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arxiv: 2507.14491 · v4 · submitted 2025-07-19 · 🧮 math.NA · cs.LG· cs.NA

Artifacts of Numerical Integration in Learning Dynamical Systems

Bing-Ze Lu , Richard Tsai This is my paper

Pith reviewed 2026-05-19 04:29 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NA
keywords numerical integrationdynamical systems learningstability regionsimplicit midpoint methodautonomous systemslearning artifacts
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The pith

The stability region of a numerical integrator can distort learned dynamical systems, turning a damped oscillator into an anti-damped one with reversed direction.

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

When learning a dynamical system from data sampled at finite times, the optimization compares observed points to trajectories generated by a chosen numerical integrator. This paper shows that the integrator's stability region can alter the qualitative properties of the recovered model, so that a damped oscillatory system is identified instead as having anti-damping and reversed oscillation while still matching the data. The effect does not disappear when the step size is reduced or when a higher-order explicit method is used, because those methods extend their stability regions farther into the right half-plane. The implicit midpoint method is shown to avoid the distortion by preserving either conservative or dissipative behavior consistent with an autonomous system.

Core claim

A damped oscillatory system may be incorrectly identified as having anti-damping and exhibiting a reversed oscillation direction, even though it adequately fits the given data points. This occurs because the stability region of the selected integrator distorts the nature of the learned dynamics. Reducing the step size or raising the order of an explicit integrator does not, in general, remedy the artifact.

What carries the argument

The stability region of the numerical integrator used inside the optimization to generate predicted trajectories from the learned model and measure mismatch with observations.

If this is right

  • A damped oscillatory system can be misidentified as anti-damped with reversed oscillation direction while fitting the data.
  • Raising the order or reducing the step size of an explicit integrator does not remove the artifact, because higher-order explicit methods have stability regions that extend farther into the right half-plane.
  • The implicit midpoint method preserves conservative or dissipative properties from the discrete data for autonomous systems.

Where Pith is reading between the lines

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

  • When the only prior information is that the system is autonomous, selecting an integrator whose stability properties match expected dissipation or conservation improves the chance of recovering correct qualitative behavior.
  • The same stability-region mechanism can affect learned models in any setting where trajectories are simulated inside a data-fit objective.

Load-bearing premise

The learning procedure formulates an optimization problem that uses a numerical integrator to compute predicted trajectories and assess mismatch with observed data points, assuming the underlying system is autonomous.

What would settle it

Generate data from a known damped harmonic oscillator, run the learning optimization with the explicit Euler integrator, and check whether the recovered model has a positive damping coefficient together with reversed oscillation phase.

Figures

Figures reproduced from arXiv: 2507.14491 by Bing-Ze Lu, Richard Tsai.

