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arxiv: 2511.06609 · v3 · submitted 2025-11-10 · 💻 cs.LG · math.DS

A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series

Xuyang Li , John Harlim , Dibyajyoti Chakraborty , Romit Maulik This is my paper

Pith reviewed 2026-05-18 00:01 UTC · model grok-4.3

classification 💻 cs.LG math.DS
keywords Neural ODEchaotic dynamicsweak formulationnoisy time seriesdynamical systemsforecastingmachine learningclimate data
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The pith

A weak formulation of the loss lets Neural ODEs filter noise and forecast chaotic dynamics accurately.

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

The paper develops a training method for Neural ODEs that replaces or augments the usual pointwise loss with a weak penalty term derived from the differential equation residual. This term is formed by integrating the residual against selected test functions over chosen domains, which the authors show acts like a built-in smoother that removes high-frequency noise while leaving the underlying flow intact. As a result, the trained model produces short-term predictions that remain accurate even when initial errors would normally explode in chaotic systems, and it also maintains correct long-term statistical behavior. A reader would care because real measurements of weather, fluids, or other nonlinear systems are almost always noisy, and this offers a direct, solver-independent way to obtain usable forecasts without separate denoising steps.

Core claim

The authors show that the weak form of the Neural ODE residual, when used as a penalty in the training objective with suitable test functions and integration intervals, filters measurement noise in a manner equivalent to fitting the model to pre-filtered data. This formulation stabilizes training for chaotic attractors, where standard L2 losses on noisy trajectories lead to rapid divergence, and produces forecasts whose short-term accuracy and long-term invariants both improve. The method is demonstrated to be computationally efficient and independent of the particular ODE integrator chosen during training or prediction.

What carries the argument

The weak penalty loss, formed by integrating the Neural ODE residual against test functions over finite domains so that high-frequency noise contributions average toward zero.

If this is right

  • The trained models achieve both short-term forecast accuracy and preservation of long-term invariant properties on standard chaotic benchmarks.
  • The same strategy produces usable forecasts on a real-world climate dataset despite observational noise.
  • Training remains efficient and works with any choice of numerical ODE solver.
  • The approach avoids the exponential error growth that plagues standard Neural ODE training on noisy chaotic time series.

Where Pith is reading between the lines

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

  • The same weak-penalty idea could be applied to other neural architectures that learn differential equations, such as neural SDEs or graph-based dynamical models.
  • Optimal selection of test functions for a given noise spectrum or system dimension remains an open design choice that future work could automate.
  • Because the method operates directly on the integral form, it may combine naturally with existing data-assimilation pipelines used in operational forecasting.

Load-bearing premise

The weak formulation with a proper choice of test function and integration domain effectively filters noisy data.

What would settle it

Train the model on a benchmark chaotic system with added noise, then verify whether the long-term statistics of simulated trajectories, such as the shape of the attractor or the leading Lyapunov exponent, match those obtained from clean data to within a small tolerance.

Figures

Figures reproduced from arXiv: 2511.06609 by Dibyajyoti Chakraborty, John Harlim, Romit Maulik, Xuyang Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Forecasting of the Lorenz-63 system using WP-NODE under 5% training data noise. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Invariant measure of the predicted Lorenz-63 system (for 100 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Performance comparison of WP-NODE and baselines models on the KS system, under 5% data noise. DeepSkip is [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the long-term behavior of the KS system under 5% data noise, using the invariant measure (i.e., joint [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. NODE predictions performance on real-world ERA5 dataset. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

The accurate forecasting of complex, high-dimensional dynamical systems from observational data is a fundamental task across numerous scientific and engineering disciplines. A significant challenge arises from noise-corrupted measurements, which severely degrade the performance of data-driven models. In chaotic dynamical systems, where small initial errors amplify exponentially, it is particularly difficult to develop a model from noisy data that achieves short-term accuracy while preserving long-term invariant properties. To overcome this, we consider the weak formulation as a complementary approach to the classical $L2$-loss function for training models of dynamical systems. We empirically verify that the weak formulation, with a proper choice of test function and integration domain, effectively filters noisy data. This insight explains why a weak form loss function is analogous to fitting a model to filtered data and provides a practical way to parameterize the weak form. Subsequently, we demonstrate how this approach overcomes the instability and inaccuracy of standard Neural ODE (NODE) in modeling chaotic systems. Through numerical examples, we show that our proposed training strategy, the Weak Penalty NODE, is computationally efficient, solver-agnostic, and yields accurate and robust forecasts across benchmark chaotic systems and a real-world climate dataset.

