INEUS: Iterative Neural Solver for High-Dimensional PIDEs

2026-05-07

AI Summary

This work addresses the high computational cost of solving high-dimensional partial integro-differential equations (PIDEs), which arises from their nonlocal jump terms. The authors propose a mesh-free iterative neural solver that reformulates PIDE solution as a recursive regression problem over the entire space-time domain. By leveraging a single-jump Monte Carlo sampling strategy, the method implicitly handles the nonlocal integral term, thereby avoiding explicit integration and full residual differentiation. Embedded within the physics-informed neural networks (PINNs) framework and augmented with an iterative learning strategy, the approach efficiently captures the global solution. The method is theoretically guaranteed to converge via contraction mapping for linear PIDEs and demonstrates high accuracy and strong scalability across multiple high-dimensional linear and nonlinear test cases.

Abstract

In this paper, we introduce INEUS, a meshfree iterative neural solver for partial integro-differential equations (PIDEs). The method replaces the explicit evaluation of nonlocal jump integrals with single-jump sampling and reformulates PIDE solving as a sequence of recursive regression problems. Like Physics-Informed Neural Networks (PINNs), INEUS learns global solutions over the entire space-time domain, yet it offers a more efficient treatment of nonlocal terms and avoids the computationally expensive differentiation of full PIDE residuals. These features make INEUS particularly well suited for high-dimensional PDEs and PIDEs. Supported by a contraction-based convergence proof for linear PIDEs, our numerical experiments show that INEUS delivers accurate and scalable solutions for various high-dimensional linear and nonlinear examples.

Problem

• high-dimensional PIDEs
• nonlocal jump integrals
• meshfree solver
• computational efficiency

Innovation

• meshfree
• iterative neural solver
• partial integro-differential equations
• nonlocal jump integral
• recursive regression