用迭代采样法高效求解高维积分微分方程，精度高且可扩展。
INEUS: Iterative Neural Solver for High-Dimensional PIDEs
- 用单次跳跃采样替代显式积分计算，降低非局部项计算开销。
- 在10维问题上仍保持高精度，求解误差低于5%且随维度增长稳定。
- 适合高维非线性积分微分方程求解，尤其适用于金融与物理建模场景。
本文提出INEUS，一种针对偏积分微分方程（PIDEs）的无网格迭代神经求解器。该方法将显式的非局部跳跃积分替换为单次跳跃采样，并将PIDE求解重构为一系列递归回归问题。与物理信息神经网络（PINNs）类似，INEUS可在整个时空域上学习全局解，但对非局部项的处理更高效，避免了完整PIDE残差的复杂求导。这些特性使其特别适用于高维PDE和PIDE。基于线性PIDE的压缩映射收敛性证明，数值实验表明，INEUS在多种高维线性和非线性示例中均能提供准确且可扩展的解。
原文摘要 · Abstract (English)
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.
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