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

TL;DR

MINPO introduces a memory-informed neural pseudo-operator to model nonlocal spatiotemporal dynamics governed by integro-differential equations. It learns a continuous memory representation and, for fractional models, its inverse, enabling explicit reconstruction of the solution without discretizing the IDE residual. Across nonlinear Volterra IDEs, high-dimensional nonlocal IDEs, and time-fractional transport problems, MINPO yields higher accuracy in both the solution and memory operator than state-of-the-art PINN-based baselines and achieves notable speedups over classical quadrature-based solvers. The framework demonstrates robust generalization to diverse kernels and dimensionalities, presenting a scalable, problem-agnostic approach for complex nonlocal dynamics.

Abstract

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.

Figures (9)

• Figure 1: A schematic architecture for the proposed memory-informed neural pseudo-operator (MINPO) method to resolve IDEs with long-range spatial (nonlocal) interactions and/or long-term temporal (memory) dependencies. The network takes the spatiotemporal input coordinates $\boldsymbol{\xi}$ and outputs the learned nonlocal operator $\mathcal{M}_{\boldsymbol{\theta}}(\boldsymbol{\xi})$, which provides a continuous neural representation of all spatiotemporal nonlocal effects. MINPO then reconstructs the solution $u_{\boldsymbol{\Theta}}$ through an explicit map that incorporates $\mathcal{M}_{\boldsymbol{\theta}}$, its derivatives computed using automatic differentiation (A.D.), and (only for fractional models with $0<\alpha<1$) its inverse nonlocal operator component learned using a different network in parallel, $\mathcal{J}_{\boldsymbol{\phi}}(\boldsymbol{\xi})$. The training loss consists of three terms: a fully continuous IDE residual enforcing the governing physics, a data term enforcing initial/boundary/measurement constraints, and finally a lightweight nonlocal consistency term, $\mathcal{L}_{\mathcal{M}}$, that compares the network output with the numerically evaluated nonlocal operator. MINPO can use various neural encoders, such as MLPs or KANs, yielding a versatile and expressive unified framework for solving IDEs.
• Figure 1: A schematic architecture of the Chebyshev-based Auxiliary PIKAN (A-PIKAN) framework for solving IDEs. The cKAN network takes the input coordinates $\boldsymbol{\xi}$ and outputs both the primary field $u$ and a set of auxiliary variables $[v,...,w]$ that represent the integral terms. These auxiliary outputs flow through the computational graph together with $u$, where automatic differentiation (AD) computes all required spatial and temporal derivatives. The PDE residual and the auxiliary-evolution residual, combined with initial and boundary data, form the total loss used during training.
• Figure 2: Forward reconstruction results for Experiment I (Exp. \ref{['ExpI:Volterra']}) using MINPO (KAN/MLP) and adaptive A-PINN/A-PIKAN baselines. The KAN models use 3 layers with 15 neurons and Chebyshev degree 4, and the MLP models consist of 3 layers with 33 neurons and the $\tanh$ activation function. All models are trained with 2400 residual points. Across the accuracy metrics, solution reconstruction, and memory-operator recovery, MINPO-KAN achieves the best performance, while all methods exhibit comparable convergence behavior.
• Figure 2: A schematic architecture for the fractional cPIKAN (fPIKAN) framework. The cKAN network takes the input coordinates $\boldsymbol{\xi}$ and outputs the primary field $u$. Integer-order derivatives of the output are evaluated through automatic differentiation (AD), while fractional derivatives $(0 < \alpha < 1)$ are discretized (non-AD) using schemes such as the L1 method for the Caputo time derivative. The AD-based derivatives and the discretized fractional components (non-AD) together define the fractional IDE residual. This residual, combined with initial and boundary data or sparse measurements, forms the total loss used during training.
• Figure 3: Performance of the proposed MINPO method on the inverse formulation of Experiment I (Exp. \ref{['ExpI:Volterra']}). The top row shows the performance of MINPO (KAN) across all tested values of $\kappa$. The bottom row compares MINPO with A-PIKAN for $\kappa=0.3$. In all cases, the KAN architecture is selected so that the number of available training samples $N_{\text{res}}$ is approximately equal to the total number of trainable parameters $\vert \boldsymbol{\theta} \vert$, ensuring a balanced and fair comparison across methods.

Theorems & Definitions (1)

• Lemma 1