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Neural Oscillators for Generalization of Physics-Informed Machine Learning
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Combines a PINN trained on a time interval with a neural oscillator (CoRNN or LEM) that extrapolates the PINN's solution sequence to unseen later times, improving generalization beyond the training domain on nonlinear PDEs and a beam equation.
Problem
Generalization of physics-informed neural networks beyond the training spatio-temporal domain (extrapolation in time) for PDEs, where PINNs fail to extrapolate and gradient-based RNNs suffer exploding/vanishing gradients.
Approach
Feed the PINN's solution snapshots as a sequence into ODE-based recurrent neural oscillators (CoRNN, LEM), whose ODE-driven hidden-state updates capture long-time dependencies and mitigate exploding/vanishing gradients, then roll the oscillator forward autoregressively into the unseen time domain.
Key results
- On viscous Burgers, LEM reaches a relative L2 error of 0.0001 versus 0.083 for DPM and 0.0044 for CoRNN in the extrapolation region.
- On the Euler-Bernoulli beam equation, LEM achieves L2 error 0.0593 and explained variance 0.9409, while RNN, LSTM and GRU have L2 errors of 2.1 to 4.7 with negative explained variance.
- On Schrödinger, LEM achieves L2 error 0.0034 and explained variance 0.9944 versus 0.0426 (CoRNN) and 0.141 (DPM).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add LEM to PINN instead of DPM | PIML generalization · Viscous Burgers extrapolation | relative L2 error: 0.083 -> 0.0001 | Helped |
| Add LEM to PINN instead of DPM | PIML generalization · Allen–Cahn extrapolation | relative L2 error: 0.182 -> 0.0049 | Helped |
| Add LEM to PINN instead of DPM | PIML generalization · Schrödinger extrapolation | relative L2 error: 0.141 -> 0.0034 | Helped |
| Add LEM to PINN instead of GRU | PIML generalization · Euler–Bernoulli beam extrapolation | relative L2 error: 2.9176 -> 0.0593 | Helped |
| Use lower oscillator Δt (ablation) | LEM and CoRNN · Viscous Burgers equation | error | Helped |
| Vary CoRNN ε and γ (ablation) | CoRNN · Viscous Burgers equation | error | No clear effect |
Surprising findings
- Plain RNN, LSTM and GRU augmented with the PINN perform far worse than the oscillator models, and on the beam equation yield negative explained variance.
Limitations
- Oscillators require evenly spaced data, so the PINN must be evaluated on a uniform grid before the oscillator is applied.
- Performance depends on PINN accuracy; errors propagate through the pipeline.
- Experiments are limited to 1D (plus one 2D Kovasznay flow in supplementary) with short extrapolation horizons (T' from 0.8 to 1).
Experimental setup
Datasets: viscous Burgers equation (simulated) Allen-Cahn equation (simulated) nonlinear Schrödinger equation (simulated) Euler-Bernoulli beam equation (analytical solution) Kovasznay flow
Benchmarks: viscous Burgers equation Allen-Cahn equation nonlinear Schrödinger equation Euler-Bernoulli beam equation
Metrics: relative L2 norm explained variance score max error mean absolute error
Baselines: DPM RNN LSTM GRU
Builds on: PINN CoRNN LEM DPM DeepXDE
Closest papers by meaning
- Number Theoretic Accelerated Learning of Physics-Informed Neural Networks (AAAI 2025 · similarity 0.86)
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- The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.86)
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