面向旋转鲁棒神经动力学的精确 SO(3) 等变各向同性核

# Exact SO(3)-Equivariant Isotropic Kernels for Rotation-Robust Neural Dynamics
Source: [https://arxiv.org/html/2610.10626](https://arxiv.org/html/2610.10626)
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Ridham Patel

Ridham PatelEmail:[ridham\.patel@iitgn\.ac\.in](mailto:[email protected])Affiliation:Indian Institute of Technology Gandhinagar, Gandhinagar, India

###### Abstract

Neural surrogates for vector\-valued partial differential equations can fit training data yet change their predictions when the same physical state is expressed in a rotated coordinate frame\. We study this failure on three\-dimensional Navier–Stokes dynamics observed at irregularly placed points\. We introduce the Invariant\-Conditioned Isotropic Kernel Neural Operator \(IKNO\), a compact graph model that builds local interactions from scalar quantities unchanged by rotation and vector directions that rotate with the data\. Consequently, rotating the positions and velocities rotates the predicted velocity change in exactly the same way\. On a held\-out test set fixed after model design, training unconstrained graph models on randomly rotated examples reduces but does not eliminate their coordinate dependence\. In contrast,IKNOis consistent to numerical precision, matches the forecasting accuracy of a general rotation\-aware Tensor Field Network with5\.65\.6times fewer parameters, and outperforms a parameter\-matched graph simulator\. These results show that a compact, PDE\-specialized model can remove coordinate dependence without sacrificing forecasting accuracy\.

††year:2026††workshop:Symmetry and Geometry in Neural Representations###### keywords

equivariant neural networks, neural PDE solvers, irregular meshes, Navier–Stokes, geometric deep learning

## 1Introduction

The same physical state can be expressed in many coordinate frames\. A quantity is invariant if a transformation leaves it unchanged; a map is equivariant if its output transforms with its input\. For a three\-dimensional velocity field, rotation equivariance requires a joint rotation of coordinates and velocities to rotate the predicted velocity change by the same amount\. The vector prediction changes, but the physical rule does not depend on the axes\. We study proper three\-dimensional rotations, denotedSO⁡\(3\)\\mathrm\{SO\}\(3\), which exclude reflections\. Without this constraint, low error in the training frame does not determine behavior in a new frame\([Villar et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib14);[Helwig et al\., 2023](https://arxiv.org/html/2610.10626#bib.bib10)\)\.

This issue is especially relevant for learned partial differential equation \(PDE\) solvers on irregular meshes\. Graph models are usually permutation\-equivariant: reordering sampled points merely reorders their outputs\. This lets graph simulators accommodate changing discretizations and supports accurate long\-horizon prediction\([Sanchez\-Gonzalez et al\., 2020](https://arxiv.org/html/2610.10626#bib.bib6);[Pfaff et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib7)\)\. However, permutation equivariance concerns point labels, not spatial orientation, and does not specify how a vector field transforms under rotation\. Treating Cartesian components as ordinary feature channels can therefore make the learned dynamics depend on the axes\([Bronstein et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib19)\)\.

Rotation\-equivariant architectures remove this dependence by prescribing how features transform\. Tensor\-field and steerable networks propagate geometric quantities according to their rotation rules\([Thomas et al\., 2018](https://arxiv.org/html/2610.10626#bib.bib2);[Fuchs et al\., 2020](https://arxiv.org/html/2610.10626#bib.bib3)\), while lighter\-weight constructions use vector\-valued features or invariant scalar weights\([Satorras et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib4);[Villar et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib14)\)\. We ask whether such an exact constraint improves forecasting under a coordinate\-frame shift without reducing accuracy in the observed frame\. We compare it with rotation augmentation, which trains a model on randomly rotated copies of the observed states\([Brandstetter et al\., 2022b](https://arxiv.org/html/2610.10626#bib.bib12);[Puny et al\., 2022](https://arxiv.org/html/2610.10626#bib.bib18)\)\.

We use a controlled one\-frame prediction experiment on three\-dimensional incompressible Navier–Stokes flow\. The model maps a velocity field sampled at irregular points to a one\-step derivative estimate\. Training and validation use orientation\-biased initial conditions; testing uses both the original states and jointly rotated coordinates, velocities, and targets\. Because the rotated target is known exactly by changing coordinates, additional disagreement isolates frame dependence without a second simulation\.

To perform this test, we introduce the Invariant\-Conditioned Isotropic Kernel Neural Operator \(IKNO\)\. Each edge coefficient is produced from scalar rotation invariants\. The resulting coefficients weight four vector bases that rotate with the data\. The construction has three properties relevant to the experiment: it is exactlySO⁡\(3\)\\mathrm\{SO\}\(3\)\-equivariant, it annihilates constant fields in its difference channels, and it contains only4,8694\{,\}869trainable parameters\.

The main result is that architectural equivariance and rotation augmentation produce measurably different operators\. Augmentation reduces the rotation sensitivity of two graph controls but does not eliminate it;IKNOremains unchanged across 1,000 sampled coordinate frames\. A Tensor Field Network provides the complementary exact\-equivariance control: it matchesIKNOin forecasting accuracy but uses5\.65\.6times more parameters\. A parameter\-matched graph simulator is substantially less accurate\. The evidence therefore supports a specific conclusion: the specialized isotropic construction provides exact rotation robustness at low parameter cost\.

