SALTO: Solver-Anchored Lifted Transfer Operators for Nonlinear Event Inference

# SALTO: Solver-Anchored Lifted Transfer Operators for Nonlinear Event Inference
Source: [https://arxiv.org/html/2610.21049](https://arxiv.org/html/2610.21049)
Qi WangAffiliation:Department of Aerospace EngineeringAffiliation:San Diego State UniversityAffiliation:San Diego, CA 92182, USAEmail:[qwang4@sdsu\.edu](mailto:)Rylie FosterAffiliation:Computational Science Research CenterAffiliation:San Diego State UniversityAffiliation:San Diego, CA 92182, USAAffiliation:Department of Mechanical and Aerospace EngineeringAffiliation:University of California, Irvine, CA 92697, USA

###### Abstract

Learned dynamical models can accurately predict trajectories while producing unreliable sensitivities when differentiated or transposed, limiting their use in inverse problems and scientific decision\-making\. Classical adjoints provide exact local derivatives, but a single adjoint cannot represent the nonlinear response of a system to a family of finite\-amplitude perturbations\. We introduceSALTO\(*Solver\-Anchored Lifted Transfer Operators*\), a Koopman\-based framework for inferring finite\-amplitude sources\. In the lifted space, the nonlinear dynamics are approximated by a linear propagator, and its transpose carries each sensor backward in a single solve\. The result is a reusable lifted adjoint that predicts the measurement produced by any candidate localized event, such as a source of given position and amplitude, with one encoder pass and an inner product, so that the search needs no repeated nonlinear simulation\. To give the model the simulator’s exact local behavior,SALTOimposes three structural constraints: it anchors the physical\-state dynamics to the simulator’s linearized propagator, centers the learned observables so that they vanish to second order at the reference state, and uses a block\-triangular lifted propagator\. These constraints guarantee that the model reduces to the simulator’s tangent dynamics and classical adjoint as the amplitude vanishes, while the learned observables carry only the finite\-amplitude correction\. On three nonlinear PDE systems,SALTOpredicts the measurements of finite\-amplitude sources to within22–8%8\\%where the classical adjoint is in error by4141–165%165\\%, recovers source position and intensity from sparse sensors nearly as accurately as inversion through the solver, and samples the joint posterior of two interacting sources, matching the solver’s posterior on reaction–diffusion at a twentieth of its cost\. One trained model serves new sensor layouts and measurement times with a single backward solve, and its exact local structure keeps both measurements and their gradients correct as the amplitude vanishes, where unconstrained Koopman models do not\. Enforcing exact local structure thus turns a learned lift into a reliable instrument for nonlinear sensitivity and inference\.

## 1Introduction

Many problems in the physical sciences ask us to identify the cause of an observed signal\. When a contaminant is detected at a downstream sensor, or a disturbance is recorded in a flow, we want to know which event produced it, where and when it occurred, and how strong it was\. Such source localization problems arise in environmental monitoring, leak detection and flow diagnosis, and they all depend on how sensitive a measurement is to a perturbation of the state\. The standard tool for computing this sensitivity is the adjoint of the tangent linear model\. A single backward solve gives the derivative of a measurement with respect to the state at every earlier time, which is why adjoints underlie source localization, data assimilation, sensor placement and gradient\-based design\([Errico, 1997](https://arxiv.org/html/2610.21049#bib.bib9);[Giles & Pierce, 2000](https://arxiv.org/html/2610.21049#bib.bib10);[Luchini & Bottaro, 2014](https://arxiv.org/html/2610.21049#bib.bib19);[Zaki, 2025](https://arxiv.org/html/2610.21049#bib.bib35)\)\. The adjoint, however, has a structural limitation\. Because it linearizes the dynamics about a base trajectory, the measurement it predicts for a source is linear in the amplitude of that source\. For a source of finite amplitude the measurement is a nonlinear functional of the source, and no single adjoint, whether about the base state or about any one nonlinear trajectory, can represent it\. Inference therefore needs a sensitivity defined over a whole family of sources rather than a single vector\.

This paper obtains that functional from a single linear adjoint solve by means of a Koopman lift\([Koopman, 1931](https://arxiv.org/html/2610.21049#bib.bib17);[Mezić, 2005](https://arxiv.org/html/2610.21049#bib.bib21);[Williams et al\., 2015](https://arxiv.org/html/2610.21049#bib.bib31);[Brunton et al\., 2022](https://arxiv.org/html/2610.21049#bib.bib2)\)\. We augment the state with a set of learned nonlinear observables, so that the augmented, or lifted, state evolves linearly under an approximate Koopman operator\. A measurement at the final time is then a linear functional of the lifted state, and carrying it one step backward in time amounts to applying the transpose of that operator\. Repeating this back to the time of the source gives what we call the*lifted adjoint*\(Fig\.[1](https://arxiv.org/html/2610.21049#S3.F1)\)\. Because the lifted adjoint does not depend on the state, it is computed once for each sensor and reused for every candidate source\. The measurement predicted for a candidate source is then an inner product between the lifted adjoint and the lifted source\. This inner product is a nonlinear functional of the source that costs a single encoder pass to evaluate, and it is exactly what finite\-amplitude source inference needs\.

The difficulty is that forward accuracy does not make a learned lift a reliable sensitivity\. An unconstrained lift must learn the linear dynamics along with the nonlinear ones, even though the solver already provides them exactly, and any error in this linear part enters every sensitivity the lift predicts, including those of vanishingly weak sources\. Such errors barely affect trajectory predictions at the training amplitudes, which is why a lift can forecast well and still transpose poorly\. We therefore take the linear part from the solver and learn only the nonlinear correction, and we enforce this by construction with three constraints\. The learned observables are centered so that they vanish to second order at the base state \(C1\), the lifted propagator uses the solver’s linearized propagator for the physical state \(C2\), and it is block\-triangular so that the learned observables evolve on their own \(C3\)\. The lifted model then reduces exactly to the solver’s tangent dynamics and the classical adjoint as the amplitude vanishes, and the learned observables carry only the finite\-amplitude correction\.

#### Contributions\.

\(i\)A solver\-anchored lifted adjoint that is exact at small amplitude\.We formulate finite\-amplitude sensitivity with a Koopman lift whose state\-independent adjoint, obtained from one backward solve, turns inference over a prescribed source family into inner products \(§[3](https://arxiv.org/html/2610.21049#S3)\)\. We impose three constraints that make its transpose usable, namely solver anchoring, closed\-form quadratic centering and a block\-triangular propagator, and show that they make the tangent linear model and the classical adjoint exact in the infinitesimal limit, leave a nonlinear remainder of second order in the amplitude, and give the learned blocks a well\-posed, closed\-form fit \(§[4](https://arxiv.org/html/2610.21049#S4)\)\. \(ii\)One backward solve for every candidate source\.One backward solve per sensor yields a lifted adjoint whose physical\-state part is exactly the classical adjoint\. The linear term of the functional is therefore the classical positional embedding\([You & Wang, 2026](https://arxiv.org/html/2610.21049#bib.bib33)\), and the learned channels add the finite\-amplitude correction\. Predicting the measurement of any candidate source, computing its gradient or sampling a full posterior then costs encoder passes instead of nonlinear solves\. \(iii\)Accurate, fast inference on three nonlinear PDE systems\.We test the framework on the viscous Burgers equation about an evolving base flow, a two\-dimensional reaction–diffusion system, and two\-dimensional Kolmogorov flow about a laminar shear, each with a prescribed low\-dimensional source family\. Against the solver, the classical adjoint and its quadratic correction, unconstrained lifts, structural ablations, a neural surrogate and networks trained on measurements, the lift is exact at small amplitude in both the measurement and its gradient, infers sources nearly as accurately as the solver, reproduces the solver’s joint posterior on reaction–diffusion at a twentieth of the cost, and serves new sensor layouts without retraining \(§[5](https://arxiv.org/html/2610.21049#S5)\)\.

## 2Related work

Three lines of research are relevant to inferring finite\-amplitude sources from sparse sensors\.Koopman and transfer operators\.Koopman theory represents nonlinear dynamics by a linear operator acting on observables of the state\([Koopman, 1931](https://arxiv.org/html/2610.21049#bib.bib17);[Mezić, 2005](https://arxiv.org/html/2610.21049#bib.bib21)\), and its data\-driven approximations are reviewed by[Brunton et al\. \(2022\)](https://arxiv.org/html/2610.21049#bib.bib2)\. Extended dynamic mode decomposition fits a finite\-dimensional Koopman matrix on a prescribed dictionary\([Williams et al\., 2015](https://arxiv.org/html/2610.21049#bib.bib31)\), and Koopman autoencoders learn the dictionary together with the matrix\([Lusch et al\., 2018](https://arxiv.org/html/2610.21049#bib.bib20);[Otto & Rowley, 2019](https://arxiv.org/html/2610.21049#bib.bib23)\)\. The transpose of the EDMD matrix acts on the coefficients of observables, and together with the dictionary’s Gram matrix it gives the Galerkin approximation of the Perron–Frobenius operator\([Klus et al\., 2016](https://arxiv.org/html/2610.21049#bib.bib15)\)\. Physics\-informed DMD constrains the fitted operator to known structure such as symmetries and conservation laws\([Baddoo et al\., 2023](https://arxiv.org/html/2610.21049#bib.bib1)\), and hybrid models add a learned correction to a known physical model\([Yin et al\., 2021](https://arxiv.org/html/2610.21049#bib.bib32);[Um et al\., 2020](https://arxiv.org/html/2610.21049#bib.bib24)\)\. Carleman linearization builds a linear embedding of a polynomial system from its equations, at the cost of an infinite hierarchy that must be truncated\([Carleman, 1932](https://arxiv.org/html/2610.21049#bib.bib3);[Kowalski & Steeb, 1991](https://arxiv.org/html/2610.21049#bib.bib18)\)\.Adjoints and learned surrogates\.Adjoint sensitivity is classical in fluid mechanics and control, where one backward solve gives the gradient of a measured functional with respect to every input\([Errico, 1997](https://arxiv.org/html/2610.21049#bib.bib9);[Giles & Pierce, 2000](https://arxiv.org/html/2610.21049#bib.bib10);[Luchini & Bottaro, 2014](https://arxiv.org/html/2610.21049#bib.bib19)\)\. In turbulent flows the adjoint itself degrades over long horizons, which has motivated regularized and latent\-space assimilation\([Ke et al\., 2026](https://arxiv.org/html/2610.21049#bib.bib14);[Wang & Zaki, 2026](https://arxiv.org/html/2610.21049#bib.bib28);[Gutierrez & Wang, 2026a](https://arxiv.org/html/2610.21049#bib.bib11);[Cleary et al\., 2025](https://arxiv.org/html/2610.21049#bib.bib4)\)\. Differentiable surrogates provide gradients through learned models\([Holl et al\., 2020](https://arxiv.org/html/2610.21049#bib.bib13);[Kochkov et al\., 2021](https://arxiv.org/html/2610.21049#bib.bib16)\), although these gradients are rarely validated against the solver\. Sobolev training supervises the derivatives of a network directly, at the cost of tangent solves during training\([Czarnecki et al\., 2017](https://arxiv.org/html/2610.21049#bib.bib6)\), and derivative\-informed neural operators train on derivatives for inverse problems\([O’Leary\-Roseberry et al\., 2024](https://arxiv.org/html/2610.21049#bib.bib22)\)\.Source inference\.Adjoint\-based methods localize sources from sparse sensors with a model that is linear in the source amplitude, and a quadratic correction extends this linear embedding to second order\([Wang & You, 2026](https://arxiv.org/html/2610.21049#bib.bib26)\)\. Amortized simulation\-based inference instead learns the posterior from simulations\([Cranmer et al\., 2020](https://arxiv.org/html/2610.21049#bib.bib5)\), and other reconstruction methods exploit the domain of dependence of each measurement\([You et al\., 2026](https://arxiv.org/html/2610.21049#bib.bib34);[Wang & Zaki, 2025](https://arxiv.org/html/2610.21049#bib.bib27);[Gutierrez & Wang, 2026b](https://arxiv.org/html/2610.21049#bib.bib12);[Du & Wang, 2026](https://arxiv.org/html/2610.21049#bib.bib8);[Domínguez\-Vázquez et al\., 2025](https://arxiv.org/html/2610.21049#bib.bib7)\)\.

