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In this paper, we propose a a machine learning approach viamodel-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven bya model to solve PDEs based on operator representation with regularization fromdata. For linear PDEs, we use a DNN to parameterize the Green's function andobtain the neural operator to approximate the solution according to the Green'smethod. To train the DNN, the empirical risk consists of the mean squared losswith the least square formulation or the variational formulation of thegoverning equation and boundary conditions. For complicated problems, theempirical risk also includes a few labels, which are computed on coarse gridpoints with cheap computation cost and significantly improves the modelaccuracy. Intuitively, the labeled dataset works as a regularization inaddition to the model constraints. The MOD-Net solves a family of PDEs ratherthan a specific one and is much more efficient than original neural operatorbecause few expensive labels are required. We numerically show MOD-Net is veryefficient in solving Poisson equation and one-dimensional radiative transferequation. For nonlinear PDEs, the nonlinear MOD-Net can be similarly used as anansatz for solving nonlinear PDEs, exemplified by solving several nonlinear PDEproblems, such as the Burgers equation.

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