Physics-Informed Neural Networks and their Implementation in MATLAB

Abstract

An analysis was made of physics-informed neural networks used to solve partial differential equations. The prospects for the implementation of physics-informed neural networks in the MATLAB system are shown. An algorithm for solving partial differential equations in MATLAB using physics-informed neural networks has been developed. On the example of a model problem described by the Poisson equation, physics-informed neural networks were implemented and studied, which showed that MATLAB can be successfully used to implement such networks. MATLAB made it possible to solve the Poisson equation up to the mean square value of the loss function equal to 0.01. The best results were obtained by networks with a small number of layers (3–4) and a sufficiently large number of neurons in each layer (50–100). Comparison with known results showed that MATLAB was inferior to TensorFlow in terms of learning speed. The application of the Optimization Toolbox MATLAB for the implementation of the L-BFGS quasi-Newtonian learning algorithm for physics-informed neural networks was studied. The quasi-Newtonian algorithm makes it possible to increase the accuracy of solving the problem, but requires a lot of training time. As further research, it is recommended to expand the capabilities of the Deep Learning Toolbox by including quasi-Newtonian learning algorithms, in particular, the Levenberg-Marquard algorithm, and new neural network architectures, for example, networks of radial basis functions.