_\{i\}\)\\,\\big\\\|^\{2\},\\,\\,\\text\{where\}\\,\\,\\bm\{\\eta\}\_\{i\}=\\widetilde\{\\mathbf\{V\}\}\_\{1\}^\{\\top\}\\widetilde\{\\mathbf\{y\}\}\_\{i\},\(20\)where𝐔~1=𝐋⊤​𝐔1\\widetilde\{\\mathbf\{U\}\}\_\{1\}=\\mathbf\{L\}^\{\\top\}\\mathbf\{U\}\_\{1\}and𝐕~1=𝐋−1​𝐕1\\widetilde\{\\mathbf\{V\}\}\_\{1\}=\\mathbf\{L\}^\{\-1\}\\mathbf\{V\}\_\{1\}are the whitened versions of𝐔1\\mathbf\{U\}\_\{1\}and𝐕1\\mathbf\{V\}\_\{1\}, and𝐡~​\(𝜼\)=𝐋⊤​𝐡​\(𝜼\)\\widetilde\{\\mathbf\{h\}\}\(\\bm\{\\eta\}\)=\\mathbf\{L\}^\{\\top\}\\mathbf\{h\}\(\\bm\{\\eta\}\)is the whitened version of𝐡\\mathbf\{h\}\. The advantage of this change of coordinates is that we can now work with the standard inner product in the whitened coordinates, which allows us to use standard SVD to solve linear∥⋅∥𝒢\\\|\\cdot\\\|\_\{\\mathcal\{G\}\}\-least\-squares problems\.

#### Step \(i\.1\): Symmetry\-adapted initialisation: fixing the similarity class ofEE

To begin with, a central property that we want to incorporate in our parametrisation of𝒲​\(E\)\\mathcal\{W\}\(E\)is the𝒢\\mathcal\{G\}\-invariance ofEE\(cf\. Theorem[2\.7](https://arxiv.org/html/2608.04239#S2.Thmtheorem7)\)\. The identification of such an invariantEEnecessarily involves a discrete choice which we fix in the symmetry adapted initialisation\. In particular, we will show in the following thatdd\-dimensional𝒢\\mathcal\{G\}\-invariant subspaces ofℝn\\mathbb\{R\}^\{n\}can be characterised through similarity classes of the representation of𝒢\\mathcal\{G\}restricted to the corresponding subspace\. Let us make this more precise: for any𝒢\\mathcal\{G\}\-invariantdd\-dimensional subspaceF⊆ℝnF\\subseteq\\mathbb\{R\}^\{n\}with basis matrix𝐔∈ℝn×d\\mathbf\{U\}\\in\\mathbb\{R\}^\{n\\times d\}\(full column rank\), the columns ofS​𝐔S\\mathbf\{U\}lie again inFF, so that

S​𝐔=𝐔​𝐑𝐔​\(S\),with​𝐑𝐔​\(S\)=\(𝐔⊤​𝐔\)−1​𝐔⊤​S​𝐔∈G​L​\(d,ℝ\),S∈𝒢,\\displaystyle S\\mathbf\{U\}=\\mathbf\{U\}\\,\\mathbf\{R\}\_\{\\mathbf\{U\}\}\(S\),\\,\\,\\text\{with\\,\\,\}\\mathbf\{R\}\_\{\\mathbf\{U\}\}\(S\)=\\big\(\\mathbf\{U\}^\{\\top\}\\mathbf\{U\}\\big\)^\{\-1\}\\mathbf\{U\}^\{\\top\}S\\,\\mathbf\{U\}\\in GL\(d,\\mathbb\{R\}\),\\,\\,S\\in\\mathcal\{G\},\(21\)and injectivity of𝐔\\mathbf\{U\}gives𝐑𝐔​\(S1​S2\)=𝐑𝐔​\(S1\)​𝐑𝐔​\(S2\)\\mathbf\{R\}\_\{\\mathbf\{U\}\}\(S\_\{1\}S\_\{2\}\)=\\mathbf\{R\}\_\{\\mathbf\{U\}\}\(S\_\{1\}\)\\mathbf\{R\}\_\{\\mathbf\{U\}\}\(S\_\{2\}\), i\.e\.𝐑𝐔\\mathbf\{R\}\_\{\\mathbf\{U\}\}corresponds to the matrix representation of𝒢\|F\\mathcal\{G\}\|\_\{F\}in the coordinates𝐔\\mathbf\{U\}onFF\. The similarity class\[𝐑F\]\[\\mathbf\{R\}\_\{F\}\]of matrices𝐑𝐔\\mathbf\{R\}\_\{\\mathbf\{U\}\}is then, by definition, invariant under a change of basis ofFF, and thus induces an equivalence relation on𝒢\\mathcal\{G\}\-invariant subspaces\.

###### Definition 4\.3\(Equivalence of𝒢\\mathcal\{G\}\-invariant subspaces\)\.

We sayF1,F2∈𝒮d,𝒢F\_\{1\},F\_\{2\}\\in\\mathcal\{S\}\_\{d,\\mathcal\{G\}\}are related,F1∼F2F\_\{1\}\\sim F\_\{2\}, if and only if the similarity classes of corresponding matrix representations of𝒢\\mathcal\{G\}are equal,\[𝐑F1\]=\[𝐑F2\]\[\\mathbf\{R\}\_\{F\_\{1\}\}\]=\[\\mathbf\{R\}\_\{F\_\{2\}\}\]\.

Writing

𝒮d,𝒢:=\{E⊂ℝn,dim​E=d,Eis𝒢\-invariant\},\\displaystyle\\mathcal\{S\}\_\{d,\\mathcal\{G\}\}:=\\\{E\\subset\\mathbb\{R\}^\{n\},\\ \\mathrm\{dim\}E=d,\\ \\text\{$E$ is $\\mathcal\{G\}$\-invariant\}\\\},it is then straightforward to show that this relation induces an equivalence relation on the set𝒮d,𝒢\\mathcal\{S\}\_\{d,\\mathcal\{G\}\}, and thus𝒮d,𝒢\\mathcal\{S\}\_\{d,\\mathcal\{G\}\}can be written as a disjoint union of such equivalence classes, henceforth referred to assimilarity classes of invariant subspaces\. In addition, we have the following result\.

###### Proposition 4\.4\(Classes of invariant subspaces\)\.

Let𝒢⊂G​L​\(n,ℝ\)\\mathcal\{G\}\\subset GL\(n,\\mathbb\{R\}\)be a finite linear group and1≤d≤n1\\leq d\\leq n\.

1. \(i\)*Finiteness\.*Up to similarity,𝒢\\mathcal\{G\}admits only finitely manydd\-dimensional matrix representations\. In particular,F↦\[𝐑F\]F\\mapsto\[\\mathbf\{R\}\_\{F\}\]partitions𝒮d,𝒢\\mathcal\{S\}\_\{d,\\mathcal\{G\}\}into finitely many similarity classes\[ℰ1\],…,\[ℰL\]\[\\mathcal\{E\}\_\{1\}\],\\dots,\[\\mathcal\{E\}\_\{L\}\]of invariant subspaces\.
2. \(ii\)*Rigidity\.*If\[0,1\]∋t↦𝐔​\(t\)∈ℝn×d\[0,1\]\\ni t\\mapsto\\mathbf\{U\}\(t\)\\in\\mathbb\{R\}^\{n\\times d\}is continuous with full column rank and everyF​\(t\):=span⁡𝐔​\(t\)F\(t\):=\\operatorname\{span\}\\mathbf\{U\}\(t\)is𝒢\\mathcal\{G\}\-invariant, then𝐑𝐔​\(t\)∼𝐑𝐔​\(0\)\\mathbf\{R\}\_\{\\mathbf\{U\}\(t\)\}\\sim\\mathbf\{R\}\_\{\\mathbf\{U\}\(0\)\}for alltt, i\.e\. a continuous path of invariant subspaces never leaves its class,\[F​\(t\)\]=\[F​\(0\)\]\[F\(t\)\]=\[F\(0\)\]\.

###### Proof\.

Since everyS∈𝒢S\\in\\mathcal\{G\}satisfiesS\|𝒢\|=𝐈S^\{\|\\mathcal\{G\}\|\}=\\mathbf\{I\}\(Lagrange’s theorem\), we have

𝐑​\(S\)\|𝒢\|=𝐑​\(S\|𝒢\|\)=𝐑​\(𝐈\)=𝐈,\\displaystyle\\mathbf\{R\}\(S\)^\{\|\\mathcal\{G\}\|\}=\\mathbf\{R\}\(S^\{\|\\mathcal\{G\}\|\}\)=\\mathbf\{R\}\(\\mathbf\{I\}\)=\\mathbf\{I\},thus all eigenvalues of𝐑​\(S\)\\mathbf\{R\}\(S\)must be\|𝒢\|\|\\mathcal\{G\}\|\-th roots of unity\. We now resort to the use of character theory of linear representations of groups as per\[[52](https://arxiv.org/html/2608.04239#bib.bib43), §2\]: firstly we writeχ𝐑​\(S\):=tr⁡𝐑​\(S\)\\chi\_\{\\mathbf\{R\}\}\(S\):=\\operatorname\{tr\}\\mathbf\{R\}\(S\)for the character of a representation𝐑\\mathbf\{R\}\. Secondly, we observe thatχ𝐑\\chi\_\{\\mathbf\{R\}\}maps𝒢\\mathcal\{G\}into the finite setΣd\\Sigma\_\{d\}of sums ofddsuch roots of unity, so only finitely many characters occur\. Now we note that Corollary 2 of\[[52](https://arxiv.org/html/2608.04239#bib.bib43), §2\.3\]states that representations with the same character are similar overℂ\\mathbb\{C\}, i\.e\. ifχ𝐑=χ𝐑′\\chi\_\{\\mathbf\{R\}\}=\\chi\_\{\\mathbf\{R\}^\{\\prime\}\}, then there is an invertible𝐗∈G​L​\(d,ℂ\)\\mathbf\{X\}\\in GL\(d,\\mathbb\{C\}\)such that𝐑′​𝐗=𝐗𝐑\\mathbf\{R\}^\{\\prime\}\\mathbf\{X\}=\\mathbf\{X\}\\mathbf\{R\}\. Writing𝐏=Re​𝐗,𝐐=Im​𝐗\\mathbf\{P\}=\\mathrm\{Re\}\\mathbf\{X\},\\mathbf\{Q\}=\\mathrm\{Im\}\\mathbf\{X\}for the real and imaginary part respectively, we have

𝐑′​𝐏=𝐏𝐑,𝐑′​𝐐=𝐐𝐑,\\displaystyle\\mathbf\{R\}^\{\\prime\}\\mathbf\{P\}=\\mathbf\{P\}\\mathbf\{R\},\\quad\\mathbf\{R\}^\{\\prime\}\\mathbf\{Q\}=\\mathbf\{Q\}\\mathbf\{R\},and sincef:t↦det⁡\(P\+t​𝐐\)f:t\\mapsto\\det\(P\+t\\mathbf\{Q\}\)is a non\-zero polynomial onℂ\\mathbb\{C\}\(f​\(i\)=det⁡\(𝐗\)≠0f\(i\)=\\operatorname\{det\}\(\\mathbf\{X\}\)\\neq 0\), there is a realt∗t^\{\*\}such thatf​\(t∗\)≠0f\(t^\{\*\}\)\\neq 0, i\.e\. for which𝐏\+t∗​𝐐\\mathbf\{P\}\+t^\{\*\}\\mathbf\{Q\}is invertible\. Therefore𝐑′\\mathbf\{R\}^\{\\prime\}and𝐑\\mathbf\{R\}are similar overℝ\\mathbb\{R\}and so\[𝐑\]=\[𝐑′\]\[\\mathbf\{R\}\]=\[\\mathbf\{R\}^\{\\prime\}\]\. This completes the proof of \(i\)\.

For \(ii\), we note that the formula in \([21](https://arxiv.org/html/2608.04239#S4.E21)\) shows thatχt​\(S\):=tr⁡𝐑𝐔​\(t\)​\(S\)\\chi\_\{t\}\(S\):=\\operatorname\{tr\}\\mathbf\{R\}\_\{\\mathbf\{U\}\(t\)\}\(S\)is continuous intt\. Moreover,𝐑𝐔​\(t\)​\(S\)\\mathbf\{R\}\_\{\\mathbf\{U\}\(t\)\}\(S\)represents the restriction ofSStoF​\(t\)F\(t\), so its spectrum consists ofddeigenvalues ofSS, counted with multiplicity, and thusχt​\(S\)\\chi\_\{t\}\(S\)can only take finitely many values\. Any continuous function taking a finite number of values on\[0,1\]\[0,1\]is constant, soχt=χ0\\chi\_\{t\}=\\chi\_\{0\}and the similarity𝐑𝐔​\(t\)∼𝐑𝐔​\(0\)\\mathbf\{R\}\_\{\\mathbf\{U\}\(t\)\}\\sim\\mathbf\{R\}\_\{\\mathbf\{U\}\(0\)\}follows as in the proof of \(i\)\. ∎

The immediate consequence of this result is that once we fix the invariant subspace similarity class\[E\]\[E\]ofEE, no continuous optimisation algorithm on𝒮d,𝒢\\mathcal\{S\}\_\{d,\\mathcal\{G\}\}can move the equivariant subspace away from\[E\]\[E\]\. Thus it makes sense to fix\[E\]\[E\]once \(at initialisation\) through the representation𝐑\\mathbf\{R\}of𝒢\\mathcal\{G\}onEEbefore using continuous optimisation in Step \(i\.3\) to move inside\[E\]\[E\]jointly with the graph parametrisation of𝒲​\(E\)\\mathcal\{W\}\(E\)\. Conveniently it is this rigidity which allows us to fix the equivariant basis in Step \(i\.2\) thus resulting in a tangible and efficient algorithm\. This initialisation is precisely the purpose of Step \(i\.1\): we identify from data the correct class\[E0\]\[E\_\{0\}\]\(equivalently, the restricted representation𝐑0\\mathbf\{R\}\_\{0\}of𝒢\\mathcal\{G\}\) together with an initial subspaceE0E\_\{0\}within it\.

To enforce𝒢\\mathcal\{G\}\-invariance ofE0E\_\{0\}given a finite data sample, we symmetrise the data by orbit augmentation\. In particular, representing the action of an elementS∈𝒢S\\in\\mathcal\{G\}in the whitened coordinates asS~:=𝐋⊤​S​𝐋−⊤∈O​\(n,ℝ\)\\widetilde\{S\}:=\\mathbf\{L\}^\{\\top\}S\\mathbf\{L\}^\{\-\\top\}\\in O\(n,\\mathbb\{R\}\)we can, for finite symmetry groups𝒢\\mathcal\{G\}define the orbit\-augmented snapshot matrix

𝐘~𝒢:=\[S~​𝐘~\]S∈𝒢=\[S~1​𝐘~​∣⋯∣​S~\|𝒢\|​𝐘~\]∈ℝn×\|𝒢\|​N\.\\displaystyle\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}:=\\big\[\\,\\widetilde\{S\}\\widetilde\{\\mathbf\{Y\}\}\\,\\big\]\_\{S\\in\\mathcal\{G\}\}=\\big\[\\,\\widetilde\{S\}\_\{1\}\\widetilde\{\\mathbf\{Y\}\}\\mid\\cdots\\mid\\widetilde\{S\}\_\{\|\\mathcal\{G\}\|\}\\widetilde\{\\mathbf\{Y\}\}\\,\\big\]\\in\\mathbb\{R\}^\{n\\times\|\\mathcal\{G\}\|N\}\.\(22\)
###### Lemma 4\.5\.

