Solving Partial Differential Equations with Neural Networks

TL;DR

This paper introduces a neural network-based method for solving PDEs that offers continuous, differentiable solutions, generalizes across parameters, and could enable real-time physics simulations without supercomputers.

Contribution

It proposes a novel neural network approach for PDEs that replaces traditional discretization methods, allowing for continuous solutions and parameter generalization.

Findings

Neural networks can approximate PDE solutions continuously over the domain.

The method reduces computational time by avoiding repeated calculations for different parameters.

Potential to enable real-time physics simulations and geometry optimization.

Abstract

Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many others. In some exceptional cases an analytical solution to the PDEs exists, but in the vast majority of the applications some kind of numerical approximation has to be computed. In this work, an alternative approach is proposed using neural networks (NNs) as the approximation function for the PDEs. Unlike traditional numerical methods, NNs have the property to be able to approximate any function given enough parameters. Moreover, these solutions are continuous and derivable over the entire domain removing the need for discretization. Another advantage that NNs as function approximations provide is the ability to include the free-parameters in the process of finding the solution. As a result, the solution can generalize to a range of situations instead of a particular case, avoiding the need of performing new calculations every time a parameter is changed dramatically decreasing the optimization time. We believe that the presented method has the potential to disrupt the physics simulation field enabling real-time physics simulation and geometry optimization without the need of big supercomputers to perform expensive and time consuming simulations