## 2Related Work

Equivariant representation learning formalizes weight sharing over transformations\([Cohen and Welling, 2016](https://arxiv.org/html/2610.10626#bib.bib1);[Bronstein et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib19)\)\. Tensor Field Networks andSE⁡\(3\)\\mathrm\{SE\}\(3\)\-Transformers use typed features that follow irreducible group representations and spherical harmonics, a basis for angular dependence\([Thomas et al\., 2018](https://arxiv.org/html/2610.10626#bib.bib2);[Fuchs et al\., 2020](https://arxiv.org/html/2610.10626#bib.bib3)\); e3nn implements their tensor products\([Geiger and Smidt, 2022](https://arxiv.org/html/2610.10626#bib.bib13)\)\. Lighter Cartesian constructions include EGNN, Vector Neurons, PaiNN, and SEGNN\([Satorras et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib4);[Deng et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib5);[Schütt et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib16);[Brandstetter et al\., 2022a](https://arxiv.org/html/2610.10626#bib.bib17)\)\. Classical isotropic representation theory and its modern scalar formulation express equivariant vector functions through invariant coefficients and equivariant generators\([Smith, 1971](https://arxiv.org/html/2610.10626#bib.bib15);[Villar et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib14)\)\. The latter also distinguishes proper rotations from reflections through cross\-product generators\.IKNOselects a compact, PDE\-oriented subset of admissible generators on an irregular mesh; it does not claim a complete parameterization\.

Learned PDE models take several complementary architectural forms\. Neural operators learn maps between function spaces and can amortize repeated PDE solution\([Li et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib9)\)\. Graph simulators instead represent local physical interactions by message passing, which naturally supports particles and irregular meshes\([Sanchez\-Gonzalez et al\., 2020](https://arxiv.org/html/2610.10626#bib.bib6);[Pfaff et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib7)\)\. Message\-passing PDE solvers connect these learned updates to classical local discretizations\([Brandstetter et al\., 2022c](https://arxiv.org/html/2610.10626#bib.bib8)\)\. Symmetry has been incorporated through data augmentation derived from Lie point symmetries\([Brandstetter et al\., 2022b](https://arxiv.org/html/2610.10626#bib.bib12)\), steerable differential operators\([Jenner and Weiler, 2022](https://arxiv.org/html/2610.10626#bib.bib11)\), and group equivariant Fourier layers\([Helwig et al\., 2023](https://arxiv.org/html/2610.10626#bib.bib10)\)\. LieConv treats arbitrary continuous data\([Finzi et al\., 2020](https://arxiv.org/html/2610.10626#bib.bib21)\), while Neural ePDOs learn adaptive equivariant differential operators\([He et al\., 2023](https://arxiv.org/html/2610.10626#bib.bib20)\)\. In contrast to Fourier methods on regular grids,IKNOacts directly on irregular three\-dimensional samples\. Our experiment also measures operator equivariance separately from forecasting error, which distinguishes exact architectural consistency from good performance on one orientation distribution\.

## 3Problem Setting

We first define the evolution problem and the symmetry to be tested\. Let\{𝒙i\}i=1N⊂ℝ3\\\{\\boldsymbol\{x\}\_\{i\}\\\}\_\{i=1\}^\{N\}\\subset\\mathbb\{R\}^\{3\}be sample coordinates and let𝒖i​\(t\)∈ℝ3\\boldsymbol\{u\}\_\{i\}\(t\)\\in\\mathbb\{R\}^\{3\}denote velocity\. A learned local operatorfθf\_\{\\theta\}predicts the derivative

𝒖˙i​\(t\)=fθ,i​\(𝑼⁡\(t\),𝑿\),𝑼=\(𝒖1,…,𝒖N\)\.\\dot\{\\boldsymbol\{u\}\}\_\{i\}\(t\)=f\_\{\\theta,i\}\(\\boldsymbol\{U\}\(t\),\\boldsymbol\{X\}\),\\qquad\\boldsymbol\{U\}=\(\\boldsymbol\{u\}\_\{1\},\\ldots,\\boldsymbol\{u\}\_\{N\}\)\.\(1\)We integrate this derivative with an explicit Euler step of durationΔ​t\\Delta tto obtain autoregressive forecasts\.

The relevant transformation rotates coordinates and vector components together\. ForR∈SO⁡\(3\)R\\in\\mathrm\{SO\}\(3\), defineR​𝑿=\{R​𝒙i\}iR\\boldsymbol\{X\}=\\\{R\\boldsymbol\{x\}\_\{i\}\\\}\_\{i\}andR​𝑼=\{R​𝒖i\}iR\\boldsymbol\{U\}=\\\{R\\boldsymbol\{u\}\_\{i\}\\\}\_\{i\}\. The operator isSO⁡\(3\)\\mathrm\{SO\}\(3\)\-equivariant when

fθ​\(R​𝑼,R​𝑿\)=R​fθ​\(𝑼,𝑿\)for every​R∈SO⁡\(3\)\.f\_\{\\theta\}\(R\\boldsymbol\{U\},R\\boldsymbol\{X\}\)=Rf\_\{\\theta\}\(\\boldsymbol\{U\},\\boldsymbol\{X\}\)\\quad\\text\{for every \}R\\in\\mathrm\{SO\}\(3\)\.\(2\)Equation[2](https://arxiv.org/html/2610.10626#S3.E2)is stronger than low rotated prediction error\. The equation is an operator identity; rotated prediction error also depends on the difficulty of the forecasting task\. We therefore report both quantities\.

Our physical system is the incompressible Navier–Stokes equation

∂t𝒖\+\(𝒖⋅∇\)𝒖\\displaystyle\\partial\_\{t\}\\boldsymbol\{u\}\+\(\\boldsymbol\{u\}\\cdot\\nabla\)\\boldsymbol\{u\}=−∇p\+ν​∇2​𝒖,\\displaystyle=\-\\nabla p\+\\nu\\nabla^\{2\}\\boldsymbol\{u\},∇⋅𝒖\\displaystyle\\nabla\\cdot\\boldsymbol\{u\}=0,\\displaystyle=0,\(3\)on a periodic three\-dimensional domain\. The differential terms in Equation[3](https://arxiv.org/html/2610.10626#S3.E3)are coordinate\-covariant\. Consequently, jointly rotating coordinates and vector values gives a valid representation of the same local dynamics, even when the observed initial\-condition distribution favors one orientation\([Olver, 1993](https://arxiv.org/html/2610.10626#bib.bib22);[Brandstetter et al\., 2022b](https://arxiv.org/html/2610.10626#bib.bib12)\)\.