Any eigenspace of𝐘~𝒢​𝐘~𝒢⊤\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}^\{\\top\}is invariant under the whitened action of the symmetry group𝒢\\mathcal\{G\}\. In particular, the leading left singular vectors of𝐘~𝒢\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}span a𝒢\\mathcal\{G\}\-invariant subspace providedσd\>σd\+1\\sigma\_\{d\}\>\\sigma\_\{d\+1\}, whereσd\\sigma\_\{d\}andσd\+1\\sigma\_\{d\+1\}are thedd\-th and\(d\+1\)\(d\+1\)\-th singular values of𝐘~𝒢\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}, respectively\.

###### Proof\.

It suffices to show that, for anyS0∈𝒢S\_\{0\}\\in\\mathcal\{G\}, the left Gram matrix𝐘~𝒢​𝐘~𝒢⊤\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}^\{\\top\}commutes withS~0\\widetilde\{S\}\_\{0\}\. We have

S~0​𝐘~𝒢​𝐘~𝒢⊤\\displaystyle\\widetilde\{S\}\_\{0\}\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}^\{\\top\}=∑S∈𝒢S~0​S~​𝐘~​𝐘~⊤​S~⊤=∑S1=S0​S∈𝒢S~1​𝐘~​𝐘~⊤​S~1⊤​S~0=𝐘~𝒢​𝐘~𝒢⊤​S~0,\\displaystyle=\\sum\_\{S\\in\\mathcal\{G\}\}\\widetilde\{S\}\_\{0\}\\widetilde\{S\}\\,\\widetilde\{\\mathbf\{Y\}\}\\widetilde\{\\mathbf\{Y\}\}^\{\\top\}\\,\\widetilde\{S\}^\{\\top\}=\\sum\_\{S\_\{1\}=S\_\{0\}S\\in\\mathcal\{G\}\}\\widetilde\{S\}\_\{1\}\\,\\widetilde\{\\mathbf\{Y\}\}\\widetilde\{\\mathbf\{Y\}\}^\{\\top\}\\,\\widetilde\{S\}\_\{1\}^\{\\top\}\\widetilde\{S\}\_\{0\}=\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}^\{\\top\}\\widetilde\{S\}\_\{0\},where in the second equality we changed the dummy variable of the sum toS1=S0​SS\_\{1\}=S\_\{0\}Swhich is still in𝒢\\mathcal\{G\}since𝒢\\mathcal\{G\}is a group, and used orthogonality ofS~0\\widetilde\{S\}\_\{0\}\. ∎

The leading left singular vectors of𝐘~𝒢\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}then span an exactly𝒢\\mathcal\{G\}\-invariant subspace, which we take as the initial tangent space of the SSM\. To obtain add\-dimensional reduced order model we thus compute thedd\-dimensional truncated SVD on the augmented snapshot matrix𝐘~𝒢\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\},

𝐘~𝒢≈𝐔~​𝚺~​𝐕~⊤,\\displaystyle\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}\\approx\\widetilde\{\\mathbf\{U\}\}\\widetilde\{\\bm\{\\Sigma\}\}\\widetilde\{\\mathbf\{V\}\}^\{\\top\},where𝐔~∈ℝn×d\\widetilde\{\\mathbf\{U\}\}\\in\\mathbb\{R\}^\{n\\times d\},𝚺~∈ℝd×d\\widetilde\{\\bm\{\\Sigma\}\}\\in\\mathbb\{R\}^\{d\\times d\}, and𝐕~∈ℝ\|𝒢\|​N×d\\widetilde\{\\mathbf\{V\}\}\\in\\mathbb\{R\}^\{\|\\mathcal\{G\}\|N\\times d\}\.

Writing𝐔~0:=𝐔~\\widetilde\{\\mathbf\{U\}\}\_\{0\}:=\\widetilde\{\\mathbf\{U\}\}for the orthonormal leading left singular vectors of𝐘~𝒢\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}, the initial primal and dual bases ofEE\(cf\. §[3\.1](https://arxiv.org/html/2608.04239#S3.SS1)\) are then chosen as

𝐔1,0=𝐋−⊤​𝐔~0,𝐕1,0=𝐋​𝐔~0\.\\displaystyle\\mathbf\{U\}\_\{1,0\}=\\mathbf\{L\}^\{\-\\top\}\\widetilde\{\\mathbf\{U\}\}\_\{0\},\\qquad\\mathbf\{V\}\_\{1,0\}=\\mathbf\{L\}\\,\\widetilde\{\\mathbf\{U\}\}\_\{0\}\.The representation𝐑0\\mathbf\{R\}\_\{0\}of𝒢\\mathcal\{G\}onEEis then fixed by𝐑0​\(S\)=𝐕1,0⊤​S​𝐔1,0\\mathbf\{R\}\_\{0\}\(S\)=\\mathbf\{V\}\_\{1,0\}^\{\\top\}S\\mathbf\{U\}\_\{1,0\}\.

#### Step \(i\.2\): Equivariant bases for the chart𝐔~1\\widetilde\{\\mathbf\{U\}\}\_\{1\}and the graph coefficients𝐖~k\\widetilde\{\\mathbf\{W\}\}\_\{k\}of𝐡~\\widetilde\{\\mathbf\{h\}\}

Step \(i\.1\) fixes the reduced representation𝐑0​\(S\)=𝐔~0⊤​S~​𝐔~0\\mathbf\{R\}\_\{0\}\(S\)=\\widetilde\{\\mathbf\{U\}\}\_\{0\}^\{\\top\}\\widetilde\{S\}\\,\\widetilde\{\\mathbf\{U\}\}\_\{0\}of𝒢\\mathcal\{G\}onEE, in this step we will construct the equivariant bases for the chart𝐔~1\\widetilde\{\\mathbf\{U\}\}\_\{1\}and the graph coefficients𝐖~k\\widetilde\{\\mathbf\{W\}\}\_\{k\}\. To keep the representation of𝒢\\mathcal\{G\}unchanged as we refine the chart,𝐔~1\\widetilde\{\\mathbf\{U\}\}\_\{1\}must satisfyS~​𝐔~1=𝐔~1​𝐑0​\(S\)\\widetilde\{S\}\\,\\widetilde\{\\mathbf\{U\}\}\_\{1\}=\\widetilde\{\\mathbf\{U\}\}\_\{1\}\\mathbf\{R\}\_\{0\}\(S\)for allS∈𝒢S\\in\\mathcal\{G\}, which is precisely the degree\-one instance of the equivariance condition of Theorem[3\.8](https://arxiv.org/html/2608.04239#S3.Thmtheorem8)\(the casek=1k=1,𝐃1=𝐑0\\mathbf\{D\}\_\{1\}=\\mathbf\{R\}\_\{0\}\)\. The chart and the graph coefficients are therefore the admissible blocks of equivariant monomial coefficients: letting𝐁~k\\widetilde\{\\mathbf\{B\}\}\_\{k\}denote a whitened orthonormal basis of the nullspace of the stacked operator𝐌k\\mathbf\{M\}\_\{k\}\([13](https://arxiv.org/html/2608.04239#S3.E13)\) built from𝐑0\\mathbf\{R\}\_\{0\}, we write

vec⁡\(𝐔~1\)=𝐁~1​𝐜,𝐜∈ℝp1,vec⁡\(𝐖~k\)=𝐁~k​𝐰k,𝐰k∈ℝpk,k=2,…,M\.\\displaystyle\\operatorname\{vec\}\(\\widetilde\{\\mathbf\{U\}\}\_\{1\}\)=\\widetilde\{\\mathbf\{B\}\}\_\{1\}\\,\\mathbf\{c\},\\quad\\mathbf\{c\}\\in\\mathbb\{R\}^\{p\_\{1\}\},\\qquad\\operatorname\{vec\}\(\\widetilde\{\\mathbf\{W\}\}\_\{k\}\)=\\widetilde\{\\mathbf\{B\}\}\_\{k\}\\,\\mathbf\{w\}\_\{k\},\\quad\\mathbf\{w\}\_\{k\}\\in\\mathbb\{R\}^\{p\_\{k\}\},\\ \\ k=2,\\dots,M\.\(24\)In this parametrisation we have, for anyS∈𝒢S\\in\\mathcal\{G\},

S~​𝐔~1=𝐔~1​𝐑0​\(S\),\\displaystyle\\widetilde\{S\}\\widetilde\{\\mathbf\{U\}\}\_\{1\}=\\widetilde\{\\mathbf\{U\}\}\_\{1\}\\mathbf\{R\}\_\{0\}\(S\),by construction, thus the reduced representation of𝒢\\mathcal\{G\}onEEis preserved\. This ensures also that the parametrisationsunvec⁡\(𝐁~k​𝐰k\),2≤k≤M,\\operatorname\{unvec\}\(\\widetilde\{\\mathbf\{B\}\}\_\{k\}\\mathbf\{w\}\_\{k\}\),2\\leq k\\leq M,satisfy the linear constraints \([11](https://arxiv.org/html/2608.04239#S3.E11)\) for any choice of𝐜\\mathbf\{c\}and, hence, the bases𝐁~k\\widetilde\{\\mathbf\{B\}\}\_\{k\}are independent of𝐜\\mathbf\{c\}and never need recomputing during the optimisation of Step \(i\.3\)\. Finally we note that the rescaling𝐃\\mathbf\{D\}introduced in \([23](https://arxiv.org/html/2608.04239#S4.E23)\) commutes with𝐑0\\mathbf\{R\}\_\{0\}, meaning that the𝒢\\mathcal\{G\}\-equivariant Taylor series expansion for the rescaled coordinate𝐃−1​𝜼\\mathbf\{D\}^\{\-1\}\\bm\{\\eta\}carries the same representation𝐑0\\mathbf\{R\}\_\{0\}, and the bases𝐁~k\\widetilde\{\\mathbf\{B\}\}\_\{k\}are unchanged by the rescaling\.

#### Step \(i\.3\): The reduced constrained least\-squares problem

Substituting \([24](https://arxiv.org/html/2608.04239#S4.E24)\) into the whitened objective \([20](https://arxiv.org/html/2608.04239#S4.E20)\) reduces the fit to a nonlinear constrained least\-squares problem in the finite coefficient vectors𝐜\\mathbf\{c\}and𝐰=\(𝐰2,…,𝐰M\)\\mathbf\{w\}=\(\\mathbf\{w\}\_\{2\},\\dots,\\mathbf\{w\}\_\{M\}\),

min𝐜,𝐰​∑i=1N‖𝐲~i−𝐔~1​𝜼i−∑k=2M𝐖~k​ϕk​\(𝐃−1​𝜼i\)‖2,𝜼i=𝐔~1⊤​𝐲~i,\\displaystyle\\min\_\{\\mathbf\{c\},\\,\\mathbf\{w\}\}\\ \\sum\_\{i=1\}^\{N\}\\Big\\\|\\,\\widetilde\{\\mathbf\{y\}\}\_\{i\}\-\\widetilde\{\\mathbf\{U\}\}\_\{1\}\\bm\{\\eta\}\_\{i\}\-\\sum\_\{k=2\}^\{M\}\\widetilde\{\\mathbf\{W\}\}\_\{k\}\\,\\bm\{\\phi\}\_\{k\}\(\\mathbf\{D\}^\{\-1\}\\bm\{\\eta\}\_\{i\}\)\\,\\Big\\\|^\{2\},\\qquad\\bm\{\\eta\}\_\{i\}=\\widetilde\{\\mathbf\{U\}\}\_\{1\}^\{\\top\}\\widetilde\{\\mathbf\{y\}\}\_\{i\},\(25\)where𝐔~1=unvec⁡\(𝐁~1​𝐜\)\\widetilde\{\\mathbf\{U\}\}\_\{1\}=\\operatorname\{unvec\}\(\\widetilde\{\\mathbf\{B\}\}\_\{1\}\\mathbf\{c\}\),𝐖~k=unvec⁡\(𝐁~k​𝐰k\)\\widetilde\{\\mathbf\{W\}\}\_\{k\}=\\operatorname\{unvec\}\(\\widetilde\{\\mathbf\{B\}\}\_\{k\}\\mathbf\{w\}\_\{k\}\)are the matrix forms of𝐁~1​𝐜\\widetilde\{\\mathbf\{B\}\}\_\{1\}\\mathbf\{c\},𝐁~k​𝐰k\\widetilde\{\\mathbf\{B\}\}\_\{k\}\\mathbf\{w\}\_\{k\}, subject to

\(a\)𝐔~1⊤​𝐔~1=𝐈d,\(b\)𝐔~1⊤​𝐖~k=𝟎,k=2,…,M,\\displaystyle\\text\{\(a\)\}\\ \\ \\widetilde\{\\mathbf\{U\}\}\_\{1\}^\{\\top\}\\widetilde\{\\mathbf\{U\}\}\_\{1\}=\\mathbf\{I\}\_\{d\},\\qquad\\text\{\(b\)\}\\ \\ \\widetilde\{\\mathbf\{U\}\}\_\{1\}^\{\\top\}\\widetilde\{\\mathbf\{W\}\}\_\{k\}=\\mathbf\{0\},\\ \\ k=2,\\dots,M,\(26\)where \(b\) is the graph condition, placing the nonlinear part off the tangent space so that𝜼i=𝐔~1⊤​𝐲~i\\bm\{\\eta\}\_\{i\}=\\widetilde\{\\mathbf\{U\}\}\_\{1\}^\{\\top\}\\widetilde\{\\mathbf\{y\}\}\_\{i\}is consistent\. We solve this using a standard quasi\-Newton constrained solver, warm\-started at the𝐜\\mathbf\{c\}representing𝐔~0\\widetilde\{\\mathbf\{U\}\}\_\{0\}and with𝐰k=𝟎\\mathbf\{w\}\_\{k\}=\\mathbf\{0\}\.

