This paper is devoted to a Feynman-Kac formula for general nonlinear time-dependent Schr\"odinger equations with applications in numerical approximations. Our formulation integrates both the Fisk-Stratonovich and It\^o integrals within the framework of backward stochastic differential equations. Utilizing this Feynman-Kac representation, we propose a deep-learning-based approach for numerical approximation. Numerical experiments are performed to validate the accuracy and efficiency of our method, and a convergence analysis is provided to support the results.

翻译：本文致力于建立一般非线性含时薛定谔方程的费曼-卡茨公式，并探讨其在数值逼近中的应用。我们的理论框架在倒向随机微分方程体系中融合了费斯克-斯特拉托诺维奇积分与伊藤积分。基于该费曼-卡茨表示，我们提出了一种基于深度学习的数值逼近方法。通过数值实验验证了所提方法的精度与效率，并提供了收敛性分析以支撑所得结论。