Machine Learning Solvers of Partial Differential Equations

Discretization-based solvers of PDEs, such as the finite element methods described in Chap. 4 , are applicable across domains, boundary conditions, and coefficients. In contrast, while it is possible to develop machine learning approaches as “solvers” of PDEs, the nature of data-informed training means that there is a limited generalizability (over domains, boundary conditions, and coefficients). One such class of machine learning approaches are the so-called surrogate models. The standard type of surrogate is trained on direct numerical simulations of PDEs—with field solutions as labels—and applied to one specific boundary value problem. A different approach is the label-free machine learning of solvers. This approach is centered on a loss function incorporating the PDE and boundary conditions in residual form. However, they too do not naturally extend to problems with unseen boundary conditions, and the accuracy of field solutions remains strongly dependent on how similar the problem domain is to that on which the solver was trained.