## 4Invariant\-Conditioned Isotropic Kernel

The method must use local directional information without tying a prediction to a global axis\. Motivated by isotropic representation results, we separate each message into invariant coefficients and equivariant vector generators\([Smith, 1971](https://arxiv.org/html/2610.10626#bib.bib15);[Villar et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib14)\)\.

local edge𝒓^i​j,𝒖i,δ​𝒖i​j\\widehat\{\\boldsymbol\{r\}\}\_\{ij\},\\boldsymbol\{u\}\_\{i\},\\delta\\boldsymbol\{u\}\_\{ij\}rotation invariants𝒔i​j∈ℝ6\\boldsymbol\{s\}\_\{ij\}\\in\\mathbb\{R\}^\{6\}scalar MLPϕθ\\phi\_\{\\theta\}ai​j,bi​j,ci​j,di​ja\_\{ij\},b\_\{ij\},c\_\{ij\},d\_\{ij\}equivariant vector basesδ​𝒖,𝒓^​\(𝒓^⊤​δ​𝒖\),\\delta\\boldsymbol\{u\},\\ \\widehat\{\\boldsymbol\{r\}\}\(\\widehat\{\\boldsymbol\{r\}\}^\{\\\!\\top\}\\delta\\boldsymbol\{u\}\),𝒖i,𝒖i×𝒓^\\ \\boldsymbol\{u\}\_\{i\},\\ \\boldsymbol\{u\}\_\{i\}\\\!\\times\\\!\\widehat\{\\boldsymbol\{r\}\}scalar–vectorweighted sum𝒎i​j\\boldsymbol\{m\}\_\{ij\}mean overjjand\+λ​𝒖i\+\\lambda\\boldsymbol\{u\}\_\{i\}fθ,i​\(𝑼,𝑿\)f\_\{\\theta,i\}\(\\boldsymbol\{U\},\\boldsymbol\{X\}\)norms, dot productstyped vectors

Figure 1:IKNOmessage construction\. Invariant coefficients weight equivariant vector bases; aggregation preserves their common rotation and therefore yields an equivariant output\.### 4\.1Invariant edge description

For a directed edge from nodeiito nodejj, define the relative coordinate, unit direction, and velocity difference

𝒓i​j=𝒙j−𝒙i,𝒓^i​j=𝒓i​j‖𝒓i​j‖2,δ​𝒖i​j=𝒖j−𝒖i\.\\boldsymbol\{r\}\_\{ij\}=\\boldsymbol\{x\}\_\{j\}\-\\boldsymbol\{x\}\_\{i\},\\qquad\\widehat\{\\boldsymbol\{r\}\}\_\{ij\}=\\frac\{\\boldsymbol\{r\}\_\{ij\}\}\{\\\|\\boldsymbol\{r\}\_\{ij\}\\\|\_\{2\}\},\\qquad\\delta\\boldsymbol\{u\}\_\{ij\}=\\boldsymbol\{u\}\_\{j\}\-\\boldsymbol\{u\}\_\{i\}\.\(4\)The edge graph connects points within radiusrmaxr\_\{\\max\}and retains at mostKKneighbors\. Because this rule depends only on distance, a rotation does not change the graph except for floating\-point tie cases\.

The coefficient network receives six scalar invariants:

𝒔i​j=\[‖𝒓i​j‖2,‖𝒖i‖2,‖δ​𝒖i​j‖2,𝒖i⊤​𝒓^i​j,δ​𝒖i​j⊤​𝒓^i​j,𝒖i⊤​δ​𝒖i​j\]\.\\boldsymbol\{s\}\_\{ij\}=\\big\[\\\|\\boldsymbol\{r\}\_\{ij\}\\\|\_\{2\},\\\|\\boldsymbol\{u\}\_\{i\}\\\|\_\{2\},\\\|\\delta\\boldsymbol\{u\}\_\{ij\}\\\|\_\{2\},\\boldsymbol\{u\}\_\{i\}^\{\\top\}\\widehat\{\\boldsymbol\{r\}\}\_\{ij\},\\delta\\boldsymbol\{u\}\_\{ij\}^\{\\top\}\\widehat\{\\boldsymbol\{r\}\}\_\{ij\},\\boldsymbol\{u\}\_\{i\}^\{\\top\}\\delta\\boldsymbol\{u\}\_\{ij\}\\big\]\.\(5\)Apart from radial scale‖𝒓i​j‖2\\\|\\boldsymbol\{r\}\_\{ij\}\\\|\_\{2\}and the fixed norm‖𝒓^i​j‖2=1\\\|\\widehat\{\\boldsymbol\{r\}\}\_\{ij\}\\\|\_\{2\}=1, these quantities encode the nonconstant Gram information of\(𝒓^i​j,𝒖i,δ​𝒖i​j\)\(\\widehat\{\\boldsymbol\{r\}\}\_\{ij\},\\boldsymbol\{u\}\_\{i\},\\delta\\boldsymbol\{u\}\_\{ij\}\)\. Inner products generate rotation\-invariant scalar descriptions of vector inputs\([Villar et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib14)\)\. A multilayer perceptronϕθ:ℝ6→ℝ4\\phi\_\{\\theta\}:\\mathbb\{R\}^\{6\}\\rightarrow\\mathbb\{R\}^\{4\}maps them to\(ai​j,bi​j,ci​j,di​j\)\(a\_\{ij\},b\_\{ij\},c\_\{ij\},d\_\{ij\}\)\.

