Regime-Aware Time Weighting for Physics-Informed Neural Networks
This addresses the challenge of handling time dimensions in PINNs for researchers and practitioners in computational physics and machine learning, offering a more robust framework for time-dependent problems, though it is an incremental improvement over heuristic methods.
The paper tackles the problem of improving solution quality for time-dependent differential equations using Physics-Informed Neural Networks (PINNs) by introducing a regime-aware time weighting method based on Lyapunov exponents, resulting in improved convergence and accuracy on benchmarks like the chaotic Lorenz system and Burgers' equation without extra hyperparameter tuning.
We introduce a novel method to handle the time dimension when Physics-Informed Neural Networks (PINNs) are used to solve time-dependent differential equations; our proposal focuses on how time sampling and weighting strategies affect solution quality. While previous methods proposed heuristic time-weighting schemes, our approach is grounded in theoretical insights derived from the Lyapunov exponents, which quantify the sensitivity of solutions to perturbations over time. This principled methodology automatically adjusts weights based on the stability regime of the system -- whether chaotic, periodic, or stable. Numerical experiments on challenging benchmarks, including the chaotic Lorenz system and the Burgers' equation, demonstrate the effectiveness and robustness of the proposed method. Compared to existing techniques, our approach offers improved convergence and accuracy without requiring additional hyperparameter tuning. The findings underline the importance of incorporating causality and dynamical system behavior into PINN training strategies, providing a robust framework for solving time-dependent problems with enhanced reliability.