物理信息随机特征神经网络求解偏微分方程
Physics Informed Random Feature Neural Networks for Solving PDEs
- Illinois Institute of Technology（伊利诺伊理工学院）
- University of Arkansas（阿肯色大学）
- University of Houston（休斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出物理信息随机特征方法求解PDE，降低计算复杂度，通过严格误差分析和数值测试验证其能有效对抗谱偏差。
AI中文摘要：
基于机器学习的偏微分方程（PDE）求解器近年来引起了广泛关注。该领域的大部分进展由深度神经网络（如物理信息神经网络（PINNs））和核方法（如物理信息高斯过程）推动。我们提出了一种物理信息随机特征方法，以对抗PINN类求解器在求解某类PDE时所面临的谱偏差问题。随机特征方法最初被提出用于近似大规模核机器，可视为一种专门的随机神经网络。与其他需要大量配点的最新PINN类求解器相比，我们提出的方法降低了计算复杂度。在本文中，我们进行了严格的逼近误差分析，并推导了关于$H^1$范数的高概率误差界。我们提供了广泛的数值测试来验证我们关于误差衰减率的理论保证，以及若干对比测试，以展示我们声称的对抗这些基于深度学习方法中谱偏差的能力。
英文摘要：
Machine learning-based partial differential equations (PDEs) solvers have attracted significant attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and kernel method (such as physics-informed Gaussian Processes). We introduce a physics-informed random feature method for countering part of the spectral bias which PINN-based solvers are facing for a certain class of PDEs. Random feature method was originally proposed to approximate large-scale kernel machines and can be viewed as a specialized randomized neural network. Compared to other state-of-the-art PINN-based solvers which require a large number of collocation points, our proposed method reduces the computational complexity. In this paper, we develop a rigorous approximation error analysis and derive high-probability error bounds on the $H^1$ norm. We provide extensive numerical tests for verifying our theoretical guarantees on error decay rates, as well as several comparison tests to showcase our claimed capability for combating spectral bias in these deep learning based methods.
