改进优化算法，让物理神经网络更快更准地求解复杂方程。
Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks
- 引入自然梯度与自缩放拟牛顿法，提升PINN收敛速度。
- 在亥姆霍兹方程、欧拉方程等复杂问题上实现高精度求解。
- 支持批量训练扩展，适合大规模数据驱动的科学建模。
高效且稳健的优化对神经网络至关重要，使科学机器学习模型能快速收敛至极高精度，忠实捕捉由微分方程支配的复杂物理行为。本文提出先进的优化策略，加速物理信息神经网络（PINNs）在挑战性偏微分方程（PDEs）和常微分方程（ODEs）上的收敛。具体而言，我们实现了自然梯度（NG）、自缩放拟牛顿法（Self-Scaling BFGS）和布罗伊登（Broyden）优化器的高效版本，并在亥姆霍兹方程、斯托克斯流、无粘伯格斯方程、高速流动欧拉方程以及药代动力学/药效学中的刚性ODE问题上验证其性能。此外，我们还提出了基于PINN的新方法求解无粘伯格斯方程和欧拉方程，并与高阶数值方法对比，进行严格公平评估。最后，我们解决了这些拟牛顿优化器在批量训练中的可扩展性问题，为大规模数据驱动问题提供高效可扩展的解决方案。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。