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arxiv: 2605.24868 · v1 · pith:7BS5MHHPnew · submitted 2026-05-24 · 💻 cs.LG · nlin.CD· physics.comp-ph

A comparative study of accuracy and rollout stability of temporal surrogate models

Rajarshi Biswas This is my paper

Pith reviewed 2026-06-30 12:29 UTC · model grok-4.3

classification 💻 cs.LG nlin.CDphysics.comp-ph
keywords temporal surrogate modelsrollout stabilitychaotic dynamical systemsintegrator-like updateslong-horizon predictionneural network architecturesdouble pendulumKuramoto-Sivashinsky
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The pith

Models with integrator-like updates achieve more stable and accurate long-horizon predictions of chaotic systems.

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

The paper compares several neural network architectures for building temporal surrogate models that forecast chaotic dynamical systems. Using matched model capacities and a shared training protocol across the double pendulum, Kuramoto-Sivashinsky equations, and Kolmogorov flow, it isolates the effect of architecture on rollout behavior. Architectures whose updates resemble numerical integrators display lower one-step bias and weaker amplification of perturbations. This leads to slower error growth during long rollouts and closer reproduction of the true system's attractor. A sympathetic reader would care because such models could replace expensive direct simulations for long times without drifting off the correct dynamics.

Core claim

The central claim is that models having integrator-like updates show lower bias and perturbation amplification yielding stable long-horizon rollout and more accurate predictions. This holds both when all models use the same training protocol and when each is individually optimized. The conclusion rests on metrics including local Jacobian norms, relative one-step bias, finite-time Lyapunov growth rates, and attractor geometry comparisons.

What carries the argument

The integrator-like update rule, which advances the state in a manner analogous to a numerical time-stepping scheme rather than a direct residual or recurrent mapping.

If this is right

  • Integrator-like models exhibit lower relative one-step bias than other architectures under matched conditions.
  • These models show reduced finite-time Lyapunov growth, indicating slower divergence from the true trajectory.
  • Long-horizon rollouts remain closer to the true attractor for integrator-like models.
  • The categorical performance differences persist even after individual hyperparameter optimization for each model.
  • An ablation isolating components of the continuous-update architecture confirms the contribution of the integrator-style step.

Where Pith is reading between the lines

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

  • Architectural bias toward stable integration may matter more than capacity or training details for long-term forecasting tasks.
  • Similar integrator-style designs could be tested on other high-dimensional chaotic systems to check if the stability advantage generalizes.
  • The attractor analysis suggests these models better preserve invariant measures, which could aid downstream tasks like uncertainty quantification.
  • Future work might combine the integrator update with explicit conservation constraints to further reduce bias.

Load-bearing premise

A common training protocol and matched model capacity produce a fair comparison of architectural effects on rollout stability independent of optimization details.

What would settle it

Finding that a non-integrator architecture achieves lower perturbation amplification and better long-horizon accuracy than an integrator-like model on the Kolmogorov flow when capacities are matched would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.24868 by Rajarshi Biswas.

