用自适应采样和神经网络求解高维福克-普朗克方程，精度高且效率不随维度下降。
Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations
- 将二阶福克-普朗克方程转为一阶连续性方程，用神经网络逼近概率流。
- 在100维问题上仍保持稳定，误差由残差损失加权控制。
- 适合高维随机动力系统模拟，尤其适用于非高斯分布场景。
高维福克-普朗克（FP）方程求解在计算物理与随机动力学中仍具挑战性，受限于维数灾难、无界域及复杂概率结构。本文提出自适应概率流残差最小化（A-PFRM）方法：将二阶FP方程重构为关联概率流常微分方程的一阶连续性方程，构建无需海森矩阵计算的损失函数以训练神经网络近似解。为进一步提升效率，采用Hutchinson迹估计器计算得分函数中的散度，使训练时间在GPU上实现维度无关。结合自适应采样策略生成配点，理论分析表明，近似解与精确解间的KL散度受残差损失乘以估计密度函数的加权上界。数值实验涵盖奥尔恩斯坦-乌伦贝克（OU）过程、时变扩散布朗运动及具有非高斯解的几何OU过程，最高达一百维。
原文摘要 · Abstract (English)
Solving high-dimensional Fokker-Planck (FP) equations remains a challenging problem in computational physics and stochastic dynamics, due to the curse of dimensionality, unbounded domains, and complex probability landscapes. In this work, we propose an adaptive probability flow residual minimization (A-PFRM) method for this problem. The second-order FP equation is reformulated as an equivalent first-order continuity equation associated with the probability flow ordinary differential equation, based on which a loss function is constructed to train neural network approximations without Hessian computation. To further improve the computational efficiency, the Hutchinson trace estimator is applied to compute the divergence in the corresponding score function, such that the training time can be dimension-independent on GPUs. Adaptive sampling strategies are employed to generate the collocation points, and our analysis shows that the Kullback-Leibler divergence between our A-PFRM approximation and the exact solution is bounded by the residual loss weighted by the estimated density function. Numerical experiments are presented to demonstrate the performance of A-PFRM, which include Ornstein-Uhlenbeck (OU) processes problems, Brownian motions with time-varying diffusion, and Geometric OU processes featuring non-Gaussian solutions up to one hundred dimensions.
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