Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective
This addresses optimization challenges in PINNs for scientific computing, with incremental implications for broader multi-task learning.
The paper tackled the problem of directional conflicts between loss terms in multi-task learning, particularly in physics-informed neural networks (PINNs), by showing that second-order optimization resolves these conflicts, achieving state-of-the-art results on 10 PDE benchmarks with 2-10x accuracy improvements and enabling applications to turbulent flows up to Reynolds numbers of 10,000.
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.