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Out-of-Domain Generalization in Dynamical Systems Reconstruction
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Formal framework for out-of-domain generalization in dynamical systems reconstruction, showing that black-box DSR models trained on one basin of attraction fail to recover other basins while SINDy with the correct library succeeds.
Problem
Dynamical systems reconstruction (DSR) models trained on time series from a limited portion of state space often fail to capture coexisting attractors (multistability) in unobserved basins, and there is no formal treatment of out-of-domain generalization in DSR.
Approach
Frames multistable basins of attraction as distinct OOD domains, defines statistical (sliced Wasserstein occupation-measure) and topological (Lyapunov-based) generalization errors, proves that black-box universal approximators cannot be strictly learnable from single-basin data while library-based SINDy can, and diagnoses causes via initialization simplicity bias and saddle-point/sharp-minimum tra
Key results
- Models trained on one Duffing basin fail to generalize to the other basin, while trained on both basins shPLRNN achieves E_stat about 1.4e-3, RC 2.7e-3, N-ODE 2.1e-3.
- SINDy with the correct function library generalizes across basins from single-basin data, per Theorem on library-based learnability.
- Retraining generalizing shPLRNN models on one basin increases reconstruction error on the other basin about 20-fold, and generalizing minima are sharper than non-generalizing ones.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| train black-box DSR models using trajectories from only one basin | RC, N-ODE, and shPLRNN · Multistable Duffing and Lorenz-like systems | OOD generalization error | Hurt |
| add training trajectories from both basins | RC, N-ODE, and shPLRNN · Duffing system | statistical error: RC \approx 2.7\cdot 10^{-3}, N-ODE \approx 2.1 \cdot 10^{-3}, shPLRNN \approx 1.4\cdot 10^{-3} | Helped |
| add training trajectories from both basins | RC, N-ODE, and shPLRNN · Multistable Lorenz-like system | statistical error: ≈ 0.131 for RC, ≈ 0.133 for N-ODE and ≈ 0.064 for the shPLRNN | Helped |
| use SINDy with the correct function library | SINDy · Multistable systems | OOD generalization | Helped |
| omit the third-order term present in the Duffing equations from the SINDy library (ablation) | SINDy · Duffing system | OOD generalization | Hurt |
| retrain a generalizing model using trajectories from only one of the two basins | Generalizing DSR models · Duffing system, unobserved basin | statistical error: about 20-fold increase | Hurt |
| increase parameter initialization gain (ablation) | shPLRNN · Dynamical complexity at initialization | mean Shannon entropy | Helped |
| increase parameter initialization gain (ablation) | shPLRNN trained with SGD · Training feasibility | trainability | Hurt |
| train on trajectories from both basins instead of one basin | shPLRNN · Duffing system, minimum radius | minimum radius | Hurt |
Surprising findings
- Generalizing solutions exist in the loss landscape of the same architectures but are saddle points that SGD-based retraining leaves, so optimization unlearns the second basin.
- Generalizing minima have smaller radius (are sharper), contrary to the common deep learning view that wider minima generalize better.
Limitations
- Theoretical results for black-box models assume universal approximation limit; empirical analysis limited to low-dimensional Duffing and a Lorenz-like system.
- SINDy guarantees rely on the correct function library being provided, which is rarely known for empirical systems.
- The paper proposes no practical method that solves OODG; it mainly diagnoses the problem and suggests directions.
Experimental setup
Datasets: Duffing oscillator simulations multistable Lorenz-like system simulations
Benchmarks: Duffing system multistable Lorenz-like system
Metrics: sliced Wasserstein-1 statistical error topological generalization error based on Lyapunov exponents MSE
Baselines: SINDy shPLRNN reservoir computing (echo state network) Neural ODE
Builds on: SINDy generalized teacher forcing shallow PLRNN Neural ODE reservoir computing
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