#### Step \(ii\): Computation of the reduced dynamics on the SSM in extended normal form style

We first estimate the linear part𝐁=D​𝐫​\(𝟎\)\\mathbf\{B\}=D\\mathbf\{r\}\(\\mathbf\{0\}\)of the reduced dynamics𝜼˙=𝐫​\(𝜼\)\\dot\{\\bm\{\\eta\}\}=\\mathbf\{r\}\(\\bm\{\\eta\}\)by regression from the projected snapshot data𝚵,𝚵′∈ℝd×\(N−1\)\\bm\{\\Xi\},\\bm\{\\Xi\}^\{\\prime\}\\in\\mathbb\{R\}^\{d\\times\(N\-1\)\}, given by

𝚵=\[𝜼1,…,𝜼N−1\],𝚵′=\[𝜼2,…,𝜼N\],\\displaystyle\\bm\{\\Xi\}=\[\\bm\{\\eta\}\_\{1\},\\dots,\\bm\{\\eta\}\_\{N\-1\}\],\\quad\\bm\{\\Xi\}^\{\\prime\}=\[\\bm\{\\eta\}\_\{2\},\\dots,\\bm\{\\eta\}\_\{N\}\],where𝜼i=𝐕1⊤​𝐲i\\bm\{\\eta\}\_\{i\}=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}\_\{i\}are the projected data points in the reduced coordinates\. Since the true \(unknown\) value of𝐁\\mathbf\{B\}is𝒢\|E\\mathcal\{G\}\|\_\{E\}\-equivariant, we proceed similarly to Step \(i\) by forming the orbit\-augmented projected snapshot matrices

𝚵𝒢=\[S\|E​𝚵\]S∈𝒢∈ℝd×\|𝒢\|​\(N−1\),𝚵𝒢′=\[S\|E​𝚵′\]S∈𝒢∈ℝd×\|𝒢\|​\(N−1\),\\displaystyle\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}=\[\\,S\|\_\{E\}\\bm\{\\Xi\}\\,\]\_\{S\\in\\mathcal\{G\}\}\\in\\mathbb\{R\}^\{d\\times\|\\mathcal\{G\}\|\(N\-1\)\},\\qquad\\bm\{\\Xi\}^\{\\prime\}\_\{\\mathcal\{G\}\}=\[\\,S\|\_\{E\}\\bm\{\\Xi\}^\{\\prime\}\\,\]\_\{S\\in\\mathcal\{G\}\}\\in\\mathbb\{R\}^\{d\\times\|\\mathcal\{G\}\|\(N\-1\)\},analogously to \([22](https://arxiv.org/html/2608.04239#S4.E22)\), whereS\|E=𝐕1⊤​S​𝐔1S\|\_\{E\}=\\mathbf\{V\}\_\{1\}^\{\\top\}S\\mathbf\{U\}\_\{1\}, and then compute the least\-squares estimate of𝐁Δ​t=D​ϕ𝐫Δ​t​\(𝟎\)=e𝐁​Δ​t\\mathbf\{B\}\_\{\\Delta t\}=D\\bm\{\\phi\}\_\{\\mathbf\{r\}\}^\{\\Delta t\}\(\\mathbf\{0\}\)=e^\{\\mathbf\{B\}\\Delta t\}, the linearised one\-step map, from the orbit\-augmented data as

𝐁Δ​t=argmin𝐁Δ​t​∑S∈𝒢∑i=1N−1‖S\|E​𝜼i\+1−𝐁Δ​t​S\|E​𝜼i∥22,𝜼i=𝐕1⊤​𝐲i,\\displaystyle\\mathbf\{B\}\_\{\\Delta t\}=\\operatorname\*\{argmin\}\_\{\\mathbf\{B\}\_\{\\Delta t\}\}\\sum\_\{S\\in\\mathcal\{G\}\}\\sum\_\{i=1\}^\{N\-1\}\\big\\\|\\,S\|\_\{E\}\\bm\{\\eta\}\_\{i\+1\}\-\\mathbf\{B\}\_\{\\Delta t\}S\|\_\{E\}\\bm\{\\eta\}\_\{i\}\\,\\big\\\|\_\{2\}^\{2\},\\qquad\\bm\{\\eta\}\_\{i\}=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}\_\{i\},i\.e\.𝐁Δ​t=𝚵𝒢′​𝚵𝒢†=𝚵𝒢′​𝚵𝒢⊤​\(𝚵𝒢​𝚵𝒢⊤\)−1\\mathbf\{B\}\_\{\\Delta t\}=\\bm\{\\Xi\}^\{\\prime\}\_\{\\mathcal\{G\}\}\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}^\{\\dagger\}=\\bm\{\\Xi\}^\{\\prime\}\_\{\\mathcal\{G\}\}\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}^\{\\top\}\\big\(\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}^\{\\top\}\\big\)^\{\-1\}is the dynamic mode decomposition in the reduced coordinates𝜼\\bm\{\\eta\}\.

Since𝐁Δ​t\\mathbf\{B\}\_\{\\Delta t\}is the linearised time\-Δ​t\\Delta tmap, we recover the continuous\-time generator𝐁=D​𝐫​\(𝟎\)\\mathbf\{B\}=D\\mathbf\{r\}\(\\mathbf\{0\}\)from \([6](https://arxiv.org/html/2608.04239#S3.E6)\) by𝐁=1Δ​t​log⁡𝐁Δ​t\\mathbf\{B\}=\\tfrac\{1\}\{\\Delta t\}\\log\\mathbf\{B\}\_\{\\Delta t\}\. In practice we compute the eigendecomposition𝐁Δ​t=𝐖​diag⁡\(μ1,…,μd\)​𝐖−1\\mathbf\{B\}\_\{\\Delta t\}=\\mathbf\{W\}\\operatorname\{diag\}\(\\mu\_\{1\},\\dots,\\mu\_\{d\}\)\\mathbf\{W\}^\{\-1\}and set

𝐁=𝐖​𝚲​𝐖−1,𝚲=1Δ​t​diag⁡\(log⁡μ1,…,log⁡μd\),\\displaystyle\\mathbf\{B\}=\\mathbf\{W\}\\bm\{\\Lambda\}\\mathbf\{W\}^\{\-1\},\\qquad\\bm\{\\Lambda\}=\\tfrac\{1\}\{\\Delta t\}\\operatorname\{diag\}\(\\log\\mu\_\{1\},\\dots,\\log\\mu\_\{d\}\),so that the eigenvectors𝐖\\mathbf\{W\}and continuous eigenvalues𝝀\\bm\{\\lambda\}form the basis of the extended normal\-form parametrisation and enter the resonance condition𝐦⋅𝝀−λj\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}directly\. We will proceed to determine𝐭\{\\mathbf\{t\}\}and𝐧\\mathbf\{n\}using truncated Taylor expansions up to orderMRODM\_\{\\text\{ROD\}\}, i\.e\.

𝐧​\(𝐳;𝐍\)\\displaystyle\\mathbf\{n\}\(\\mathbf\{z\};\\mathbf\{N\}\)=𝚲​𝐳\+∑k=2MROD𝐍k​ϕk​\(𝐳\),\\displaystyle=\\bm\{\\Lambda\}\\mathbf\{z\}\+\\sum\_\{k=2\}^\{M\_\{\\text\{ROD\}\}\}\\mathbf\{N\}\_\{k\}\\,\\bm\{\\phi\}\_\{k\}\(\\mathbf\{z\}\),\(27\)𝐭​\(𝐳;𝐓\)\\displaystyle\\mathbf\{t\}\(\\mathbf\{z\};\\mathbf\{T\}\)=𝐖​\(𝐳\+∑k=2MROD𝐓k​ϕk​\(𝐳\)\),𝐭−1​\(𝜼;𝐓⋆\)=𝐖−1​𝜼\+∑k=2MROD𝐓k⋆​ϕk​\(𝐖−1​𝜼\)\.\\displaystyle=\\mathbf\{W\}\\Big\(\\mathbf\{z\}\+\\sum\_\{k=2\}^\{M\_\{\\text\{ROD\}\}\}\\mathbf\{T\}\_\{k\}\\,\\bm\{\\phi\}\_\{k\}\(\\mathbf\{z\}\)\\Big\),\\quad\\mathbf\{t\}^\{\-1\}\(\\bm\{\\eta\};\\mathbf\{T\}^\{\\star\}\)=\\mathbf\{W\}^\{\-1\}\\bm\{\\eta\}\+\\sum\_\{k=2\}^\{M\_\{\\text\{ROD\}\}\}\\mathbf\{T\}^\{\\star\}\_\{k\}\\,\\bm\{\\phi\}\_\{k\}\(\\mathbf\{W\}^\{\-1\}\\bm\{\\eta\}\)\.\(28\)To determine the coefficients𝐍k\\mathbf\{N\}\_\{k\}and𝐓k⋆\\mathbf\{T\}\_\{k\}^\{\\star\}we proceed in three steps: \(a\) support selection, \(b\) equivariant reparametrisation, and \(c\) coefficient fit\.

\(a\) Support selection\.In the above expansions, the support of the coefficients𝐍k,𝐓k\\mathbf\{N\}\_\{k\},\\mathbf\{T\}\_\{k\}and𝐓k⋆\\mathbf\{T\}\_\{k\}^\{\\star\}is determined by the near resonance condition \([9](https://arxiv.org/html/2608.04239#S3.E9)\), i\.e\. we set

supp⁡\(𝐍k\)=ℐδ,supp⁡\(𝐓k\)=supp⁡\(𝐓k⋆\)=ℐδc,k=2,…,M,\\displaystyle\\operatorname\{supp\}\(\\mathbf\{N\}\_\{k\}\)=\\mathcal\{I\}\_\{\\delta\},\\qquad\\operatorname\{supp\}\(\\mathbf\{T\}\_\{k\}\)=\\operatorname\{supp\}\(\\mathbf\{T\}\_\{k\}^\{\\star\}\)=\\mathcal\{I\}\_\{\\delta\}^\{\\,c\},\\qquad k=2,\\dots,M,whereℐδ=\{\(j,𝐦\):\|Im⁡\(𝐦⋅𝝀−λj\)\|<δ\}\\mathcal\{I\}\_\{\\delta\}=\\\{\(j,\\mathbf\{m\}\):\|\\operatorname\{Im\}\(\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}\)\|<\\delta\\\}\.

\(b\) Equivariant reparametrisation\.Proposition[3\.5](https://arxiv.org/html/2608.04239#S3.Thmtheorem5)shows that, if the support of a Taylor series is chosen according to the near\-resonance condition \([9](https://arxiv.org/html/2608.04239#S3.E9)\), then the support of the coefficients is invariant under the action of the symmetry group𝒢\\mathcal\{G\}\. This means, that the support restriction and the equivariance constraint are compatible and may be imposed simultaneously\. Concretely, fix a degreekkand split the degree\-kkcoefficient matrices according to the partition of the index set\{1,…,d\}×\{\|𝐦\|=k\}\\\{1,\\dots,d\\\}\\times\\\{\|\\mathbf\{m\}\|=k\\\}into resonant and non\-resonant entries,

ℂd×Nk=Vk𝐧⊕Vk𝐭,\\displaystyle\\mathbb\{C\}^\{d\\times N\_\{k\}\}=V\_\{k\}^\{\\mathbf\{n\}\}\\oplus V\_\{k\}^\{\\mathbf\{t\}\},whereVk𝐧V\_\{k\}^\{\\mathbf\{n\}\}\(Vk𝐭V\_\{k\}^\{\\mathbf\{t\}\}\) consists of those𝐂\\mathbf\{C\}supported on the resonant indicesℐδ\\mathcal\{I\}\_\{\\delta\}\(on their complement\)\. By the proof of Proposition[3\.5](https://arxiv.org/html/2608.04239#S3.Thmtheorem5)the diagonalised actionS^\\widehat\{S\}commutes with𝚲\\bm\{\\Lambda\}, hence preserves the resonance value𝐦⋅𝝀−λj\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}of every monomial; the equivariance operator

𝐌k​\[S^\]​\(𝐂\)=S^​𝐂−𝐂​𝐃k​\(S^\)\\displaystyle\\mathbf\{M\}\_\{k\}\[\\widehat\{S\}\]\(\\mathbf\{C\}\)=\\widehat\{S\}\\mathbf\{C\}\-\\mathbf\{C\}\\,\\mathbf\{D\}\_\{k\}\(\\widehat\{S\}\)therefore maps each ofVk𝐧V\_\{k\}^\{\\mathbf\{n\}\}andVk𝐭V\_\{k\}^\{\\mathbf\{t\}\}into itself\. We may consequently restrict𝐌k​\[S^\]\\mathbf\{M\}\_\{k\}\[\\widehat\{S\}\]to each support subspace and impose equivariance there, so that the admissible coefficients of𝐧\\mathbf\{n\}and𝐭−1\\mathbf\{t\}^\{\-1\}are

𝐍k∈⋂S∈𝒢ker⁡\(𝐌k​\[S^\]\|Vk𝐧\),𝐓k⋆∈⋂S∈𝒢ker⁡\(𝐌k​\[S^\]\|Vk𝐭\)\.\\displaystyle\\mathbf\{N\}\_\{k\}\\in\\bigcap\_\{S\\in\\mathcal\{G\}\}\\ker\\\!\\big\(\\mathbf\{M\}\_\{k\}\[\\widehat\{S\}\]\\big\|\_\{V\_\{k\}^\{\\mathbf\{n\}\}\}\\big\),\\qquad\\mathbf\{T\}^\{\\star\}\_\{k\}\\in\\bigcap\_\{S\\in\\mathcal\{G\}\}\\ker\\\!\\big\(\\mathbf\{M\}\_\{k\}\[\\widehat\{S\}\]\\big\|\_\{V\_\{k\}^\{\\mathbf\{t\}\}\}\\big\)\.If we write𝐌k​\[S^\]\\mathbf\{M\}\_\{k\}\[\\widehat\{S\}\]in vectorised form as𝐈Nk⊗S^−𝐃k​\(S^\)⊤⊗𝐈d\\mathbf\{I\}\_\{N\_\{k\}\}\\otimes\\widehat\{S\}\-\\mathbf\{D\}\_\{k\}\(\\widehat\{S\}\)^\{\\\!\\top\}\\otimes\\mathbf\{I\}\_\{d\}, then the restriction to the support subspaces is simply the deletion of the rows and columns indexed outside the relevant support, and the admissible coefficients are the nullspace of the resulting smaller matrix, computed by SVD analogously to Step \(ii\)\. Writing𝐁k𝐧\\mathbf\{B\}\_\{k\}^\{\\mathbf\{n\}\}and𝐁k𝐭\\mathbf\{B\}\_\{k\}^\{\\mathbf\{t\}\}for bases of these nullspaces we have, similarly to \([24](https://arxiv.org/html/2608.04239#S4.E24)\),

vec⁡\(𝐍k\)=𝐁k𝐧​𝐜k𝐧,vec⁡\(𝐓k\)=𝐁k𝐭​𝐜k𝐭,vec⁡\(𝐓k⋆\)=𝐁k𝐭​𝐜k𝐭,⋆,\\displaystyle\\operatorname\{vec\}\(\\mathbf\{N\}\_\{k\}\)=\\mathbf\{B\}\_\{k\}^\{\\mathbf\{n\}\}\\,\\mathbf\{c\}\_\{k\}^\{\\mathbf\{n\}\},\\qquad\\operatorname\{vec\}\(\\mathbf\{T\}\_\{k\}\)=\\mathbf\{B\}\_\{k\}^\{\\mathbf\{t\}\}\\,\\mathbf\{c\}\_\{k\}^\{\\mathbf\{t\}\},\\qquad\\operatorname\{vec\}\(\\mathbf\{T\}^\{\\star\}\_\{k\}\)=\\mathbf\{B\}\_\{k\}^\{\\mathbf\{t\}\}\\,\\mathbf\{c\}\_\{k\}^\{\\mathbf\{t\},\\star\},\(29\)with free coefficient vectors𝐜k𝐧,𝐜k𝐭\\mathbf\{c\}\_\{k\}^\{\\mathbf\{n\}\},\\mathbf\{c\}\_\{k\}^\{\\mathbf\{t\}\}counting the equivariant resonant, respectively non\-resonant, monomials at degreekk\.

\(c\) Coefficient fit\.The final step is to use this reduced parametrisation together with the trajectory data to infer the free coefficients𝐜k𝐧,𝐜k𝐭,⋆\\mathbf\{c\}\_\{k\}^\{\\mathbf\{n\}\},\\mathbf\{c\}\_\{k\}^\{\\mathbf\{t\},\\star\}of the reduced dynamics and the normal form transformation\. We do this by firstly plugging the equivariant parametrisation \([29](https://arxiv.org/html/2608.04239#S4.E29)\) into the expansions \([27](https://arxiv.org/html/2608.04239#S4.E27)\)\-\([28](https://arxiv.org/html/2608.04239#S4.E28)\), and then minimising the error in the conjugacy equation \([7](https://arxiv.org/html/2608.04239#S3.E7)\) over the data:

\(𝐜𝐧,𝐜𝐭,⋆\)=argmin𝐜𝐧,𝐜𝐭,⋆​∑i‖D​𝐭−1​\(𝜼i\)​𝜼˙i−𝐧​\(𝐭−1​\(𝜼i\)\)‖22,\\displaystyle\(\\mathbf\{c\}^\{\\mathbf\{n\}\},\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}\)=\\operatorname\*\{argmin\}\_\{\\mathbf\{c\}^\{\\mathbf\{n\}\},\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}\}\\sum\_\{i\}\\Big\\\|D\\mathbf\{t\}^\{\-1\}\(\\bm\{\\eta\}\_\{i\}\)\\,\\dot\{\\bm\{\\eta\}\}\_\{i\}\-\\mathbf\{n\}\\big\(\\mathbf\{t\}^\{\-1\}\(\\bm\{\\eta\}\_\{i\}\)\\big\)\\Big\\\|\_\{2\}^\{2\},where the time derivatives𝜼˙i\\dot\{\\bm\{\\eta\}\}\_\{i\}are approximated with finite differences from data \(our implementation is using a sixth\-order central stencil\)\. This is a nonlinear least\-squares problem in the free coefficients\(𝐜𝐧,𝐜𝐭,⋆\)\(\\mathbf\{c\}^\{\\mathbf\{n\}\},\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}\), which we solve using Gauss–Newton, with initial condition𝐜𝐧,𝐜𝐭,⋆=𝟎\\mathbf\{c\}^\{\\mathbf\{n\}\},\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}=\\mathbf\{0\}\. Finally, we recover the coefficients𝐜𝐭\\mathbf\{c\}^\{\\mathbf\{t\}\}of the forward transformation𝐭\\mathbf\{t\}by regression on the following linear least\-squares problem:

𝐜𝐭=argmin𝐜𝐭​∑i‖𝜼i−𝐭​\(𝐭−1​\(𝜼i;𝐜𝐭,⋆\);𝐜𝐭\)‖22\.\\displaystyle\\mathbf\{c\}^\{\\mathbf\{t\}\}=\\operatorname\*\{argmin\}\_\{\\mathbf\{c\}^\{\\mathbf\{t\}\}\}\\sum\_\{i\}\\Big\\\|\\bm\{\\eta\}\_\{i\}\-\\mathbf\{t\}\(\\mathbf\{t\}^\{\-1\}\(\\bm\{\\eta\}\_\{i\};\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}\);\\mathbf\{c\}^\{\\mathbf\{t\}\}\)\\Big\\\|\_\{2\}^\{2\}\.

