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

arXiv: https://arxiv.org/abs/2512.17273 · HTML: https://mlsanity.com/paper/2512.17273/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 99 references that are in the map.
- First posted on arXiv 19 Dec 2025.

## Builds on (most cited first, 22 of 22)

- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [Wav-KAN: Wavelet Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.12832/) (2024)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems](https://mlsanity.com/paper/2405.19143/) (2024)
- [KAN or MLP: A Fairer Comparison](https://mlsanity.com/paper/2407.16674/) (2024)
- [KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics](https://mlsanity.com/paper/2407.04192/) (2024)
- [A Practitioner's Guide to Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2510.25781/) (2025)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2411.06286/) (2024)
- [Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architectures](https://mlsanity.com/paper/2501.00016/) (2024)
- [ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification](https://mlsanity.com/paper/2410.16528/) (2024)
- [EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomials](https://mlsanity.com/paper/2410.10897/) (2024)
- [Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs](https://mlsanity.com/paper/2506.07958/) (2025)
- [GMC-PINNs: A new general Monte Carlo PINNs method for solving fractional partial differential equations on irregular domains](https://mlsanity.com/paper/2405.00217/) (2024)
- [Scientific Machine Learning with Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2507.22959/) (2025)
- [Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equations](https://mlsanity.com/paper/2310.09434/) (2023)
- [RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion](https://mlsanity.com/paper/2509.21659/) (2025)
- [FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equations](https://mlsanity.com/paper/2412.11010/) (2024)
- [PMNN:Physical Model-driven Neural Network for solving time-fractional differential equations](https://mlsanity.com/paper/2310.04788/) (2023)
- [Identifying Memory Effects in Epidemics via a Fractional SEIRD Model and Physics-Informed Neural Networks](https://mlsanity.com/paper/2509.22760/) (2025)

## Built on by (most cited first, 1 of 1)

- [Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov--Arnold Networks](https://mlsanity.com/paper/2607.11110/) (2026)

## Abstract (arXiv)

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.