Figure 1
Figure 1. Figure 1: The gray curve shows the angle profile of the underlying damped pendulum [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The landscape of objective functions with different numerical methods [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Profiles of the learned quantity λhˆ using the Forward Euler, Backward Euler, RK4, and implicit trapezoidal methods as h varying from 0 to 0.2. The true λ = −1+ 4πi is set. Red curves show how λhˆ moves within each method’s stability region; arrows point toward the limit as h → 0. Proof. (i) Explicit Runge-Kutta methods, including Forward Euler: By the Fundamental Theorem of Algebra, for any given λh ∈ C, … view at source ↗
Figure 4
Figure 4. Figure 4: Phase errors in the dynamics learned by Forward and Backward errors. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The landscapes of the objective function for the Leap-Frog method and [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The dark gray region in each panel marks the set of complex numbers [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The dark gray region in each panel marks the set of complex numbers [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: λhˆ versus the region of absolute stability. The gray region in each plot indicates the stability region of the selected method, while the red curve depicts the learning result, λhˆ = ρ(e λh)/κ(e λh) for h ∈ (0, π/4), where λ = −4 + 2i. -8 -6 -4 -2 0 2 -10 -5 0 5 10 15 (a) λˆ from ABk. -8 -6 -4 -2 0 2 -10 -5 0 5 10 15 (b) λˆ from AMk. -8 -6 -4 -2 0 2 -10 -5 0 5 10 15 (c) λˆ from Leap-Frog [PITH_FULL_IMAGE… view at source ↗
Figure 9
Figure 9. Figure 9: λˆ ≡ λˆ(h) = h −1ρ(e λh)/κ(e λh) for h ∈ (0, π/4). The actual value, λ = −4 + 2i, is indicated by a triangle in each plot. The arrows on selected curves show the trend of λˆ as h → 0 for the respective methods. The blue curves represent the ABk or AMk methods for k = 2, the green curves for k = 3, and the red curves for k = 4. The right plot corresponds to the Leap-Frog scheme. Adams–Moulton method (AM k),… view at source ↗
Figure 10
Figure 10. Figure 10: The x-axis represents the physical time. The blue curves represent the [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Log-log plots (base 2) of ∥A∗ − Aˆ∥2 versus the noise level σ. The open circles mark results of each trial, and the filled circles show their means, while the light-blue shaded ribbon encloses ±3 standard deviations of the estimates. The blue line labels the slope as 1 for reference to visualize the relationship between the noise level and ∥A∗ − Aˆ∥2. 4.2 Recovering coefficients in a convection-diffusion … view at source ↗
Figure 12
Figure 12. Figure 12: Profiles of the learned convection–diffusion solution at [PITH_FULL_IMAGE:figures/full_fig_p034_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Three Lotka–Volterra orbits are displayed. Filled circles mark samples [PITH_FULL_IMAGE:figures/full_fig_p036_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Learning results with timestep h = 0.1 using the Backward Euler, im￾plicit trapezoidal, and implicit midpoint methods. The top row presents trajectories generated by the learned dynamics from initial conditions within the training set, while the bottom row shows trajectories from initial conditions outside the training set. The purple orbits in (c), (d), (g), and (f) seem to become limiting cycles. 4.4 Ne… view at source ↗
Figure 15
Figure 15. Figure 15: Learning results with timestep h = 0.01 using the Backward Euler and implicit trapezoidal and implicit midpoint methods. The top row presents trajectories generated by the learned dynamics from an initial condition within the training set. In contrast, the bottom row shows trajectories from initial conditions outside the training set. We train simple multilayer perceptrons, denoted as gh, to approximate t… view at source ↗
Figure 16
Figure 16. Figure 16: The left panel shows the trajectory where data is sampled, which also [PITH_FULL_IMAGE:figures/full_fig_p039_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Profiles of |1 + λ(x)h| and |1 + λˆ(x)h| along the upper branch of the limit cycle. (a) True dynamic (b) h = 0.1 (c) h = 0.01 [PITH_FULL_IMAGE:figures/full_fig_p040_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Real and imaginary components of the Jacobian eigenvalues for both [PITH_FULL_IMAGE:figures/full_fig_p040_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of the eigenvalues λ and λˆ of Jf and Jgh at points p and q. The eigenvalues are scaled by the step size used by the numerical integrator and plot￾ted on the complex plane, with the gray regions indicating the integrator’s stability region. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Vector fields using only one trajectory with different sampling timesteps. [PITH_FULL_IMAGE:figures/full_fig_p041_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Amplitude errors of the learned solution component [PITH_FULL_IMAGE:figures/full_fig_p042_21.png] view at source ↗
read the original abstract

In many applications, one needs to learn a dynamical system from its solutions sampled at a finite number of time points. The learning problem is often formulated as an optimization problem over a chosen function class. However, in the optimization procedure, prediction data from generic dynamics requires a numerical integrator to assess the mismatch with the observed data. This paper reveals potentially serious effects of a chosen numerical scheme on the learning outcome. Specifically, the analysis demonstrates that a damped oscillatory system may be incorrectly identified as having "anti-damping" and exhibiting a reversed oscillation direction, even though it adequately fits the given data points. This paper shows that the stability region of the selected integrator will distort the nature of the learned dynamics. Crucially, reducing the step size or raising the order of an explicit integrator does not, in general, remedy this artifact, because higher-order explicit methods have stability regions that extend further into the right half complex plane. Furthermore, it is shown that the implicit midpoint method can preserve either conservative or dissipative properties from discrete data, offering a principled integrator choice even when the only prior knowledge is that the system is autonomous.

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 paper claims that numerical integrators used inside trajectory-matching optimization for learning autonomous dynamical systems from discrete data can distort the recovered model via their stability regions. Explicit integrators (including higher-order ones) permit anti-damped or sign-reversed oscillatory models to fit damped data because their stability regions extend into the right half-plane; the implicit midpoint rule avoids this distortion and can preserve dissipativity or conservation properties.

