Analysis of the Physics-Informed Neural Network Approach to Solving Diffusion Equation

The application of physics-informed neural networks for solving the differential equation of parabolic type is considered. The influence of the neural network structure, optimization algorithms, software and processors’ types on the learning process and accuracy of the solution of the two-dimensional diffusion problem is investigated using computational experiments. The accuracy of the neural network solution is evaluated on the basis of comparison with the numerical solution. Based on the analysis of the results of multivariate calculations, it is shown that if the initial condition is included into the loss function expression, the accuracy of the solution increases significantly. The choice of the activation functions affects both the dynamics of the learning process and the accuracy of the solution, with the hyperbolic tangent being the most effective for the considered case. The structure of the neural network, which contains four hidden layers with thirty-two neurons in each layer, turned out to be the most optimal. Structures with more or fewer layers are less amenable to machine learning and their use does not improve (and in some cases worsens) the accuracy of the solution construction. It is also shown that the LBFGS optimization algorithm is the most suitable for training multilayer neural networks in this problem. When the structure of the neural network is optimal, the use of the PyTorch library and its algorithms of parallel computations on the integrated and external graphics processors allows to accelerate the training process ten times. The optimal size of the training set of about one hundred thousand points has been established, which gives the smallest values of the loss function and error of the neural network solution of the diffusion problem are achieved.