Physics-informed neural networks exploiting weak formulations of PDEs

Neural networks can be useful tools for approximating solutions of partial derivative equations (PDEs). The literature on numerical solutions to these problems has produced robust methods, such as finite element methods (FEM), which are characterized by high computational cost. By constructing suitable neural networks and implementing the differential operator structure of the equation in the training process, it is possible to obtain very efficient surrogate models that are independent of the input functions.