We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solut...

Abstract We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once framework. The underlying PDE model is formulated rather general setting with three unknowns: physical parameter, state nonlinearity. Inspired by advances machine learning, we approximate the nonlinearity via neural network, whose parameters are le...

Many solutions of nonlinear time dependent partial differential equations show particular spatio-temporal patterns, such as traveling waves in one space dimension or spiral and scroll waves in higher space dimensions. The purpose of this paper is to review some recent progress on the analytical and the numerical treatment of such patterns. Particular emphasis is put on symmetries and on the dyn...