MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs

TL;DR

MOD-Net introduces a machine learning framework that combines model-driven regularization with data to efficiently solve both linear and nonlinear PDEs, reducing computational cost and improving accuracy.

Contribution

The paper presents a novel MOD-Net framework that leverages operator representation and limited labeled data to solve PDEs more efficiently than existing neural operators.

Findings

Efficiently solves Poisson and radiative transfer equations

Requires fewer labels for training, reducing computational cost

Successfully applies to nonlinear PDEs like Burgers equation

Abstract

In this paper, we propose a a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. For linear PDEs, we use a DNN to parameterize the Green's function and obtain the neural operator to approximate the solution according to the Green's method. To train the DNN, the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions. For complicated problems, the empirical risk also includes a few labels, which are computed on coarse grid points with cheap computation cost and significantly improves the model accuracy. Intuitively, the labeled dataset works as a regularization in addition to the model constraints. The MOD-Net solves a family of PDEs rather than a specific one and is much more efficient than original neural operator because few expensive labels are required. We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional radiative transfer equation. For nonlinear PDEs, the nonlinear MOD-Net can be similarly used as an ansatz for solving nonlinear PDEs, exemplified by solving several nonlinear PDE problems, such as the Burgers equation.