### 4\.2Equivariant message and update

Guided by the same representation results, we choose a compact PDE\-oriented subset of admissibleSO⁡\(3\)\\mathrm\{SO\}\(3\)\-equivariant generators rather than a universal basis\([Smith, 1971](https://arxiv.org/html/2610.10626#bib.bib15);[Villar et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib14)\):

𝒎i​j=\\displaystyle\\boldsymbol\{m\}\_\{ij\}=\{\}ai​j​δ​𝒖i​j\+bi​j​𝒓^i​j​\(𝒓^i​j⊤​δ​𝒖i​j\)\+ci​j​𝒖i\+di​j​\(𝒖i×𝒓^i​j\)\.\\displaystyle a\_\{ij\}\\delta\\boldsymbol\{u\}\_\{ij\}\+b\_\{ij\}\\widehat\{\\boldsymbol\{r\}\}\_\{ij\}\(\\widehat\{\\boldsymbol\{r\}\}\_\{ij\}^\{\\top\}\\delta\\boldsymbol\{u\}\_\{ij\}\)\+c\_\{ij\}\\boldsymbol\{u\}\_\{i\}\+d\_\{ij\}\(\\boldsymbol\{u\}\_\{i\}\\times\\widehat\{\\boldsymbol\{r\}\}\_\{ij\}\)\.\(6\)The first two terms span direct and longitudinal difference responses\. The third permits state\-dependent local dynamics\. The cross product is orientation\-sensitive but rotation\-consistent; as a pseudovector under reflections, it distinguishesSO⁡\(3\)\\mathrm\{SO\}\(3\)fromO⁡\(3\)\\mathrm\{O\}\(3\)\([Villar et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib14)\)\.

Messages are mean\-aggregated and combined with a scalar self\-interaction:

fθ,i​\(𝑼,𝑿\)=λ​𝒖i\+1\|𝒩⁡\(i\)\|​∑j∈𝒩⁡\(i\)𝒎i​j\.f\_\{\\theta,i\}\(\\boldsymbol\{U\},\\boldsymbol\{X\}\)=\\lambda\\boldsymbol\{u\}\_\{i\}\+\\frac\{1\}\{\|\\mathcal\{N\}\(i\)\|\}\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\boldsymbol\{m\}\_\{ij\}\.\(7\)The difference channels vanish on constant fields, matching the nullspace of local differential responses\([Brandstetter et al\., 2022c](https://arxiv.org/html/2610.10626#bib.bib8);[Jenner and Weiler, 2022](https://arxiv.org/html/2610.10626#bib.bib11)\)\. The coefficient network has two hidden layers of 64 SiLU units\. For optimization stability, its final layer andλ\\lambdaare initialized to zero, so training begins from a stationary derivative\.

###### Proposition 1\.

If𝒩⁡\(i\)\\mathcal\{N\}\(i\)depends only on pairwise distances, then for everyR∈SO⁡\(3\)R\\in\\mathrm\{SO\}\(3\)and𝐚∈ℝ3\\boldsymbol\{a\}\\in\\mathbb\{R\}^\{3\},fθ​\(R​𝐔,R​𝐗\+𝐚\)=R​fθ​\(𝐔,𝐗\)f\_\{\\theta\}\(R\\boldsymbol\{U\},R\\boldsymbol\{X\}\+\\boldsymbol\{a\}\)=Rf\_\{\\theta\}\(\\boldsymbol\{U\},\\boldsymbol\{X\}\)\.

###### Proof\.

Under a joint rotation, each entry of𝒔i​j\\boldsymbol\{s\}\_\{ij\}is unchanged, so the four scalar coefficients are unchanged\. Each vector basis in Equation[6](https://arxiv.org/html/2610.10626#S4.E6)is multiplied byRR: this follows from preservation of inner products and from\(R​𝒂\)×\(R​𝒃\)=R⁡\(𝒂×𝒃\)\(R\\boldsymbol\{a\}\)\\times\(R\\boldsymbol\{b\}\)=R\(\\boldsymbol\{a\}\\times\\boldsymbol\{b\}\)forR∈SO⁡\(3\)R\\in\\mathrm\{SO\}\(3\)\. Distance\-based neighborhoods are unchanged, and summation and scalar self\-interaction commute withRR\. Therefore each output is multiplied byRR\. A global translation cancels in every𝒓i​j=𝒙j−𝒙i\\boldsymbol\{r\}\_\{ij\}=\\boldsymbol\{x\}\_\{j\}\-\\boldsymbol\{x\}\_\{i\}, proving the translation claim\. ∎

## 5Experimental Design

The experiments test one question: does exact operator equivariance improve prediction under an unseen coordinate frame while preserving useful accuracy in the observed frame? We first describe the data and controls, then define metrics that distinguish forecasting accuracy from operator consistency\.

### 5\.1Data and split protocol

We generate three\-dimensional incompressible flow on\[0,2​π\]3\[0,2\\pi\]^\{3\}using a16316^\{3\}pseudo\-spectral solver with two\-thirds dealiasing, Leray projection, and Strang\-split diffusion\([Canuto et al\., 1988](https://arxiv.org/html/2610.10626#bib.bib23);[Strang, 1968](https://arxiv.org/html/2610.10626#bib.bib24)\)\. A fixed\-axis large\-scale forcing is used for 50 warm\-up steps to create an orientation\-biased initial state\. Each trajectory adds an independent10%10\\%divergence\-free perturbation, chosen to diversify trajectories while retaining the warm\-up orientation bias, and is then evolved without forcing or observation noise\. This separation is deliberate: the forecast dynamics obey rotational covariance, while the observed initial\-condition distribution supplies only a restricted set of orientations\.

Each trajectory contains 20 snapshots separated byΔ​t=0\.02\\Delta t=0\.02\. The spectral velocity is trilinearly interpolated to a separately sampled irregular cloud of 500 points in normalized coordinates\. Training uses random 128\-point subsamples\. The supervised derivative target is\(𝑼⁡\(t\+Δ​t\)−𝑼⁡\(t\)\)/Δ​t\(\\boldsymbol\{U\}\(t\+\\Delta t\)\-\\boldsymbol\{U\}\(t\)\)/\\Delta t\. Evaluation uses a fixed 128\-point subsample per trajectory\.

The development data contain 25 trajectories with seeds 342–366\. The first 20 form the model\-development pool: a deterministic split assigns 16 to optimization and four to validation, and validation error alone selects checkpoints\. Five additional trajectories with seeds 362–366 screen which new controls warrant final evaluation; no screening result appears in the paper\. The proposed architecture was frozen before generating the final test set\. The added controls were specified and screened only on development data, then fixed before their single final evaluation\. The reported test set contains five trajectories with seeds 367–371 and new point clouds\.

