Extensions of the deep Galerkin method

• An extension of the deep Galerkin method (DGM) to solve Fokker–Planck PDEs keeping probability density constraints automatically satisfied. A novel application policy iteration algorithm (PIA) together with DGM HJB equations. Additional applications system coupled equations (arising from stochastic games) and mean-field Game PDE (coupled Fokker Planck). Both extensions are applicable multidimensional PDEs. We extend Deep Method introduced in Sirignano Spiliopoulos(2018)[25] a number partial differential (PDEs) that arise context optimal control mean field games. First, we consider where function is constrained be positive integrate unity, as case Our approach involves reparameterizing solution exponential neural network appropriately normalized ensure both requirements This then gives rise nonlinear integro-differential equation (PIDE) integral appearing handled by importance sampling. Secondly, tackle Hamilton–Jacobi–Bellman (HJB) appear problems. The key contribution these approached their unsimplified primal form which includes an optimization problem part equation. for value simultaneously characterizing networks. Training networks performed taking alternating gradient descent steps two functions, technique inspired improvement algorithms (PIA).