Neural Control of Parametric Solutions for High-dimensional Evolution PDEs

TL;DR

This paper introduces a neural network-based control framework to efficiently approximate solution operators for high-dimensional evolution PDEs, significantly reducing computational costs while maintaining accuracy.

Contribution

The paper presents a novel control-based neural network approach to approximate PDE solution operators, enabling efficient handling of high-dimensional evolution PDEs with arbitrary initial conditions.

Findings

Effective approximation of solution operators for high-dimensional PDEs.

Reduced computational cost compared to traditional methods.

Successful numerical experiments demonstrating accuracy and efficiency.

Abstract

We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the solution of a given PDE, we realize that the evolution of the model parameter is a control problem in the parameter space. Based on this observation, we propose to approximate the solution operator of the PDE by learning the control vector field in the parameter space. From any initial value, this control field can steer the parameter to generate a trajectory such that the corresponding reduced-order model solves the PDE. This allows for substantially reduced computational cost to solve the evolution PDE with arbitrary initial conditions. We also develop comprehensive error analysis for the proposed method when solving a large class of semilinear parabolic PDEs. Numerical experiments on different high-dimensional evolution PDEs with various initial conditions demonstrate the promising results of the proposed method.