Regime-Aware Time Weighting for Physics-Informed Neural Networks

TL;DR

This paper presents a theoretically grounded time-weighting strategy for Physics-Informed Neural Networks that adapts to system stability regimes, improving solution accuracy for time-dependent differential equations.

Contribution

It introduces a Lyapunov exponent-based weighting method that automatically adjusts to system dynamics, enhancing PINN performance without extra hyperparameters.

Findings

Improved convergence and accuracy on chaotic and stable systems

Robustness across challenging benchmarks like Lorenz and Burgers' equation

Effective incorporation of system stability into PINN training

Abstract

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.