#### Summary of the eSSM algorithm

For clarity of presentation we summarise the above procedure in Algorithm[1](https://arxiv.org/html/2608.04239#alg1)\. The analogous discrete\-time version of the algorithm is presented in Algorithm[2](https://arxiv.org/html/2608.04239#alg2)in Appendix[A](https://arxiv.org/html/2608.04239#A1)\. We note that, by construction, the resulting SSM reduction is exactly𝒢\\mathcal\{G\}\-equivariant\.

Algorithm 1eSSM reduction method1:Inputs:snapshot matrix

𝐘∈ℝn×N\\mathbf\{Y\}\\in\\mathbb\{R\}^\{n\\times N\}, symmetry group

𝒢\\mathcal\{G\}, reduced dimension

dd, manifold degree

MM, time step

Δ​t\\Delta tin

𝐘\\mathbf\{Y\}, resonance tolerance

δ\\delta,

MRODM\_\{\\text\{ROD\}\}for reduced\-order dynamics\.

2:

3:*Step \(i\): Spectral submanifold identification*

4:Computethe Cholesky factorisation

𝐏𝒢=𝐋𝐋⊤\\mathbf\{P\}\_\{\\mathcal\{G\}\}=\\mathbf\{L\}\\mathbf\{L\}^\{\\top\}of

𝐏𝒢=1\|𝒢\|​∑S∈𝒢S⊤​S\\mathbf\{P\}\_\{\\mathcal\{G\}\}=\\frac\{1\}\{\|\\mathcal\{G\}\|\}\\sum\_\{S\\in\\mathcal\{G\}\}S^\{\\top\}S\.

5:Whiten

𝐘~=𝐋⊤​𝐘\\widetilde\{\\mathbf\{Y\}\}=\\mathbf\{L\}^\{\\top\}\\mathbf\{Y\}andstack

𝐘~𝒢=\[S~​𝐘~\]S∈𝒢\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}=\\big\[\\,\\widetilde\{S\}\\widetilde\{\\mathbf\{Y\}\}\\,\\big\]\_\{S\\in\\mathcal\{G\}\}\.

6:Computethe

dd\-truncated SVD

𝐘~𝒢≈𝐔~0​𝚺~​𝐕~⊤\\widetilde\{\\mathbf\{Y\}\}\_\{\\mathcal\{G\}\}\\approx\\widetilde\{\\mathbf\{U\}\}\_\{0\}\\widetilde\{\\bm\{\\Sigma\}\}\\widetilde\{\\mathbf\{V\}\}^\{\\top\}\.

7:Freeze

𝐑0​\(S\)=𝐕1,0⊤​S​𝐔1,0\\mathbf\{R\}\_\{0\}\(S\)=\\mathbf\{V\}\_\{1,0\}^\{\\top\}S\\mathbf\{U\}\_\{1,0\},

S∈𝒢S\\in\\mathcal\{G\}andset

𝐃=diag⁡\(𝚺~\)/\|𝒢\|​N\\mathbf\{D\}=\\operatorname\{diag\}\(\\widetilde\{\\bm\{\\Sigma\}\}\)/\\sqrt\{\|\\mathcal\{G\}\|N\}\.

8:for

k=1,…,Mk=1,\\dots,Mdo

9:Computethe nullspace basis

𝐁k\\mathbf\{B\}\_\{k\}of

𝐌k\\mathbf\{M\}\_\{k\}via SVD and whiten it,

𝐁~k=\(𝐈Nk⊗𝐋⊤\)​𝐁k\\widetilde\{\\mathbf\{B\}\}\_\{k\}=\(\\mathbf\{I\}\_\{N\_\{k\}\}\\otimes\\mathbf\{L\}^\{\\top\}\)\\mathbf\{B\}\_\{k\}\.

10:endfor

11:Warm start

𝐜←𝐁~1⊤​vec⁡\(𝐔~0\)\\mathbf\{c\}\\leftarrow\\widetilde\{\\mathbf\{B\}\}\_\{1\}^\{\\top\}\\operatorname\{vec\}\(\\widetilde\{\\mathbf\{U\}\}\_\{0\}\),

𝐰k←𝟎\\mathbf\{w\}\_\{k\}\\leftarrow\\mathbf\{0\},

k=2,…,Mk=2,\\dots,M
12:Solvethe constrained nonlinear least\-squares problem \([25](https://arxiv.org/html/2608.04239#S4.E25)\)–\([26](https://arxiv.org/html/2608.04239#S4.E26)\)\.

13:Recover

𝐔1=𝐋−⊤​𝐔~1\\mathbf\{U\}\_\{1\}=\\mathbf\{L\}^\{\-\\top\}\\widetilde\{\\mathbf\{U\}\}\_\{1\},

𝐕1=𝐋​𝐔~1\\mathbf\{V\}\_\{1\}=\\mathbf\{L\}\\widetilde\{\\mathbf\{U\}\}\_\{1\},

𝐖k=𝐋−⊤​𝐖~k\\mathbf\{W\}\_\{k\}=\\mathbf\{L\}^\{\-\\top\}\\widetilde\{\\mathbf\{W\}\}\_\{k\}\.

14:

15:*Step \(ii\): Compute extended normal form on𝒲​\(E\)\\mathcal\{W\}\(E\)*

16:Formthe orbit\-augmented reduced snapshots

𝚵𝒢=\[S\|E​𝚵\]S∈𝒢\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}=\[\\,S\|\_\{E\}\\bm\{\\Xi\}\\,\]\_\{S\\in\\mathcal\{G\}\},

𝚵𝒢′=\[S\|E​𝚵′\]S∈𝒢\\bm\{\\Xi\}^\{\\prime\}\_\{\\mathcal\{G\}\}=\[\\,S\|\_\{E\}\\bm\{\\Xi\}^\{\\prime\}\\,\]\_\{S\\in\\mathcal\{G\}\}\.

17:Estimatethe linearised one\-step map

𝐁Δ​t=𝚵𝒢′​𝚵𝒢†\\mathbf\{B\}\_\{\\Delta t\}=\\bm\{\\Xi\}^\{\\prime\}\_\{\\mathcal\{G\}\}\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}^\{\\dagger\}, andset

𝐁=1Δ​t​log⁡𝐁Δ​t\\mathbf\{B\}=\\tfrac\{1\}\{\\Delta t\}\\log\\mathbf\{B\}\_\{\\Delta t\}\.

18:for

k=2,…,Mk=2,\\dots,Mdo

19:Selectthe resonant support

ℐδ=\{\(j,𝐦\):\|Im⁡\(𝐦⋅𝝀−λj\)\|<δ\}\\mathcal\{I\}\_\{\\delta\}=\\\{\(j,\\mathbf\{m\}\):\|\\operatorname\{Im\}\(\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}\)\|<\\delta\\\}\.

20:Computethe equivariant bases

𝐁k𝐧,𝐁k𝐭\\mathbf\{B\}\_\{k\}^\{\\mathbf\{n\}\},\\mathbf\{B\}\_\{k\}^\{\\mathbf\{t\}\}of

𝐌k​\[S^\]\\mathbf\{M\}\_\{k\}\[\\widehat\{S\}\]restricted to

supp⁡\(ℐδ\),supp⁡\(ℐδc\)\\operatorname\{supp\}\(\\mathcal\{I\}\_\{\\delta\}\),\\operatorname\{supp\}\(\\mathcal\{I\}\_\{\\delta\}^\{\\,c\}\)\.

21:endfor

22:Fit

\(𝐜𝐧,𝐜𝐭,⋆\)\(\\mathbf\{c\}^\{\\mathbf\{n\}\},\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}\)on the conjugacy residual \([7](https://arxiv.org/html/2608.04239#S3.E7)\), initialised at

𝟎\\mathbf\{0\}\.

23:Recover

𝐜𝐭\\mathbf\{c\}^\{\\mathbf\{t\}\}by linear least squares andassemble

𝐍k,𝐓k,𝐓k⋆\\mathbf\{N\}\_\{k\},\\mathbf\{T\}\_\{k\},\\mathbf\{T\}\_\{k\}^\{\\star\}via \([29](https://arxiv.org/html/2608.04239#S4.E29)\)\.

24:

25:Outputs:SSM parametrisation

\(𝐔1,𝐕1,\{𝐖k\}k=2M,𝐃\)\(\\mathbf\{U\}\_\{1\},\\mathbf\{V\}\_\{1\},\\\{\\mathbf\{W\}\_\{k\}\\\}\_\{k=2\}^\{M\},\\mathbf\{D\}\), and reduced dynamics

\(𝐖,𝚲,\{𝐍k,𝐓k,𝐓k⋆\}k=2MROD\)\(\\mathbf\{W\},\\bm\{\\Lambda\},\\\{\\mathbf\{N\}\_\{k\},\\mathbf\{T\}\_\{k\},\\mathbf\{T\}^\{\\star\}\_\{k\}\\\}\_\{k=2\}^\{M\_\{\\text\{ROD\}\}\}\)on

𝒲​\(E\)\\mathcal\{W\}\(E\)in extended normal form\.

### 4\.3Simulation using the reduced order model

With the output of Algorithm[1](https://arxiv.org/html/2608.04239#alg1), we can then simulate the reduced\-order dynamics from a given initial condition𝐲0∈ℝn\\mathbf\{y\}\_\{0\}\\in\\mathbb\{R\}^\{n\}as follows\. Firstly,𝐲0\\mathbf\{y\}\_\{0\}is projected to the reduced coordinates through the chart,𝜼0=𝐕1⊤​𝐲0\\bm\{\\eta\}\_\{0\}=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{y\}\_\{0\}, and mapped to normal\-form coordinates,𝐳0=𝐭−1​\(𝜼0\)\\mathbf\{z\}\_\{0\}=\\mathbf\{t\}^\{\-1\}\(\\bm\{\\eta\}\_\{0\}\), using \([28](https://arxiv.org/html/2608.04239#S4.E28)\)\. Secondly, we integrate thedd\-dimensional normal\-form dynamics𝐳˙=𝐧​\(𝐳\)\\dot\{\\mathbf\{z\}\}=\\mathbf\{n\}\(\\mathbf\{z\}\)from \([27](https://arxiv.org/html/2608.04239#S4.E27)\)\. Thirdly, the trajectory is mapped back to the original coordinates𝜼​\(t\)=𝐭​\(𝐳​\(t\)\)\\bm\{\\eta\}\(t\)=\\mathbf\{t\}\(\\mathbf\{z\}\(t\)\)via \([28](https://arxiv.org/html/2608.04239#S4.E28)\) onEE\. Finally, the full state is recovered through the graph parametrisation \([19](https://arxiv.org/html/2608.04239#S4.E19)\),

𝐲​\(t\)=𝐔1​𝜼​\(t\)\+∑k=2M𝐖k​ϕk​\(𝐃−1​𝜼​\(t\)\)\.\\displaystyle\\mathbf\{y\}\(t\)=\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\(t\)\+\\sum\_\{k=2\}^\{M\}\\mathbf\{W\}\_\{k\}\\,\\bm\{\\phi\}\_\{k\}\\big\(\\mathbf\{D\}^\{\-1\}\\bm\{\\eta\}\(t\)\\big\)\.

## 5Numerical examples

Following the above exposition of the eSSM reduction method, we now present several examples comparing this new, equivariant method to standard SSM reduction and similar data\-driven methods introduced in earlier work\. All of the following experiments were conducted on an Apple M2 Max with 64 GB RAM\.

### 5\.1Example 1: chain of oscillators

In this first example we apply the eSSM reduction method to the chain of oscillators described in Example[2\.6](https://arxiv.org/html/2608.04239#S2.Thmtheorem6)\. We use this simple example to understand how much the free parameter count can be reduced by enforcing equivariance and to provide a direct comparison against a non\-equivariant SSM reduction method\. For this we compare the following three methods:

- •eSSM:Our new method as described in §[4](https://arxiv.org/html/2608.04239#S4)and Algorithm[1](https://arxiv.org/html/2608.04239#alg1)\. The corresponding Python implementation used in the following examples is available at[https://github\.com/GeorgAUT/eSSM](https://github.com/GeorgAUT/eSSM)\.
- •
- •
- •SSMLearn \(Python\):For fairness of runtime comparison, we also compare against our eSSM Python implementation with trivial group𝒢=\{I\}\\mathcal\{G\}=\\\{I\\\}which effectively reduces to the original SSMLearn algorithm\.

We will compare the performance of these methods in terms of accuracy, number of free parameters and wall\-clock time\.

#### Experimental setup

We consider the chain of Example[2\.6](https://arxiv.org/html/2608.04239#S2.Thmtheorem6)with up to100100masses, i\.e\. state\-space dimensionn=200n=200\. The dataset consists of four trajectories released from four different initial conditions integrated overt∈\[0,2000\]t\\in\[0,2000\]and sampled atΔ​t=0\.05\\Delta t=0\.05\. The first500500samples of every trajectory \(2525time units\) are discarded as the off\-manifold transient towards the slow SSM\. We train the SSM methods on two trajectories and evaluate on two unseen trajectories at new initial conditions\. The symmetry group supplied to eSSM is the parity group𝒢=\{𝐈,−𝐈\}\\mathcal\{G\}=\\\{\\mathbf\{I\},\-\\mathbf\{I\}\\\}of Example[2\.6](https://arxiv.org/html/2608.04239#S2.Thmtheorem6)\. Configurations of all methods are chosen identically where possible: reduced dimensiond=2d=2, manifold orderMM, and resonance toleranceδ=10−4\\delta=10^\{\-4\}\. The largest singular values of the orbit\-augmented training data and a fitted spectral subspace and SSM are shown in Figure[3](https://arxiv.org/html/2608.04239#S5.F3)\.

![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_spectral_gap.png)\(a\)Leading singular valuesσi\\sigma\_\{i\}of the orbit\-augmented training data\.
![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_manifold_fit.png)\(b\)Fitted spectral subspace \(purple plane\) and SSM \(orange surface\), together with a training trajectory \(blue\)\.

