Deep learning for high-dimensional PDEs with fat-tailed Lévy measure

The partial differential equations (PDEs) for jump process with Lévy measure have wide applications. When the measure has fat tails, it will bring big challenges for both computational cost and accuracy. In this work, we develop a deep learning method for high-dimensional PDEs related to fat-tailed Lévy measure, which can be naturally extended to the general case. Building on the theory of backward stochastic differential equations for Lévy processes, our deep learning method avoids the need for neural network differentiation and introduces a novel technique to address the singularity of fat-tailed Lévy measures. The developed method is used to solve four kinds of high-dimensional PDEs: the diffusion equation with fractional Laplacian; the advective diffusion equation with fractional Laplacian; the advective diffusion reaction equation with fractional Laplacian; and the nonlinear reaction diffusion equation with fractional Laplacian. The parameter $\beta$ in fractional Laplacian is an indicator of the strength of the singularity of Lévy measure. Specifically, for $\beta \in (0,1)$, the model describes super-ballistic diffusion; while for $\beta \in (1,2)$, it characterizes super-diffusion. In addition, we experimentally verify that the developed algorithm can be easily extended to solve fractional PDEs with finite general Lévy measures. Our method achieves a relative error of $O\left({10}^{-3}\right)$ for low-dimensional problems and $O\left({10}^{-2}\right)$ for high-dimensional ones. We also investigate three factors that influence the algorithm’s performance: the number of hidden layers; the number of Monte Carlo samples; and the choice of activation functions. Furthermore, we test the efficiency of the algorithm in solving problems in 3D, 10D, 20D, 50D, and 100D. Our numerical results demonstrate that the algorithm achieves excellent performance with deeper hidden layers, a larger number of Monte Carlo samples, and the Softsign activation function.