## 3Lifted sensitivity

![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/Salto_v2.png)Figure 1:SALTOin one picture\. Localized sourcesS⁡\(𝒑\)S\(\\bm\{p\}\)seed a nonlinear flow observed by a few probes\. Offline, the solver supplies𝑳k\\bm\{L\}\_\{k\}and the data supply only the finite\-amplitude correction, so the lifted model is exact at small amplitude\. Online, one backward solve per sensor gives the lifted adjoint𝒁j†\\bm\{Z\}^\{\\dagger\}\_\{j\}, and every candidate source is scored by an encoder pass and inner products\.This section poses the inference problem and shows how a Koopman lift turns its nonlinear measurement functional into an inner product with a lifted adjoint\. Let𝒒⁡\(t\)∈ℝN\\bm\{q\}\(t\)\\in\\mathbb\{R\}^\{N\}be the discretized perturbation of the state about a base solution\. A nonlinear solver advances it over a snapshot intervalΔ​s\\Delta s, and we write this one\-step map as𝒒k\+1=𝒩k​\(𝒒k\)\\bm\{q\}\_\{k\+1\}=\\mathcal\{N\}\_\{k\}\(\\bm\{q\}\_\{k\}\),k=0,…,K−1k=0,\\dots,K\-1, withT=K​Δ​sT=K\\Delta s\. We call an initial perturbation a source and write it as𝒒0=S⁡\(𝒑\)\\bm\{q\}\_\{0\}=S\(\\bm\{p\}\), where𝒑\\bm\{p\}collects a few parameters such as position, intensity, width, orientation and injection time\. Because the dynamics are written for the perturbation about a base solution, the system is homogeneous, so a zero source produces a zero response at all times\. We assume throughout that the base solution is known and that the solver can reproduce it\. The measurements are linear functionals of the final state,mj=⟨𝒆j,𝒒K⟩m\_\{j\}=\\langle\\bm\{e\}\_\{j\},\\bm\{q\}\_\{K\}\\rangleforj=1,…,Nmj=1,\\dots,N\_\{m\}, where𝒆j\\bm\{e\}\_\{j\}is a point or footprint kernel\. The solver also provides the linearized one\-step propagators𝑳k=∂𝒩k/∂𝒒\\bm\{L\}\_\{k\}=\\partial\\mathcal\{N\}\_\{k\}/\\partial\\bm\{q\}about the base trajectory, which together form the discrete tangent linear model\. The*linear adjoint*𝒒L†\\bm\{q\}^\{\\dagger\}\_\{\\mathrm\{L\}\}carries a kernel backward through𝑳k⊤\\bm\{L\}\_\{k\}^\{\\\!\\top\}, and it is what a classical adjoint code returns\([Wang et al\., 2019a](https://arxiv.org/html/2610.21049#bib.bib25);[Wang et al\., 2022](https://arxiv.org/html/2610.21049#bib.bib30);[Wang et al\., 2019b](https://arxiv.org/html/2610.21049#bib.bib29)\)\. It is a single vector, so it cannot express the measurementmj​\(𝒑\)m\_\{j\}\(\\bm\{p\}\), which is a nonlinear function of𝒑\\bm\{p\}across the source family\. We form the lifted adjoint once, about the base state, because the perturbation itself is the unknown that the inverse problem seeks\. Section[5](https://arxiv.org/html/2610.21049#S5)tests the functional and its gradient away from the base\.

Let𝚽θ:ℝN→ℝc​N\\bm\{\\Phi\}\_\{\\theta\}:\\mathbb\{R\}^\{N\}\\to\\mathbb\{R\}^\{cN\}be a learned map toccchannels, and let𝚿⁡\(𝒒\)=\[𝒒;𝚽θ​\(𝒒\)\]∈ℝd\\bm\{\\Psi\}\(\\bm\{q\}\)=\[\\bm\{q\};\\bm\{\\Phi\}\_\{\\theta\}\(\\bm\{q\}\)\]\\in\\mathbb\{R\}^\{d\}, withd=N⁡\(1\+c\)d=N\(1\+c\), be the lifted state\. The lifted linear model is𝒁k\+1=𝑮k​𝒁k\\bm\{Z\}\_\{k\+1\}=\\bm\{G\}\_\{k\}\\bm\{Z\}\_\{k\}with the block\-triangular propagator

𝑮k=\[𝑳k𝑮q​Φ𝟎𝑮Φ​Φ\],\\bm\{G\}\_\{k\}=\\begin\{bmatrix\}\\bm\{L\}\_\{k\}&\\bm\{G\}\_\{q\\Phi\}\\\\ \\bm\{0\}&\\bm\{G\}\_\{\\Phi\\Phi\}\\end\{bmatrix\},\(1\)where𝑳k\\bm\{L\}\_\{k\}is the solver’s propagator, and𝑮q​Φ\\bm\{G\}\_\{q\\Phi\}and𝑮Φ​Φ\\bm\{G\}\_\{\\Phi\\Phi\}are learned, time\-invariant blocks that couple the features to the state and advance the features\. Fitting all of𝑮k\\bm\{G\}\_\{k\}by least squares on snapshot pairs would amount to extended dynamic mode decomposition\([Williams et al\., 2015](https://arxiv.org/html/2610.21049#bib.bib31)\)\. Here, by contrast, two of its blocks are imposed rather than fitted \(§[4](https://arxiv.org/html/2610.21049#S4)\)\. The lifted model starts from the encoded source,𝒁0=𝚿⁡\(𝒒0\)\\bm\{Z\}\_\{0\}=\\bm\{\\Psi\}\(\\bm\{q\}\_\{0\}\), and predicts the measurement as a linear functional of the lifted state,m^j=⟨\[𝒆j;𝟎\],𝒁K⟩\\hat\{m\}\_\{j\}=\\langle\[\\bm\{e\}\_\{j\};\\bm\{0\}\],\\bm\{Z\}\_\{K\}\\rangle\. Carrying this functional backward in time gives the*lifted adjoint*

𝒁j†​\(τk\)\\displaystyle\\bm\{Z\}^\{\\dagger\}\_\{j\}\(\\tau\_\{k\}\)=𝑮K−k⊤⋯𝑮K−1⊤\[𝒆j𝟎\],\\displaystyle=\\bm\{G\}\_\{K\-k\}^\{\\\!\\top\}\\cdots\\bm\{G\}\_\{K\-1\}^\{\\\!\\top\}\\begin\{bmatrix\}\\bm\{e\}\_\{j\}\\\\ \\bm\{0\}\\end\{bmatrix\},\(2\)m^j\\displaystyle\\hat\{m\}\_\{j\}=⟨\[𝒆j𝟎\],𝒁K⟩=⟨𝒁j†​\(τk\),𝒁K−k⟩\\displaystyle=\\big\\langle\\begin\{bmatrix\}\\bm\{e\}\_\{j\}\\\\ \\bm\{0\}\\end\{bmatrix\},\\bm\{Z\}\_\{K\}\\big\\rangle=\\big\\langle\\bm\{Z\}^\{\\dagger\}\_\{j\}\(\\tau\_\{k\}\),\\bm\{Z\}\_\{K\-k\}\\big\\rangle
for every backward timeτk=k​Δ​s\\tau\_\{k\}=k\\Delta s\. The lifted adjoint holds the coefficients of the observable whose value on the lifted state at timeT−τT\-\\tauequals the final measurement, just as the classical adjoint field is the sensor kernel carried backward by the transposed tangent model\. The identity equation[2](https://arxiv.org/html/2610.21049#S3.E2)holds exactly for the lifted rollout, so the only error of the lifted model relative to the solver is the error of that rollout, which Section[5](https://arxiv.org/html/2610.21049#S5)measures\. Three properties follow\. First,𝒁j†\\bm\{Z\}^\{\\dagger\}\_\{j\}does not depend on the state, so it is computed once per sensor and reused for every source\. Second, settingτ=T\\tau=Tand𝒁0=𝚿⁡\(S⁡\(𝒑\)\)\\bm\{Z\}\_\{0\}=\\bm\{\\Psi\}\(S\(\\bm\{p\}\)\)in equation[2](https://arxiv.org/html/2610.21049#S3.E2)gives the predicted measurement of a finite source directly,

m^j​\(𝒑\)=⟨𝒁j,q†​\(T\),S⁡\(𝒑\)⟩\+⟨𝒁j,Φ†​\(T\),𝚽θ​\(S⁡\(𝒑\)\)⟩,\\hat\{m\}\_\{j\}\(\\bm\{p\}\)=\\big\\langle\\bm\{Z\}^\{\\dagger\}\_\{j,q\}\(T\),S\(\\bm\{p\}\)\\big\\rangle\+\\big\\langle\\bm\{Z\}^\{\\dagger\}\_\{j,\\Phi\}\(T\),\\bm\{\\Phi\}\_\{\\theta\}\(S\(\\bm\{p\}\)\)\\big\\rangle,\(3\)a nonlinear functional of the source, which we call the*nonlinear duality relation*\. Its first term is the linear positional embedding of the classical adjoint, and its second term is the finite\-amplitude correction\. Evaluating it for a candidate costs one encoder pass and two inner products, with no time stepping\. Third, by \(C2\) and \(C3\) the state block satisfies𝒁j,q†\(τk\)=𝑳K−k⊤⋯𝑳K−1⊤𝒆j=𝒒L†\\bm\{Z\}^\{\\dagger\}\_\{j,q\}\(\\tau\_\{k\}\)=\\bm\{L\}\_\{K\-k\}^\{\\\!\\top\}\\cdots\\bm\{L\}\_\{K\-1\}^\{\\\!\\top\}\\bm\{e\}\_\{j\}=\\bm\{q\}^\{\\dagger\}\_\{\\mathrm\{L\}\}, which is the linear adjoint itself\. Every inference in this paper scores candidates by evaluating the duality relation equation[3](https://arxiv.org/html/2610.21049#S3.E3)directly\. Only the two\-source point estimates of Appendix[B\.3](https://arxiv.org/html/2610.21049#A2.SS3)are afterwards refined, by gradient descent on the same functional\.

