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Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective
🤖 AI Summary
To address training failure in Physics-Informed Neural Networks (PINNs) caused by gradient direction conflicts among multi-task loss components, this work theoretically establishes—under a second-order optimization perspective—that Hessian preconditioning inherently alleviates such conflicts. Building on this insight, we propose SOAP, a quasi-Newton method that efficiently approximates the Hessian preconditioner, and introduce a Multi-Gradient Alignment Score (MGAS), an extended cosine similarity metric quantifying alignment across task gradients. Evaluated on ten challenging PDE benchmarks, SOAP achieves state-of-the-art performance: it is the first method to stably solve turbulent flow problems at Reynolds numbers up to Re = 10,000, and delivers 2–10× higher accuracy than existing approaches. These advances substantially broaden the applicability of PINNs to strongly nonlinear, complex physical systems.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Multi-task Learning
Conflicting Objective Functions
Gradient Alignment
Innovation
Methods, ideas, or system contributions that make the work stand out.
Multi-task Learning
Physical Information Neural Networks (PINNs)
Gradient Alignment
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