论文标题

一种用于物理信息神经网络的新型自适应因果抽样方法

A Novel Adaptive Causal Sampling Method for Physics-Informed Neural Networks

论文作者

Guo, Jia, Wang, Haifeng, Hou, Chenping

论文摘要

物理信息神经网络(PINN)已成为一种有吸引力的机器学习方法,用于获得偏微分方程(PDES)的解决方案。训练PINN可以看作是一项半监督的学习任务,其中仅在解决前向问题时只能获得初始点和边界点的精确值,在整个时空域结构域搭配点中,可以在没有精确标签的情况下进行采样,这带来了训练困难。因此,搭配点和抽样方法的选择对于训练PINN非常重要。现有的采样方法包括固定类型和动态类型,在更流行的后者中,采样通常由PDE残留损失控制。我们指出,仅考虑自适应抽样和抽样中的残余损失不足以遵守时间因果关系。我们进一步将时间因果关系引入自适应抽样中,并提出了一种新型的适应性因果采样方法,以提高PINN的性能和效率。具有高阶导数和强非线性(包括Cahn Hilliard和KDV方程)的几个PDE的数值实验表明,所提出的采样方法可以改善PINN的性能,而PINN的性能很少。我们证明,通过使用这种相对简单的采样方法,与最新的结果相比,可以提高预测性能高达两个数量级,而几乎没有额外的计算成本,尤其是当点有限时。

Physics-Informed Neural Networks (PINNs) have become a kind of attractive machine learning method for obtaining solutions of partial differential equations (PDEs). Training PINNs can be seen as a semi-supervised learning task, in which only exact values of initial and boundary points can be obtained in solving forward problems, and in the whole spatio-temporal domain collocation points are sampled without exact labels, which brings training difficulties. Thus the selection of collocation points and sampling methods are quite crucial in training PINNs. Existing sampling methods include fixed and dynamic types, and in the more popular latter one, sampling is usually controlled by PDE residual loss. We point out that it is not sufficient to only consider the residual loss in adaptive sampling and sampling should obey temporal causality. We further introduce temporal causality into adaptive sampling and propose a novel adaptive causal sampling method to improve the performance and efficiency of PINNs. Numerical experiments of several PDEs with high-order derivatives and strong nonlinearity, including Cahn Hilliard and KdV equations, show that the proposed sampling method can improve the performance of PINNs with few collocation points. We demonstrate that by utilizing such a relatively simple sampling method, prediction performance can be improved up to two orders of magnitude compared with state-of-the-art results with almost no extra computation cost, especially when points are limited.

扫码加入交流群

加入微信交流群

微信交流群二维码

扫码加入学术交流群,获取更多资源