The aim of this paper is to provide a survey on recent advances on probabilistic numerical methods for nonlinear PDEs, which serve as an alternative to classical deterministic schemes and allow to handle a large class of multidimensional nonlinear problems. These probabilistic schemes are based on the stochastic representation of semilinear PDEs by means of backward SDEs, which can be viewed as...

In this paper, the discrete gradient methods are investigated for ODEs with first integral, and the recursive formula is presented for deriving the high-order numerical methods. We generalize the idea of discrete gradient methods to PDEs and construct the high-order energypreserving numerical methods for multi-symplectic Hamiltonian PDEs. By integrating nonlinear Schrödinger equation, some nume...

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First, the PDE is rewritten its dual stochastic control representation form, and corresponding optimal feedback estimated using a neural network. Next, three different presented to approximate associated value function, i.e., s...

Numerical approximations of fractional PDEs in unbounded domains are considered in this paper. Since their solutions decay slowly with power laws at infinity, a domain truncation approach is not effective as no transparent boundary condition is available. We develop efficient Hermite-collocation and Hermite–Galerkin methods for solving a class of fractional PDEs in unbounded domains directly, a...