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 introduces the Weak Penalty Neural ODE (WP-NODE), which augments standard Neural ODE training with a weak-form loss term. The central claim is that, for a suitable choice of test function and integration domain, the weak formulation filters noise in time-series observations of chaotic systems, yielding short-term forecast accuracy while preserving long-term invariants; the method is further asserted to be computationally efficient and solver-agnostic, with supporting numerical results on benchmark chaotic systems and a real-world climate dataset.

Significance. If the noise-filtering property and long-term stability claims hold under broader conditions, the work would offer a practical, preprocessing-free route to training NODEs on noisy chaotic data. This could be useful in climate modeling and other domains where explicit denoising is costly or distorts invariants, and the solver-agnostic character would be a clear practical advantage.

major comments (2)
  1. [Abstract and weak-formulation section] Abstract and method description of the weak formulation: the assertion that 'the weak formulation, with a proper choice of test function and integration domain, effectively filters noisy data' is load-bearing for the entire approach, yet no general rule, stability analysis, or selection criterion is supplied for these choices. Without such guidance the reported success on the chosen benchmarks may reflect case-specific tuning rather than a structural guarantee, leaving the robustness claim for varying noise statistics or system dimensionality unproven.
  2. [Numerical experiments] Numerical experiments on long-term forecasts: while short-term accuracy and computational efficiency are shown, quantitative evidence that long-term invariants (e.g., Lyapunov exponents or attractor geometry) are preserved independently of the specific test-function parameters is limited. This weakens the claim that the method reliably overcomes standard NODE instability for chaotic systems.
minor comments (2)
  1. [Method] The notation used for the weak penalty loss should be contrasted explicitly with the classical L2 NODE loss to make the algorithmic difference immediately clear.
  2. [Numerical experiments] A short table or paragraph quantifying the sensitivity of forecast error to the integration-domain length would help readers assess practical implementation cost.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their constructive and insightful comments. We address each major comment point by point below, indicating where revisions have been made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and weak-formulation section] Abstract and method description of the weak formulation: the assertion that 'the weak formulation, with a proper choice of test function and integration domain, effectively filters noisy data' is load-bearing for the entire approach, yet no general rule, stability analysis, or selection criterion is supplied for these choices. Without such guidance the reported success on the chosen benchmarks may reflect case-specific tuning rather than a structural guarantee, leaving the robustness claim for varying noise statistics or system dimensionality unproven.

    Authors: We appreciate this observation. The manuscript presents the approach as an empirical method and supplies a practical parameterization by showing that the weak-form loss corresponds to fitting filtered data, along with guidance on matching test-function support to system time scales (Section 3). We agree that more explicit selection criteria would improve clarity. In the revised manuscript we have added a dedicated paragraph in the method section that outlines concrete rules: choose the integration domain to exceed the noise correlation time but remain shorter than the system's Lyapunov time, and select test functions whose frequency content overlaps the dominant dynamics while attenuating high-frequency noise. A full stability analysis under arbitrary noise statistics and dimensionality is not derived, as it would require additional theoretical assumptions and development beyond the paper's empirical focus; robustness is instead demonstrated across multiple chaotic benchmarks with varying noise levels and dimensions. revision: partial

  2. Referee: [Numerical experiments] Numerical experiments on long-term forecasts: while short-term accuracy and computational efficiency are shown, quantitative evidence that long-term invariants (e.g., Lyapunov exponents or attractor geometry) are preserved independently of the specific test-function parameters is limited. This weakens the claim that the method reliably overcomes standard NODE instability for chaotic systems.

    Authors: We concur that stronger quantitative support for parameter independence would reinforce the long-term stability claim. The original experiments reported results for effective parameter choices. To address the concern, the revised manuscript now includes a sensitivity study: Lyapunov exponents and attractor geometry metrics (e.g., correlation dimension) are computed over a grid of test-function widths and integration lengths for each benchmark system. The results show that these invariants remain consistent within a practically useful range of parameters, with degradation only outside that range. Corresponding figures and tables have been added to the numerical experiments section. revision: yes

standing simulated objections not resolved
  • A rigorous theoretical stability analysis of the weak formulation for arbitrary noise statistics and system dimensions is not provided.

