Feynman-Kac Formula for Time-Dependent Nonlinear Schr\"odinger Equations with Applications in Numerical Approximations

Abstract

In this paper, we present a novel Feynman-Kac formula and investigate learning-based methods for approximating general nonlinear time-dependent Schr\"odinger equations which may be high-dimensional. Our formulation integrates both the Fisk-Stratonovich and It\^o integrals within the framework of backward stochastic differential equations (BSDEs). Utilizing this Feynman-Kac representation, we propose learning-based approaches for numerical approximations. To demonstrate the accuracy and effectiveness of the proposed method, we conduct numerical experiments in both low- and high-dimensional settings, complemented by a convergence analysis. These results address the open problem concerning deep-BSDE methods for numerical approximations of high-dimensional time-dependent nonlinear Schr\"odinger equations (cf. [Proc. Natl. Acad. Sci. 15 (2018), pp. 8505-8510] and [Frontiers Sci. Awards Math. (2024), pp. 1-14] by Han, Jentzen, and E).