Paper: arXiv 2605.06281

Authors: Jean-Loup Dupret, Davide Gallon, Patrick Cheridito

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous mathematical framework, including a contraction-based convergence proof and fixed-point operator analysis for high-dimensional PIDEs. It also provides empirical validation through numerical experiments on linear and nonlinear examples, showing efficacy in overcoming the curse of dimensionality compared to existing PINN and BSDE methods.

Research Flowchart

  flowchart TD
    A[<b>Research Goal:</b> Efficiently solve high-dimensional PIDEs] --> B[<b>Key Inputs:</b> PIDE formulation & Jump data]
    B --> C[<b>Core Method:</b> Iterative Neural Solver (INEUS)]
    C --> D{<b>Process:</b> Single-jump sampling & recursive regression}
    D --> E[<b>Avoids:</b> Expensive full residual differentiation]
    E --> F[<b>Outcomes:</b> Scalable, accurate solutions for high-D problems]
    F --> G[<b>Proof:</b> Contraction-based convergence]

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3 of 81 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.