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When are dynamical systems learned from time series data statistically accurate?
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Proposes an ergodic-theoretic notion of generalization for learned chaotic dynamics, showing that Jacobian-matching (C^1) regression yields statistically accurate neural ODE surrogates via shadowing theory, while MSE-trained models can fail to reproduce invariant measures and Lyapunov exponents.
Problem
Standard generalization (small test error on the vector field or one-step map) does not guarantee that learned chaotic dynamical systems reproduce physical invariant statistics and Lyapunov exponents.
Approach
Adds first-derivative (Jacobian) matching to the regression loss and uses shadowing theory to show that C^1 strong generalization plus typical shadowing implies Wasserstein-accurate ergodic statistics of the learned map, explaining why MSE-trained Neural ODEs fail while Jacobian-trained ones succeed.
Key results
- On Lorenz '63, the JAC ResNet model reduces W1 distance to 0.1433 versus 1.3567 for the MSE ResNet, and its leading and stable Lyapunov exponents match truth (0.88, -14.54 vs true 0.9, -14.52) while the MSE model gives -4.82 for the stable exponent.
- On Kuramoto-Sivashinsky (127D), the best JAC model learns more than half of the first 64 Lyapunov exponents, while the best MSE model learns only 2 of 64 with under 10% relative error.
- Across tent, Lorenz, Rossler, and hyperchaos systems, Jacobian-matching models reproduce true Lyapunov spectra more accurately than MSE models in most cases.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add Jacobian-matching loss | Lorenz '63 Neural ODE with MLP · Lorenz '63 | Wasserstein-1 distance: 18.9711 -> 0.6800 | Helped |
| add Jacobian-matching loss | Lorenz '63 Neural ODE with ResNet · Lorenz '63 | Lyapunov exponent norm difference: 10.8516 -> 0.0106 | Helped |
| add Jacobian-matching loss | Neural ODE · Lorenz '63 | Lyapunov spectrum: [0.87, 0.0091, -4.82] -> [0.88, -0.0012, -14.54] | Helped |
| add Jacobian-matching loss | Neural ODE · Tent maps (tilted, pinched, and plucked; s=0.2 and s=0.8) | Lyapunov spectrum accuracy | Mixed |
| replace one-step loss with unrolled loss using k=50 (ablation) | Lorenz '63 Neural ODE · Lorenz '63 | Lyapunov exponent norm difference: 4.4824 (MLP); 2.9456 (ResNet) | Helped |
Surprising findings
- Models with small vector-field test error (under 5% relative error) can produce atypical orbits and wrong statistics, so generalization error does not imply statistical accuracy.
- For the KS system the MSE models overpredict the number of positive Lyapunov exponents.
Limitations
- Theoretical results on C^1 generalization assume typicality of shadowing orbits, which may fail for some systems.
- Computing Jacobians can be expensive for high-dimensional scientific applications.
- Most studied systems are not uniformly hyperbolic, so the theory does not fully cover the experiments.
Experimental setup
Datasets: Lorenz '63 simulated trajectories (RK4) tent map orbits Baker's map orbits Rossler system orbits hyperchaos system orbits Kuramoto-Sivashinsky trajectories
Benchmarks: Lorenz '63 attractor statistics tent maps Baker's map Rossler hyperchaos Kuramoto-Sivashinsky
Metrics: Wasserstein-1 distance Lyapunov exponents relative vector-field error time-average error
Baselines: MSE-trained Neural ODE (MLP) MSE-trained Neural ODE (ResNet) MSE-trained Neural ODE (FNO) latent SDE unrolled multi-step loss
Builds on: Neural ODE latent SDE Fourier neural operator
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