  Solving high dimensional partial differential equations (PDEs) has
historically posed a considerable challenge when utilizing conventional
numerical methods, such as those involving domain meshes. Recent advancements
in the field have seen the emergence of neural PDE solvers, leveraging deep
networks to effectively tackle high dimensional PDE problems. This study
introduces Inf-SupNet, a model-based unsupervised learning approach designed to
acquire solutions for a specific category of elliptic PDEs. The fundamental
concept behind Inf-SupNet involves incorporating the inf-sup formulation of the
underlying PDE into the loss function. The analysis reveals that the global
solution error can be bounded by the sum of three distinct errors: the
numerical integration error, the duality gap of the loss function (training
error), and the neural network approximation error for functions within Sobolev
spaces. To validate the efficacy of the proposed method, numerical experiments
conducted in high dimensions demonstrate its stability and accuracy across
various boundary conditions, as well as for both semi-linear and nonlinear
PDEs.
