MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs (Communications in Computational Physics, 2022)
In this paper, we propose a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solvePDEs 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 toapproximate 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 thevariational formulation of the governing equation and boundary conditions. For complicated problems, the empirical risk also includes a few labels, which are computed oncoarse grid points with cheap computation cost and significantly improves the modelaccuracy. Intuitively, the labeled dataset works as a regularization in addition to themodel constraints. The MOD-Net solves a family of PDEs rather than a specific oneand is much more efficient than original neural operator because few expensive labels are required. We numerically show MOD-Net is very efficient in solving Poissonequation and one-dimensional radiative transfer equation. For nonlinear PDEs, thenonlinear 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.