Past Event: Oden Institute Seminar

Understanding and mitigating gradient flow pathologies in physics-informed neural networks

Paris Perdikaris, Assistant Professor, Mechanical Engineering and Applied Mechanics (MEAM), Penn

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 also identify and analyze a fundamental mode of failure of such approaches that is related to numerical stiffness in the gradient flow dynamics leading to unbalanced back-propagated gradients during model training via gradient descent. 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. Bio Paris Perdikaris is an Assistant Professor in the Department of Mechanical Engineering and Applied Mechanics at the University of Pennsylvania. His work spans a wide range of areas in computational science and engineering, with a particular focus on the analysis and design of complex physical and biological systems using machine learning, stochastic modeling, computational mechanics, and high-performance computing. Current research thrusts include physics-informed machine learning, uncertainty quantification in deep learning, engineering design optimization, and data-driven non-invasive medical diagnostics.