  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.
