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arxiv: 2604.19465 · v2 · submitted 2026-04-21 · ⚛️ physics.flu-dyn · cs.AI

Recognition: unknown

A neural operator framework for data-driven discovery of stability and receptivity in physical systems

Chengyun Wang , Liwei Chen , Nils Thuerey
Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:23 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.AI
keywords neural operatordata-driven stability analysisresolvent analysiseigenmodesfluid dynamicsautomatic differentiationdynamical systemsnonlinear systems
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The pith

A neural network trained on data alone can compute eigenmodes and resolvent modes via automatic differentiation on its Jacobian.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a data-driven method to extract stability properties and optimal forcing responses from observational data without any governing equations. A neural network is trained to emulate the system's time evolution; automatic differentiation then supplies the Jacobian, from which eigenmodes and resolvent modes are obtained directly. This matters for nonlinear or high-dimensional systems such as chaotic models and fluid flows where equations are incomplete or unavailable, allowing identification of dominant instabilities and input-output structures even in strongly nonlinear regimes. The same emulator also supplies a nonlinear representation of the dynamics alongside the linear modal information.

Core claim

By training a neural network as a dynamics emulator and using automatic differentiation to extract its Jacobian, eigenmodes and resolvent modes can be computed directly from data. The method identifies dominant instability modes and input-output structures on canonical chaotic models and high-dimensional fluid flows, even in strongly nonlinear regimes, while also providing a nonlinear representation of system dynamics.

What carries the argument

A neural network trained as a dynamics emulator, from whose output the Jacobian is obtained by automatic differentiation to furnish eigenmodes and resolvent modes.

If this is right

  • Stability and receptivity analysis becomes feasible for systems whose governing equations are unknown or incomplete.
  • The same trained emulator supplies both nonlinear dynamics and linear modal information.
  • Dominant modes can be recovered in strongly nonlinear and high-dimensional regimes where classical linearization is difficult.
  • The approach applies to observational datasets in climate science, neuroscience, and fluid engineering without requiring model derivation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be tested on experimental time-series data where no full model exists, to check whether predicted modes align with observed perturbation growth.
  • Embedding the emulator in a control loop would allow real-time adjustment of forcings based on the extracted resolvent information.
  • Accuracy in recovering known modes from data with added noise would quantify the method's robustness for practical measurements.

Load-bearing premise

The trained neural network must approximate the true underlying dynamics closely enough that derivatives of its outputs with respect to inputs recover meaningful stability and receptivity information about the physical system.

What would settle it

In a system with known equations, such as the Lorenz attractor or a simple shear flow, the eigenmodes or resolvent modes extracted from the neural Jacobian would fail to match those computed analytically or numerically from the equations.

Figures

Figures reproduced from arXiv: 2604.19465 by Chengyun Wang, Liwei Chen, Nils Thuerey.

