Understanding and mitigating gradient pathologies in physics-informed neural networks

TL;DR

This paper investigates the training challenges of physics-informed neural networks, identifies issues caused by numerical stiffness, and proposes new training algorithms and architectures that significantly enhance predictive accuracy in physical systems.

Contribution

It introduces a gradient-based learning rate annealing algorithm and a resilient neural network architecture to address gradient pathologies in physics-informed neural networks.

Findings

Improved predictive accuracy by 50-100x across various physics problems.

Identified numerical stiffness as a key failure mode in PINNs.

Proposed methods effectively mitigate gradient pathologies.

Abstract

The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers of the empirical risk loss. Physics-informed neural networks is an example of this philosophy in which the outputs of deep neural networks are constrained to approximately satisfy a given set of partial differential equations. In this work we review recent advances in scientific machine learning with a specific focus on the effectiveness of physics-informed neural networks in predicting outcomes of physical systems and discovering hidden physics from noisy data. We will also identify and analyze a fundamental mode of failure of such approaches that is related to numerical stiffness leading to unbalanced back-propagated gradients during model training. To address this limitation we present a learning rate annealing algorithm that utilizes gradient statistics during model training to balance the interplay between different terms in composite loss functions. We also propose a novel neural network architecture that is more resilient to such gradient pathologies. Taken together, our developments provide new insights into the training of constrained neural networks and consistently improve the predictive accuracy of physics-informed neural networks by a factor of 50-100x across a range of problems in computational physics. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/GradientPathologiesPINNs}.