# Regime-Aware Time Weighting for Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2407.21642 · HTML: https://mlsanity.com/paper/2407.21642/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 26 references that are in the map.
- First posted on arXiv 31 Jul 2024.

## Builds on (most cited first, 12 of 12)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems](https://mlsanity.com/paper/2006.11144/) (2020)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations](https://mlsanity.com/paper/2205.09332/) (2022)
- [An extended physics informed neural network for preliminary analysis of parametric optimal control problems](https://mlsanity.com/paper/2110.13530/) (2021)
- [Parareal with a physics-informed neural network as coarse propagator](https://mlsanity.com/paper/2303.03848/) (2023)
- [Huber-energy measure quantization](https://mlsanity.com/paper/2212.08162/) (2022)
- [Optimal time sampling in physics-informed neural networks](https://mlsanity.com/paper/2404.18780/) (2024)

## Abstract (arXiv)

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.