Posterior density of a source\.The duality relation equation[3](https://arxiv.org/html/2610.21049#S3.E3)scores one candidate at a time, but the experiments also ask how precisely a source can be located, which a point estimate cannot tell us\. We writep=\(xs,ys,Is\)p=\(x\_\{s\},y\_\{s\},I\_\{s\}\)for the parameters of one source and𝒎^​\(p1,…,pns\)\\widehat\{\\bm\{m\}\}\(p\_\{1\},\\dots,p\_\{n\_\{s\}\}\)for the measurement that a model predicts whennsn\_\{s\}sources are present together\. Every model, including the solver, is evaluated on the combined field∑aS⁡\(pa\)\\sum\_\{a\}S\(p\_\{a\}\), so that the prediction includes the interaction between the sources\. With independent Gaussian noise of widthσ\\sigmaand a flat prior,

ℙ⁡\(p1,…,pns\)\\displaystyle\\mathbb\{P\}\(p\_\{1\},\\dots,p\_\{n\_\{s\}\}\)∝exp⁡\(−12​σ2​‖𝒎obs−𝒎^​\(p1,…,pns\)‖2\),\\displaystyle\\propto\\;\\exp\\\!\\Big\(\-\\tfrac\{1\}\{2\\sigma^\{2\}\}\\big\\\|\\bm\{m\}\_\{\\mathrm\{obs\}\}\-\\widehat\{\\bm\{m\}\}\(p\_\{1\},\\dots,p\_\{n\_\{s\}\}\)\\big\\\|^\{2\}\\Big\),\(4\)ρ⁡\(xs,ys\)\\displaystyle\\rho\(x\_\{s\},y\_\{s\}\)∝∫d​Is​∫∏b≥2d​pb​ℙ​\(\(xs,ys,Is\),p2,…,pns\),\\displaystyle\\propto\\;\\int\\\!\\mathrm\{d\}I\_\{s\}\\int\\\!\\prod\_\{b\\geq 2\}\\mathrm\{d\}p\_\{b\}\\,\\mathbb\{P\}\\big\(\(x\_\{s\},y\_\{s\},I\_\{s\}\),p\_\{2\},\\dots,p\_\{n\_\{s\}\}\\big\),whereρ\\rhois the density shown in the figures\. Because the sources are exchangeable,ρ\\rhohas one peak per resolved source, and we estimate it from Markov chain Monte Carlo samples \(Appendix[A](https://arxiv.org/html/2610.21049#A1)\)\.

The widthσ\\sigmaaccounts for more than the observation noise\. A surrogate’s own error in predicting the measurement adds to the noise in the residual, and when this error dominates, a likelihood based on the noise alone is overconfident and can be confidently wrong\. We therefore setσ2=σn2\+σmod2\\sigma^\{2\}=\\sigma\_\{n\}^\{2\}\+\\sigma\_\{\\mathrm\{mod\}\}^\{2\}, whereσmod\\sigma\_\{\\mathrm\{mod\}\}is the model’s root\-mean\-square discrepancy from the solver on an independent set of combined pair fields\. This makes models of different accuracy comparable, and for the solverσmod=0\\sigma\_\{\\mathrm\{mod\}\}=0\.

## 4Trustworthy structure

The lifted adjoint of Section[3](https://arxiv.org/html/2610.21049#S3)is only as reliable as the transpose of the learned propagator\. A lift fitted only for forward accuracy leaves its transpose unconstrained away from the states visited in training, and it overwrites the linear dynamics that the solver already knows, so it need not reduce to the classical adjoint at small amplitude\. On Burgers about a quiescent base, for example, an unconstrained lift of the same size matches our forward accuracy, yet its rollout error stays between4%4\\%and53%53\\%as the amplitude vanishes instead of vanishing with it \(Appendix[C](https://arxiv.org/html/2610.21049#A3)\)\. The three constraints below remove this failure by construction rather than through training\.

\(C1\) Quadratic centering\.The learned channels must be at least second order in the source perturbation, or they would corrupt the linear response that the solver already gives exactly\. A nonzero𝚽θ​\(𝟎\)\\bm\{\\Phi\}\_\{\\theta\}\(\\bm\{0\}\)injects a constant through the coupling block, and a nonzero Jacobian at the base state adds a spurious linear term to the adjoint that does not vanish with the amplitude\. We therefore require𝚽θ​\(𝟎\)=𝟎\\bm\{\\Phi\}\_\{\\theta\}\(\\bm\{0\}\)=\\bm\{0\}and∂𝚽θ/∂𝒒\|𝟎=𝟎\\partial\\bm\{\\Phi\}\_\{\\theta\}/\\partial\\bm\{q\}\|\_\{\\bm\{0\}\}=\\bm\{0\}, so that𝚽θ​\(𝒒\)=O⁡\(‖𝒒‖2\)\\bm\{\\Phi\}\_\{\\theta\}\(\\bm\{q\}\)=O\(\\\|\\bm\{q\}\\\|^\{2\}\)\. Given any smooth network𝚽~θ\\widetilde\{\\bm\{\\Phi\}\}\_\{\\theta\}, which is the encoder before centering, we set

𝚽θ​\(𝒒\)=𝚽~θ​\(𝒒\)−𝚽~θ​\(𝟎\)−𝑱0​𝒒,𝑱0=∂𝚽~θ/∂𝒒\|𝟎,\\bm\{\\Phi\}\_\{\\theta\}\(\\bm\{q\}\)=\\widetilde\{\\bm\{\\Phi\}\}\_\{\\theta\}\(\\bm\{q\}\)\-\\widetilde\{\\bm\{\\Phi\}\}\_\{\\theta\}\(\\bm\{0\}\)\-\\bm\{J\}\_\{0\}\\bm\{q\},\\qquad\\bm\{J\}\_\{0\}=\\partial\\widetilde\{\\bm\{\\Phi\}\}\_\{\\theta\}/\\partial\\bm\{q\}\|\_\{\\bm\{0\}\},\(5\)which is exact and differentiable inθ\\theta\. Both correction terms have closed forms for a tanh network \(Appendix[A](https://arxiv.org/html/2610.21049#A1)\)\. For a translation\-equivariant convolutional encoder,𝚽~θ​\(𝟎\)\\widetilde\{\\bm\{\\Phi\}\}\_\{\\theta\}\(\\bm\{0\}\)is one constant per channel and𝑱0\\bm\{J\}\_\{0\}is a fixed convolution, so the centering needs no extra pass of the encoder\.

\(C2\) Solver anchoring\.The state row of equation[1](https://arxiv.org/html/2610.21049#S3.E1)uses the exact linearized solver𝑳k\\bm\{L\}\_\{k\}\.

\(C3\) Block triangularity\.The feature rows depend on the features alone\. The lower\-left block is𝟎\\mathbf\{0\}\.

#### Identities\.

Under \(C1\)–\(C3\), four identities hold for everykk\. \(i\) Thekk\-step state block of the lifted propagator is exactly the solver’s,\(𝑮k−1⋯𝑮0\)q​q=𝑳k−1⋯𝑳0\(\\bm\{G\}\_\{k\-1\}\\cdots\\bm\{G\}\_\{0\}\)\_\{qq\}=\\bm\{L\}\_\{k\-1\}\\cdots\\bm\{L\}\_\{0\}, so at every horizon the linearization of the lifted model at the base state is the tangent linear model\. \(ii\) The lifted rollout from𝒒0=ϵ​𝒗\\bm\{q\}\_\{0\}=\\epsilon\\bm\{v\}equals the tangent linear rollout up toO⁡\(ϵ2\)O\(\\epsilon^\{2\}\), and the learned channels contribute only theO⁡\(ϵ2\)O\(\\epsilon^\{2\}\)remainder\. \(iii\) The state block of the lifted adjoint is the linear adjoint,𝒁j,q†=𝒒L†\\bm\{Z\}^\{\\dagger\}\_\{j,q\}=\\bm\{q\}^\{\\dagger\}\_\{\\mathrm\{L\}\}, so the lifted functional reduces to the classical positional embedding asϵ→0\\epsilon\\to 0, since∂𝚽θ/∂𝒒=O⁡\(ϵ\)\\partial\\bm\{\\Phi\}\_\{\\theta\}/\\partial\\bm\{q\}=O\(\\epsilon\)\. \(iv\) The fit of the learned blocks is a constrained ridge least\-squares problem with a closed\-form solution\. Each proof follows in one line from the block structure and from𝚽θ=O⁡\(‖𝒒‖2\)\\bm\{\\Phi\}\_\{\\theta\}=O\(\\\|\\bm\{q\}\\\|^\{2\}\), and we verify the identities numerically to machine precision relative to the supplied𝑳k\\bm\{L\}\_\{k\}, which on Burgers and Kolmogorov flow are central\-difference Jacobians of the solver\. None of \(i\)–\(iii\) holds for an unconstrained lift, whose linearization at the base state is a fitted matrix that matches the tangent model only where the data constrain it\.

