Physics-informed neural networks (PINNs) usually confront significant difficulties to accurately solve partial differential equations (PDEs) due to many pathologies caused by gradient failures during training process. Particularly, the regular PINN approaches can obtain good predictions for trivial problems but may fail to learn solutions when their PDE coefficients and/or physical domain sizes vary. To overcome the abovementioned restrictions, a hierarchically normalized PINN (hnPINN) is devised. The key idea of the proposed hnPINN is, on the one hand, to transform the original PDE system into one of two proposed dimensionless forms to alleviate the negative effects of PDE coefficients and domain size; on the other hand, to use secondary output scalers to flexibly calibrate the gradient flow for training effectively and improving the solution preciseness. The determination of the secondary output scaler is formulated by a heuristic framework inspired from theoretical analyses on the hnPINN gradient flow. The obtained results from some typical PDEs and common problems in solid mechanics strongly confirm the high effectiveness of the hnPINN in practice.
Source codes of each example in our manuscript are provided in each folder. To implement the traditional ndPINN and the proposed hnPINN, the user just needs to change the parameter "alpha" at the head of main file. In particular, alpha = 1 is for the ndPINN and alpha = specific value is for the hnPINN, in that how to determine the specific value was described problem-by-problem in the manuscript. It is suggested that the users should read thoroughly our paper presented in the reference below before using the codes. The structure of our codes is inspired from a demo of MathWorks, the users can additionally refer the link as below
Thang Le-Duc, Deep Learning Architecture Research Center, Sejong University, email: le.duc.thang0312@gmail.com; thangld@sju.ac.kr
Thang Le-Duc, Seunghye Lee, H. Nguyen-Xuan, and Jaehong Lee. "A hierarchically normalized physics-informed neural network for solving differential equations: application for solid mechanics problems." Engineering Applications of Artificial Intelligence (2024).