Convergence Analysis of PINNs for Fractional Diffusion Equations in Bounded Domains

TL;DR

This paper proves the convergence of physics-informed neural networks (PINNs) for solving time-dependent fractional diffusion equations on bounded domains, addressing nonlocal boundary conditions with a novel mollification strategy.

Contribution

It introduces a spectrally-defined mollification method that ensures boundary compatibility and enables rigorous energy estimates for PINNs solving nonlocal PDEs.

Findings

PINNs converge in Sobolev norms for fractional diffusion equations.

The mollification strategy preserves nonlocal operator structure.

The approach provides theoretical guarantees for neural network solutions.

Abstract

We establish the convergence of physics-informed neural networks (PINNs) for time-dependent fractional diffusion equations posed on bounded domains. The presence of fractional Laplacian operators introduces nonlocal behavior and regularity constraints, and standard neural network approximations do not naturally enforce the associated spectral boundary conditions. To address this challenge, we introduce a spectrally-defined mollification strategy that preserves the structure of the nonlocal operator while ensuring boundary compatibility. This enables the derivation of rigorous energy estimates in Sobolev spaces. Our results rely on analytical tools from PDE theory, highlighting the compatibility of PINN approximations with classical energy estimates for nonlocal equations. We prove convergence of the PINN approximation in any space-time Sobolev norm $H^k$ (with $k \in \N)$. The analysis highlights the role of mollified residuals in enabling theoretical guarantees for neural-network-based solvers of nonlocal PDEs.