### 5\.2Controls and optimization

We compare with persistence and three learned architecture families\. Persistence copies the initial field and establishes whether a learned derivative improves over zero dynamics\. Graph Network\-based Simulator \(GNS\) uses six message\-passing layers following[Sanchez\-Gonzalez et al\. \(2020\)](https://arxiv.org/html/2610.10626#bib.bib6)\. The second control adapts the EGNN implementation of[Satorras et al\. \(2021\)](https://arxiv.org/html/2610.10626#bib.bib4)by passing the three velocity components as node\-feature channels; we call it component\-EGNN\. This is an unconstrained vector\-field control, not an equivariant implementation of typed vector features: the original EGNN guarantee applies to coordinate updates with invariant node features\.

To distinguish the proposed specialization from general equivariant message passing, we implement a Tensor Field Network \(TFN\)\([Thomas et al\., 2018](https://arxiv.org/html/2610.10626#bib.bib2)\)with e3nn\([Geiger and Smidt, 2022](https://arxiv.org/html/2610.10626#bib.bib13)\)\. It carries even scalars and odd vectors, and applies distance\-conditioned tensor products with spherical harmonics through degree one\. The TFN isO⁡\(3\)\\mathrm\{O\}\(3\)\-equivariant, and therefore alsoSO⁡\(3\)\\mathrm\{SO\}\(3\)\-equivariant\. UnlikeIKNO, it propagates learned scalar and vector hidden channels through tensor products\.

We introduce two controls for possible confounds\. First, GNS and component\-EGNN are retrained with a fresh Haar\-random joint rotation of coordinates, velocities, and derivative targets for every training sample\. We sample these rotations by Gaussian QR factorization\([Mezzadri, 2007](https://arxiv.org/html/2610.10626#bib.bib25)\)\. These variants test approximate symmetry learned through augmentation\([Brandstetter et al\., 2022b](https://arxiv.org/html/2610.10626#bib.bib12);[Puny et al\., 2022](https://arxiv.org/html/2610.10626#bib.bib18)\)\. Second, a six\-layer GNS with hidden width nine contains 5,352 parameters and controls for the capacity difference between the original GNS andIKNO\.

All learned methods use the same radiusrmax=0\.4r\_\{\\max\}=0\.4, at most 64 neighbors, the same trajectory split, and the same temporal\-window sampling protocol\.IKNOuses 4,869 parameters, TFN 27,297, component\-EGNN 30,058, and GNS 63,174\. We train three seeds for 100 epochs with Adam and cosine learning\-rate decay\([Kingma and Ba, 2015](https://arxiv.org/html/2610.10626#bib.bib26)\), using validation\-selected checkpoints\. The initial learning rate is3×10−33\\times 10^\{\-3\}forIKNOand2×10−32\\times 10^\{\-3\}for the controls\. Additional implementation details and the two\-frame extrapolation control are reported in Appendix[B](https://arxiv.org/html/2610.10626#A2)\.

### 5\.3Metrics and rotation intervention

Forecasting error at horizonhhis the relative Frobenius error

eh=‖𝑼^h−𝑼h‖F‖𝑼h‖F\.e\_\{h\}=\\frac\{\\\|\\widehat\{\\boldsymbol\{U\}\}\_\{h\}\-\\boldsymbol\{U\}\_\{h\}\\\|\_\{F\}\}\{\\\|\\boldsymbol\{U\}\_\{h\}\\\|\_\{F\}\}\.\(8\)We reporth=1h=1andh=10h=10\. To measure coordinate\-frame generalization, we draw 1,000 fixed Haar\-random rotations for one\-step prediction and use the first 20 for ten\-step rollouts, concentrating the denser stress test on the cheaper one\-step evaluation\. The same rotations are applied to every method, coordinate set, input velocity, and target velocity\. The resulting rotated error tests the same physical state in an unseen frame\.

Average rotated error can conceal a sensitive coordinate frame\. We therefore also measure one\-step degradation relative to the original frame\. Heree1​\(𝑿,𝑼\)e\_\{1\}\(\\boldsymbol\{X\},\\boldsymbol\{U\}\)denotes the one\-step error evaluated on the specified coordinates and input velocity:

Δ1​\(R\)=e1​\(R​𝑿,R​𝑼\)−e1​\(𝑿,𝑼\),\\Delta\_\{1\}\(R\)=e\_\{1\}\(R\\boldsymbol\{X\},R\\boldsymbol\{U\}\)\-e\_\{1\}\(\\boldsymbol\{X\},\\boldsymbol\{U\}\),\(9\)and report its mean, 95th percentile, and sampled maximum over the 1,000 rotations\. The maximum is a finite\-sample stress test, not a certified worst\-case bound\.

Forecasting error can remain small even when the learned derivative is not equivariant\. We therefore define our normalized operator equivariance diagnostic

eeq=𝔼R​\[RMS⁡\(fθ​\(R​𝑼,R​𝑿\)−R​fθ​\(𝑼,𝑿\)\)RMS⁡\(fθ​\(R​𝑼,R​𝑿\)\)\+RMS⁡\(R​fθ​\(𝑼,𝑿\)\)\]\.e\_\{\\mathrm\{eq\}\}=\\mathbb\{E\}\_\{R\}\\left\[\\frac\{\\operatorname\{RMS\}\\\!\\left\(f\_\{\\theta\}\(R\\boldsymbol\{U\},R\\boldsymbol\{X\}\)\-Rf\_\{\\theta\}\(\\boldsymbol\{U\},\\boldsymbol\{X\}\)\\right\)\}\{\\operatorname\{RMS\}\\\!\\left\(f\_\{\\theta\}\(R\\boldsymbol\{U\},R\\boldsymbol\{X\}\)\\right\)\+\\operatorname\{RMS\}\\\!\\left\(Rf\_\{\\theta\}\(\\boldsymbol\{U\},\\boldsymbol\{X\}\)\\right\)\}\\right\]\.\(10\)A value near zero means that the learned derivative is coordinate\-consistent: rotating the input rotates the derivative by the same amount\. We average each test metric over five trajectories within a run and report mean±\\pmsample standard deviation over three training seeds\.