Figure 1
Figure 1. Figure 1: Schematic of our data-driven algorithm. (1) Time-resolved state snapshots col￾lected from a dynamical system are used. (2) An NN emulator is trained using a rollout strategy to accurately learn the system’s evolution map. (3) The Jacobian of the trained emulator is then extracted via automatic differentiation and used to approximate the local Jacobian. This data-driven operator is the key ingredient for (4… view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of the true Lorenz system (red solid) compared with the NN emulator prediction (green dotted) from an unseen initial point (blue circle). (a) Trajectories in phase space; (b) Component-wise time evolution series. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the real part of the Jacobians between the NN-based operator and the operator-based ground truth. The first and second rows correspond to the linear and nonlinear scenarios, respectively. We aim to demonstrate the applicability of the NN emulator approach to both linear and nonlinear regimes even in complex number space. For this purpose, we consider two datasets derived respectively from the… view at source ↗
Figure 4
Figure 4. Figure 4: Linear stability analysis results of the linear and nonlinear Ginzburg-Landau sys￾tems. (a)(b) Eigenspectra obtained from the NN-based Jacobian (◦), DMD-based Jacobian (+) and operator-based ground truth (•). The dashed line denotes the stability boundary in the complex plane. (c)(d) The leading three eigenmodes of the NN-based Jacobians in both scenarios, where solid and dashed lines show the real part an… view at source ↗
Figure 5
Figure 5. Figure 5: Resolvent analysis results of the complex Ginzburg–Landau systems. The first and the second row correspond to the linear and nonlinear scenarios, respectively. (a)(c) Resolvent gain distribution for the first three modes with respect to the frequency. (b)(d) The first three forcing and response modes at the peak gain frequency. The thick gray lines in the background show operator-based ground-truth for com… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the eigenspectra obtained from the NN-based Jacobian (◦), DMD￾based Jacobian (+) and operator-based ground truth (•) for the 2D channel flow system. The dashed line denotes the stability boundary in the complex plane. (a) Weakly nonlinear dataset. (b) Nonlinear dataset. 1st A mode Operator-based Weakly nonlinear scenario Nonlinear scenario 1st P mode [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the first-order A and P eigenmodes of the streamwise velocity field for the 2D channel flow system. The first column is the reference results obtained from the OS operator. The second and third columns are the NN-based Jacobian obtained from weakly nonlinear and nonlinear datasets. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Resolvent analysis results of the 2D channel flow system. (a) Resolvent gain curves of the analytical operator and the NN-based resolvent operators derived from weakly nonlinear and nonlinear datasets. (b) The first-order forcing and response modes of the streamwise velocity field at the ω = 0.31. identify the two most dominant perturbation structures near the wall and centerline, i.e., the least-damped A … view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the (a) eigenspectra and (b) resolvent gain curves from the operator￾based Jacobian and NN-based Jacobian for the 3D channel flow system [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the first-order A and P eigenmodes of the streamwise velocity field for the 3D channel flow system. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the first-order forcing and response modes of the streamwise velocity field at the ω = 0.38 for the reduced 3D channel flow system. where ϕ(y) is the profile function introduced previously, and ϵ controls the magnitude of the initial perturbation. In this three-dimensional case, we only generate a strongly nonlinear dataset with ϵ = 10−1 . The original three-dimensional data correspond to a … view at source ↗
read the original abstract

Understanding how complex systems respond to perturbations, such as whether they will remain stable or what their most sensitive patterns are, is a fundamental challenge across science and engineering. Traditional stability and receptivity (resolvent) analyses are powerful but rely on known equations and linearization, limiting their use in nonlinear or poorly modeled systems. Here, we introduce a data-driven framework that automatically identifies stability properties and optimal forcing responses from observation data alone, without requiring governing equations. By training a neural network as a dynamics emulator and using automatic differentiation to extract its Jacobian, we can compute eigenmodes and resolvent modes directly from data. We demonstrate the method on both canonical chaotic models and high-dimensional fluid flows, successfully identifying dominant instability modes and input-output structures even in strongly nonlinear regimes. By leveraging a neural network-based emulator, we readily obtain a nonlinear representation of system dynamics while additionally retrieving intricate dynamical patterns that were previously difficult to resolve. This equation-free methodology establishes a broadly applicable tool for analyzing complex, high-dimensional datasets, with immediate relevance to grand challenges in fields such as climate science, neuroscience, and fluid engineering.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a data-driven framework for discovering stability and receptivity properties in physical systems. It trains a neural network to emulate the system's dynamics from observational data and employs automatic differentiation to extract the Jacobian matrix, enabling computation of eigenmodes for stability analysis and resolvent modes for receptivity analysis without requiring explicit governing equations. Demonstrations are provided on canonical chaotic models and high-dimensional fluid flows, showing identification of dominant modes even in nonlinear regimes.

Significance. If the extracted modes prove accurate, the method would offer a broadly applicable equation-free tool for stability and receptivity analysis in complex, high-dimensional systems where governing equations are unavailable, with relevance to fluid dynamics, climate science, and neuroscience. The combination of neural emulation for nonlinear dynamics with automatic differentiation for linear operators is a notable strength, as is the demonstration on both low-dimensional chaotic systems and fluid data.

