一般化
偏微分方程
数学
非线性系统
应用数学
形式主义(音乐)
人工神经网络
误差分析
近似误差
泛化误差
理论(学习稳定性)
计算机科学
数学分析
人工智能
机器学习
物理
艺术
视觉艺术
量子力学
音乐剧
作者
Siddhartha Mishra,Roberto Molinaro
出处
期刊:Ima Journal of Numerical Analysis
日期:2022-01-14
卷期号:43 (1): 1-43
被引量:51
标识
DOI:10.1093/imanum/drab093
摘要
Abstract Physics-informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of partial differential equations (PDEs). We provide upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.
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