## 6Results

### 6\.1Main result: architecture and augmentation produce different symmetry

The main experiment asks whether rotation augmentation recovers the same operator\-level consistency as an equivariant architecture\. It does not\. Figure[2](https://arxiv.org/html/2610.10626#S6.F2)a shows normalized equivariance errors of0\.1630\.163for GNS and0\.2380\.238for component\-EGNN\. Augmentation reduces these errors by45%45\\%and43%43\\%, respectively, but leaves residual errors of0\.0900\.090and0\.1360\.136\. In contrast,IKNOreaches5\.60×10−85\.60\\times 10^\{\-8\}and TFN reaches2\.59×10−72\.59\\times 10^\{\-7\}\. These values match the numerical precision observed for the theoretically equivariant implementations \(Appendix[B](https://arxiv.org/html/2610.10626#A2)\)\.

Figure 2:Architectural equivariance differs from rotation augmentation\. Bars and error bars show mean and sample standard deviation over three training seeds on the post\-design holdout\. \(a\) Augmentation reduces but does not eliminate operator inconsistency;IKNOand TFN are equivariant to numerical precision\. \(b\) The sampled maximum ofΔ1​\(R\)/e1\\Delta\_\{1\}\(R\)/e\_\{1\}over 1,000 rotations remains nonzero after augmentation, whereas both exact architectures are stable up to numerical precision\.The 1,000\-rotation sweep establishes that the residual operator error affects prediction in particular frames\. Figure[2](https://arxiv.org/html/2610.10626#S6.F2)b reports the sampled maximum degradation as a percentage of original\-frame error\. The value is1\.41%1\.41\\%for GNS and2\.45%2\.45\\%for component\-EGNN\. Augmentation reduces these values to0\.27%0\.27\\%and0\.72%0\.72\\%, but does not eliminate them\. The corresponding values are4\.3×10−5%4\.3\\times 10^\{\-5\}\\%forIKNOand4\.5×10−5%4\.5\\times 10^\{\-5\}\\%for TFN\. Thus the distinction between approximate and exact operator consistency is visible even when their average forecasting errors are close\.

The lower panel makes the pairwise forecasting comparisons explicit rather than retaining only global column winners\. Component\-EGNN is more accurate in the original orientation, butIKNOlowers its rotated errors at both horizons\. Against its augmented counterpart,IKNOlowers all four forecasting errors\. These descriptive differences accompany reductions by several orders of magnitude in operator inconsistency and sampled worst\-frame degradation\.

Tableanswers a second question: does the specialized operator sacrifice accuracy relative to a general equivariant network? TFN andIKNOdiffer by only0\.03%0\.03\\%at one step and less than0\.001%0\.001\\%at ten steps\. Exploratory two\-sided pairedtt\-tests givep=0\.706p=0\.706andp=0\.994p=0\.994, respectively\. Their forecasting performance is therefore indistinguishable in this experiment, while TFN uses5\.65\.6times more parameters and spherical harmonics\. This comparison supports an efficiency claim, not an accuracy advantage over TFN\.

### 6\.2Supporting result: controlling for parameter count

The preceding comparison leaves open whether the large unconstrained controls are disadvantaged because capacity permits orientation\-specific fitting\. We isolate capacity with a six\-layer, 5,352\-parameter GNS\. This parameter\-matched model obtains rotated errors of0\.0119030\.011903at one step and0\.1165000\.116500at ten steps\.IKNOreduces these errors by3\.45%3\.45\\%and4\.30%4\.30\\%, respectively; exploratory paired tests across seeds givep=0\.014p=0\.014at both horizons\. The corresponding original\-frame reductions are3\.22%3\.22\\%and4\.03%4\.03\\%\. The advantage therefore persists when parameter count and message\-passing depth are controlled\.

### 6\.3Supporting result: learning nontrivial dynamics

Exact equivariance would be uninformative if obtained by predicting no dynamics\. Persistence provides this control because it is exactly equivariant but copies the input field\.IKNOreduces persistence error by5\.95%5\.95\\%at one step and7\.22%7\.22\\%at ten steps\. Per\-seed results in Appendix[A](https://arxiv.org/html/2610.10626#A1)further show that the aggregate comparisons are not driven by a single run\.

## 7Discussion and Limitations

Low in\-frame error measures fit to an observed orientation distribution, whereas Equation[10](https://arxiv.org/html/2610.10626#S5.E10)tests whether the derivative is a coordinate\-consistent map\. Augmentation improves both quantities but leaves nonzero operator error and worst\-frame degradation, consistent with the distinction between sampled augmentation and exact constructions\([Brandstetter et al\., 2022b](https://arxiv.org/html/2610.10626#bib.bib12);[Puny et al\., 2022](https://arxiv.org/html/2610.10626#bib.bib18)\)\.IKNOand TFN remove that degradation by construction\.

The scope of this conclusion is limited\. First, the final test contains five trajectories from one low\-resolution Navier–Stokes generator\. The evidence does not establish superiority across PDEs, Reynolds numbers, geometries, or boundary conditions\. Second, the warm\-up uses a fixed\-axis forcing, whereas forecast trajectories are unforced\. This choice creates the orientation shift needed for the controlled test, but it is not an isotropic sampling of initial conditions\. Third, arbitrary rotations change the coordinate description of the periodic box; the test is a joint coordinate\-frame transformation of sampled states rather than a second solver run on an independently rotated grid\.