Figure 3:Manifold identification for the oscillator chain \(n=100n=100,d=2d=2,M=3M=3\)\.All iterative fits are run to first\-order optimality tolerance10−910^\{\-9\}with an iteration cap of20002000, where possible \(in the SSMLearn Matlab implementation these handles were not fully available as is discussed further below\)\. Our quality metric is the normalised mean trajectory error \(NMTE\) on the test set,

NMTE=‖𝐲^i−𝐲i‖2maxi⁡‖𝐲i‖2×100%,\\displaystyle\\mathrm\{NMTE\}=\\frac\{\\big\\\|\\hat\{\\mathbf\{y\}\}\_\{i\}\-\\mathbf\{y\}\_\{i\}\\big\\\|\_\{2\}\}\{\\max\_\{i\}\\\|\\mathbf\{y\}\_\{i\}\\\|\_\{2\}\}\\times 100\\,\\%,\(30\)where𝐲^i\\hat\{\\mathbf\{y\}\}\_\{i\}is the model prediction of theii\-th test trajectory𝐲i\\mathbf\{y\}\_\{i\}\.

#### Experimental results

In our first experiment we examine the behaviour of the methods as the manifold orderMMis varied\. The results of this experiment can be seen in Figure[4](https://arxiv.org/html/2608.04239#S5.F4)\. As demonstrated in Example[3\.12](https://arxiv.org/html/2608.04239#S3.Thmtheorem12)the parity symmetry of the chain of oscillators eliminates all even\-degree monomials from the Taylor expansions of the SSM parametrisation and the reduced dynamics\. This reduces the number of free parameters in the model significantly \(cf\. Table[1](https://arxiv.org/html/2608.04239#S5.T1)\)\. In practice, we observe that this reduced parameter count leads to a significant reduction in wall\-clock time of the fit \(cf\. Figure[4\(a\)](https://arxiv.org/html/2608.04239#S5.F4.sf1)\), while maintaining the accuracy of the reduced dynamics on the test trajectory \(cf\. Figure[4\(b\)](https://arxiv.org/html/2608.04239#S5.F4.sf2)\)\.

Table 1:Total number of fitted parameters in the casen=100,d=2n=100,d=2as a function ofM=MRODM=M\_\{\\text\{ROD\}\}\.![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_prediction_M_time.png)\(a\)Wall\-clock time of the full fit\.
![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_prediction_M_nmte.png)\(b\)NMTE on the held\-out trajectories\.

Figure 4:Prediction benchmark versus the manifold orderMM\(n=100n=100,d=2d=2,MROD=MM\_\{\\text\{ROD\}\}=M\)\.In our second experiment we fix the manifold order toM=3M=3and vary the ambient dimensionnnof the chain of oscillators\. The results of this experiment can be seen in Figure[5](https://arxiv.org/html/2608.04239#S5.F5)\. As expected, the wall\-clock time of the fit increases with increasing ambient dimension \(cf\. Figure[5\(a\)](https://arxiv.org/html/2608.04239#S5.F5.sf1)\), while the accuracy of the reduced dynamics on the test trajectory remains largely unaffected \(cf\. Figure[5\(b\)](https://arxiv.org/html/2608.04239#S5.F5.sf2), noting the scale of y\-axis\)\. As in the previous experiment, the reduced parameter count of the eSSM method leads to a significant reduction in wall\-clock time of the fit, while maintaining the accuracy of the reduced dynamics on the test trajectory\.

![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_prediction_n_time.png)\(a\)Wall\-clock time\.
![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_prediction_n_nmte.png)\(b\)NMTE on the test set\.

Figure 5:Performance of the methods as a function of the ambient dimensionnnat fixedM=3M=3\(d=2d=2,MROD=3M\_\{\\text\{ROD\}\}=3\)\.

### 5\.2Example 2: dissipative shallow\-water equations on the sphere

Our second example is a two\-dimensional PDE whose symmetry is inherited from the geometry of the underlying domain: the viscous shallow\-water equations on a rotating sphere,S2=\{𝐱∈ℝ3:\|𝐱\|=1\}S^\{2\}=\\\{\\mathbf\{x\}\\in\\mathbb\{R\}^\{3\}:\|\\mathbf\{x\}\|=1\\\}, in the formulation of\[[23](https://arxiv.org/html/2608.04239#bib.bib6)\],

∂t𝐮\+ν​Δ2​𝐮\+g​∇h\+f​𝐤^×𝐮\+γ​𝐮\\displaystyle\\partial\_\{t\}\\mathbf\{u\}\+\\nu\\Delta^\{2\}\\mathbf\{u\}\+g\\nabla h\+f\\,\\hat\{\\mathbf\{k\}\}\\times\\mathbf\{u\}\+\\gamma\\,\\mathbf\{u\}=−\(𝐮⋅∇\)​𝐮,\\displaystyle=\-\(\\mathbf\{u\}\\cdot\\nabla\)\\mathbf\{u\},\(31\)∂th\+ν​Δ2​h\+H​∇⋅𝐮\\displaystyle\\partial\_\{t\}h\+\\nu\\Delta^\{2\}h\+H\\nabla\\cdot\\mathbf\{u\}=−∇⋅\(h​𝐮\),\\displaystyle=\-\\nabla\\cdot\(h\\,\\mathbf\{u\}\),where𝐮\\mathbf\{u\}is the tangential velocity field,hhthe perturbation of the surface fluid about the constant mean depthHH,f=2​Ω​sin⁡φf=2\\Omega\\sin\\varphithe Coriolis parameter at latitudeφ\\varphi,ggis the gravitational constant,𝐤^\\hat\{\\mathbf\{k\}\}is the outward unit normal, and∇\\nabla,∇⋅\\nabla\\cdot,Δ\\Deltathe intrinsic surface differential operators\. The state of rest\(𝐮,h\)=\(𝟎,0\)\(\\mathbf\{u\},h\)=\(\\mathbf\{0\},0\)is a fixed point and the geometry induces a natural symmetry group of rotations about the polar axis\. Our time series data are generated by observing the height perturbationhhand the relative vorticityζ=𝐤^⋅\(∇×𝐮\)\\zeta=\\hat\{\\mathbf\{k\}\}\\cdot\(\\nabla\\times\\mathbf\{u\}\)at a finite number of3636sensor locations on the sphere \(cf\. Figure[6](https://arxiv.org/html/2608.04239#S5.F6)\)\.

![Refer to caption](https://arxiv.org/html/2608.04239v1/images/swe_ring_sensors.png)Figure 6:Sensor locations in our SWE experiment at four longitudes and nine latitudinal rings\.The system \([31](https://arxiv.org/html/2608.04239#S5.E31)\) is equivariant under the group of rotations about the polar axis, which implies that the observations inherit a discrete cyclic symmetry, in the discrete group of rotations𝒢=C4\\mathcal\{G\}=C\_\{4\}, from our sensor placement\.

#### Experimental setup

Data is generated with the spectral solver Dedalus\[[11](https://arxiv.org/html/2608.04239#bib.bib5)\], using the publicly available spherical shallow\-water example`vp\_sphere\_shallow\_water`with minor modifications, including the implementation of the drag term and the modified initial conditions

h0\\displaystyle h\_\{0\}=h~0−∫S2h~0​𝑑σ,h~0=A​\[cos⁡φ​cos⁡λ\+ε​e−\(1−cos⁡d​\(λ,φ\)\)/w\]\\displaystyle=\\tilde\{h\}\_\{0\}\-\\int\_\{\{S\}^\{2\}\}\\tilde\{h\}\_\{0\}\\,d\\sigma,\\quad\\tilde\{h\}\_\{0\}=A\\Big\[\\cos\\varphi\\cos\\lambda\+\\varepsilon\\,e^\{\-\(1\-\\cos d\(\\lambda,\\varphi\)\)/w\}\\Big\]𝐮0\\displaystyle\\qquad\\mathbf\{u\}\_\{0\}=𝟎,\\displaystyle=\\mathbf\{0\},withddthe great\-circle distance to\(λ0,φ0\)=\(0,π/6\)\(\\lambda\_\{0\},\\varphi\_\{0\}\)=\(0,\\pi/6\)andA=0\.4​HA=0\.4\\,H\. Our parameter choices areΩ=0\.2625\\Omega=0\.2625,g=19\.95g=19\.95,H=1\.570×10−3H=1\.570\\times 10^\{\-3\},ν=1\.75×10−3\\nu=1\.75\\times 10^\{\-3\},γ=2×10−3\\gamma=2\\times 10^\{\-3\},ε=0\.15\\varepsilon=0\.15,w=0\.15w=0\.15, and the spectral method resolution used in Dedalus is\(Nλ,Nθ\)=\(128,64\)\(N\_\{\\lambda\},N\_\{\\theta\}\)=\(128,64\)\. A visualisation of the states of this system can be seen in Figure[7](https://arxiv.org/html/2608.04239#S5.F7)\.

![Refer to caption](https://arxiv.org/html/2608.04239v1/images/swe_ring_height_j00.png)\(a\)hhatt=0t=0\.
![Refer to caption](https://arxiv.org/html/2608.04239v1/images/swe_ring_height_j01.png)\(b\)hhatt=8t=8\.

Figure 7:Visualisation of thehh\-perturbation in \([31](https://arxiv.org/html/2608.04239#S5.E31)\)\.The trajectory is integrated overt∈\[0,400\]t\\in\[0,400\]and sampled atΔ​t=0\.25\\Delta t=0\.25; the first600600steps are discarded as the off\-manifold transient towards the slow SSM\. Our observations live inℝ72\\mathbb\{R\}^\{72\}\(two scalar fields\) and𝒢≅C4\\mathcal\{G\}\\cong C\_\{4\}\. Configurations of all methods are again chosen identically where possible\. Given the quadratic nature of the nonlinearity in \([31](https://arxiv.org/html/2608.04239#S5.E31)\) the reduced\-dynamics orderMRODM\_\{\\text\{ROD\}\}was fixed to22throughout this example\. We fit the SSM dynamics on a single trajectory as above, and evaluate on two metrics \(using the NMTE \([30](https://arxiv.org/html/2608.04239#S5.E30)\) as in Example 1\):

- •Self:corresponding to the training trajectory, which is used to assess the accuracy of the reduced dynamics on the SSM\.
- •Rotated:corresponding to a trajectory obtained by applying a 90\-degree rotation to the training trajectory, which is used to assess the accuracy of the reduced dynamics on the SSM under symmetry transformations\.

Given the more complex nature of this example we commence with a sweep over the manifold dimensionddto identify a suitable reduced dimension for the SSM, fixingM=2M=2given the quadratic nature of the nonlinearity in \([31](https://arxiv.org/html/2608.04239#S5.E31)\)\. The results of this sweep can be seen in Figure[8](https://arxiv.org/html/2608.04239#S5.F8)\. We observe that the NMTE appears to be smallest atd=6d=6thus suggesting thatd=6d=6is a suitable reduced dimension for the SSM\. As in Example 1 we notice that eSSM achieves comparable accuracy to SSMLearn with a significantly reduced CPU time\. We note in this example SSMLearn Matlab is underperforming in terms of accuracy \(withd=8d=8the method did not converge\)\. This is most likely due to the fact that the Matlab implementation of SSMLearn enforcesMROD≥3M\_\{\\text\{ROD\}\}\\geq 3\(which is a sensible constraint for exact normal forms\) and thus is unable to fit the reduced dynamics on the extended normal form withMROD=2M\_\{\\text\{ROD\}\}=2directly\. Apparently, this dynamics fitting problem becomes badly conditioned whenMROD≥3M\_\{\\text\{ROD\}\}\\geq 3thus leading to the poor fit observed in Figure[8\(b\)](https://arxiv.org/html/2608.04239#S5.F8.sf2)\.

![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_benchmark_swe_d_time.png)\(a\)Wall\-clock time of the full fit\.
![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_benchmark_swe_d_orbit.png)\(b\)Prediction error on the training trajectory\.

Figure 8:Performance of the methods as a function of the reduced dimensionddat fixedM=2M=2\.In our second experiment we fix the reduced dimension tod=6d=6and vary the manifold orderMM\(withMROD=2M\_\{\\text\{ROD\}\}=2\)\. The results of this experiment can be seen in Figure[9](https://arxiv.org/html/2608.04239#S5.F9)\. In this experiment the fourth order symmetry groupC4C\_\{4\}leads to a nearly 75% reduction in the number of free parameters at every manifold order \(cf\. Table[2](https://arxiv.org/html/2608.04239#S5.T2)\)\.

Table 2:Total number of fitted parameters for the shallow\-water example \(n=72n=72,d=6d=6\) as a function of the manifold orderMM\.The result in terms of practical performance can be seen in Figure[9](https://arxiv.org/html/2608.04239#S5.F9)\. We observe that the reduced parameter count leads to a significant reduction in wall\-clock time of the fit at matched prediction accuracy on the training trajectory \(Figure[9](https://arxiv.org/html/2608.04239#S5.F9)\), in particular we observe a roughly 50% cost reduction throughout and a more significant reduction atM=1,2M=1,2where the unconstrained SSMLearn method struggles to fit the reduced dynamics on the extended normal form withMROD=2M\_\{\\text\{ROD\}\}=2directly\. We note that the accuracy of the reduced dynamics on the training trajectory remains largely unaffected by the manifold orderMM\(Figure[9\(b\)](https://arxiv.org/html/2608.04239#S5.F9.sf2)\), however the accuracy of the reduced dynamics on the symmetry\-transformed trajectory is significantly improved by enforcing equivariance\.

![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_benchmark_swe_M_time.png)\(a\)Wall\-clock time of the full fit\.
![Refer to caption](https://arxiv.org/html/2608.04239v1/images/paper_benchmark_swe_M_orbit.png)\(b\)Prediction error on training and rotated trajectories\.

Figure 9:Prediction benchmark versus the manifold orderMM\(n=72n=72,d=6d=6\)\.

### 5\.3Example 3: KS equation on periodic domain \(CTF4Science benchmark\)

In this final example we will benchmark the eSSM method on the Kuramoto–Sivashinsky \(KS\) equation challenge of the CTF4Science project\[[55](https://arxiv.org/html/2608.04239#bib.bib13)\]\. The KS equation,

∂tu\+u​∂xu\+∂x​xu\+μ​∂x​x​x​xu=0,x∈\[0,32​π\],\\displaystyle\\partial\_\{t\}u\+u\\,\\partial\_\{x\}u\+\\partial\_\{xx\}u\+\\mu\\,\\partial\_\{xxxx\}u=0,\\qquad x\\in\[0,32\\pi\],\(32\)with periodic boundary conditions, is a canonical example of spatio\-temporal chaos in one dimension\. The benchmark provides training trajectories of \([32](https://arxiv.org/html/2608.04239#S5.E32)\) on an10241024\-point grid \(time step and initial conditions undisclosed\) and evaluates predictions on hidden test data\. Strictly speaking, this setting lies outside the scope of our theory: although the originu=0u=0is a fixed point of \([32](https://arxiv.org/html/2608.04239#S5.E32)\), the CTF4Science data explores a chaotic attractor rather than a decaying transient towards a stable equilibrium\. We include this example to test the robustness of SSM\-based forecasting of dynamical systems on a standardised benchmark against a broad field of data\-driven methods\. We focus on the forecasting task \(Test 1 of\[[55](https://arxiv.org/html/2608.04239#bib.bib13)\]\), which is scored by a short\-time \(“weather”\) and a long\-time \(“climate”\) metric,

E1=100​\(1−SST\),E2=100​\(1−SLT\),\\displaystyle E\_\{1\}=100\\,\\big\(1\-S\_\{\\mathrm\{ST\}\}\\big\),\\qquad E\_\{2\}=100\\,\\big\(1\-S\_\{\\mathrm\{LT\}\}\\big\),\(33\)whereSSTS\_\{\\mathrm\{ST\}\}is the relative error of the predicted state over themmsnapshots of the forecast window,

SST=\(∑i=1m‖𝐮^i−𝐮i‖22\)1/2\(∑i=1m‖𝐮i‖22\)1/2,\\displaystyle S\_\{\\mathrm\{ST\}\}=\\frac\{\\Big\(\\sum\_\{i=1\}^\{m\}\\big\\\|\\hat\{\\mathbf\{u\}\}\_\{i\}\-\\mathbf\{u\}\_\{i\}\\big\\\|\_\{2\}^\{2\}\\Big\)^\{1/2\}\}\{\\Big\(\\sum\_\{i=1\}^\{m\}\\big\\\|\\mathbf\{u\}\_\{i\}\\big\\\|\_\{2\}^\{2\}\\Big\)^\{1/2\}\},with𝐮i\\mathbf\{u\}\_\{i\}theii\-th test snapshot and𝐮^i\\hat\{\\mathbf\{u\}\}\_\{i\}its prediction, andSLTS\_\{\\mathrm\{LT\}\}is the corresponding relative error of the log power spectral density restricted to the lowestkmax=100k\_\{\\max\}=100wavenumbers,

SLT=\(∑i=1m∑\|k\|≤kmax\(p^i,k−pi,k\)2\)1/2\(∑i=1m∑\|k\|≤kmaxpi,k2\)1/2,pi,k=ln⁡\|\(ℱ​𝐮i\)k\|2,\\displaystyle S\_\{\\mathrm\{LT\}\}=\\frac\{\\Big\(\\sum\_\{i=1\}^\{m\}\\sum\_\{\|k\|\\leq k\_\{\\max\}\}\\big\(\\hat\{p\}\_\{i,k\}\-p\_\{i,k\}\\big\)^\{2\}\\Big\)^\{1/2\}\}\{\\Big\(\\sum\_\{i=1\}^\{m\}\\sum\_\{\|k\|\\leq k\_\{\\max\}\}p\_\{i,k\}^\{2\}\\Big\)^\{1/2\}\},\\qquad p\_\{i,k\}=\\ln\|\(\\mathcal\{F\}\\mathbf\{u\}\_\{i\}\)\_\{k\}\\big\|^\{2\},where\(ℱ​𝐮i\)k\(\\mathcal\{F\}\\mathbf\{u\}\_\{i\}\)\_\{k\}denotes thekk\-th discrete Fourier coefficient of theii\-th snapshot andp^i,k\\hat\{p\}\_\{i,k\}the same quantity for the prediction\. The scores are normalised such that a forecast of zeros leads to a score of0in both metrics, and a score of100100corresponds to a perfect match with the hidden test data\. For further details the reader is referred to\[[55](https://arxiv.org/html/2608.04239#bib.bib13)\]\.