Circularity Check

0 steps flagged

No significant circularity; method relies on empirical verification of weak-form filtering rather than self-referential derivation

full rationale

The paper presents the Weak Penalty NODE as a training strategy that leverages the weak formulation of dynamical systems to handle noisy data. The key assertion—that the weak form with suitable test functions and domains filters noise and is analogous to fitting filtered data—is stated as an empirical insight verified through numerical examples on benchmark chaotic systems and climate data. No load-bearing step reduces by construction to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled via prior work. The derivation chain consists of proposing the penalty-augmented loss, demonstrating solver-agnostic training, and reporting forecast accuracy; these rest on external benchmarks and observed performance rather than tautological reparameterization of inputs. Self-contained against the provided numerical validation, this yields a normal non-finding of circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach depends on the effectiveness of the weak form in filtering noise, which is taken as given and verified empirically rather than derived from first principles.

free parameters (1)
  • test function and integration domain
    Proper choice is required to filter noise effectively as stated in the abstract.
axioms (1)
  • domain assumption Weak formulation filters noisy data in dynamical systems when test functions and domains are chosen properly
    Invoked to explain why the weak form loss works analogously to fitting filtered data.

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    For the endpoints: ℓ0(s) =    s1 −s s1 −s 0 , s∈[s 0, s1] 0,otherwise ℓM−1(s) =    s−s M−2 sM−1 −s M−2 , s∈[s M−2 , sM−1] 0,otherwise

    Linear Basis Functions.Define the piecewise linear Lagrange basis functionsℓ i(s), ℓi(s) =    s−s i−1 si −s i−1 , s∈[s i−1, si] si+1 −s si+1 −s i , s∈[s i, si+1] 0,otherwise for 1≤i≤M−2. For the endpoints: ℓ0(s) =    s1 −s s1 −s 0 , s∈[s 0, s1] 0,otherwise ℓM−1(s) =    s−s M−2 sM−1 −s M−2 , s∈[s M−2 , sM−1] 0,otherwise

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    Note that ifx i, fi ∈R D are vectors andℓ i(s)∈Ris a scalar, the termsx iℓi(s) andf iℓi(s) represent the standard scalar-vector products

    Interpolant Construction.We define the linear interpolants of the functionsx(s) andf(u(s), s;θ) as: ˜x(s) = M−1X i=0 xiℓi(s), ˜f(s) = M−1X i=0 fiℓi(s).(S2) These functions are continuous and piecewise linear. Note that ifx i, fi ∈R D are vectors andℓ i(s)∈Ris a scalar, the termsx iℓi(s) andf iℓi(s) represent the standard scalar-vector products

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    Projection Against a Test Function.We approximate the following integrals using Eq. (S2) as follows, L 2 Z sM−1 s0 f(x(s), s;θ)ϕ(s)ds≈ L 2 Z sM−1 s0 ˜f(s)ϕ(s)ds= M−1X i=0 fi L 2 Z sM−1 s0 ℓi(s)ϕ(s)ds = M−1X i=0 fiwrhs,i, where the weights are defined as: wrhs,i = L 2 Z sM−1 s0 ℓi(s)ϕ(s)ds. 22 Similarly, Z sM−1 s0 x(s)ϕ′(s)ds≈ Z sM−1 s0 ˜x(s)ϕ′(s)ds= M−1X ...

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    Explicit Expression for Weights.For interior indices 1≤i≤M−2, we have: wrhs,i = L 2 Z si si−1 s−s i−1 si −s i−1 ϕ(s)ds+ L 2 Z si+1 si si+1 −s si+1 −s i ϕ(s)ds = L 2(si −s i−1) [Ψ(si)−Ψ(s i−1)−s i−1(Φ(si)−Φ(s i−1))] + L 2(si+1 −s i) [si+1(Φ(si+1)−Φ(s i))−Ψ(s i+1) + Ψ(si)], where Φp,d(s) = R ϕ(d) p (s)dsand Ψ p,d(s) = R sϕ(d) p (s)ds. For the endpoints: wrh...

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