Using neural networks to approximate the solutions of differential equations

Abstract

In this work neural networks are used to approximate the solutions of multiple differential equa- tions. Three main differential equations are investigated: the mass-spring equation, the heat equation and the Navier-Stokes equations coupled with the continuity equation. Each of these equations are approached differently with the current methods, therefore these equations test the applicability of the newly proposed method on a general differential equation. The results of various experiments performed in this work show that the method is able to approximate to great precision the mass-spring equation as well as the heat equation. However, the method does not perform well on the Navier-Stokes equations coupled with the continuity equation. The main advantage of this method is that meshes used by this method can often be very small compared to other approximation algorithms. When a slight decrease in performance is allowed, the method can be made completely mesh free. A huge drawback of the method is that there is no guarantee that the method will generate a reasonable approximation, for a given neural network configuration. This makes it necessary to look for a good configuration of the neural network. At this moment, this can only be done using trial and error and is therefore very time-consuming. Sometimes no good configuration is found at all. Another drawback is that the generated approximation does not satisfy the boundary and initial conditions. Multiple ways are proposed in this work to tackle this problem, but they introduce much more complexity and a performance decrease. The neural networks used in this work are generated using software called PyTorch, which comes with an automatic analytic differentiation tool. This part of PyTorch makes it possible to approximate differential equations using neural networks.