#### Experimental setup

We use the discrete\-time eSSM method of Algorithm[2](https://arxiv.org/html/2608.04239#alg2)for this forecasting task\. The training trajectory is of size𝐘∈ℝ10000×1024\\mathbf\{Y\}\\in\\mathbb\{R\}^\{10000\\times 1024\}and we are asking the model to continue this same trajectory for an additional10001000steps\. In our setup the state is subsampled to annxn\_\{x\}\-point spatial grid before the fit and interpolated back to the full grid by trigonometric interpolation\. We use a time\-delay embedding \(cf\. §[4\.1](https://arxiv.org/html/2608.04239#S4.SS1)\) withqqcopies at lagqlagq\_\{\\mathrm\{lag\}\}steps\. Since the CTF measures error directly in the full state space starting at the final step of the training trajectory, the metric is sensitive to the off\-manifold residual that arises when projecting the initial condition onto the fitted SSM\. To mitigate this, we use a simple exponential decay of the off\-manifold residual during the forecast: letting𝜼0=𝐕1⊤​𝐮0\\bm\{\\eta\}\_\{0\}=\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{u\}\_\{0\}be the initial condition on the fitted SSM, we define the initial off\-manifold residual

𝐫0:=𝐮0−𝐔1​𝜼0−𝐡​\(𝜼0\),\\displaystyle\\mathbf\{r\}\_\{0\}\\;:=\\;\\mathbf\{u\}\_\{0\}\\;\-\\;\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\_\{0\}\\;\-\\;\\mathbf\{h\}\(\\bm\{\\eta\}\_\{0\}\),i\.e\. the part of𝐮0\\mathbf\{u\}\_\{0\}that is not captured by the manifold parametrisation𝜼↦𝐔1​𝜼\+𝐡​\(𝜼\)\\bm\{\\eta\}\\mapsto\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\+\\mathbf\{h\}\(\\bm\{\\eta\}\)\. We then correct the forecast𝐮^i\\hat\{\\mathbf\{u\}\}\_\{i\}at stepiiby adding the decayed residual𝐫0​e−i/τ\\mathbf\{r\}\_\{0\}\\,e^\{\-i/\\tau\}, whereτ\\tauis a time constant \(in steps\) that controls the decay rate\. This ensures that the forecast matches the observed initial condition exactly ati=0i=0and relaxes onto the SSM prediction over𝒪​\(τ\)\\mathcal\{O\}\(\\tau\)steps, which removes the initial jump that would otherwise be introduced by projecting onto the manifold\. On the periodic grid, \([32](https://arxiv.org/html/2608.04239#S5.E32)\) is equivariant under the cyclic group of grid translations and we supply to eSSM the subgroup of shifts by multiples of2k2^\{k\}grid points,

𝒢≅Cnx/2k,k=0,…,log2⁡\(nx\)−1,\\displaystyle\\mathcal\{G\}\\cong C\_\{n\_\{x\}/2^\{k\}\},k=0,\\dots,\\log\_\{2\}\(n\_\{x\}\)\-1,with𝐒\\mathbf\{S\}the elementary cyclic shift\. To evaluate our method fairly against the prepopulated CTF4Science leaderboard\[[55](https://arxiv.org/html/2608.04239#bib.bib13)\], we follow the original evaluation methodology of\[[55](https://arxiv.org/html/2608.04239#bib.bib13)\]and use Ray Tune\[[38](https://arxiv.org/html/2608.04239#bib.bib4)\]as a hyperparameter tuner to select the optimal configuration of the eSSM method based on an 80/20 split of the training data \(Ray Tune does not see the held\-out test data used to compute the scores in Table[4](https://arxiv.org/html/2608.04239#S5.T4)\)\. We also compare against our Python implementation of SSMLearn \(the original Matlab implementation is not compatible with the CTF Python codebase\)\. The optimal configurations found in this way are given in Table[3](https://arxiv.org/html/2608.04239#S5.T3)\.

Table 3:Ray\-Tune\-selected configurations \(τ=500\\tau=500steps for both\):ddthe SSM dimension,MMthe manifold order,MRODM\_\{\\text\{ROD\}\}the reduced\-dynamics order,nxn\_\{x\}the subsampled grid size,qqdelay copies at lagqlagq\_\{\\mathrm\{lag\}\}, stride is the temporal subsampling of the fit samples, and𝒢\\mathcal\{G\}the symmetry group supplied to the method\.
#### Experimental results

Table[4](https://arxiv.org/html/2608.04239#S5.T4)reports the scores against the CTF4Science field\. Both SSM\-based entries rank near the top of the leaderboard on this first CTF task, with only the reservoir computing method clearly ahead\. We note that the equivariant and non\-equivariant fits reach comparable accuracy, but, consistent with the parameter\-count reductions of the previous examples, the eSSM fit takes1\.481\.48\\,s against3\.693\.69\\,s for SSMLearn\. We note in particular, that eSSM clearly outperforms linear methods such as DMD and PyKoopman\.

Table 4:CTF4Science KS forecasting task \(Test 1\): short\-time scoreE1E\_\{1\}, long\-time scoreE2E\_\{2\}\([33](https://arxiv.org/html/2608.04239#S5.E33)\) and their average, ranked by average\. Bold rows correspond to scores obtained in the present work, the remaining scores are those reported in\[[55](https://arxiv.org/html/2608.04239#bib.bib13)\]\.

## 6Conclusions

In this work we introduced equivariant spectral submanifold \(eSSM\) reduction as a means of computing accurate nonlinear reduced order models of high\-dimensional systems with symmetries\. We showed that SSMs of equivariant systems are themselves equivariant submanifolds, that suitably chosen charts, the reduced dynamics and the extended normal form all inherit induced actions of the symmetry group, and we characterised the admissible Taylor coefficients of equivariant maps\. Building on these results, we developed the eSSM reduction algorithm \(Algorithm[1](https://arxiv.org/html/2608.04239#alg1)and its discrete\-time counterpart, Algorithm[2](https://arxiv.org/html/2608.04239#alg2)\), whose output is exactly equivariant by construction, for any input data\. In our numerical experiments the resulting reduction in free parameters translated into significantly faster fits at matched predictive accuracy, improved fidelity of the reduced model under symmetry transformations of the data, and competitive performance on the CTF4Science Kuramoto–Sivashinsky benchmark\[[55](https://arxiv.org/html/2608.04239#bib.bib13)\]at a fraction of the computational cost of the unconstrained method\.

## Appendix ADescription of the method for discrete dynamical systems

The eSSM reduction method extends, with only minor modifications, to discrete dynamical systems of the form

𝐱k\+1=𝐅​\(𝐱k\)=𝐀​𝐱k\+𝐟​\(𝐱k\),𝐱k∈ℝn,\\displaystyle\\mathbf\{x\}\_\{k\+1\}=\\mathbf\{F\}\(\\mathbf\{x\}\_\{k\}\)=\\mathbf\{A\}\\,\\mathbf\{x\}\_\{k\}\+\\mathbf\{f\}\(\\mathbf\{x\}\_\{k\}\),\\qquad\\mathbf\{x\}\_\{k\}\\in\\mathbb\{R\}^\{n\},\(34\)where𝐟=𝒪​\(‖𝐱‖2\)\\mathbf\{f\}=\\mathcal\{O\}\(\\\|\\mathbf\{x\}\\\|^\{2\}\)is smooth and𝐱=𝟎\\mathbf\{x\}=\\mathbf\{0\}is a hyperbolic fixed point, i\.e\.Spect⁡\(𝐀\)\\operatorname\{Spect\}\(\\mathbf\{A\}\)does not intersect the unit circle\. The discrete system \([34](https://arxiv.org/html/2608.04239#A1.E34)\) is said to be equivariant with respect to a linear symmetry group𝒢\\mathcal\{G\}if

S​𝐅​\(𝐱\)=𝐅​\(S​𝐱\),∀S∈𝒢,𝐱∈ℝn\.\\displaystyle S\\mathbf\{F\}\(\\mathbf\{x\}\)=\\mathbf\{F\}\(S\\mathbf\{x\}\),\\qquad\\forall S\\in\\mathcal\{G\},\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\}\.The theory of equivariant spectral submanifolds and the associated reduced dynamics carries over to the discrete setting with only minor straightforward modifications and is therefore, in the interest of brevity, not repeated here\. Instead we focus on presenting the main differences in the data\-driven eSSM reduction method described in §[4](https://arxiv.org/html/2608.04239#S4)and Algorithm[1](https://arxiv.org/html/2608.04239#alg1)in this discrete setting\. Indeed, Step \(i\) of Algorithm[1](https://arxiv.org/html/2608.04239#alg1)only relies on point\-values in the observed data and is therefore identical in the discrete setting\. The only modifications occur in Step \(ii\) of Algorithm[1](https://arxiv.org/html/2608.04239#alg1)\. In the discrete setting the reduced dynamics is the one\-step map

𝜼k\+1=𝐫​\(𝜼k\)=𝐕1⊤​𝐅​\(𝐔1​𝜼k\+𝐡​\(𝜼k\)\),\\displaystyle\\bm\{\\eta\}\_\{k\+1\}=\\mathbf\{r\}\(\\bm\{\\eta\}\_\{k\}\)=\\mathbf\{V\}\_\{1\}^\{\\top\}\\,\\mathbf\{F\}\\big\(\\mathbf\{U\}\_\{1\}\\bm\{\\eta\}\_\{k\}\+\\mathbf\{h\}\(\\bm\{\\eta\}\_\{k\}\)\\big\),\(35\)whose linear part𝐁=D​𝐫​\(𝟎\)\\mathbf\{B\}=D\\mathbf\{r\}\(\\mathbf\{0\}\)is estimated, exactly as in Step \(ii\), by the orbit\-augmented dynamic mode decomposition

𝐁=𝚵𝒢′​𝚵𝒢†,𝚵=\[𝜼1,…,𝜼N−1\],𝚵′=\[𝜼2,…,𝜼N\]\.\\displaystyle\\mathbf\{B\}=\\bm\{\\Xi\}^\{\\prime\}\_\{\\mathcal\{G\}\}\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}^\{\\dagger\},\\qquad\\bm\{\\Xi\}=\[\\bm\{\\eta\}\_\{1\},\\dots,\\bm\{\\eta\}\_\{N\-1\}\],\\quad\\bm\{\\Xi\}^\{\\prime\}=\[\\bm\{\\eta\}\_\{2\},\\dots,\\bm\{\\eta\}\_\{N\}\]\.However, in contrast to the continuous case, no matrix logarithm is required\. To derive the extended normal form of the discrete reduced dynamics \([35](https://arxiv.org/html/2608.04239#A1.E35)\) we follow the same steps as in §[3](https://arxiv.org/html/2608.04239#S3), starting with the eigendecomposition of the linear part𝐁\\mathbf\{B\}:

𝐁=𝐖​𝚲μ​𝐖−1,𝚲μ=diag⁡\(μ1,…,μd\)\.\\displaystyle\\mathbf\{B\}=\\mathbf\{W\}\\bm\{\\Lambda\}\_\{\\mu\}\\mathbf\{W\}^\{\-1\},\\qquad\\bm\{\\Lambda\}\_\{\\mu\}=\\operatorname\{diag\}\(\\mu\_\{1\},\\dots,\\mu\_\{d\}\)\.The normal form is now sought as a conjugate one\-step map: with the same truncated expansions \([27](https://arxiv.org/html/2608.04239#S4.E27)\)–\([28](https://arxiv.org/html/2608.04239#S4.E28)\), but with𝚲\\bm\{\\Lambda\}replaced by𝚲μ\\bm\{\\Lambda\}\_\{\\mu\}, we seek

𝐳k\+1=𝐧​\(𝐳k\),𝜼=𝐭​\(𝐳\),\\displaystyle\\mathbf\{z\}\_\{k\+1\}=\\mathbf\{n\}\(\\mathbf\{z\}\_\{k\}\),\\qquad\\bm\{\\eta\}=\\mathbf\{t\}\(\\mathbf\{z\}\),where the differential conjugacy \([7](https://arxiv.org/html/2608.04239#S3.E7)\) is replaced by its composition form

𝐭​\(𝐧​\(𝐳\)\)=𝐫​\(𝐭​\(𝐳\)\)\.\\displaystyle\\mathbf\{t\}\\big\(\\mathbf\{n\}\(\\mathbf\{z\}\)\\big\)=\\mathbf\{r\}\\big\(\\mathbf\{t\}\(\\mathbf\{z\}\)\\big\)\.\(36\)Matching coefficients order by order in \([36](https://arxiv.org/html/2608.04239#A1.E36)\) yields the discrete homological equations

\(𝝁𝐦−μj\)​t~j,𝐦\+nj,𝐦=gj,𝐦,𝝁𝐦:=∏l=1dμlml,\\displaystyle\\big\(\\bm\{\\mu\}^\{\\mathbf\{m\}\}\-\\mu\_\{j\}\\big\)\\,\\widetilde\{t\}\_\{j,\\mathbf\{m\}\}\+n\_\{j,\\mathbf\{m\}\}=g\_\{j,\\mathbf\{m\}\},\\qquad\\bm\{\\mu\}^\{\\mathbf\{m\}\}:=\\prod\_\{l=1\}^\{d\}\\mu\_\{l\}^\{m\_\{l\}\},in place of \([8](https://arxiv.org/html/2608.04239#S3.E8)\)\. The discrete multipliers can be regarded as the discrete\-time analogue of the continuous\-time eigenvalues withμj=eλj​Δ​t\\mu\_\{j\}=e^\{\\lambda\_\{j\}\\Delta t\}and, as a result, the additive resonance quantity𝐦⋅𝝀−λj\\mathbf\{m\}\\cdot\\bm\{\\lambda\}\-\\lambda\_\{j\}is replaced by its multiplicative counterpart𝝁𝐦−μj\\bm\{\\mu\}^\{\\mathbf\{m\}\}\-\\mu\_\{j\}, and a monomial can be removed from the reduced dynamics precisely when𝝁𝐦≠μj\\bm\{\\mu\}^\{\\mathbf\{m\}\}\\neq\\mu\_\{j\}, with small denominators arising whenever𝝁𝐦≈μj\\bm\{\\mu\}^\{\\mathbf\{m\}\}\\approx\\mu\_\{j\}\. Thus it is natural to characterise the near\-resonant monomials in the discrete setting by the index set

