Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights

Robust Variational Physics-Informed Neural Networks

“…Thus, a more general framework allowing for a less restrictive assumption must be considered. To this end, following [9], we start by considering the following definition:…”

“…The difference with classical VPINNs is that our strategy is robust and independent of the choice of the basis functions. Other works employ similar ideas closely related to minimum residual methods [11,9]. In [2], the authors approximate the solution of symmetric and positive definite problems employing the discrete weak residual of the variational problem.…”

A Deep Double Ritz Method (D2rm) for Solving Partial Differential Equations Using Neural Networks

“…In the last decade, Neural Networks (NNs) have emerged as a powerful alternative for solving Partial Differential Equations (PDEs). For example, [1][2][3][4] employ NNs to generate optimal meshes for later solving PDEs by a Finite Element Method (FEM), [5] proposes a Deep-FEM method that mimics mesh-refinements within the NN architecture, [6] employs a NN to generate the mesh via r -adaptivity, and [7,8] use NNs to improve discrete weak formulations. Alternatively, there exist approaches to directly represent the PDE solutions via NNs.…”

Memory-Based Monte Carlo Integration for Solving Partial Differential Equations Using Neural Networks

“…In general, VΘ is non-convex and non-closed, possibly preventing the uniqueness and existence of minimizers (e.g., see Example 2.3 in[1]). …”