We introduce an innovative approach for solving high-dimensional Fokker-Planck-L\'evy (FPL) equations in modeling non-Brownian processes across disciplines such as physics, finance, and ecology. We utilize a fractional score function and Physical-informed neural networks (PINN) to lift the curse of dimensionality (CoD) and alleviate numerical overflow from exponentially decaying solutions with dimensions. The introduction of a fractional score function allows us to transform the FPL equation into a second-order partial differential equation without fractional Laplacian and thus can be readily solved with standard physics-informed neural networks (PINNs). We propose two methods to obtain a fractional score function: fractional score matching (FSM) and score-fPINN for fitting the fractional score function. While FSM is more cost-effective, it relies on known conditional distributions. On the other hand, score-fPINN is independent of specific stochastic differential equations (SDEs) but requires evaluating the PINN model's derivatives, which may be more costly. We conduct our experiments on various SDEs and demonstrate numerical stability and effectiveness of our method in dealing with high-dimensional problems, marking a significant advancement in addressing the CoD in FPL equations.

翻译：我们提出了一种创新方法，用于求解高维Fokker-Planck-Lévy（FPL）方程，该方程在物理学、金融学和生态学等学科中用于建模非布朗过程。我们利用分数分数函数和物理信息神经网络（PINN）来缓解维度灾难（CoD），并减轻因解随维度指数衰减而导致的数值溢出问题。引入分数分数函数使我们能够将FPL方程转化为不含分数拉普拉斯算子的二阶偏微分方程，从而可以直接使用标准物理信息神经网络（PINN）进行求解。我们提出了两种获取分数分数函数的方法：分数分数匹配（FSM）和用于拟合分数分数函数的score-fPINN。虽然FSM更具成本效益，但它依赖于已知的条件分布。另一方面，score-fPINN不依赖于特定的随机微分方程（SDE），但需要评估PINN模型的导数，这可能成本更高。我们在多种SDE上进行了实验，证明了我们的方法在处理高维问题时的数值稳定性和有效性，标志着在解决FPL方程中的维度灾难方面取得了重要进展。