A derivative-free localized stochastic method for very high-dimensional semilinear parabolic PDEs
Shuixin Fang, Changtao Sheng, Bihao Su, Tao Zhou
TL;DR
The paper presents a mesh-free, derivative-free, matrix-free localized stochastic framework to solve very high-dimensional semilinear parabolic PDEs by leveraging a martingale-based FBSDE formulation, stochastic forward particle dynamics, local linear regression for gradient reconstruction, and Newton iterations for nonlinear scalar updates. It provides a rigorous error bound of the form $O(\Delta t) + O(\Delta t\,e^{-cM\varepsilon^d})$, showing first-order temporal accuracy with exponential decay in the particle count $M$, and demonstrates scalable performance up to dimension $d=10{,}000$ with linear cost in $N\cdot M$ and $d$. Numerical experiments on Allen–Cahn, Burgers, and Hamilton–Jacobi type equations confirm accuracy, stability, and the method’s robustness to dimensionality, with strong results achieved on a standard laptop. The approach offers a practical, interpretable alternative to mesh-based or neural-network–driven methods, avoiding global bases and gradient computations while enabling high-dimensional PDE solvability and rigorous error control.
Abstract
We develop a mesh-free, derivative-free, matrix-free, and highly parallel localized stochastic method for high-dimensional semilinear parabolic PDEs. The efficiency of the proposed method is built upon four essential components: (i) a martingale formulation of the forward backward stochastic differential equation (FBSDE); (ii) a small scale stochastic particle method for local linear regression (LLR); (iii) a decoupling strategy with a matrix-free solver for the weighted least-squares system used to compute $\nabla u$; (iv) a Newton iteration for solving the univariate nonlinear system in $u$. Unlike traditional deterministic methods that rely on global information, this localized computational scheme not only provides explicit pointwise evaluations of $u$ and $\nabla u$ but, more importantly, is naturally suited for parallelization across particles. In addition, the algorithm avoids the need for spatial meshes and global basis functions required by classical deterministic approaches, as well as the derivative-dependent and lengthy training procedures often encountered in machine learning. More importantly, we rigorously analyze the error bound of the proposed scheme, which is fully explicit in both the particle number $M$ and the time step size $Δt$. Numerical results conducted for problem dimensions ranging from $d=100$ to $d=10000$ consistently verify the efficiency and accuracy of the proposed method. Remarkably, all computations are carried out efficiently on a standard personal computer, without requiring any specialized hardware. These results confirm that the proposed method is built upon a principled design that not only extends the practically solvable range of ultra-high-dimensional PDEs but also maintains rigorous error control and ease of implementation.