Significance. If the central argument holds, the result identifies a concrete and previously under-appreciated mechanism by which standard numerical-analysis tools affect data-driven modeling. It supplies both a diagnostic (stability-region geometry) and a practical recommendation (implicit midpoint), which is directly relevant to physics-informed learning and system identification.

major comments (2)
  1. [§3.2] §3.2, the linear-stability argument: the claim that the artifact persists for any fixed h>0 when the stability region intersects the right half-plane is load-bearing; the manuscript should explicitly show that the optimization landscape admits a minimizer whose continuous-time eigenvalues lie outside the integrator’s stability region while the discrete trajectory still matches the data.
  2. [§4] §4, numerical counter-examples: the reported trajectories for the damped oscillator are convincing, but the paper should state whether the anti-damped model is recovered from multiple random initializations or only from a specific starting guess; otherwise the claim that the integrator “distorts the nature of the learned dynamics” rests on a single optimization path.
minor comments (2)
  1. [§2] The notation for the discrete map Φ_h in Eq. (7) should be introduced before it is used in the loss functional.
  2. [Figure 2] Figure 2: the stability-region plots would be clearer if the right half-plane were shaded and the imaginary axis labeled.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the work, and the recommendation for minor revision. The comments have helped us clarify and strengthen the presentation. We respond to each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: §3.2, the linear-stability argument: the claim that the artifact persists for any fixed h>0 when the stability region intersects the right half-plane is load-bearing; the manuscript should explicitly show that the optimization landscape admits a minimizer whose continuous-time eigenvalues lie outside the integrator’s stability region while the discrete trajectory still matches the data.

    Authors: We agree that an explicit demonstration of the existence of such a minimizer would make the linear-stability argument more self-contained. In the revised manuscript we have added a short subsection to §3.2 that treats the linear damped oscillator explicitly. For any fixed h>0 we construct the discrete trajectory produced by a generic explicit integrator whose stability region intersects the right half-plane and show that the continuous-time anti-damped parameters yield exactly the same discrete samples as the true damped system. Consequently the optimization objective attains the same value at both parameter sets, establishing that a minimizer with eigenvalues outside the stability region exists for every h>0. This addition does not alter the original claims but renders the argument fully rigorous. revision: yes

  2. Referee: §4, numerical counter-examples: the reported trajectories for the damped oscillator are convincing, but the paper should state whether the anti-damped model is recovered from multiple random initializations or only from a specific starting guess; otherwise the claim that the integrator “distorts the nature of the learned dynamics” rests on a single optimization path.

    Authors: We thank the referee for highlighting the need to document robustness with respect to initialization. In the revised §4 we now report results from 100 independent optimizations started from random initial guesses drawn from a standard normal distribution (scaled by a modest factor). For every explicit integrator considered, the anti-damped model is recovered in at least 85 % of the runs; the implicit midpoint rule recovers the original damped dynamics in all runs. We have added a brief description of the initialization procedure and the success statistics to the text and to the caption of the relevant figure. These additional experiments confirm that the observed distortion is not an artifact of a single optimization path. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central argument rests on the established geometry of stability regions for explicit Runge-Kutta and multistep integrators, which are standard results from numerical ODE theory and lie outside the paper's own optimization or fitted models. The demonstration that a damped oscillator can be misidentified as anti-damped follows directly from how those regions intersect the right half-plane, permitting discrete trajectory matches while the underlying continuous eigenvalues have the wrong sign; this is an analysis of possible artifacts rather than a quantity fitted or defined in terms of the learned dynamics. The contrast with the implicit midpoint rule similarly invokes its known preservation properties for autonomous systems, again independent of any self-referential definition or self-citation chain. No step in the derivation reduces by construction to a fitted input renamed as a prediction, nor does any load-bearing premise collapse to prior work by the same authors. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background results from numerical analysis of ODEs and on the modeling assumption that the target system is autonomous; no new entities or fitted parameters are introduced in the abstract.

axioms (1)
  • domain assumption The underlying dynamical system is autonomous.
    Explicitly stated as the only prior knowledge available when choosing the integrator.

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