Figure 1
Figure 1. Figure 1: Training and inference setup for the autoregressive temporal models with initial-state conditioning. This conditioning provides global trajectory context throughout the rollout. Training uses a one-step teacher￾forcing objective, while evaluation is performed through closed-loop autoregressive rollouts [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Left) Mean relative reconstruction error over time on validation trajectories. (Right) Energy spectra of raw data and autoencoder reconstructions. 3.2.3 Evaluation Metrics Trajectory-level space–time MSE is used as the primary criteria for gauging predictive performances as given in Eqn. 21, MSE(i) traj = 1 NtNx N Xt−1 n=0 N Xx−1 j=0 (ˆu (i) j,n − u (i) j,n) 2 (21) Error growth across rollout horizon is f… view at source ↗
Figure 3
Figure 3. Figure 3: (Left) Mean relative reconstruction error over time on validation trajectories. (Right) Comparison of spatial energy spectra between simulated vorticity fields and autoencoder reconstructions. 3.4 Hyperparameter Summary [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Double pendulum: Mean state RMSE vs. time averaged across validation trajectories. 4.1.1 Error Growth Over Time [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Double Pendulum Energy-binned regime analysis. Left: fraction of validation trajectories on which each model attains the lowest full-horizon MSE within each E(0) tercile. Right: distributions of log10 full-horizon state MSE for each model and regime. 4.2 KS Equation Identical metrics are evaluated for this problem. In this setting, small temporal prediction errors can spread across spatial modes and amplif… view at source ↗
Figure 6
Figure 6. Figure 6: KS Eqn.: Space-time prediction contours for a representative KS validation trajectory [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: KS Eqn.: Mean state RMSE vs. time averaged over validation trajectories. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: KS Eqn.: Regime analysis using time-averaged spatial variance σ 2 . 4.2.3 Regime-dependent Performance Time-averaged spatial variance (Eqn. 26) is used to group the trajectories into different regimes, σ 2 i = 1 Nt X Nt k=1 Varx [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Kolmogorov flow: Slices of ω(x, ymid, t) from a representative validation trajectory at four time instants [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Kolmogorov flow: Mean state RMSE vs. time averaged over validation trajectories. Metric Model Mean Median P90 Full-horizon state MSE MLP 0.182 0.158 0.270 LSTM 0.323 0.300 0.484 TCN 0.207 0.189 0.246 NeuralODE 0.136 0.130 0.162 CoRD 0.136 0.130 0.162 Early-horizon state MSE MLP 0.179 0.156 0.280 LSTM 0.289 0.256 0.409 TCN 0.214 0.192 0.286 NeuralODE 0.137 0.132 0.172 CoRD 0.137 0.132 0.172 [PITH_FULL_IMA… view at source ↗
Figure 11
Figure 11. Figure 11: Kolmogorov flow: regime analysis based on time-averaged enstrophy ⟨E⟩. (a) Double pendulum (b) KS equation (c) Kolmogorov flow [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Mean state RMSE vs. time for individually optimized models across the three benchmark systems. 4.3.3 Regime-dependent Performance To examine how performance varies with flow complexity, validation trajectories are grouped by their time-averaged enstrophy, [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Distributions of the Jacobian spectral radius ρ(J) for the KS equation (left) and Kolmogorov flow (right). Points denote median values and error bars indicate the interquartile range. (a) KS equation (b) Kolmogorov flow [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Relative one-step bias log10 B(θ) for the latent PDE models. Figures 14a and 14b show the one-step bias for both systems. Continuous-time models operate with the lowest bias. This is likely due to the neural architecture. Instead of learning a discrete jump from one step to the other, this class of models integrate a learned vector field, which in turn keeps each step update closely anchored to the underl… view at source ↗
Figure 15
Figure 15. Figure 15: Finite-time model Lyapunov exponent Λ (10) K (θ). measured in raw latent space, Λ (10) K (θm) = 1 K log10 ∥z raw,pert i,t+K − z raw i,t+K∥2 ε ! . (34) A value of Λ (10) K = 0 means the perturbation neither grew nor shrank over the rollout; a value of 1 indicates a tenfold amplification per step on average; negative values indicate that the model is contracting perturbations. Figures 15a and 15b show the r… view at source ↗
Figure 16
Figure 16. Figure 16: KS attractor diagnostics: Top: kernel density estimates of the snapshot spatial variance σ 2 . Bottom: PCA projections of the attractor clouds in a basis fit on the reference AE reconstruction cloud. the directions of maximum variance and to construct an optimal low-dimensional projection for comparison. All model attractors are projected into this common space. For the KS Equation, [PITH_FULL_IMAGE:figu… view at source ↗
Figure 17
Figure 17. Figure 17: Kolmogorov attractor diagnostics: Top: kernel density estimates of the snapshot enstrophy ⟨ω 2 ⟩x,y. Bottom: PCA projections of the attractor clouds in a basis fit on the reference AE reconstruction cloud. 6 Architecture Ablation The previous sections established the continuous-time models’ capability to closely approximate complex chaotic systems. For the CoRD architecture, several design choices such as… view at source ↗
Figure 18
Figure 18. Figure 18: KS architecture ablation: Left: median trajectory-wise log10(MSEtraj) for each variant, with error bars denoting the interquartile range. Right: kernel density estimates of log10(MSEvariant/MSECoRD). to learn a larger single-step map, which is a harder function to approximate accurately. Eliminating global conditioning (CoRD_v3) also increases error and broadens the distribution considerably. The initial … view at source ↗
read the original abstract

Temporal surrogate models are effective for predicting chaotic dynamical systems where computational cost can be prohibitive. Several deep neural network architectures can be used for such purposes. In this work, a few commonly used architectures are compared using a common training protocol. The objective is to fairly assess the impact of model architectures for long-horizon prediction stability. Experiments are carried out for three problems, the double pendulum, the Kuramoto-Sivashinsky equations, and the Kolmogorov flow. The experiments are carried out with matching model capacity. Analysis is also carried out for a scenario where each model is individually optimized. It is observed that in both scenarios, the models exhibit categorical differences in long-horizon rollouts. For a concrete quantification, stepwise error injections and perturbation amplifications are analyzed using metrics such as local jacobian, relative one-step bias, and finite-time Lyapunov growth. Additionally, an attractor analysis is also conducted to assess how well the learned models replicate the underlying system geometry. An ablation study to isolate the impact of each component of a continuous-update architecture is also carried out. It is concluded that models that having integrator-like updates show lower bias and perturbation amplification yielding stable long-horizon rollout and more accurate predictions.

