TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method

doi:10.1088/1674-1056/ad21f3

Abstract Physics-informed neural networks (PINNs) have become an attractive machine learning framework for obtaining solutions to partial differential equations (PDEs). PINNs embed initial, boundary, and PDE constraints into the loss function. The performance of PINNs is generally affected by both training and sampling. Specifically, training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs, and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished. However, a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category, namely, time-dependent PDEs, where temporal information plays a key role in the algorithms used. There is one method, called Causal PINN, that considers temporal causality at the training level but not special temporal utilization at the sampling level. Incorporating temporal knowledge into sampling remains to be studied. To fill this gap, we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality. By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain, we provide a practical solution by incorporating temporal information into sampling. Numerical experiments of several nonlinear time-dependent PDEs, including the Cahn-Hilliard, Korteweg-de Vries, Allen-Cahn and wave equations, show that our proposed sampling method can improve the performance. We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods, especially when points are limited.

Keywords: partial differential equation physics-informed neural networks residual-based adaptive sampling temporal causality

Received: 18 October 2023
Revised: 08 January 2024
Accepted manuscript online: 24 January 2024

PACS:	07.05.Mh	(Neural networks, fuzzy logic, artificial intelligence)
	02.60.Cb	(Numerical simulation; solution of equations)
	02.30.Jr	(Partial differential equations)
	84.35.+i	(Neural networks)

Fund: Project supported by the Key National Natural Science Foundation of China (Grant No. 62136005) and the National Natural Science Foundation of China (Grant Nos. 61922087, 61906201, and 62006238).

Corresponding Authors: Chenping Hou
E-mail: hcpnudt@hotmail.com

Cite this article:

Jia Guo(郭嘉), Haifeng Wang(王海峰), Shilin Gu(古仕林), and Chenping Hou(侯臣平) TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method 2024 Chin. Phys. B 33 050701

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TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method

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