2024/09/25 by Cheung, Hang, Qiu, Jinniao, Yang, Yang
#35Q41 #60H30 #65C05 #65M12 #65M75 #81Q05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Probability (math.PR)
paper · doi:10.48550/arxiv.2409.16519
In this paper, we present a novel Feynman-Kac formula and investigate learning-based methods for approximating general nonlinear time-dependent Schrödinger equations which may be high-dimensional. Our formulation integrates both the Fisk-Stratonovich and Itô 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 Schrödinger 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).