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.

我们介绍了一种用于解决高维 Fokker-Planck-Lévy (FPL) 方程的创新方法，该方程被广泛应用于物理学、金融学和生态学等多个领域中的非布朗运动建模。我们利用分数评分函数和物理信息的神经网络（PINN）来解决由维度增加引起的数字溢出和指数衰减解的问题。引入分数评分函数使得 FPL 方程可以转化为一个没有分数 Laplacian 的二阶偏微分方程，从而可以使用标准的物理信息神经网络（PINN）进行求解。我们提出了两种方法来获取分数评分函数：分数评分匹配（FSM）和得分 - fPINN 用于拟合分数评分函数。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，得分 - fPINN 则不依赖于特定的随机微分方程（SDEs），但需要计算 PINN 模型的导数，这可能更加昂贵。我们在各种 SDEs 上进行了实验，证明了我们的方法在处理高维问题时具有数值稳定性和效果，从而在解决 FPL 方程中的维度诅咒方面取得了重要进展。