#### Closed\-form fit and bilevel training\.

Given snapshot pairs\(𝒒k,𝒒k\+1\)\(\\bm\{q\}\_\{k\},\\bm\{q\}\_\{k\+1\}\)along the training trajectories, with𝑳k​𝒒k\\bm\{L\}\_\{k\}\\bm\{q\}\_\{k\}supplied by the solver, the learned blocks solve

𝑮q​Φ\\displaystyle\\bm\{G\}\_\{q\\Phi\}=argmin∑k∥𝒒k\+1−𝑳k𝒒k−𝑮q​Φ𝚽θ\(𝒒k\)∥2\+λ∥⋅∥2,\\displaystyle=\\arg\\min\\textstyle\\sum\_\{k\}\\\|\\bm\{q\}\_\{k\+1\}\-\\bm\{L\}\_\{k\}\\bm\{q\}\_\{k\}\-\\bm\{G\}\_\{q\\Phi\}\\bm\{\\Phi\}\_\{\\theta\}\(\\bm\{q\}\_\{k\}\)\\\|^\{2\}\+\\lambda\\\|\\cdot\\\|^\{2\},\(6\)𝑮Φ​Φ\\displaystyle\\bm\{G\}\_\{\\Phi\\Phi\}=argmin∑k∥𝚽θ\(𝒒k\+1\)−𝑮Φ​Φ𝚽θ\(𝒒k\)∥2\+λ∥⋅∥2,\\displaystyle=\\arg\\min\\textstyle\\sum\_\{k\}\\\|\\bm\{\\Phi\}\_\{\\theta\}\(\\bm\{q\}\_\{k\+1\}\)\-\\bm\{G\}\_\{\\Phi\\Phi\}\\bm\{\\Phi\}\_\{\\theta\}\(\\bm\{q\}\_\{k\}\)\\\|^\{2\}\+\\lambda\\\|\\cdot\\\|^\{2\},Each is a single Gram solve on the feature block, with rows scaled to unit RMS and a ridge parameterλ\\lambdaset relative to the trace\. The data only have to explain the residual𝒒k\+1−𝑳k​𝒒k\\bm\{q\}\_\{k\+1\}\-\\bm\{L\}\_\{k\}\\bm\{q\}\_\{k\}, because the linear transport is not learned\. We train the encoder parametersθ\\thetaby gradient descent on the multi\-step rollout error of the lifted model, differentiating through the closed\-form solve of the inner loop, and we lengthen the rollout gradually during training\. Every problem and every ablation uses the same trainer\.

#### Baselines with the same code\.

Removing \(C1\) leaves raw, uncentered features \(anchoring only\)\. Removing \(C2\) and \(C3\) fits the full𝑮\\bm\{G\}by ridge EDMD \(centering only\)\. Removing all three gives the unconstrained Koopman autoencoder\. A neural surrogate uses the same encoder with a linear head as a residual one\-step map,𝒒k\+1=𝑳k​𝒒k\+𝑾​𝚽~θ​\(𝒒k\)\+𝒃\\bm\{q\}\_\{k\+1\}=\\bm\{L\}\_\{k\}\\bm\{q\}\_\{k\}\+\\bm\{W\}\\widetilde\{\\bm\{\\Phi\}\}\_\{\\theta\}\(\\bm\{q\}\_\{k\}\)\+\\bm\{b\}, trained on the same rollout loss\. Its gradient is exact by automatic differentiation, but scoring each candidate requires a rollout\.

## 5Experiments

We evaluateSALTOon three systems, each showing a different way in which a nonlinearity defeats a linear adjoint\. The first has self\-advection that carries the response away from where the linear model places it, the second has a saturating reaction, and the third has a vortex that advects itself inside a sheared Navier–Stokes flow\. Each system has a prescribed low\-dimensional source family, and we compare against the solver and two classical models\. The linear embedding predicts a measurement as⟨𝒒L†,S⁡\(𝒑\)⟩\\langle\\bm\{q\}^\{\\dagger\}\_\{\\mathrm\{L\}\},S\(\\bm\{p\}\)\\rangle, and the quadratic correction adds the second\-order term of the expansion about the base, computed with second\-order tangent and adjoint solves\([Wang & You, 2026](https://arxiv.org/html/2610.21049#bib.bib26)\)\. Throughout,εm\\varepsilon\_\{m\}is the relative error of a model’s predicted measurements at the top of the training range\. The median position errorδx\\delta\_\{x\}and the median relative intensity errorδI\\delta\_\{I\}describe the one\-shot inversion at1%1\\%noise, withδx\\delta\_\{x\}in units of the candidate spacing so that it compares across domains\. The cost is the time for one estimate, which is a single pass over the candidate grid\. Because candidate measurements do not depend on the data, a dictionary of solver measurements could also be cached once per sensor layout and horizon\. The lift’s cost advantage therefore lies in what no finite dictionary covers, such as the continuous joint posterior of several sources\. Table[1](https://arxiv.org/html/2610.21049#S5.T1)collects accuracy and cost, with bootstrap intervals in Appendix[B](https://arxiv.org/html/2610.21049#A2), for one lift per system, withc=24c=24channels on Burgers andc=4c=4on the two\-dimensional systems\. The paper uses three kinds of lift, namely thec=4c=4dense lifts of Table[1](https://arxiv.org/html/2610.21049#S5.T1), Fig\.[4](https://arxiv.org/html/2610.21049#S5.F4)and the reuse study, thec=2c=2dense lifts of the ablations, evaluated on a separate draw that extends down to zero intensity, and an eight\-channel translation\-equivariant lift trained on random fields for Fig\.[3](https://arxiv.org/html/2610.21049#S5.F3)\. Appendix[A](https://arxiv.org/html/2610.21049#A1)gives the equations, discretizations, source families, sensors, architectures and hardware\.

Table 1:Accuracy and cost on the three systems, with thec=24c=24lift on Burgers and thec=4c=4lifts on the two\-dimensional systems\. Each block uses one common draw of test sources\. Bold marks the best of the three models, the solver serves as the reference, and timings are wall\-clock times\.### 5\.1Self\-advection: Burgers about an evolving base flow

![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/burgers_results.png)Figure 2:\(a–d\)Space–time evolution of the base flow and of one release, from the solver, the lifted model and the linearized solver\. The dashed line is the half\-maximum contour of the ground truth\.\(e\)Five representative blocks of the lift, forward in time \(top\) and adjoint in backward timeτ\\tau\(bottom\)\.\(f\)Median inversion error for 800 sources at1%1\\%noise, withδx\\delta\_\{x\}in black andδI\\delta\_\{I\}in red, and linear dotted, quadratic dashed and lifted solid\. The dotted horizontal line marks one grid cell\.The first system is the viscous Burgers equation perturbed about an evolving base flow \(Appendix[A](https://arxiv.org/html/2610.21049#A1)\)\. The linearized solver misses both the steepening and the slowing of the front \(Fig\.[2](https://arxiv.org/html/2610.21049#S5.F2)\)\. Its error atTTgrows from about a tenth of the response for the weakest source to four fifths for the strongest, whereas the lifted model keeps it below a few per cent\. When we invert\(xs,Is\)\(x\_\{s\},I\_\{s\}\)in one shot from the five sensors by a grid search with1%1\\%noise, the lifted functional matches brute\-force inversion through the solver in intensity in every band and is at least as accurate in position \(Appendix[B](https://arxiv.org/html/2610.21049#A2)\)\. The quadratic correction is the stronger classical baseline for inference\. It halves the position error of the linear embedding and reduces its intensity error from15%15\\%to9%9\\%, although its measurement error at the top of the training range is larger than the linear embedding’s\. It still falls short of the lifted functional, because it expands about the base, and the front carries the release beyond the amplitude at which that expansion is accurate\.

### 5\.2Saturation: two\-dimensional reaction–diffusion

The second system saturates rather than advects, and we use it to study inference in full, from a single source to the joint posterior of two \(Appendix[A](https://arxiv.org/html/2610.21049#A1)\)\.Forward, measurement, small amplitude\.The four\-channel lifted functional predicts the nine measurements to within2\.4%2\.4\\%at the top of the training range and4%4\\%beyond it, where the error of the linear embedding exceeds the measurement itself\. As the amplitude decreases, the error of the two\-channel ablation lift decreases in proportion, reaching0\.008%0\.008\\%atIs=10−3I\_\{s\}=10^\{\-3\}, below the linear embedding’s0\.06%0\.06\\%\. There the unconstrained lift is off by23%23\\%, and the neural surrogate by more than the measurement itself \(Appendix[B](https://arxiv.org/html/2610.21049#A2)\)\.

Inference and cost\.We infer each source by a grid search over13,82413\{,\}824candidates, scoring each with the duality relation equation[3](https://arxiv.org/html/2610.21049#S3.E3), with1%1\\%noise added to the measurements\. Over 60 test sources the lifted functional recovers the position to0\.550\.55cells at the median, close to the0\.460\.46cells of brute\-force inversion through the solver, and matches its intensity error of14%14\\%\. It places 48 of the 60 sources within one cell of the truth, against 49 for brute force and 39 for the linear embedding \(Appendix[B](https://arxiv.org/html/2610.21049#A2)\)\. Both classical baselines fall behind\. The linear embedding cannot represent saturation and underestimates the intensity by40%40\\%at the median, reading a saturated response as a weaker source, and the quadratic correction helps only within its radius of convergence, since the expansion diverges for the stronger sources\. On this problem, scoring by inner products is the most accurate of the surrogates but not the cheapest\. It takes twelve seconds per estimate, against two for the quadratic correction and twenty\-one for the solver \(Table[1](https://arxiv.org/html/2610.21049#S5.T1)\)\.

Two sources at once\.When two sources overlap, one to three source widths apart, their joint measurement differs from the sum of their separate measurements by about a sixth \(Appendix[B\.3](https://arxiv.org/html/2610.21049#A2.SS3)\)\. The lift used here is more general than the one reported in the tables\. Its learned blocks are translation\-equivariant \(Appendix[A\.3](https://arxiv.org/html/2610.21049#A1.SS3)\), it has eight channels, and it is trained on 1600 band\-limited positive random fields rather than on the blob family that the inference searches over, so it never sees a source of the kind it is asked to find\. The training fields must stay positive, because the reaction−κ​u−γ​u2\-\\kappa u\-\\gamma u^\{2\}drivesuuwithout bound onceu<−κ/γu<\-\\kappa/\\gamma\.

