MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamics

Many physical systems exhibit nonlocal spatiotemporal behaviors described by integro-differential equations (IDEs). Classical methods for solving IDEs require repeatedly evaluating convolution integrals, whose cost increases quickly with kernel complexity and dimensionality. Existing neural solvers can accelerate selected instances of these computations, yet they do not generalize across diverse nonlocal structures. In this work, we introduce the Memory-Informed Neural Pseudo-Operator (MINPO), a unified framework for modeling nonlocal dynamics arising from long-range spatial interactions and/or long-term temporal memory. MINPO, employing either Kolmogorov-Arnold Networks (KANs) or multilayer perceptron networks (MLPs) as encoders, learns the nonlocal operator and its inverse directly through neural representations, and then explicitly reconstruct the unknown solution fields. The learning is guarded by a lightweight nonlocal consistency loss term to enforce coherence between the learned operator and reconstructed solution. The MINPO formulation allows to naturally capture and efficiently resolve nonlocal spatiotemporal dependencies governed by a wide spectrum of IDEs and their subsets, including fractional PDEs. We evaluate the efficacy of MINPO in comparison with classical techniques and state-of-the-art neural-based strategies based on MLPs, such as A-PINN and fPINN, along with their newly-developed KAN variants, A-PIKAN and fPIKAN, designed to facilitate a fair comparison. Our study offers compelling evidence of the accuracy of MINPO and demonstrates its robustness in handling (i) diverse kernel types, (ii) different kernel dimensionalities, and (iii) the substantial computational demands arising from repeated evaluations of kernel integrals. MINPO, thus, generalizes beyond problem-specific formulations, providing a unified framework for systems governed by nonlocal operators.

💡 Research Summary

The paper introduces MINPO (Memory‑Informed Neural Pseudo‑Operator), a unified neural framework designed to solve integro‑differential equations (IDEs) that feature both long‑range spatial interactions and long‑term temporal memory. Traditional numerical methods for IDEs suffer from rapidly increasing computational cost due to repeated evaluation of convolution‑type integrals, especially when kernels are complex, high‑dimensional, or singular. Existing physics‑informed neural solvers such as A‑PINN and fPINN mitigate some of these issues by learning auxiliary fields or embedding quadrature rules, but they remain tied to specific kernel structures and often require explicit discretization of the integral operator, limiting their generality and scalability.

MINPO addresses these limitations by directly learning a continuous neural representation of the memory operator (M