Numerical Solution Method for Stochastic Partial Differential Equations Based on the Feynman-Kac Formula and Random Forest Regression

The Feynman-Kac formula establishes a crucial connection between stochastic processes and partial differential equations (PDEs). By expressing the solution of a PDE as the expected value of a stochastic variable, this formula provides an innovative approach to numerically solving PDEs and can be extended to compute the expected solutions of stochastic partial differential equations (SPDEs). However, in practical applications, the numerical stability of this formula is often affected by the range of stochastic variable values. When the stochastic variable takes excessively large or small values, computational errors may increase sharply, potentially rendering the formula ineffective. To address this issue, this paper proposes an adaptive algorithm that dynamically adjusts computational parameters, enhancing the applicability of the Feynman-Kac formula to a broader class of problems. Specifically, the algorithm adaptively optimizes the sampling strategy based on the range of stochastic variable values, ensuring computational stability and convergence. Furthermore, this paper extends the adaptive algorithm to the solution of SPDEs and integrates it with the random forest method to achieve data-driven fitting and approximation of the expected solution.