PDE-aware Optimizer for Physics-informed Neural Networks

TL;DR

This paper introduces a PDE-aware optimizer for Physics-Informed Neural Networks that improves training stability and accuracy by adaptively balancing PDE residual gradients without high computational costs.

Contribution

The proposed optimizer adaptively adjusts parameter updates based on PDE residual gradient variance, enhancing PINNs training stability and accuracy compared to standard methods.

Findings

Achieves smoother convergence and lower errors than Adam and SOAP.

Performs well on 1D Burgers', Allen-Cahn, and KdV equations.

Improves stability in PINNs training, especially in regions with sharp gradients.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical constraints into the loss function. However, standard optimizers such as Adam often struggle to balance competing loss terms, particularly in stiff or ill-conditioned systems. In this work, we propose a PDE-aware optimizer that adapts parameter updates based on the variance of per-sample PDE residual gradients. This method addresses gradient misalignment without incurring the heavy computational costs of second-order optimizers such as SOAP. We benchmark the PDE-aware optimizer against Adam and SOAP on 1D Burgers', Allen-Cahn and Korteweg-de Vries(KdV) equations. Across both PDEs, the PDE-aware optimizer achieves smoother convergence and lower absolute errors, particularly in regions with sharp gradients. Our results demonstrate the effectiveness of PDE residual-aware adaptivity in enhancing stability in PINNs training. While promising, further scaling on larger architectures and hardware accelerators remains an important direction for future research.