major comments (3)
  1. Demonstrations section: The reported results on canonical models and fluid flows show only qualitative agreement with expected modes; no quantitative metrics are supplied, such as eigenvalue errors relative to analytical or linearized-operator references, eigenvalue perturbation under retraining, or comparison of resolvent gains to known values. This leaves the central claim that the NN Jacobian yields meaningful physical stability/receptivity information unverified.
  2. Method section (neural dynamics emulator and Jacobian extraction): State-prediction loss (e.g., trajectory MSE) is used to train the emulator, but no analysis, bounds, or regularization is provided to ensure that the automatically differentiated Jacobian approximates the true linearized vector field DF sufficiently closely; small state errors can produce large derivative discrepancies, especially in chaotic or high-dimensional regimes.
  3. Fluid-flow demonstrations: No direct validation is performed against the linearized Navier-Stokes operator or known resolvent spectra for the same base flows, so it is unclear whether the data-driven modes recover the physical input-output structures rather than artifacts of the emulator.
minor comments (2)
  1. The notation for the neural operator and its embedding of the dynamics emulator could be made more explicit with additional equations relating the network output to the discrete-time map.
  2. Figure captions and legends in the results section would benefit from clearer indication of which curves or fields correspond to the neural-operator modes versus reference solutions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive review of our manuscript. We address each major comment below and indicate the revisions we will implement to improve the clarity and rigor of the work.

read point-by-point responses

  1. Referee: Demonstrations section: The reported results on canonical models and fluid flows show only qualitative agreement with expected modes; no quantitative metrics are supplied, such as eigenvalue errors relative to analytical or linearized-operator references, eigenvalue perturbation under retraining, or comparison of resolvent gains to known values. This leaves the central claim that the NN Jacobian yields meaningful physical stability/receptivity information unverified.

    Authors: We agree that quantitative validation would strengthen the central claims. In the revised manuscript we will add explicit metrics for the canonical models, including eigenvalue errors against analytical references and sensitivity of the extracted eigenvalues to retraining with different random initializations. For the fluid cases we will report comparisons of resolvent gains against literature values where such references exist. revision: yes

  2. Referee: Method section (neural dynamics emulator and Jacobian extraction): State-prediction loss (e.g., trajectory MSE) is used to train the emulator, but no analysis, bounds, or regularization is provided to ensure that the automatically differentiated Jacobian approximates the true linearized vector field DF sufficiently closely; small state errors can produce large derivative discrepancies, especially in chaotic or high-dimensional regimes.

    Authors: The referee correctly identifies a gap between the training objective and the accuracy of the extracted Jacobian. We will revise the methods section to include empirical checks that compare the automatically differentiated Jacobian against finite-difference approximations evaluated on the trained network. We will also add a brief discussion of the limitations that may arise in strongly chaotic regimes. Adding a Jacobian-regularization term during training is feasible and will be explored; however, deriving rigorous a-priori bounds on the derivative error without further assumptions on the data distribution remains outside the scope of the present framework. revision: partial

  3. Referee: Fluid-flow demonstrations: No direct validation is performed against the linearized Navier-Stokes operator or known resolvent spectra for the same base flows, so it is unclear whether the data-driven modes recover the physical input-output structures rather than artifacts of the emulator.

    Authors: The method is intended for equation-free settings where the linearized operator is unavailable by construction. Nevertheless, we will strengthen the fluid-flow section by comparing the extracted modes against well-documented resolvent and stability results from the literature for the specific base flows examined. Where computational resources permit, we will also generate reference resolvent spectra from the linearized Navier-Stokes equations for at least one canonical case to enable direct side-by-side validation. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper trains a neural network emulator on observed trajectory data to approximate system dynamics, then applies automatic differentiation to obtain the Jacobian of this emulator and computes its eigenmodes or resolvent modes via standard linear algebra. This chain does not reduce any claimed result to a fitted parameter by construction, nor does it rely on self-definitional mappings, self-citation load-bearing premises, or imported uniqueness theorems. The central output (data-driven modes) is obtained by post-processing the trained emulator rather than being statistically forced by the training loss itself. The method is self-contained against external benchmarks once the emulator is trained, with no evidence of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on the neural network learning an accurate dynamics emulator from data and the Jacobian of that emulator providing valid linear stability information even for nonlinear systems.

axioms (1)
  • domain assumption A neural network can be trained to emulate the dynamics of a physical system from observation data alone.
    This underpins the entire pipeline for Jacobian extraction and mode computation.
invented entities (1)
  • Neural dynamics emulator no independent evidence
    purpose: To serve as a differentiable surrogate for system evolution without explicit governing equations
    Core component introduced to enable data-driven Jacobian-based analysis.

pith-pipeline@v0.9.0 · 5497 in / 1275 out tokens · 49924 ms · 2026-05-10T01:23:00.374147+00:00 · methodology

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