This paper is devoted to a Feynman-Kac formula for general nonlinear time-dependent Schr\"odinger equations. 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。该公式在倒向随机微分方程框架下，统一融入了Fisk-Stratonovich积分与Itô积分。基于此费曼-卡茨表示，我们提出了一种基于深度学习的数值近似方法。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。