Feynman-Kac Formula for Nonlinear Schr\"odinger Equations with Applications in Numerical Approximations
<|reference_start|>Feynman-Kac Formula for Nonlinear Schr\"odinger Equations with Applications in Numerical Approximations: 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.<|reference_end|>
@article{cheung2024feynman-kac,
    title={Feynman-Kac Formula for Nonlinear Schr\"odinger Equations with
  Applications in Numerical Approximations},
    author={Hang Cheung, Jinniao Qiu, Yang Yang},
    journal={arXiv preprint arXiv:2409.16519},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.16519},
    primaryClass={math.AP cs.NA math.NA math.PR}
}