A point estimate cannot tell whether a pair has been resolved, so we sample the joint posterior of both sources \(Eq\.[4](https://arxiv.org/html/2610.21049#S3.E4)\), applying every model to the combined field \(Fig\.[3](https://arxiv.org/html/2610.21049#S5.F3)a–d\)\. Over 17 overlapping pairs the lifted posterior reproduces the solver’s\. Its mean lies a median of0\.40\.4solver posterior standard deviations from the solver’s mean\. The central95%95\\%interval of each of the six parameters contains the true value in87%87\\%of the 102 cases, against85%85\\%for the solver under the same sampler, a difference within the binomial standard error of about3\.53\.5points\. Each lifted posterior costs1313seconds, against295295for the solver\. The linear and quadratic posteriors are1313to2727times wider, with means3\.63\.6and5\.65\.6solver standard deviations away, so their full coverage only reflects that they are uninformative\. Although this lift was trained on generic random fields and never on the blob family, it matches the solver on that family\.

![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/rd_results.png)Figure 3:Two\-dimensional reaction–diffusion, with the eight\-channel equivariant lift\. The lifted posterior reproduces the solver’s, while the classical ones spread over the whole window\.\(a–d\)Joint posterior density of source position \(Eq\.[4](https://arxiv.org/html/2610.21049#S3.E4)\) for a pair1\.41\.4source widths apart, on a logarithmic scale\. Each model is evaluated on the combined field, and its likelihood is widened by its own discrepancy\. Crosses mark the true sources and the dot marks a probe\.\(e\)Error of the one\-shot inversion against the true intensity, withδx\\delta\_\{x\}in black andδI\\delta\_\{I\}in red, and linear dotted, quadratic dashed and lifted solid\. Shading marks where each classical model still predicts the measurements to within three per cent, and the narrow unlabelled band at the left is where the linear embedding does\.
### 5\.3Sheared advection: Kolmogorov flow

![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/kolmo_results.png)Figure 4:Kolmogorov flow\.\(a–d\)Perturbation vorticity at±35%\\pm 35\\%of its initial maximum, time upward to2​T2T, for the most strongly interacting pair, from the solver, the lifted rollout, the quadratic expansion and the linear superposition\.\(e\)The state and one learned channel along that pair’s solver trajectory \(top\), and the same blocks of the centre probe’s lifted adjoint𝒁†\\bm\{Z\}^\{\\dagger\}\(circle\) in backward time \(bottom\), each on its own scale\. The state block is exactly𝒒L†\\bm\{q\}^\{\\dagger\}\_\{\\mathrm\{L\}\}\. The channel block is spread out because a measurement uses only its projection on the channels\.\(f\)Inversion error over 600 further sources, as in Fig\.[3](https://arxiv.org/html/2610.21049#S5.F3)e on linear axes; the narrow band is where the linear embedding holds\.The third system is the two\-dimensional Navier–Stokes equations in vorticity form about a steady laminar shear flow, with a localized injection of vorticity as the source \(Appendix[A](https://arxiv.org/html/2610.21049#A1)\)\. The linearized propagator already carries the injected vorticity downstream and tilts it with the shear, so the lift only has to learn how the injected vortex advects itself \(Fig\.[4](https://arxiv.org/html/2610.21049#S5.F4)e\)\. At the top of the training range, missing this self\-advection costs the linear embedding41%41\\%of the measurement and the quadratic correction23%23\\%, against7\.7%7\.7\\%for the lifted functional \(Table[1](https://arxiv.org/html/2610.21049#S5.T1)\)\.

Inference and cost\.The one\-shot inversion of the 60 test sources matches the solver in position \(0\.410\.41cells\) and nearly matches it in intensity \(3\.9%3\.9\\%against2\.8%2\.8\\%\), whereas the linear embedding reaches0\.630\.63cells and7\.2%7\.2\\%\(Table[1](https://arxiv.org/html/2610.21049#S5.T1), Fig\.[4](https://arxiv.org/html/2610.21049#S5.F4)f\)\. Scoring the13,82413\{,\}824candidates takes1212seconds with the lifted functional, against214214for the solver and3131for the quadratic correction\. For separated strong pairs the lift places both sources within one cell in38%38\\%of cases, against13%13\\%for the linear embedding and all pairs for brute force \(Appendix[B\.3](https://arxiv.org/html/2610.21049#A2.SS3)\)\.

Table 2:What the constraints buy \(two\-channel lifts\)\. Only the anchored, centered lift is exact as the amplitude vanishes, in the measurement atIs=10−3I\_\{s\}=10^\{\-3\}\(a ratio\) and in the gradient forIs∈\[0\.01,0\.1\)I\_\{s\}\\in\[0\.01,0\.1\), and on Kolmogorov flow it is the most accurate learned model throughout\. The neural surrogate did not reach the lift’s training loss on reaction–diffusion\. Bold marks the best per column\.reaction–diffusionKolmogorov flowεm\\varepsilon\_\{m\}\(%\)εm\\varepsilon\_\{m\}gradient \(%\)εm\\varepsilon\_\{m\}\(%\)εm\\varepsilon\_\{m\}gradient \(%\)modeltop bandIs=10−3I\_\{s\}\{=\}10^\{\-3\}weaktraintop bandIs=10−3I\_\{s\}\{=\}10^\{\-3\}weaktrainlifted \(ours\)6\.08×10−58\\times 10^\{\-5\}0\.721139×10−59\\times 10^\{\-5\}0\.617unconstrained3\.20\.234\.97\.8140\.671625centering only4\.83×10−23\\times 10^\{\-2\}2\.917146×10−26\\times 10^\{\-2\}1020anchoring only9\.21\.91123228\.21727neural surrogate5112466943491837linear embedding1476×10−46\\times 10^\{\-4\}5\.1205402×10−42\\times 10^\{\-4\}1\.143Ablations\.The constraints make the lift exact at small amplitude, where every unconstrained variant fails, while keeping its finite\-amplitude accuracy competitive \(Table[2](https://arxiv.org/html/2610.21049#S5.T2)\)\. AtIs=10−3I\_\{s\}=10^\{\-3\}on reaction–diffusion the unconstrained lift is off by23%23\\%against0\.008%0\.008\\%in the current study, because its fitted linear part does not reduce to the solver’s\. On Kolmogorov flow ours is the most accurate learned model in every column\. Its measurement gradient is also within0\.7%0\.7\\%of the solver’s for the weakest sources, which no other learned model achieves\.

What the lifted dynamics add\.The lift learns its features through the dynamics, and these features carry the finite\-amplitude response\. On Burgers about a quiescent base, a Carleman lift built on twelve POD modes is five times less accurate than the learned lift \(Appendix[C](https://arxiv.org/html/2610.21049#A3)\)\. Trained on the same simulations, a network of the same architecture fitted to the measurements alone is less accurate in every band on both systems, by up to a factor of twenty, and a network from the source parameters is off by1313–130%130\\%for weak sources and cannot represent two sources \(Appendix[B](https://arxiv.org/html/2610.21049#A2)\)\. Because the propagator is shared, one lift serves any sensor layout and time with a single backward solve, even beyond its training horizon, where at1\.5​T1\.5Tit stays within13%13\\%on reaction–diffusion and17%17\\%on Kolmogorov flow against up to196%196\\%and63%63\\%for the linear embedding \(Appendix[B](https://arxiv.org/html/2610.21049#A2)\)\.

## 6Discussion and outlook

We have proposedSALTO, which performs linear sensitivity analysis in the lifted Koopman space to obtain a reusable nonlinear sensitivity in the original space, exact at small amplitude by construction, and which opens the way to efficient and accurate causal inference\. Open questions include how many lifted channels and which architecture a problem requires, how the lift converges as it grows, and how far the idea extends to chaotic turbulent flows, which are closer to real applications\.

## AI Use Statement

Generative AI tools were used to assist with editing and polishing the manuscript, including improving clarity, grammar, presentation, and refining the visualization and drawing part of the schematics\. All AI\-assisted text was reviewed and verified by the authors, who take full responsibility for the final content of the paper\.

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## Appendix AProblem setups and protocol

This appendix gives everything needed to reproduce the results, namely the three systems, the inference protocol and training, the posterior sampler and the translation\-equivariant blocks\. We chose the systems for the kind of nonlinearity each one presents rather than for their physics, and they appear in the same order as in Section[5](https://arxiv.org/html/2610.21049#S5)\.

### A\.1The three systems

Self\-advection: Burgers about an evolving base flow\.We perturb the viscous Burgers equationut\+u​ux=ν​ux​xu\_\{t\}\+uu\_\{x\}=\\nu u\_\{xx\}on\[0,2​π\)\[0,2\\pi\), withν=0\.1\\nu=0\.1, about the evolving base flow that starts fromub​\(x,0\)=1\+sin⁡3​xu\_\{b\}\(x,0\)=1\+\\sin 3x\. This is the configuration of an earlier linear\-adjoint localization study\([Wang & You, 2026](https://arxiv.org/html/2610.21049#bib.bib26)\)\. The grid hasN=256N=256points, the horizonT=1T=1is divided intoK=50K=50intervals, and the time\-varying propagators𝑳k\\bm\{L\}\_\{k\}are obtained from the solver by central differences\. The source is a Gaussian of unit width at positionxsx\_\{s\}with intensityIsI\_\{s\}, and five point sensors spread across the domain record it\. The encoder is a periodic dilated convolutional network withc=24c=24channels, trained on 800 releases of one to three bumps withIs≤0\.7I\_\{s\}\\leq 0\.7\.

Saturation: two\-dimensional reaction–diffusion\.The dynamics areut=D​∇2u−κ​u−γ​u2\+𝒄⋅∇uu\_\{t\}=D\\nabla^\{2\}u\-\\kappa u\-\\gamma u^\{2\}\+\\bm\{c\}\\cdot\\nabla u, withD=0\.05D=0\.05,κ=0\.2\\kappa=0\.2,γ=1\\gamma=1and𝒄=\(0\.5,0\)\\bm\{c\}=\(0\.5,0\)\. The base state isub=0u\_\{b\}=0, and𝑳\\bm\{L\}is exact in Fourier space\. The quadratic reaction saturates the response, so that atTTa blob of unit intensity departs from the linear response by four fifths, and training usesIs≤2I\_\{s\}\\leq 2\. The problem is posed on the periodic square\[0,2π\)2\[0,2\\pi\)^\{2\}with a48×4848\\times 48grid \(N=2304N=2304\) and a horizonT=1\.5T=1\.5divided intoK=30K=30intervals, and nine point probes on a3×33\\times 3lattice observe it\. The source is a Gaussian with position\(xs,ys\)\(x\_\{s\},y\_\{s\}\), intensityIsI\_\{s\}and widthσs∈\[0\.35,0\.6\]\\sigma\_\{s\}\\in\[0\.35,0\.6\]\. The encoder is a periodic two\-dimensional dilated convolutional network with tanh activations and width 16\. The ablations usec=2c=2channels \(lifted dimensiond=6,912d=6\{,\}912\)\. Table[1](https://arxiv.org/html/2610.21049#S5.T1), Fig\.[4](https://arxiv.org/html/2610.21049#S5.F4)and the reuse study usec=4c=4, and Fig\.[3](https://arxiv.org/html/2610.21049#S5.F3), including both the posterior and the inversion sweep, uses the eight\-channel equivariant lift of Appendix[A\.3](https://arxiv.org/html/2610.21049#A1.SS3)\. The two\-channel and four\-channel lifts are trained on 400 random members of a one\-or\-two\-blob family\. In this family a single blob carries the whole intensity budget or, with probability one half, two blobs share it, so the same lift serves both single\-source and multi\-source inference\.

