Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective

TL;DR

This paper introduces second-order optimization techniques, especially the SOAP method, to effectively resolve gradient conflicts in physics-informed neural networks, achieving state-of-the-art results on complex PDE benchmarks including turbulent flows.

Contribution

It provides a theoretical analysis of gradient conflicts in PINNs and demonstrates how second-order methods like SOAP can resolve these conflicts for improved performance.

Findings

SOAP approximates the Hessian preconditioner efficiently

Achieved 2-10x accuracy improvements on PDE benchmarks

Successfully applied to turbulent flows with Reynolds numbers up to 10,000

Abstract

Multi-task learning through composite loss functions is fundamental to modern deep learning, yet optimizing competing objectives remains challenging. We present new theoretical and practical approaches for addressing directional conflicts between loss terms, demonstrating their effectiveness in physics-informed neural networks (PINNs) where such conflicts are particularly challenging to resolve. Through theoretical analysis, we demonstrate how these conflicts limit first-order methods and show that second-order optimization naturally resolves them through implicit gradient alignment. We prove that SOAP, a recently proposed quasi-Newton method, efficiently approximates the Hessian preconditioner, enabling breakthrough performance in PINNs: state-of-the-art results on 10 challenging PDE benchmarks, including the first successful application to turbulent flows with Reynolds numbers up to 10,000, with 2-10x accuracy improvements over existing methods. We also introduce a novel gradient alignment score that generalizes cosine similarity to multiple gradients, providing a practical tool for analyzing optimization dynamics. Our findings establish frameworks for understanding and resolving gradient conflicts, with broad implications for optimization beyond scientific computing.