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Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations

Zheyuan Hu, Zhongqiang Zhang, George Em Karniadakis, Kenji Kawaguchi

TL;DR

This work addresses the computational challenge of high-dimensional Fokker-Planck-Lévy equations by introducing a fractional score function that converts the fractional Laplacian into a divergence form, enabling standard PINNs to solve a second-order FP equation for the log-density $q_t(\mathbf{x})=\log p_t(\mathbf{x})$. It presents two fitting strategies, Fractional Score Matching (FSM) and Score-fPINN, to obtain the fractional score and then solves for LL/PDF via a two-stage procedure. Across a range of high-dimensional SDEs with Brownian and Lévy noise (up to 100D) and various diffusion/drift structures, the fractional-score solvers achieve low relative $L_2$ errors with sublinear scaling, demonstrating robustness to dimensionality and model complexity. The results indicate that the proposed approach effectively lifts the curse of dimensionality in FPL equations and provides a practical framework for LL/PDF inference in high-dimensional stochastic systems, with future work aimed at accelerating training and extending to broader SDE classes.

Abstract

We introduce an innovative approach for solving high-dimensional Fokker-Planck-Lévy (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.

Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations

TL;DR

This work addresses the computational challenge of high-dimensional Fokker-Planck-Lévy equations by introducing a fractional score function that converts the fractional Laplacian into a divergence form, enabling standard PINNs to solve a second-order FP equation for the log-density . It presents two fitting strategies, Fractional Score Matching (FSM) and Score-fPINN, to obtain the fractional score and then solves for LL/PDF via a two-stage procedure. Across a range of high-dimensional SDEs with Brownian and Lévy noise (up to 100D) and various diffusion/drift structures, the fractional-score solvers achieve low relative errors with sublinear scaling, demonstrating robustness to dimensionality and model complexity. The results indicate that the proposed approach effectively lifts the curse of dimensionality in FPL equations and provides a practical framework for LL/PDF inference in high-dimensional stochastic systems, with future work aimed at accelerating training and extending to broader SDE classes.

Abstract

We introduce an innovative approach for solving high-dimensional Fokker-Planck-Lévy (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.
Paper Structure (38 sections, 1 theorem, 20 equations, 1 figure, 8 tables, 1 algorithm)

This paper contains 38 sections, 1 theorem, 20 equations, 1 figure, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. PINN for FP and FPL equations
  4. High-Dimensional PDE Solvers
  5. Fractional PDE Solvers
  6. Score-Based Generative Models
  7. Proposed Method
  8. Problem Definition and Background
  9. Stable Distribution
  10. Lévy Process
  11. Fractional SDE and Fokker-Planck-Lévy Equation
  12. Fractional Score Function
  13. Fractional Score Matching (FSM)
  14. Score Fractional PINN (Score-fPINN)
  15. Methods Summary and Comparison
  16. ...and 23 more sections

Key Result

Theorem 3.1

(Yoon et al. yoon2023scorebased) If a random variable $\boldsymbol{x} \sim \mathcal{S}\alpha\mathcal{S}^d(\gamma) \in \mathbb{R}^d$ and its PDF is denote $p(\boldsymbol{x})$, then its fractional score is $\boldsymbol{S}^{(\alpha)}(\boldsymbol{x}) = \frac{(-\Delta)^{\frac{\alpha-2}{2}}\nabla p(\bolds

Figures (1)

  • Figure 1: We generate 100,000 random samples from stable distributions with various $\alpha$ in 2D for visualization. The long tail of $\alpha < 2$ stable distributions exhibit a much larger support than the short tail of Gaussian with $\alpha = 2$. Rare events far from the original point are also common for stable distributions with $\alpha < 2$.

Theorems & Definitions (4)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.1