PDE analysis for sampling dynamics and generative models

Jianfeng Lu (Simons Participant)
Duke University
Mathematics

We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator, whose solution can be represented by Feynman-Kac formula
in terms of forward-backward stochastic differential equations. The method shares a similar spirit with diffusion Monte Carlo but augments a direct approximation to the eigenfunction through neural-network ansatz. The criterion of fixed point provides a natural loss function to search for parameters via optimization. Our approach is able to provide accurate eigenvalue and eigenfunction approximations in several numerical examples, including Fokker-Planck operator, linear and nonlinear Schrödinger operators in high dimensions.
(joint work with Jiequn Han and Mo Zhou)

Presentation (PDF File)

Back to Workshop II: PDE and Inverse Problem Methods in Machine Learning