Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks

2026-04-06

AI Summary

This work addresses the slow convergence and low accuracy of physics-informed neural networks (PINNs) when solving complex partial differential equations and stiff ordinary differential equations. To overcome these limitations, the authors propose a curvature-aware optimization framework that introduces natural gradient, self-scaled BFGS, and Broyden-class quasi-Newton optimizers tailored for PINNs, along with an efficient extension to batch training. The proposed approach significantly accelerates convergence and enhances solution accuracy across a range of challenging problems—including the Helmholtz equation, Stokes flow, inviscid Burgers’ equation, high-speed Euler equations, and stiff pharmacokinetic ODEs—yielding results in close agreement with high-order numerical methods.

Abstract

Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self-Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high-speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN-based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high-order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi-Newton optimizers for batched training, enabling efficient and scalable solutions for large data-driven problems.

Problem

• physics-informed neural networks
• optimization
• differential equations
• convergence
• high-accuracy

Innovation

• Physics-Informed Neural Networks
• Natural Gradient
• Quasi-Newton Optimization
• Curvature-Aware Optimization
• Scalable Training