基于数值微分的物理信息神经网络方法在非矩形区域上的应用
Application of Physics Informed Neural Net-work Method Based on Numerical Differentiation in Non-Rectangular Regions

物理信息神经网络(PINN)是一种新兴的数据驱动的偏微分方程数值求解框架。该类方法不需要进行网格划分，从而避免多余的计算消耗，这使得其对高维问题有更好的拓展性。但传统神经网络技术基于自动微分实现，依赖于大量的训练配点，容易引起梯度爆炸或梯度消失现象，且不能在非矩形区域上直接进行求解。文章简要阐述了基于数值微分的物理信息神经网络的基本原理，使用数值微分替代自动微分以避免使用大量的训练配点，避免梯度爆炸或梯度消失发生。同时，利用区域分解思想对非矩形区域进行划分，使划分后的子区域可以使用PINN直接求解，最终用一个算例验证了该方法的可行性。结果表明，基于数值微分的物理信息神经网络可以求解非矩形区域上的问题。
Physics-Informed Neural Networks (PINN) is a novel data-driven numerical framework for solving partial differential equations. Since this type of technique does not require meshing, it consumes fewer computer resources and is more scalable for high-dimensional issues. Traditional neural network technology, on the other hand, relies on a high number of training points and depends on automatic differentiation, which makes it easy to produce gradient expansion or gradient disappearance and prevents direct solution on non-rectangular areas. In order to avoid the requirement for many training points and to prevent gradient explosion or gradient disappearance, this paper uses numerical differentiation rather than automatic differentiation to briefly explain the fundamental concepts of Physics-Informed Neural Networks. In order for the divided subregion to be di-rectly solved using PINN, the non-rectangular area is simultaneously divided using the domain decomposition idea. Finally, a calculation scenario is utilized to confirm the viability of the method. The outcomes demonstrate the ability of the numerically differentiated, Physics-Informed Neural Networks to address issues in non-rectangular locations.