Sheared advection: two\-dimensional Kolmogorov flow\.The dynamics are the Navier–Stokes equations in vorticity form,ωt\+𝒖⋅∇ω=ν​∇2ω\+f\\omega\_\{t\}\+\\bm\{u\}\\cdot\\nabla\\omega=\\nu\\nabla^\{2\}\\omega\+f, with forcingf=−nfcosnfyf=\-n\_\{f\}\\cos n\_\{f\}y,nf=4n\_\{f\}=4andν=0\.1\\nu=0\.1\. We perturb them about the steady laminar baseωb=f/\(ν​nf2\)\\omega\_\{b\}=f/\(\\nu n\_\{f\}^\{2\}\)on the same periodic square, grid, horizon and3×33\\times 3probe lattice as reaction–diffusion\. This base is weakly unstable at this viscosity, but its fastest\-growing linear mode grows by only0\.2%0\.2\\%over the horizon\. The solver is pseudo\-spectral with2/32/3dealiasing and uses a Strang\-split step with fourth\-order Runge–Kutta substeps, which is second order overall\. We form the propagator𝑳\\bm\{L\}by central differences ofNNbatched perturbations with a step of10−610^\{\-6\}\. Because the base is steady,𝑳\\bm\{L\}is the same at every step, but unlike in reaction–diffusion it is not diagonal in Fourier space\. The source is a Gaussian injection of vorticity with the same parameters as in reaction–diffusion, and the finite\-amplitude effect is the self\-advection of the injected vortex within the shear\. Training usesIs≤4I\_\{s\}\\leq 4on 400 members of the same one\-or\-two\-blob family, and the encoder and channel counts are those of reaction–diffusion\.

### A\.2Inference protocol, training and sampling

Inference protocol\.The training amplitudeImaxI\_\{\\max\}is the largest source intensity in the training family, which is22on reaction–diffusion,44on Kolmogorov flow and0\.70\.7on Burgers\. The measurement error of a band isεm=∑‖𝒎^−𝒎‖/∑‖𝒎‖\\varepsilon\_\{m\}=\\sum\\\|\\hat\{\\bm\{m\}\}\-\\bm\{m\}\\\|/\\sum\\\|\\bm\{m\}\\\|over the sources in that band\. Table[1](https://arxiv.org/html/2610.21049#S5.T1)draws its test intensities from\[0\.1​Imax,Imax\]\[0\.1\\,I\_\{\\max\},I\_\{\\max\}\], whereas the ablation tables draw them from\[0,Imax\]\[0,I\_\{\\max\}\], which is why their medians differ\. Every probe carries independent Gaussian noise whose standard deviation is1%1\\%of the root\-mean\-square measurement over the test draw\. The one\-shot inversion picks the candidate nearest to the observation in measurement space, on a lattice of24×2424\\times 24positions \(every second grid point\) and2424intensities evenly spaced from0\.050\.05toImaxI\_\{\\max\}\. The source width is known and fixed at0\.450\.45, although training draws it from\[0\.35,0\.6\]\[0\.35,0\.6\]\.

Training and hardware\.We train the encoder with Adam at a learning rate of10−310^\{\-3\}on a cosine schedule for 300 iterations, raising the rollout length fromK/3K/3to2​K/32K/3and then toKK\. The ridge parameter in equation[6](https://arxiv.org/html/2610.21049#S4.E6)isλ=10−5\\lambda=10^\{\-5\}relative to the trace of the Gram matrix\. The two\-dimensional lifts were trained on a single RTX 3090 GPU\. The encoder runs in single precision, the Gram products and the ridge solve run in double precision, and the trained encoder is cast to double precision for evaluation\. The scripts in the supplementary code generate every table entry from the saved result files\.

Centering in closed form\.For a tanh network, both corrections in equation[5](https://arxiv.org/html/2610.21049#S4.E5)have closed forms\. The offset𝚽~θ​\(𝟎\)\\widetilde\{\\bm\{\\Phi\}\}\_\{\\theta\}\(\\bm\{0\}\)is obtained by passing the biases through the activations layer by layer, and𝑱0\\bm\{J\}\_\{0\}is the product of the layer weight matrices, each scaled bytanh′\\tanh^\{\\prime\}evaluated at the corresponding activation for zero input\.

Posterior sampling\.We sample the joint posterior of two sources by differential\-evolution Markov chain Monte Carlo with 96 chains\. The chains start from a Levenberg–Marquardt refinement of the best lattice pairs under the same model\. During a burn\-in of 800 steps the likelihood is annealed from2%2\\%of its strength to its full strength, and the following 1600 steps are retained\. Because the two sources are exchangeable, we relabel the chains at every step, so that proposals are not wasted on differences between chains that label the sources in opposite orders\. The prior is uniform over the periodic domain for each position and over\[0,Imax\]\[0,I\_\{\\max\}\]for each intensity\. For the coverage statistics, the two sources of each sample are labelled so as to minimize their summed periodic distance to the true pair\. Two independent sampler seeds agree on the posterior of a test pair to within its standard deviation, and the lifted functional is evaluated with the encoder in single precision\.

### A\.3Translation\-equivariant learned blocks

On a periodic domain with a uniform base state, as in reaction–diffusion, the solver, the anchor𝑳\\bm\{L\}and the convolutional encoder are all translation equivariant\. Dense learned blocks are not, so they spend capacity relearning that a source at one place behaves like the same source elsewhere\. A translation\-equivariant linear map is a convolution, so imposing this symmetry makes each learned block diagonal in wavenumber\. This shrinks the feature block by three orders of magnitude, splits the fit into onec×cc\\times csystem per wavenumber, and makes stability easy to enforce, because capping the spectral radius becomesNNindependentc×cc\\times cproblems\. On reaction–diffusion, whose linearized propagator is a Fourier multiplier, the equivariant blocks reduce the forward error to0\.19%0\.19\\%with four channels and0\.05%0\.05\\%with sixteen, compared with116%116\\%for the linearized model\.

## Appendix BStructural evidence and inference results

This appendix gives the full results for each problem\. It first presents the structural evidence that the constraints and the lift deliver what Section[4](https://arxiv.org/html/2610.21049#S4)promises, and then the inference of one source, of two sources, and with a single lift reused across sensor layouts and horizons\.

### B\.1Structural evidence

This part isolates what the constraints and the lift contribute before any inference is run, namely the small\-amplitude limit, the forward accuracy in each intensity band and the measurement gradient\.

Table 3:The constrained lift is the most accurate model at small amplitude on all three systems, and it infers the intensity nearly as well as brute force\. The columns give the measurement errorεm\\varepsilon\_\{m\}over the training range and atI=10−3I=10^\{\-3\}for our lift, the unconstrained lift and the linear embedding, followed by the median intensity error of the inference\. The two\-dimensional rows use the two\-channel lifts of Tables[5](https://arxiv.org/html/2610.21049#A2.T5)and[6](https://arxiv.org/html/2610.21049#A2.T6)on their own draw, so they differ from the four\-channel values in Table[1](https://arxiv.org/html/2610.21049#S5.T1)\. The unconstrained lift was not trained on Burgers\. Bold marks the best value in each group, excluding brute force\.Summary\.Table[3](https://arxiv.org/html/2610.21049#A2.T3)gathers the three systems\. On each of them, the constrained lift predicts the measurement of a source of intensity10−310^\{\-3\}more accurately than the linear embedding, whereas the unconstrained lift is off by tens of per cent, and its median intensity error in the inversion is within two percentage points of brute force\.

Burgers\.Table[4](https://arxiv.org/html/2610.21049#A2.T4)and Fig\.[5](https://arxiv.org/html/2610.21049#A2.F5)give the results in each band on Burgers for a separate draw of 60 sources\. The lifted model keeps the forward error atTTbelow2%2\\%, whereas the error of the linearized solver reaches four fifths of the response\. Its one\-shot inversion matches brute force through the solver in intensity in every band and is at least as accurate in position\. Figure[5](https://arxiv.org/html/2610.21049#A2.F5)f–g shows how the results depend on the number of channels\.

Table 4:On Burgers the lifted model matches brute force in every band, while the errors of the linear embedding grow with the intensity\. Top, relative error atTTof the forward models\. Bottom, median position error \(in length units, with grid cell0\.0250\.025\) and relative intensity error of the one\-shot inversion in each intensity band \(60 sources,1%1\\%noise\), on a draw separate from the 800 sources of Table[1](https://arxiv.org/html/2610.21049#S5.T1)\.
![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/burgers_appendix.png)Figure 5:Burgers about the base flow1\+sin⁡3​x1\+\\sin 3x\. The lifted model follows the solver where the linearized solver does not\. Its forward error falls steadily as the number of learned channelsccgrows, whereas its inversion error stops improving at four channels\.\(a–c\)The perturbation released atxs=3x\_\{s\}=3with the three test intensities, att=T/3t=T/3\(light grey\) and atTT\(black\)\. The solver is the broad grey band, the lifted model the solid line and the linearized solver the dotted line\.\(d\)Relative error over time at the three test intensities for the lifted model \(solid\) and the linearized solver \(dotted\), labelled byIsI\_\{s\}\.\(e\)Mean validation error over time for the linearized solver and for the lift before and after training, on the same vertical axis as \(d\)\.\(f\)Forward error atTTagainstccfor four test intensities, labelled byIsI\_\{s\}, together with the mean validation error \(open circles\)\.\(g\)Median inversion error in positionδx\\delta\_\{x\}\(black\) and intensityδI\\delta\_\{I\}\(red\) againstcc\. The dotted line marked “linear” is the linear embedding, whose two errors differ by less than one per cent and are drawn as one line, and the fine dotted line marks one grid cell\.Ablation of the constraints\.Tables[5](https://arxiv.org/html/2610.21049#A2.T5)and[6](https://arxiv.org/html/2610.21049#A2.T6)remove the constraints one at a time at two channels, the size at which every variant was trained\. Only the full lift, which is both anchored and centered, has a measurement error below10−410^\{\-4\}atIs=10−3I\_\{s\}=10^\{\-3\}on both systems, which matches or improves on the accuracy of the linear adjoint\. Centering alone leaves an error of a few per cent at that amplitude, and anchoring alone leaves an error larger than the measurement itself\. At finite amplitude the full lift is the most accurate model in every band on Kolmogorov flow\. On reaction–diffusion the unconstrained lift fits the two strongest bands about twice as closely, and the constraints trade this for the exact small\-amplitude limit\. Figures[6](https://arxiv.org/html/2610.21049#A2.F6)and[7](https://arxiv.org/html/2610.21049#A2.F7)show the same comparison as curves, together with one probe followed over twice the training horizon\.