ℐδ:=\{\(j,𝐦\):\|arg⁡\(𝝁𝐦/μj\)\|≤δ\}\.\\displaystyle\\mathcal\{I\}\_\{\\delta\}:=\\Big\\\{\(j,\\mathbf\{m\}\):\\big\|\\arg\\big\(\\bm\{\\mu\}^\{\\mathbf\{m\}\}/\\mu\_\{j\}\\big\)\\big\|\\leq\\delta\\Big\\\}\.\(37\)We can show analogously to the continuous case \(cf\. Proposition[3\.5](https://arxiv.org/html/2608.04239#S3.Thmtheorem5)\) that the near\-resonant classification \([37](https://arxiv.org/html/2608.04239#A1.E37)\) is compatible with the symmetry group𝒢\\mathcal\{G\}, so that the equivariance constraints on the normal form coefficients can be imposed in the same way as in §[4\.2](https://arxiv.org/html/2608.04239#S4.SS2)\. Finally, the coefficient fit of Step \(ii\.c\) from §[4\.2](https://arxiv.org/html/2608.04239#S4.SS2)is replaced by the discrete conjugacy fit

\(𝐜𝐧,𝐜𝐭,⋆\)=argmin𝐜𝐧,𝐜𝐭,⋆​∑i=1N−1‖𝐭−1​\(𝜼i\+1\)−𝐧​\(𝐭−1​\(𝜼i\)\)‖22\.\\displaystyle\(\\mathbf\{c\}^\{\\mathbf\{n\}\},\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}\)=\\operatorname\*\{argmin\}\_\{\\mathbf\{c\}^\{\\mathbf\{n\}\},\\,\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}\}\\sum\_\{i=1\}^\{N\-1\}\\Big\\\|\\,\\mathbf\{t\}^\{\-1\}\(\\bm\{\\eta\}\_\{i\+1\}\)\-\\mathbf\{n\}\\big\(\\mathbf\{t\}^\{\-1\}\(\\bm\{\\eta\}\_\{i\}\)\\big\)\\Big\\\|\_\{2\}^\{2\}\.\(38\)
The discrete\-time eSSM reduction method is summarised in Algorithm[2](https://arxiv.org/html/2608.04239#alg2)\.

Algorithm 2Discrete\-time eSSM reduction method1:Inputs:as in Algorithm[1](https://arxiv.org/html/2608.04239#alg1)without

Δ​t\\Delta t\.

2:

3:*Step \(i\): identical to Step \(i\) of Algorithm[1](https://arxiv.org/html/2608.04239#alg1)\.*

4:

5:*Step \(ii\): Compute reduced one\-step dynamics on𝒲​\(E\)\\mathcal\{W\}\(E\)in extended normal form\.*

6:Formthe orbit\-augmented reduced snapshots

𝚵𝒢=\[S\|E​𝚵\]S∈𝒢\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}=\[\\,S\|\_\{E\}\\bm\{\\Xi\}\\,\]\_\{S\\in\\mathcal\{G\}\},

𝚵𝒢′=\[S\|E​𝚵′\]S∈𝒢\\bm\{\\Xi\}^\{\\prime\}\_\{\\mathcal\{G\}\}=\[\\,S\|\_\{E\}\\bm\{\\Xi\}^\{\\prime\}\\,\]\_\{S\\in\\mathcal\{G\}\}\.

7:Estimatethe one\-step map

𝐁=𝚵𝒢′​𝚵𝒢†\\mathbf\{B\}=\\bm\{\\Xi\}^\{\\prime\}\_\{\\mathcal\{G\}\}\\bm\{\\Xi\}\_\{\\mathcal\{G\}\}^\{\\dagger\}andeigendecompose

𝐁=𝐖​𝚲μ​𝐖−1\\mathbf\{B\}=\\mathbf\{W\}\\bm\{\\Lambda\}\_\{\\mu\}\\mathbf\{W\}^\{\-1\}\.

8:for

k=2,…,MRODk=2,\\dots,M\_\{\\text\{ROD\}\}do

9:Selectthe resonant support

ℐδ=\{\(j,𝐦\):\|arg⁡\(𝝁𝐦/μj\)\|≤δ​Δ​t\}\\mathcal\{I\}\_\{\\delta\}=\\\{\(j,\\mathbf\{m\}\):\|\\arg\(\\bm\{\\mu\}^\{\\mathbf\{m\}\}/\\mu\_\{j\}\)\|\\leq\\delta\\,\\Delta t\\\}\.

10:Computethe equivariant bases

𝐁k𝐧,𝐁k𝐭\\mathbf\{B\}\_\{k\}^\{\\mathbf\{n\}\},\\mathbf\{B\}\_\{k\}^\{\\mathbf\{t\}\}of

𝐌k​\[S^\]\\mathbf\{M\}\_\{k\}\[\\widehat\{S\}\]restricted to

supp⁡\(ℐδ\),supp⁡\(ℐδc\)\\operatorname\{supp\}\(\\mathcal\{I\}\_\{\\delta\}\),\\operatorname\{supp\}\(\\mathcal\{I\}\_\{\\delta\}^\{\\,c\}\)\.

11:endfor

12:Fit

\(𝐜𝐧,𝐜𝐭,⋆\)\(\\mathbf\{c\}^\{\\mathbf\{n\}\},\\mathbf\{c\}^\{\\mathbf\{t\},\\star\}\)on the one\-step conjugacy residual \([38](https://arxiv.org/html/2608.04239#A1.E38)\), initialised at

𝟎\\mathbf\{0\}\.

13:Recover

𝐜𝐭\\mathbf\{c\}^\{\\mathbf\{t\}\}by linear least squares andassemble

𝐍k,𝐓k,𝐓k⋆\\mathbf\{N\}\_\{k\},\\mathbf\{T\}\_\{k\},\\mathbf\{T\}\_\{k\}^\{\\star\}via \([29](https://arxiv.org/html/2608.04239#S4.E29)\)\.

14:

15:Outputs:SSM parametrisation

\(𝐔1,𝐕1,\{𝐖k\}k=2M,𝐃\)\(\\mathbf\{U\}\_\{1\},\\mathbf\{V\}\_\{1\},\\\{\\mathbf\{W\}\_\{k\}\\\}\_\{k=2\}^\{M\},\\mathbf\{D\}\)and reduced one\-step dynamics

\(𝐖,𝚲μ,\{𝐍k,𝐓k,𝐓k⋆\}k=2MROD\)\(\\mathbf\{W\},\\bm\{\\Lambda\}\_\{\\mu\},\\\{\\mathbf\{N\}\_\{k\},\\mathbf\{T\}\_\{k\},\\mathbf\{T\}^\{\\star\}\_\{k\}\\\}\_\{k=2\}^\{M\_\{\\text\{ROD\}\}\}\)on

𝒲​\(E\)\\mathcal\{W\}\(E\)in extended normal form\.

Using the outputs of Algorithm[2](https://arxiv.org/html/2608.04239#alg2), we can then simulate the dynamics of \([34](https://arxiv.org/html/2608.04239#A1.E34)\) similarly to the continuous case, by iterating the reduced normal form𝐳k\+1=𝐧​\(𝐳k\)\\mathbf\{z\}\_\{k\+1\}=\\mathbf\{n\}\(\\mathbf\{z\}\_\{k\}\), after mapping the initial condition𝐱0\\mathbf\{x\}\_\{0\}into normal\-form coordinates via𝐳0=𝐭−1​\(𝐕1⊤​𝐱0\)\\mathbf\{z\}\_\{0\}=\\mathbf\{t\}^\{\-1\}\(\\mathbf\{V\}\_\{1\}^\{\\top\}\\mathbf\{x\}\_\{0\}\), and then mapping the trajectory back to the full state space via𝐱k=𝐔1​𝐭​\(𝐳k\)\+𝐡​\(𝐭​\(𝐳k\)\)\\mathbf\{x\}\_\{k\}=\\mathbf\{U\}\_\{1\}\\mathbf\{t\}\(\\mathbf\{z\}\_\{k\}\)\+\\mathbf\{h\}\\big\(\\mathbf\{t\}\(\\mathbf\{z\}\_\{k\}\)\\big\)\.

## Acknowledgments

The author gratefully acknowledges funding in form of a Henslow Fellowship of the Cambridge Philosophical Society\. The author thanks Matt Colbrook \(University of Cambridge\) for helpful feedback on an early draft of the manuscript\.

