Feynman-Kac Formula for Time-Dependent Nonlinear Schrödinger Equations with Applications in Numerical Approximations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915351129751552 |
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| author | Cheung, Hang Qiu, Jinniao Yang, Yang |
| author_facet | Cheung, Hang Qiu, Jinniao Yang, Yang |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Feynman-Kac Formula for Time-Dependent Nonlinear Schrödinger Equations with Applications in Numerical Approximations Cheung, Hang Qiu, Jinniao Yang, Yang Analysis of PDEs Numerical Analysis Probability 81Q05, 60H30, 35Q41, 65C05, 65M12, 65M75 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). |
| title | Feynman-Kac Formula for Time-Dependent Nonlinear Schrödinger Equations with Applications in Numerical Approximations |
| topic | Analysis of PDEs Numerical Analysis Probability 81Q05, 60H30, 35Q41, 65C05, 65M12, 65M75 |
| url | https://arxiv.org/abs/2409.16519 |