A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks

TL;DR

This paper introduces Neural IVP, a novel neural network-based method for solving initial value PDEs that maintains stability and scalability by preventing ill-conditioning and reducing computational complexity.

Contribution

The paper presents Neural IVP, an ODE-based solver that overcomes stability and scalability issues of previous methods for neural PDE solutions.

Findings

Neural IVP prevents ill-conditioning during PDE evolution.

The method scales linearly with network size, enabling larger models.

Neural IVP effectively solves complex PDE initial value problems.

Abstract

Unlike conventional grid and mesh based methods for solving partial differential equations (PDEs), neural networks have the potential to break the curse of dimensionality, providing approximate solutions to problems where using classical solvers is difficult or impossible. While global minimization of the PDE residual over the network parameters works well for boundary value problems, catastrophic forgetting impairs the applicability of this approach to initial value problems (IVPs). In an alternative local-in-time approach, the optimization problem can be converted into an ordinary differential equation (ODE) on the network parameters and the solution propagated forward in time; however, we demonstrate that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors. Second, as the ODE methods scale cubically with the number of model parameters, they are restricted to small neural networks, significantly limiting their ability to represent intricate PDE initial conditions and solutions. Building on these insights, we develop Neural IVP, an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters, enabling us to evolve the dynamics of challenging PDEs with neural networks.