## References

- \[1\]\(1983\)Geometrical methods in the theory of ordinary differential equations\.Grundlehren der mathematischen Wissenschaften, Vol\.250,Springer,Berlin\.Cited by:[§3](https://arxiv.org/html/2608.04239#S3.p4.14)\.
- \[2\]T\. Askham and J\. N\. Kutz\(2018\)Variable projection methods for an optimized dynamic mode decomposition\.SIAM Journal on Applied Dynamical Systems17\(1\),pp\. 380–416\.Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.8.7.1)\.
- \[3\]J\. Axås, M\. Cenedese, and G\. Haller\(2023/05/01\)Fast data\-driven model reduction for nonlinear dynamical systems\.Nonlinear Dynamics111\(9\),pp\. 7941–7957\.External Links:[Document](https://dx.doi.org/10.1007/s11071-022-08014-0),ISBN 1573\-269XCited by:[§1](https://arxiv.org/html/2608.04239#S1.p2.1),[§4\.1](https://arxiv.org/html/2608.04239#S4.SS1.p1.3),[Remark 4\.7](https://arxiv.org/html/2608.04239#S4.Thmtheorem7.p1.19.13),[Remark 4\.7](https://arxiv.org/html/2608.04239#S4.Thmtheorem7.p1.6.6)\.
- \[4\]J\. Axås and G\. Haller\(2023/12/01\)Model reduction for nonlinearizable dynamics via delay\-embedded spectral submanifolds\.Nonlinear Dynamics111\(24\),pp\. 22079–22099\.External Links:[Document](https://dx.doi.org/10.1007/s11071-023-08705-2),ISBN 1573\-269XCited by:[Example 2\.6](https://arxiv.org/html/2608.04239#S2.Thmtheorem6.p1.1.1)\.
- \[5\]P\. J\. Baddoo, B\. Herrmann, B\. J\. McKeon, J\. Nathan Kutz, and S\. L\. Brunton\(2023\)Physics\-informed dynamic mode decomposition\.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences479\(2271\)\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[6\]P\. Benner, M\. Ohlberger, A\. Cohen, and K\. Willcox \(Eds\.\)\(2017\)Model reduction and approximation\.edition,Society for Industrial and Applied Mathematics,Philadelphia, PA\.External Links:[Document](https://dx.doi.org/10.1137/1.9781611974829)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1)\.
- \[7\]L\. Bettini, B\. Kaszás, B\. Zybach, J\. Dual, and G\. Haller\(2025\-04\)Data\-driven nonlinear model reduction to spectral submanifolds via oblique projection\.Chaos: An Interdisciplinary Journal of Nonlinear Science35\(4\),pp\. 043135\.External Links:ISSN 1054\-1500,[Document](https://dx.doi.org/10.1063/5.0243849)Cited by:[§3](https://arxiv.org/html/2608.04239#S3.p1.13)\.
- \[8\]Y\. A\. Bronsard, Y\. Bruned, G\. Maierhofer, and K\. Schratz\(2026\)Symmetric resonance based integrators and forest formulae\.Foundations of Computational Mathematics,pp\. 1–77\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[9\]S\. L\. Brunton, M\. Budišić, E\. Kaiser, and J\. N\. Kutz\(2022\)Modern Koopman Theory for Dynamical Systems\.SIAM Review64\(2\),pp\. 229–340\.External Links:[Document](https://dx.doi.org/10.1137/21M1401243)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1),[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.9.8.1)\.
- \[10\]S\. L\. Brunton, J\. L\. Proctor, and J\. N\. Kutz\(2016\)Discovering governing equations from data by sparse identification of nonlinear dynamical systems\.Proceedings of the National Academy of Sciences113\(15\),pp\. 3932–3937\.External Links:[Document](https://dx.doi.org/10.1073/pnas.1517384113)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1),[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.7.6.1)\.
- \[11\]K\. J\. Burns, G\. M\. Vasil, J\. S\. Oishi, D\. Lecoanet, and B\. P\. Brown\(2020\-04\)Dedalus: A flexible framework for numerical simulations with spectral methods\.Physical Review Research2\(2\),pp\. 023068\.External Links:[Document](https://dx.doi.org/10.1103/PhysRevResearch.2.023068)Cited by:[§5\.2](https://arxiv.org/html/2608.04239#S5.SS2.SSS0.Px1.p1.12)\.
- \[12\]X\. Cabré, E\. Fontich, and R\. de la Llave\(2003\)The parameterization method for invariant manifolds i: manifolds associated to non\-resonant subspaces\.Indiana Univ\. Math\. J\.52,pp\. 283–328\.External Links:ISSN 0022\-2518Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p2.1)\.
- \[13\]X\. Cabré, E\. Fontich, and R\. de la Llave\(2003\)The parameterization method for invariant manifolds ii: regularity with respect to parameters\.Indiana Univ\. Math\. J\.52,pp\. 329–360\.External Links:ISSN 0022\-2518Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p2.1)\.
- \[14\]X\. Cabré, E\. Fontich, and R\. de la Llave\(2005\)The parameterization method for invariant manifolds iii: overview and applications\.Journal of Differential Equations218\(2\),pp\. 444–515\.External Links:ISSN 0022\-0396,[Document](https://dx.doi.org/10.1016/j.jde.2004.12.003)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p2.1)\.
- \[15\]M\. Cenedese, J\. Axås, B\. Bäuerlein, K\. Avila, and G\. Haller\(2022/02/15\)Data\-driven modeling and prediction of non\-linearizable dynamics via spectral submanifolds\.Nature Communications13\(1\),pp\. 872\.External Links:[Document](https://dx.doi.org/10.1038/s41467-022-28518-y),ISBN 2041\-1723Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p2.1),[§3](https://arxiv.org/html/2608.04239#S3.p4.14),[§3](https://arxiv.org/html/2608.04239#S3.p4.31),[§3](https://arxiv.org/html/2608.04239#S3.p4.34),[§4\.1](https://arxiv.org/html/2608.04239#S4.SS1.p1.3),[§4\.2](https://arxiv.org/html/2608.04239#S4.SS2.p1.5),[Remark 4\.7](https://arxiv.org/html/2608.04239#S4.Thmtheorem7.p1.19.13),[Remark 4\.7](https://arxiv.org/html/2608.04239#S4.Thmtheorem7.p1.6.6),[2nd item](https://arxiv.org/html/2608.04239#S5.I1.i2.p1.1)\.
- \[16\]R\. T\. Q\. Chen, Y\. Rubanova, J\. Bettencourt, and D\. Duvenaud\(2018\)Neural Ordinary Differential Equations\.InAdvances in Neural Information Processing Systems,External Links:[Link](https://papers.nips.cc/paper_files/paper/2018/hash/69386f6bb1dfed68692a24c8686939b9-Abstract.html)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.14.13.1)\.
- \[17\]C\. Coelho, M\. F\. P\. Costa, and L\. L\. Ferrás\(2024\)Enhancing continuous time series modelling with a latent ODE\-LSTM approach\.Applied Mathematics and Computation475,pp\. 128727\.Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.5.4.1)\.
- \[18\]M\. Colbrook, Z\. Drmač, and A\. Horning\(2016\)An introductory guide to koopman learning\.InOperator Theory,D\. Alpay, F\. Colombo, and I\. Sabadini \(Eds\.\),pp\. 1–49\.External Links:[Document](https://dx.doi.org/10.1007/978-3-0348-0692-3%5F126-1)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1)\.
- \[19\]E\. R\. Deyle and G\. Sugihara\(2011\)Generalized theorems for nonlinear state space reconstruction\.PLoS ONE6\(3\),pp\. e18295\.External Links:[Document](https://dx.doi.org/10.1371/journal.pone.0018295),[Link](https://doi.org/10.1371/journal.pone.0018295)Cited by:[§4\.1](https://arxiv.org/html/2608.04239#S4.SS1.p1.3)\.
- \[20\]E\. Faou, G\. Maierhofer, and K\. Schratz\(2026\)Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schrödinger equation\.Numerische Mathematik\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[21\]U\. Fasel, J\. N\. Kutz, B\. W\. Brunton, and S\. L\. Brunton\(2022\)Ensemble\-SINDy: Robust sparse model discovery in the low\-data, high\-noise limit, with active learning and control\.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences478\(2260\),pp\. 20210904\.External Links:[Document](https://dx.doi.org/10.1098/rspa.2021.0904)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.7.6.1)\.
- \[22\]Y\. Feng, G\. Maierhofer, and C\. Wang\(2025\)Explicit symmetric low\-regularity integrators for the nonlinear schrödinger equation\.SIAM Journal on Scientific Computing47\(4\),pp\. A2154–A2179\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[23\]J\. Galewsky, R\. K\. Scott, and L\. M\. Polvani\(2004\)An initial\-value problem for testing numerical models of the global shallow\-water equations\.Tellus A: Dynamic Meteorology and Oceanography56\(5\),pp\. 429–440\.Cited by:[§5\.2](https://arxiv.org/html/2608.04239#S5.SS2.p1.1)\.
- \[24\]J\. Guckenheimer and P\. Holmes\(1983\)Nonlinear oscillations, dynamical systems, and bifurcations of vector fields\.Applied Mathematical Sciences, Vol\.42,Springer,New York\.Cited by:[§3](https://arxiv.org/html/2608.04239#S3.p4.14)\.
- \[25\]E\. Hairer, C\. Lubich, and G\. Wanner\(2013\)Geometric Numerical Integration: Structure\-Preserving Algorithms for Ordinary Differential Equations\.Springer\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[26\]N\. Halko, P\. Martinsson, and J\. A\. Tropp\(2011\)Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions\.SIAM review53\(2\),pp\. 217–288\.Cited by:[Remark 4\.6](https://arxiv.org/html/2608.04239#S4.Thmtheorem6.p1.3.3)\.
- \[27\]G\. Haller, B\. Kaszás, A\. Liu, and J\. Axås\(2023\-06\)Nonlinear model reduction to fractional and mixed\-mode spectral submanifolds\.Chaos: An Interdisciplinary Journal of Nonlinear Science33\(6\),pp\. 063138\.External Links:ISSN 1054\-1500,[Document](https://dx.doi.org/10.1063/5.0143936)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1),[§1](https://arxiv.org/html/2608.04239#S1.p2.1),[Theorem 2\.3](https://arxiv.org/html/2608.04239#S2.Thmtheorem3),[Remark 2\.4](https://arxiv.org/html/2608.04239#S2.Thmtheorem4.p1.1.1),[§2](https://arxiv.org/html/2608.04239#S2.p3.12),[§2](https://arxiv.org/html/2608.04239#S2.p4.1),[§3](https://arxiv.org/html/2608.04239#S3.p4.14)\.
- \[28\]G\. Haller and S\. Ponsioen\(2016/11/01\)Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction\.Nonlinear Dynamics86\(3\),pp\. 1493–1534\.External Links:[Document](https://dx.doi.org/10.1007/s11071-016-2974-z),ISBN 1573\-269XCited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1),[§1](https://arxiv.org/html/2608.04239#S1.p2.1),[Remark 2\.4](https://arxiv.org/html/2608.04239#S2.Thmtheorem4.p1.1.1),[§2](https://arxiv.org/html/2608.04239#S2.p3.12)\.
- \[29\]H\. Harder, F\. Nüske, F\. M\. Philipp, M\. Schaller, K\. Worthmann, and S\. Peitz\(2025\)Group\-convolutional extended dynamic mode decomposition\.Physica D: Nonlinear Phenomena480,pp\. 134725\.External Links:ISSN 0167\-2789,[Document](https://dx.doi.org/https%3A//doi.org/10.1016/j.physd.2025.134725)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[30\]S\. Hochreiter and J\. Schmidhuber\(1997\)Long short\-term memory\.Neural computation9\(8\),pp\. 1735–1780\.Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.3.2.1)\.
- \[31\]H\. Jaeger\(2001\)The ‘echo state’ approach to analyzing and training recurrent neural networks\.Technical reportGerman National Research Center for Information Technology, Technical Report GMD 148\.Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.2.1.1)\.
- \[32\]S\. Jain and G\. Haller\(2022\)How to compute invariant manifolds and their reduced dynamics in high\-dimensional finite element models\.Nonlinear dynamics107\(2\),pp\. 1417–1450\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p2.1),[Remark 3\.1](https://arxiv.org/html/2608.04239#S3.Thmtheorem1.p1.5.5),[§3](https://arxiv.org/html/2608.04239#S3.p1.13),[§3](https://arxiv.org/html/2608.04239#S3.p4.14)\.
- \[33\]SSMTool\-2\.0: Computation of invariant manifolds & their reduced dynamics in high\-dimensional mechanics problemsExternal Links:[Document](https://dx.doi.org/10.5281/zenodo.4614202)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p2.1)\.
- \[34\]B\. Kaszás and G\. Haller\(2024\)Capturing the edge of chaos as a spectral submanifold in pipe flows\.Journal of Fluid Mechanics979,pp\. A48\.External Links:[Document](https://dx.doi.org/10.1017/jfm.2023.956)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p4.1)\.
- \[35\]J\. N\. Kutz, S\. L\. Brunton, B\. W\. Brunton, and J\. L\. Proctor\(2016\)Dynamic mode decomposition: data\-driven modeling of complex systems\.Society for Industrial and Applied Mathematics,Philadelphia, PA\.External Links:[Document](https://dx.doi.org/10.1137/1.9781611974508)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1)\.
- \[36\]S\. Le Clainche and J\. M\. Vega\(2017\)Higher order dynamic mode decomposition\.SIAM Journal on Applied Dynamical Systems16\(2\),pp\. 882–925\.External Links:[Document](https://dx.doi.org/10.1137/15M1054924)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.16.15.1)\.
- \[37\]Z\. Li, N\. B\. Kovachki, K\. Azizzadenesheli, B\. Liu, K\. Bhattacharya, A\. Stuart, and A\. Anandkumar\(2021\)Fourier neural operator for parametric partial differential equations\.InThe Ninth International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=c8P9NQVtmnO)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.13.12.1)\.
- \[38\]R\. Liaw, E\. Liang, R\. Nishihara, P\. Moritz, J\. E\. Gonzalez, and I\. Stoica\(2018\)Tune: a research platform for distributed model selection and training\.arXiv:1807\.05118\.External Links:1807\.05118Cited by:[§5\.3](https://arxiv.org/html/2608.04239#S5.SS3.SSS0.Px1.p1.16)\.
- \[39\]Z\. Liu, Y\. Wang, S\. Vaidya, F\. Ruehle, J\. Halverson, M\. Soljacic, T\. Y\. Hou, and M\. Tegmark\(2025\)KAN: kolmogorov–arnold networks\.InThe Thirteenth International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=Ozo7qJ5vZi)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.11.10.1)\.
- \[40\]L\. Lu, P\. Jin, G\. Pang, Z\. Zhang, and G\. E\. Karniadakis\(2021\)Learning nonlinear operators via deeponet based on the universal approximation theorem of operators\.Nature Machine Intelligence3\(3\),pp\. 218–229\.Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.10.9.1)\.
- \[41\]W\. Maass and H\. Markram\(2004\-12\)On the computational power of circuits of spiking neurons\.Journal of Computer and System Sciences69\(4\)\.External Links:ISSN 0022\-0000,[Document](https://dx.doi.org/10.1016/j.jcss.2004.04.001)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.2.1.1)\.
- \[42\]G\. Maierhofer and K\. Schratz\(2025\)Bridging the gap: symplecticity and low regularity in Runge–Kutta resonance\-based schemes\.Mathematics of Computation\.External Links:[Document](https://dx.doi.org/10.1090/mcom/4105)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[43\]R\. I\. McLachlan and A\. Stern\(2024/02/01\)Functional equivariance and conservation laws in numerical integration\.Foundations of Computational Mathematics24\(1\),pp\. 149–177\.External Links:[Document](https://dx.doi.org/10.1007/s10208-022-09590-8),ISBN 1615\-3383Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[44\]S\. Pan, E\. Kaiser, B\. M\. de Silva, J\. N\. Kutz, and S\. L\. Brunton\(2024\)PyKoopman: A Python Package for Data\-Driven Approximation of the Koopman Operator\.Journal of Open Source Software9\(94\),pp\. 5881\.External Links:[Document](https://dx.doi.org/10.21105/joss.05881),[Link](https://doi.org/10.21105/joss.05881)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.9.8.1)\.
- \[45\]J\. Pathak, B\. Hunt, M\. Girvan, Z\. Lu, and E\. Ott\(2018\-01\)Model\-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach\.Physical Review Letters120\(2\),pp\. 024102\.External Links:[Document](https://dx.doi.org/10.1103/PhysRevLett.120.024102)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.2.1.1)\.
- \[46\]H\. Poincaré\(1892\)Les méthodes nouvelles de la mécanique céleste\.Vol\.1,Gauthier\-Villars et Fils,Paris\.Cited by:[§3](https://arxiv.org/html/2608.04239#S3.p4.14)\.
- \[47\]S\. Riva, C\. Introini, A\. Cammi, D\. Price, A\. Yermakov, Y\. Zhao, P\. M\. Wyder, J\. Goldfeder, J\. Williams, A\. S\. Rude, M\. Tomasetto, J\. Germany, J\. Bakarji, G\. Maierhofer, M\. Cranmer, and J\. N\. Kutz\(2026\)CTF4Nuclear: Common Task Framework for Nuclear Fission and Fusion Models\.arXiv preprint arXiv:2605\.15549\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1)\.
- \[48\]S\. H\. Rudy, S\. L\. Brunton, J\. L\. Proctor, and J\. N\. Kutz\(2017\)Data\-driven discovery of partial differential equations\.Science advances3\(4\),pp\. e1602614\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[49\]A\. Salova, J\. Emenheiser, A\. Rupe, J\. P\. Crutchfield, and R\. M\. D’Souza\(2019\-09\)Koopman operator and its approximations for systems with symmetries\.Chaos: An Interdisciplinary Journal of Nonlinear Science29\(9\),pp\. 093128\.External Links:ISSN 1054\-1500,[Document](https://dx.doi.org/10.1063/1.5099091)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[50\]T\. Sauer, J\. A\. Yorke, and M\. Casdagli\(1991/11/01\)Embedology\.Journal of Statistical Physics65\(3\),pp\. 579–616\.External Links:[Document](https://dx.doi.org/10.1007/BF01053745),ISBN 1572\-9613Cited by:[§4\.1](https://arxiv.org/html/2608.04239#S4.SS1.p1.10),[§4\.1](https://arxiv.org/html/2608.04239#S4.SS1.p1.3)\.
- \[51\]P\. J\. Schmid\(2010\)Dynamic mode decomposition of numerical and experimental data\.Journal of Fluid Mechanics656,pp\. 5–28\.External Links:[Document](https://dx.doi.org/10.1017/S0022112010001217)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1)\.
- \[52\]J\. Serre\(1977\)Linear representations of finite groups\.1 edition,Graduate Texts in Mathematics, Vol\.42,Springer,New York, NY\.Note:Translation of the original French edition:Représentations linéaires des groupes finisExternal Links:[Document](https://dx.doi.org/10.1007/978-1-4684-9458-7),ISBN 978\-0\-387\-90190\-9,ISSN 0072\-5285Cited by:[§4\.2](https://arxiv.org/html/2608.04239#S4.SS2.SSS0.Px2.2.p1.15)\.
- \[53\]F\. Takens\(1981\)Detecting strange attractors in turbulence\.InDynamical Systems and Turbulence, Warwick 1980,D\. Rand and L\. Young \(Eds\.\),pp\. 366–381\.Cited by:[§4\.1](https://arxiv.org/html/2608.04239#S4.SS1.p1.10),[§4\.1](https://arxiv.org/html/2608.04239#S4.SS1.p1.3)\.
- \[54\]J\. H\. Tu, C\. W\. Rowley, D\. M\. Luchtenburg, S\. L\. Brunton, and J\. N\. Kutz\(2014\)On dynamic mode decomposition: theory and applications\.Journal of Computational Dynamics1,pp\. 391–421\.External Links:[Document](https://dx.doi.org/10.3934/jcd.2014.1.391)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1)\.
- \[55\]P\. Wyder, J\. Goldfeder, A\. Yermakov, Y\. Zhao, S\. Riva, J\. Williams, D\. Zoro, A\. Rude, M\. Tomasetto, J\. Germany, J\. Bakarji, G\. Maierhofer, M\. Cranmer, and J\. N\. Kutz\(2026\)Common task framework for a critical evaluation of scientific machine learning algorithms\.Advances in Neural Information Processing Systems38\.Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1),[§5\.3](https://arxiv.org/html/2608.04239#S5.SS3.SSS0.Px1.p1.16),[§5\.3](https://arxiv.org/html/2608.04239#S5.SS3.p1.15),[§5\.3](https://arxiv.org/html/2608.04239#S5.SS3.p1.16),[§5\.3](https://arxiv.org/html/2608.04239#S5.SS3.p1.2),[Table 4](https://arxiv.org/html/2608.04239#S5.T4),[Table 4](https://arxiv.org/html/2608.04239#S5.T4.4.2),[§6](https://arxiv.org/html/2608.04239#S6.p1.1)\.
- \[56\]J\. Yang, W\. Rao, N\. Dehmamy, R\. Walters, and R\. Yu\(2024\)Symmetry\-informed governing equation discovery\.InAdvances in Neural Information Processing Systems,A\. Globerson, L\. Mackey, D\. Belgrave, A\. Fan, U\. Paquet, J\. Tomczak, and C\. Zhang \(Eds\.\),Vol\.37,pp\. 65297–65327\.External Links:[Document](https://dx.doi.org/10.52202/079017-2085)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p3.1)\.
- \[57\]A\. Yermakov, Y\. Zhao, M\. Denolle, Y\. Ni, P\. M\. Wyder, J\. Goldfeder, S\. Riva, J\. P\. Williams, D\. Zoro, A\. S\. Rude, M\. Tomasetto, J\. Germany, J\. Bakarji, G\. Maierhofer, M\. Cranmer, and J\. N\. Kutz\(2026\)The seismic wavefield common task framework\.InThe Fourteenth International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=u4N7Kl6gzE)Cited by:[§1](https://arxiv.org/html/2608.04239#S1.p1.1)\.
- \[58\]M\. Zhang, K\. K\. Saab, M\. Poli, T\. Dao, K\. Goel, and C\. Ré\(2023\)Effectively modeling time series with simple discrete state spaces\.InThe Eleventh International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=2EpjkjzdCAa)Cited by:[Table 4](https://arxiv.org/html/2608.04239#S5.T4.6.15.14.1)\.