The baselines also delimit the claim\. Component\-EGNN is intentionally an unconstrained adaptation for vector fields and should not be interpreted as a failure of typed scalar–vector models such as PaiNN or SEGNN\([Schütt et al\., 2021](https://arxiv.org/html/2610.10626#bib.bib16);[Brandstetter et al\., 2022a](https://arxiv.org/html/2610.10626#bib.bib17)\)\. The TFN supplies typed scalar and vector features, but is restricted to degree\-one spherical harmonics and one chosen width; it does not exhaust the space of steerable architectures\. Its matched accuracy supports the parameter\-efficiency of the specialized basis on this task, not a general efficiency ordering\. The two\-frame extrapolator in Appendix[B](https://arxiv.org/html/2610.10626#A2)is much more accurate because the trajectories are temporally smooth; the present result is therefore specific to one\-frame operator identification, where the derivative must be inferred from spatial structure\.

## 8Conclusion

IKNOcombines invariant scalar edge functions with an equivariant vector basis to guaranteeSO⁡\(3\)\\mathrm\{SO\}\(3\)equivariance\. On an untouched Navier–Stokes holdout, it matches a tensor\-field network with5\.65\.6times fewer parameters, removes coordinate dependence left by augmentation, and beats a parameter\-matched control\. This supports compact, PDE\-specialized exact operator consistency\.

## References

- J\. Brandstetter, R\. Hesselink, E\. van der Pol, E\. J\. Bekkers, and M\. WellingGeometric and physical quantities improve E\(3\) equivariant message passing\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p1.1),[§7](https://arxiv.org/html/2610.10626#S7.p3.1)\.
- Brandstetteret al\.\(2022b\)J\. Brandstetter, M\. Welling, and D\. E\. WorrallLie point symmetry data augmentation for neural PDE solvers\.InInternational Conference on Machine Learning,pp\. 2241–2256\.Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p3.1),[§2](https://arxiv.org/html/2610.10626#S2.p2.1),[§3](https://arxiv.org/html/2610.10626#S3.p3.2),[§5\.2](https://arxiv.org/html/2610.10626#S5.SS2.p3.1),[§7](https://arxiv.org/html/2610.10626#S7.p1.1)\.
- Brandstetteret al\.\(2022c\)J\. Brandstetter, D\. E\. Worrall, and M\. WellingMessage passing neural PDE solvers\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p2.1),[§4\.2](https://arxiv.org/html/2610.10626#S4.SS2.p2.2)\.
- Bronsteinet al\.\(2021\)M\. M\. Bronstein, J\. Bruna, T\. Cohen, and P\. VelickovicGeometric deep learning: grids, groups, graphs, geodesics, and gauges\.arXiv preprint arXiv:2104\.13478\.Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p2.1),[§2](https://arxiv.org/html/2610.10626#S2.p1.1)\.
- Canutoet al\.\(1988\)C\. Canuto, M\. Y\. Hussaini, A\. Quarteroni, and T\. A\. ZangSpectral methods in fluid dynamics\.Springer\.Cited by:[§5\.1](https://arxiv.org/html/2610.10626#S5.SS1.p1.1)\.
- Cohen and Welling \(2016\)T\. Cohen and M\. WellingGroup equivariant convolutional networks\.InInternational Conference on Machine Learning,pp\. 2990–2999\.Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p1.1)\.
- Denget al\.\(2021\)C\. Deng, O\. Litany, Y\. Duan, A\. Poulenard, A\. Tagliasacchi, and L\. J\. GuibasVector neurons: a general framework for SO\(3\)\-equivariant networks\.InIEEE/CVF International Conference on Computer Vision,pp\. 12200–12209\.Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p1.1)\.
- Finziet al\.\(2020\)M\. Finzi, S\. Stanton, P\. Izmailov, and A\. G\. WilsonGeneralizing convolutional neural networks for equivariance to lie groups on arbitrary continuous data\.InInternational Conference on Machine Learning,pp\. 3165–3176\.Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p2.1)\.
- Fuchset al\.\(2020\)F\. Fuchs, D\. Worrall, V\. Fischer, and M\. WellingSE\(3\)\-transformers: 3d roto\-translation equivariant attention networks\.InAdvances in Neural Information Processing Systems,Vol\.33,pp\. 1970–1981\.Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p3.1),[§2](https://arxiv.org/html/2610.10626#S2.p1.1)\.
- Geiger and Smidt \(2022\)M\. Geiger and T\. Smidte3nn: euclidean neural networks\.arXiv preprint arXiv:2207\.09453\.Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p1.1),[§5\.2](https://arxiv.org/html/2610.10626#S5.SS2.p2.1)\.
- Heet al\.\(2023\)L\. He, Y\. Chen, Z\. Shen, Y\. Yang, and Z\. LinNeural ePDOs: spatially adaptive equivariant partial differential operator based networks\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p2.1)\.
- Helwiget al\.\(2023\)J\. Helwig, X\. Zhang, C\. Fu, J\. Kurtin, S\. Wojtowytsch, and S\. JiGroup equivariant Fourier neural operators for partial differential equations\.InProceedings of the 40th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.202,pp\. 12907–12930\.Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p1.1),[§2](https://arxiv.org/html/2610.10626#S2.p2.1)\.
- Jenner and Weiler \(2022\)E\. Jenner and M\. WeilerSteerable partial differential operators for equivariant neural networks\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p2.1),[§4\.2](https://arxiv.org/html/2610.10626#S4.SS2.p2.2)\.
- Kingma and Ba \(2015\)D\. P\. Kingma and J\. BaAdam: a method for stochastic optimization\.InInternational Conference on Learning Representations,Cited by:[§5\.2](https://arxiv.org/html/2610.10626#S5.SS2.p4.1)\.
- Liet al\.\(2021\)Z\. Li, N\. Kovachki, K\. Azizzadenesheli, B\. Liu, K\. Bhattacharya, A\. Stuart, and A\. AnandkumarFourier neural operator for parametric partial differential equations\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p2.1)\.
- Mezzadri \(2007\)F\. MezzadriHow to generate random matrices from the classical compact groups\.External Links:math\-ph/0609050Cited by:[§5\.2](https://arxiv.org/html/2610.10626#S5.SS2.p3.1)\.
- Olver \(1993\)P\. J\. OlverApplications of lie groups to differential equations\.2 edition,Springer\.Cited by:[§3](https://arxiv.org/html/2610.10626#S3.p3.2)\.
- Pfaffet al\.\(2021\)T\. Pfaff, M\. Fortunato, A\. Sanchez\-Gonzalez, and P\. BattagliaLearning mesh\-based simulation with graph networks\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p2.1),[§2](https://arxiv.org/html/2610.10626#S2.p2.1)\.
- Punyet al\.\(2022\)O\. Puny, M\. Atzmon, H\. Ben\-Hamu, I\. Misra, A\. Grover, E\. J\. Smith, and Y\. LipmanFrame averaging for invariant and equivariant network design\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p3.1),[§5\.2](https://arxiv.org/html/2610.10626#S5.SS2.p3.1),[§7](https://arxiv.org/html/2610.10626#S7.p1.1)\.
- Sanchez\-Gonzalezet al\.\(2020\)A\. Sanchez\-Gonzalez, J\. Godwin, T\. Pfaff, R\. Ying, J\. Leskovec, and P\. BattagliaLearning to simulate complex physics with graph networks\.InInternational Conference on Machine Learning,pp\. 8459–8468\.Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p2.1),[§2](https://arxiv.org/html/2610.10626#S2.p2.1),[§5\.2](https://arxiv.org/html/2610.10626#S5.SS2.p1.1)\.
- Satorraset al\.\(2021\)V\. G\. Satorras, E\. Hoogeboom, and M\. WellingE\(n\) equivariant graph neural networks\.InInternational Conference on Machine Learning,pp\. 9323–9332\.Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p3.1),[§2](https://arxiv.org/html/2610.10626#S2.p1.1),[§5\.2](https://arxiv.org/html/2610.10626#S5.SS2.p1.1)\.
- Schüttet al\.\(2021\)K\. Schütt, O\. Unke, and M\. GasteggerEquivariant message passing for the prediction of tensorial properties and molecular spectra\.InInternational Conference on Machine Learning,pp\. 9377–9388\.Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p1.1),[§7](https://arxiv.org/html/2610.10626#S7.p3.1)\.
- Smith \(1971\)G\. F\. SmithOn isotropic functions of symmetric tensors, skew\-symmetric tensors and vectors\.International Journal of Engineering Science9,pp\. 899–916\.External Links:[Document](https://dx.doi.org/10.1016/0020-7225%2871%2990023-1)Cited by:[§2](https://arxiv.org/html/2610.10626#S2.p1.1),[§4\.2](https://arxiv.org/html/2610.10626#S4.SS2.p1.1),[§4](https://arxiv.org/html/2610.10626#S4.p1.1)\.
- Strang \(1968\)G\. StrangOn the construction and comparison of difference schemes\.SIAM Journal on Numerical Analysis5\(3\),pp\. 506–517\.Cited by:[§5\.1](https://arxiv.org/html/2610.10626#S5.SS1.p1.1)\.
- Thomaset al\.\(2018\)N\. Thomas, T\. Smidt, S\. Kearnes, L\. Yang, L\. Li, K\. Kohlhoff, and P\. RileyTensor field networks: rotation\- and translation\-equivariant neural networks for 3d point clouds\.arXiv preprint arXiv:1802\.08219\.Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p3.1),[§2](https://arxiv.org/html/2610.10626#S2.p1.1),[§5\.2](https://arxiv.org/html/2610.10626#S5.SS2.p2.1)\.
- Villaret al\.\(2021\)S\. Villar, D\. W\. Hogg, K\. Storey\-Fisher, W\. Yao, and B\. Blum\-SmithScalars are universal: equivariant machine learning, structured like classical physics\.InAdvances in Neural Information Processing Systems,Vol\.34\.Cited by:[§1](https://arxiv.org/html/2610.10626#S1.p1.1),[§1](https://arxiv.org/html/2610.10626#S1.p3.1),[§2](https://arxiv.org/html/2610.10626#S2.p1.1),[§4\.1](https://arxiv.org/html/2610.10626#S4.SS1.p2.2),[§4\.2](https://arxiv.org/html/2610.10626#S4.SS2.p1.1),[§4\.2](https://arxiv.org/html/2610.10626#S4.SS2.p1.2),[§4](https://arxiv.org/html/2610.10626#S4.p1.1)\.