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

2 major / 2 minor

Summary. The manuscript compares several deep neural network architectures as temporal surrogate models for chaotic dynamical systems (double pendulum, Kuramoto-Sivashinsky equations, Kolmogorov flow). Using a shared training protocol with matched model capacity, plus a second scenario with per-model hyperparameter optimization, the authors report categorical differences in long-horizon rollout behavior. They quantify these via local Jacobian norms, relative one-step bias, finite-time Lyapunov growth, and attractor geometry analysis, concluding that architectures with integrator-like update rules exhibit lower bias and perturbation amplification, yielding more stable and accurate long-term predictions. An ablation study on continuous-update components is also presented.

Significance. If the attribution to update structure can be isolated from training artifacts, the work would supply actionable guidance for selecting surrogate architectures that maintain stability over long rollouts in physics-informed machine learning. The dual-protocol design (common and individually optimized) and use of multiple dynamical systems are positive features; however, the absence of tabulated quantitative results, error bars, or convergence diagnostics in the reported text limits immediate impact.

major comments (2)
  1. [Experimental protocol] Experimental protocol section: the central claim attributes observed differences in bias and Lyapunov growth to integrator-like updates rather than optimization artifacts. The manuscript states that both a common protocol and individually optimized scenarios were examined, yet provides no explicit evidence (identical random-search budgets, wall-clock limits, or convergence tolerances) that per-model optimization effort was equalized. If loss-landscape conditioning differs across architectures, under-optimized models could exhibit inflated effective bias or growth rates, undermining the architectural attribution.
  2. [Results and metrics] Results and metrics sections: the abstract and text assert categorical differences supported by local Jacobian, one-step bias, and finite-time Lyapunov metrics, but the provided manuscript text contains no quantitative tables, error bars, or statistical significance tests. Without these, the strength of the reported differences cannot be assessed and the conclusion that integrator-like models are superior remains unverified.
minor comments (2)
  1. [Abstract] Abstract, line 1: grammatical error ('models that having integrator-like updates').
  2. [Metrics definitions] Notation for finite-time Lyapunov growth and relative bias should be defined explicitly with equations rather than described only in prose.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and will revise the manuscript accordingly to improve clarity and verifiability of the results.

read point-by-point responses

  1. Referee: [Experimental protocol] Experimental protocol section: the central claim attributes observed differences in bias and Lyapunov growth to integrator-like updates rather than optimization artifacts. The manuscript states that both a common protocol and individually optimized scenarios were examined, yet provides no explicit evidence (identical random-search budgets, wall-clock limits, or convergence tolerances) that per-model optimization effort was equalized. If loss-landscape conditioning differs across architectures, under-optimized models could exhibit inflated effective bias or growth rates, undermining the architectural attribution.

    Authors: We employed an identical random-search budget of 50 trials per architecture in the individually optimized scenario, using the same convergence tolerance (validation loss plateau for 20 epochs without improvement) and monitoring wall-clock time per trial. Hyperparameter ranges were architecture-specific but the overall search effort and stopping criteria were matched across models. We will revise the Experimental protocol section to explicitly document these budgets, search spaces, and tolerances to eliminate any ambiguity regarding optimization artifacts. revision: yes

  2. Referee: [Results and metrics] Results and metrics sections: the abstract and text assert categorical differences supported by local Jacobian, one-step bias, and finite-time Lyapunov metrics, but the provided manuscript text contains no quantitative tables, error bars, or statistical significance tests. Without these, the strength of the reported differences cannot be assessed and the conclusion that integrator-like models are superior remains unverified.

    Authors: We agree that the absence of tabulated quantitative results with error bars and significance tests limits assessment of the effect sizes. The current manuscript presents trends via figures, but we will add tables in the Results section reporting means and standard deviations of Jacobian norms, one-step bias, and finite-time Lyapunov exponents over five independent runs, along with p-values from paired statistical tests comparing architectures. This will allow direct evaluation of the reported categorical differences. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical comparison of architectures

full rationale

The paper conducts an empirical study comparing neural architectures for temporal surrogate modeling on three dynamical systems, using shared training protocols and matched capacities (plus an individually optimized scenario). The central claim—that integrator-like updates yield lower bias, reduced perturbation amplification, and more stable long-horizon rollouts—is presented as an observation from direct metrics (local Jacobian, relative one-step bias, finite-time Lyapunov growth, attractor analysis, and ablation). No equations, fitted parameters, or self-citations are invoked that would reduce this claim to a definitional tautology, a renamed input, or a load-bearing self-reference. The derivation chain consists of experimental measurements rather than any mathematical reduction to prior assumptions or fits.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

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