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arXiv · 2401.01292

Solving Fokker-Planck equations using the zeros of Fokker-Planck operators and the Feynman-Kac formula

Pinak Mandal·Amit Apte
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Abstract

First we show that physics-informed neural networks are not suitable for a large class of parabolic partial differential equations including the Fokker-Planck equation. Then we devise an algorithm to compute solutions of the Fokker-Planck equation using the zeros of Fokker-Planck operator and the Feynman-Kac formula. The resulting algorithm is mesh-free, highly parallelizable and able to compute solutions pointwise, thus mitigating the curse of dimensionality in a practical sense. We analyze various nuances of this algorithm that are determined by the drift term in the Fokker-Planck equation. We work with problems ranging in dimensions from 2 to 10. We demonstrate that this algorithm requires orders of magnitude fewer trajectories for each point in space when compared to Monte-Carlo. We also prove that under suitable conditions the error that is caused by letting some trajectories (associated with the Feynman-Kac expectation) escape our domain of knowledge is proportional to the fraction of trajectories that escape.

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Pinak Mandal, Amit Apte. 2024-01-02. Solving Fokker-Planck equations using the zeros of Fokker-Planck operators and the Feynman-Kac formula. https://arxiv.org/abs/2401.01292

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