Table 5:On reaction–diffusion only the full lift reduces to the linear adjoint at small amplitude, while the unconstrained lift fits the strongest bands more closely\. The columns give the measurement error in each intensity band, with the last band beyond the training amplitudes, and atIs=10−3I\_\{s\}=10^\{\-3\}, followed by the median position error in length units \(candidate lattice0\.260\.26\) and the relative intensity error, over 60 sources with1%1\\%noise and two channels\. The neural surrogate did not converge on this problem and is reported as trained\. Bold marks the best value in each column, excluding brute force\.Table 6:On Kolmogorov flow the full lift is the most accurate learned model in every band and at small amplitude\. Columns as in Table[5](https://arxiv.org/html/2610.21049#A2.T5)\.![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/rd_curves.png)Figure 6:On reaction–diffusion the measurement error of the full lift falls in proportion to the source intensity, whereas that of the unconstrained lift levels off at small amplitude\.\(a\)Measurement error in each intensity band for the six models\.\(b\)Measurement error against the source intensity\.\(c\)The measurement of one probe over twice the training horizon for a source of intensity1\.51\.5, from the solver, the lifted functional, the linear embedding and the unconstrained lift\.![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/kolmo_curves.png)Figure 7:On Kolmogorov flow the full lift is the most accurate model in every intensity band, and its measurement error falls with the source intensity\. Panels as in Fig\.[6](https://arxiv.org/html/2610.21049#A2.F6), with an injection of intensity33in\(c\)\.Channel count\.Table[7](https://arxiv.org/html/2610.21049#A2.T7)varies the number of channels of the full lift\. Going from two to four channels reduces the measurement error in the top band from6\.0%6\.0\\%to1\.6%1\.6\\%on reaction–diffusion and from12\.6%12\.6\\%to7\.6%7\.6\\%on Kolmogorov flow, while keeping the small\-amplitude limit and improving the inference\. This is why Table[1](https://arxiv.org/html/2610.21049#S5.T1)and the inference studies use four channels\.

Table 7:With four channels the anchored, centered lift is the most accurate of the three sizes on both two\-dimensional systems\. The columns give the feature dimensiondΦ=c​Nd\_\{\\Phi\}=cN, the number of training iterations, the best validation rollout MSE, the measurement error in the top training band and beyond it, the error atIs=10−3I\_\{s\}=10^\{\-3\}, and the median inference errors\. Every row uses the same schedule and training pairs, although the cost of the fit grows withcc\. We varycconly for the anchored lift, because the full\-propagator fits of the other variants exceed the GPU memory atc=4c=4\. Bold marks the best value in each column\.val\.εm\\varepsilon\_\{m\}\(%\)εm\\varepsilon\_\{m\}\(ratio\) atinferenceccdΦd\_\{\\Phi\}itersMSEtop bandbeyondIs=10−3I\_\{s\}=10^\{\-3\}δx\\delta\_\{x\}δI\\delta\_\{I\}\(%\)*reaction–diffusion*12,3043001\.7×10−31\.7\\times 10^\{\-3\}11\.49\.09\.3×10−59\.3\\times 10^\{\-5\}0\.19213\.724,6083007\.8×10−47\.8\\times 10^\{\-4\}6\.013\.98\.4×10−58\.4\\times 10^\{\-5\}0\.14512\.649,2163007\.2×𝟏𝟎−𝟓7\.2\\times 10^\{\-5\}1\.64\.08\.3×𝟏𝟎−𝟔8\.3\\times 10^\{\-6\}0\.1138\.7*Kolmogorov flow*12,3043005\.5×10−35\.5\\times 10^\{\-3\}14\.425\.42\.9×10−42\.9\\times 10^\{\-4\}0\.1064\.324,6083002\.9×10−32\.9\\times 10^\{\-3\}12\.620\.79\.0×10−59\.0\\times 10^\{\-5\}0\.1113\.849,2163001\.3×𝟏𝟎−𝟑1\.3\\times 10^\{\-3\}7\.616\.26\.7×𝟏𝟎−𝟓6\.7\\times 10^\{\-5\}0\.1063\.7Measurement gradient\.Table[8](https://arxiv.org/html/2610.21049#A2.T8)compares the gradient of the predicted measurements with respect to the source parameters against the solver’s\. For the weakest sources the constrained lift gives the most accurate gradient among the learned models on both systems, as the constraints require\. At finite amplitude the ranking follows the forward errors of Tables[5](https://arxiv.org/html/2610.21049#A2.T5)and[6](https://arxiv.org/html/2610.21049#A2.T6), with the unconstrained lift more accurate on reaction–diffusion and the constrained lift more accurate on Kolmogorov flow\.

Table 8:For the weakest sources the constrained lift gives the most accurate measurement gradient among the learned models\. We difference every model and the solver in the same way, centrally, with steps of10−310^\{\-3\}in position and10−310^\{\-3\}\(relative\) in intensity, and reducing the steps threefold changes no entry\. Each entry is the median, over twelve sources in the band, of‖𝑱−𝑱solver‖F/‖𝑱solver‖F\\\|\\bm\{J\}\-\\bm\{J\}\_\{\\mathrm\{solver\}\}\\\|\_\{F\}/\\\|\\bm\{J\}\_\{\\mathrm\{solver\}\}\\\|\_\{F\}in per cent, where𝑱\\bm\{J\}is the Jacobian of the nine probe measurements with respect to\(xs,ys,Is\)\(x\_\{s\},y\_\{s\},I\_\{s\}\), made dimensionless by the source width and intensity\. The models are those of Tables[5](https://arxiv.org/html/2610.21049#A2.T5)and[6](https://arxiv.org/html/2610.21049#A2.T6), and the last band of each system lies beyond the training range\. Bold marks the most accurate model in each band\.
### B\.2Single\-source inference

This part shows how the lifted functional locates a single source on the two\-dimensional systems, and how its estimates compare with brute force through the solver\.

![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/sens_maps.png)Figure 8:The lifted functional reproduces the finite\-amplitude sensitivity map of the centre probe at the cost of one inner product per source position, whereas the linear embedding misses both its amplitude and its shape\. Reaction–diffusion is shown in\(a–e\)and Kolmogorov flow in\(f–j\), atIs=0\.75​ImaxI\_\{s\}=0\.75\\,I\_\{\\max\}with the four\-channel lifts\. Each row shows the exact map, which costs one nonlinear solve per source position, the lifted functional and the linear embedding, followed by the difference of each approximation from the exact map\. Every panel in a row is divided by the largest value of that row’s exact map, and all ten panels share one colour scale, so the differences read directly as fractions of the exact map\. Values beyond±1\\pm 1saturate, and the crosses mark the probes\.Sensitivity maps\.Figure[8](https://arxiv.org/html/2610.21049#A2.F8)shows the object that the inference searches, namely the measurement of one probe as a function of the source position at a fixed finite intensity\. The exact map costs one nonlinear solve per position, whereas the lifted functional evaluates it with one inner product per position\.

Misfit landscapes and uncertainty\.Figure[9](https://arxiv.org/html/2610.21049#A2.F9)shows the misfit over the candidate lattice for one test source, and the position error of all 60 test sources against their intensity\. Table[9](https://arxiv.org/html/2610.21049#A2.T9)gives95%95\\%bootstrap intervals for the medians in Table[1](https://arxiv.org/html/2610.21049#S5.T1)\. The paired interval for the difference between the lifted model and the solver contains zero for position on all three systems, and for intensity on Burgers and reaction–diffusion\. On Kolmogorov flow the lifted model is less accurate in intensity by0\.20\.2to2\.12\.1percentage points\.

Table 9:The lifted inference is statistically indistinguishable from the solver in position on all three systems\. Medians of Table[1](https://arxiv.org/html/2610.21049#S5.T1)with95%95\\%bootstrap intervals over the test sources \(10,00010\{,\}000resamples\)\.![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/inference2d.png)Figure 9:The misfit landscape of the lifted functional has its minimum at the true source, as for brute force through the solver, whereas the minimum of the linear embedding is displaced on Kolmogorov flow\. The panels show single\-source inference with the four\-channel lifts, for reaction–diffusion in\(a–d\)and Kolmogorov flow in\(e–h\)\. The first three panels of each row show the misfit over the candidate lattice for the lifted functional, the linear embedding and brute force through the solver\. At each position the misfit is minimized over the intensity, normalized by its own minimum and drawn on one logarithmic scale, on which red indicates a good fit\. The cross marks the true source and the circle the estimate\. Panels \(d\) and \(h\) show the position error of the 60 test sources against their intensity, with the candidate\-lattice spacing dotted\.
### B\.3Two sources

This part reports the two\-source study for every pair family on both two\-dimensional systems\. We use three families\. In the first, the two sources are more than4​σs4\\sigma\_\{s\}apart with0\.350\.35–0\.5​Imax0\.5\\,I\_\{\\max\}each\. In the second, strong family they are also separated but carry0\.50\.5–1\.0​Imax1\.0\\,I\_\{\\max\}each, so the combined field reaches up to twice the training budget\. In the third they overlap, at11–3​σs3\\sigma\_\{s\}apart\. Each family has 30 pairs on reaction–diffusion and 24 on Kolmogorov flow, read by a4×44\\times 4probe lattice with1%1\\%noise\. Every model scores all pairs of single\-source candidates under superposition, and the best lattice pairs are then refined by L\-BFGS on the model’s own misfit, built from the duality relation equation[3](https://arxiv.org/html/2610.21049#S3.E3)\. The study uses the four\-channel dense lifts of Table[7](https://arxiv.org/html/2610.21049#A2.T7)rather than the eight\-channel equivariant lift of Section[5\.2](https://arxiv.org/html/2610.21049#S5.SS2), which makes the comparison against the linear embedding a conservative one\.