## Appendix APer\-Seed Results

Tablereports the values underlying the aggregate main result for the exact architectures, augmented controls, and parameter\-matched GNS\. The same 1,000 one\-step rotations and 20 ten\-step rotations are used for every method and seed\.

\\FloatBarrier

## Appendix BImplementation and Additional Controls

All models follow the same optimization and selection protocol\. Every training sample contributes four randomly selected one\-step temporal windows per epoch\. Models minimize mean\-squared derivative error, clip gradient norms at 10, and use weight decay10−510^\{\-5\}\. Checkpoint selection minimizes validation one\-step relative error\. The test split is evaluated once after freezing the architecture and selecting the checkpoint\.

Two simple temporal controls characterize the predictability of the dataset\. Persistence receives one frame and obtainse1=0\.012219e\_\{1\}=0\.012219ande10=0\.120160e\_\{10\}=0\.120160\. A two\-frame linear extrapolator estimates the increment from the first two observations and obtainse1=0\.000259e\_\{1\}=0\.000259ande10=0\.014188e\_\{10\}=0\.014188\. The latter is not a like\-for\-like baseline because it receives an additional frame that nearly reveals the time derivative\. Its performance shows that this dataset is temporally smooth and that the main experiment specifically tests one\-frame spatial operator learning\.

Numerical equivariance can differ from the exact algebraic result because the graph is constructed in floating\-point arithmetic\. The theoretical graph is unchanged by rotation because pairwise distances are preserved\. In floating\-point arithmetic, points exactly at the radius boundary could change adjacency\. No such instability is observed in the reported graphs:IKNOprediction error is unchanged to the displayed precision and the residual operator error is on the order of single\-precision rounding\.