Multiscale Modeling and Computation Seminar by Senbao Jiang: Numerical Analysis and Deep Learning Solver of the Non-local Fokker-Planck Equations

Speaker: Senbao Jiang

Title: Numerical Analysis and Deep Learning Solver of the Non-local Fokker-Planck Equations

Abstract: In this talk, we firstly propose and analyze a general arbitrarily high-order modified trapezoidal rule for a class of weakly singular integrals in n dimensions. The admissible class requires the singular part of the integrand in the weakly singular integral satisfies two simple hypotheses and is large enough to contain many fractional type singular kernels. The modified trapezoidal rule is the singularity-punctured trapezoidal rule plus correction terms involving the correction weights for grid points around singularity. Correction weights are determined by enforcing the quadrature rule to exactly evaluate some monomials and solving corresponding linear systems. A long-standing difficulty of these types of methods is establishing the non-singularity of the linear system, despite strong numerical evidence. By using an algebraic-combinatorial argument, we show the non-singularity always holds and prove the general order of convergence of the modified quadrature rule. We present numerical experiments to validate the order of convergence. 

Using the modified trapezoidal rule, we propose trapz-PiNN, a physics-informed neural network for solving the space-fractional Fokker-Planck equations in 2D and 3D. We demonstrate trapz-PiNNs have high expressive power through predicting solutions with low L2 relative error on a variety of numerical examples. Applications to backward problems are also presented.

Multiscale Modeling and Computation Seminar