Table[10](https://arxiv.org/html/2610.21049#A2.T10)collects the results for all families\. On reaction–diffusion the lifted functional places both sources within one lattice cell in8080–100%100\\%of pairs, against3333–97%97\\%for the linear embedding\. It infers the intensities to within10%10\\%, where the linear embedding misses them by3434–49%49\\%, and it runs 18 to 26 times faster than brute force\. On Kolmogorov flow the two sources interact strongly, changing the measurement by4343–48%48\\%, and the lifted functional still resolves both sources more often than the linear embedding in every family\. Figure[10](https://arxiv.org/html/2610.21049#A2.F10)shows one strong separated pair on reaction–diffusion, with the forward problem on top and the inverse problem below\. Figure[11](https://arxiv.org/html/2610.21049#A2.F11)follows the same pair past the training horizon to2​T2T, and Fig\.[4](https://arxiv.org/html/2610.21049#S5.F4)a–d in the main text shows the most strongly interacting pair on Kolmogorov flow\.

Table 10:The lifted functional resolves two sources more often and more accurately than the linear embedding in every pair family\. Interaction is the median departure of the exact response from the sum of the single\-source responses\. For each model the table gives the median per\-source position errorδx\\delta\_\{x\}\(lattice spacing0\.260\.26\), the median intensity errorδI\\delta\_\{I\}and the fraction of pairs with both sources within one lattice cell, followed by the ratio of brute\-force to lifted inference time\. Bold marks the better of the lifted functional and the linear embedding\.![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/twosource_rd_strong.png)Figure 10:The lifted functional recovers both sources of a separated pair on reaction–diffusion, together with their intensities, as accurately as brute force, whereas the linear embedding underestimates every intensity\. The pairs carry0\.50\.5–1\.0​Imax1\.0\\,I\_\{\\max\}per source, up to twice the training budget in total, and the frame is shifted to centre the pair shown\.\(a–d\)The forward problem\. Panel \(a\) shows the initial field with the4×44\\times 4probes as crosses, \(b\) the exact field atTT, and \(c\) the superposition of the linearized responses on the same colour scale as \(b\)\. Panel \(d\) shows the sixteen measurements, from the solver as bars and with1%1\\%noise as dashes, together with the predictions of the lifted functional and of the linear embedding at the true parameters\.\(e–h\)The inverse problem\. Panels \(e\) and \(f\) show the misfit of the second source over the domain at its fitted intensity, given the fitted first source, for the lifted functional and for the linear embedding, with crosses at the true sources and circles at the estimates\. Panel \(g\) shows the estimated against the true intensity of every source over all pairs, and \(h\) the cumulative distribution of the per\-source position error, with the candidate\-lattice spacing dotted\.![Refer to caption](https://arxiv.org/html/2610.21049v1/figures/spacetime_rd_strong.png)Figure 11:Beyond the training horizon, the lifted rollout reproduces the decay of the strong separated pair of Fig\.[10](https://arxiv.org/html/2610.21049#A2.F10), whereas the linear superposition stays twice as strong\. The panels show isosurfaces ofu⁡\(x,y,t\)u\(x,y,t\)at25%25\\%of the initial maximum, with time running vertically to2​T2T, the translucent plane at the training horizonTTand the probes marked by crosses, for\(a\)the solver,\(b\)the lifted rollout and\(c\)the superposition of the linearized responses\. Over the whole space–time field the lifted rollout is off by3\.5%3\.5\\%and the linear superposition by81%81\\%\.
### B\.4Reuse across sensor layouts and horizons

This part tests the claim that one trained lift serves any sensor layout and measurement time with a single backward solve and no new data\. Table[11](https://arxiv.org/html/2610.21049#A2.T11)compares the four\-channel lift with models trained directly on the same simulations, using the paper’s3×33\\times 3probes, two held\-out layouts and a held\-out horizon\. The lift is more accurate than the direct CNN in every column on both systems\. The parameter MLP is off by1313–130%130\\%for the weakest sources and cannot represent pairs\. The direct readout, a ridge regression on the lift’s own features that is refitted for each layout and horizon, stays close to the lift throughout\. This shows that the features the lift learns through the dynamics carry the finite\-amplitude response\.

Table 11:One lift serves new sensor layouts and horizons as accurately as models refitted for each of them\. The table gives the relative measurement error \(%\) of the four\-channel lift and of models trained directly on the same simulations, for single sources and for overlapping pairs\. The direct readout and the direct CNN both keep the exact linear term⟨𝒒L†,S⟩\\langle\\bm\{q\}^\{\\dagger\}\_\{\\mathrm\{L\}\},S\\rangleand add a linear readout of centered features, so they are anchored and centered like the lift\. The readout is a ridge regression on the lift’s own features, fitted to the stored training fields at each layout and horizon\. The direct CNN, which has the encoder’s architecture but is trained on measurements alone, and the parameter MLP must be retrained for each layout\.Horizons beyond training and scarce data\.The lift was trained on trajectories that end atTT\. When we iterate its propagator to1\.5​T1\.5T, its measurement error stays within13%13\\%on reaction–diffusion and17%17\\%on Kolmogorov flow in every intensity band, against up to196%196\\%and63%63\\%for the linear embedding, whereas a direct readout has no data at that horizon\. With 50 instead of 400 training trajectories, the lift is up to four times more accurate than a readout fitted to the same trajectories on reaction–diffusion \(Table[12](https://arxiv.org/html/2610.21049#A2.T12)\)\. Anchoring and centering alone do not explain this accuracy\. An anchored and centered CNN with the lift’s architecture, trained end to end on the same trajectories but without the propagator, is55to1616times less accurate than the lift on reaction–diffusion, and less accurate in every band on Kolmogorov flow, with 400 trajectories\.

Table 12:The lift predicts beyond its training horizon, where the linear embedding fails\. The table gives the relative measurement error \(%\) on the3×33\\times 3probes in the four intensity bands, at the training horizonTTand at1\.5​T1\.5T\. It compares the four\-channel lift trained on 400 and on 50 trajectories, a ridge readout fitted on each lift’s own features and trajectories, an anchored and centered CNN of the lift’s architecture trained end to end on the same trajectories, and the linear embedding\. The readout and the CNN have no training data beyondTT\.reaction–diffusionKolmogorov flowTT1\.5​T1\.5TTT1\.5​T1\.5Tband1234123412341234lifted, 400 traj\.0\.20\.31\.54\.20\.30\.84\.7131\.21\.66\.0121\.42\.26\.617readout, 400 traj\.0\.50\.72\.35\.2––––1\.12\.14\.610––––anchored CNN, 400 traj\.3\.43\.31267––––2\.34\.09\.114––––lifted, 50 traj\.3\.98\.832846\.215561402\.85\.517383\.16\.52041readout, 50 traj\.16315296––––2\.14\.31528––––anchored CNN, 50 traj\.183967161––––1\.94\.81839––––linear embedding277010616931861341963\.27\.532573\.58\.33563

## Appendix CBurgers about a quiescent base: where the constraints came from

This appendix reports a Burgers study about a quiescent base state,ub=0u\_\{b\}=0, in which the field is itself the perturbation and𝑳\\bm\{L\}is the exact heat propagator over one interval\. It is not one of the three systems of Section[5](https://arxiv.org/html/2610.21049#S5)\. We keep it because the structural constraints were developed in this configuration, and because it contains the comparison with the unconstrained lift and with fitted dictionaries that motivates them\. The horizon isT=1\.5T=1\.5inK=75K=75intervals, and the source width isσ=0\.5\\sigma=0\.5\. The test intensities areIs∈\{0\.5,1\.5,3\}I\_\{s\}\\in\\\{0\.5,1\.5,3\\\}atxs=2x\_\{s\}=2, and training uses 800 sums of one to three Gaussians with amplitudes in\[0,2\]\[0,2\], so the strongest test lies beyond the training range\. Every number below is read from the saved outputs listed in the results manifest of the repository\.

#### Forward accuracy does not certify a sensitivity\.

At equal size, with eight channels and 300 iterations, the unconstrained Koopman autoencoder and the anchored, centered lift are indistinguishable on the validation set, both with a mean rollout error of1\.3%1\.3\\%\. A Carleman lift fitted on twelve POD modes reaches6\.4%6\.4\\%, and a local polynomial dictionary\[u,u2,u3\]\[u,u^\{2\},u^\{3\}\]reaches6\.8%6\.8\\%, while the physics\-built Carleman generator diverges onceIs≥1\.5I\_\{s\}\\geq 1\.5\. The two neural lifts nevertheless differ in what they transport backward, and the difference shows up in that object alone, without reference to any adjoint taken at the perturbed state\. When a sensor kernel of width0\.250\.25atxm=4x\_\{m\}=4is propagated backward over the horizon through the transposed propagator, the unconstrained lift amplifies it by a factor of217217in norm, and its state channel alone by6\.16\.1, whereas the heat adjoint, to which that channel should reduce, decays\. The anchored, centered lift keeps the same channel at0\.640\.64, which by \(C2\) and \(C3\) is exactly the linear adjoint\. The backward field of the unconstrained model also carries oscillations at wavenumbers that the training data never excite, which is the signature of directions of the feature Gram matrix that the data leave unconstrained\.

#### Small\-amplitude limit\.

The observables of the unconstrained lift do not vanish at the zero field, where their norm is of order𝒪⁡\(10\)\\mathcal\{O\}\(10\)in the trained models, and its fitted propagator cancels this offset only up to a bias of10−510^\{\-5\}per step\. That is accurate relative to the training amplitudes but not below them\. Its Jacobian at the origin matches the heat propagator to0\.1%0\.1\\%at the lowest wavenumbers but deviates from aboutk=8k=8onward, with spurious, slowly decaying gains neark=16k=16to3232\. The result is a floor in the rollout error, which stays between4%4\\%and53%53\\%as the amplitude falls through three decades instead of falling with it\. For the anchored, centered lift the bias is at double\-precision machine epsilon, the Jacobian equals the heat propagator to the same tolerance at every wavenumber, the nonlinear remainder scales asIs2\.02I\_\{s\}^\{2\.02\}, and the rollout error decreases without a floor\. Centering without anchoring removes the offset but not the floor, which stays at3%3\\%at all small amplitudes, because its multi\-step linearization is fitted rather than exact\.