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.
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