Neural Discovery of Memory and Nonlocal Kernels in Integro Differential Equations with Constrained Kolmogorov Arnold Networks

Jul 16, 2026

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Discovering the memory or nonlocal kernel governing an integro-differential equation from sparse and noisy observations is a difficult, ill-posed inverse problem. The work introduces a differentiable-solver framework that learns both kernels directly from spatiotemporal data by parameterizing the unknown kernel with constrained Kolmogorov–Arnold Networks. Two constraint strategies are used: a Bernstein-polynomial monotone–convex KAN with built-in positivity, monotonic decrease, and convexity, and a Chebyshev-based KAN with soft penalties. Trained kernels are extracted via symbolic regression.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages30

Reading Time1 h 16 min

Introduction

Abstract

Keywords

Method: Differentiable Solver and Constrained KAN Parameterization

Monotone–Convex KAN (MC-KAN)

Chebyshev-based KAN (Cheb-KAN)

Kernel Extraction via Symbolic Regression

Experimental Evaluation on IDE Benchmarks

arXiv :2607 . 1 1 1 10v 1 [ cs .LG] 13 Jul 2026 Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov–Arnold Networks Aruzhan Tleubeka , Salah A Faroughia,b,∗ a Energy & Intelligence Lab, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah 84112, USA b Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah 84112, USA Abstract Discovering the memory or nonlocal kernel governing an integro-differential equation (IDE) from sparse and noisy observations is an ill-posed inverse problem. Existing identification methods often rely on problemspecific analytical derivations, specialized observation requirements, or restrictive assumptions about the kernel, limiting their applicability across different classes of IDEs. In this work, we propose a differentiablesolver-based framework for discovering memory and nonlocal kernels directly from spatiotemporal observations. Within the solver, the unknown kernel is represented using a constrained Kolmogorov–Arnold Network (KAN) parameterization, with the physical constraints imposed through two different approaches: a Bernsteinpolynomial-based Monotone–Convex KAN (MC-KAN), whose coefficient constraints enforce positivity, monotonic decrease, and convexity by construction, and a Chebyshev-based KAN (Cheb-KAN), in which the same properties are encouraged through soft penalty terms. After training, symbolic regression is applied to the learned kernels to obtain interpretable closed-form representations. We evaluate both methods on benchmarks spanning a one-dimensional Volterra equation, a one-dimensional viscoelastic wave partial integro-differential equation, and a two-dimensional nonlocal reaction-diffusion equation with an anisotropic coupled kernel. For the 1D problems, both methods recover the correct kernel functional form and achieve comparable solutionreconstruction accuracy. In contrast, for the sparse and noisy 2D nonlocal problem, the hard-constrained MC-KAN consistently achieves lower kernel reconstruction errors than the soft-constrained Cheb-KAN. Our results demonstrate that enforcing physically motivated shape constraints by construction provides greater robustness than soft penalties for multidimensional kernel discovery from sparse and noisy observations. Keywords: Scientific Machine Learning, Integro-Differential Equations, Kernel Discovery, Differentiable Solvers, Inverse Problems, Nonlocal Models, Kolmogorov–Arnold Networks 1. Introduction Many physical systems exhibit memory effects or spatial nonlocality, where the state variables depend on the entire history of the system’s evolution and/or on long-range spatial interactions, including viscoelastic polymer relaxation [1, 2], anomalous diffusion in biological membranes [3–5], and heat conduction with thermal memory [6, 7] . These responses are modeled by integro-differential equations (IDEs), where a differential operator, as in standard ordinary or partial differential equations (ODEs/PDEs), is coupled with an integral operator. The key element in IDEs is a kernel that weights contributions from past states or spatially distant regions. Often, the kernel is unknown, and identifying it is crucial for solving IDEs efficiently [8, 9], incorporating it into reduced-order models [10, 11], and constructing physically interpretable surrogates. However, kernel identification from data remains a challenging problem. The inverse problem of identifying memory and nonlocal kernels from observational data has a long history in applied mathematics. Classical approaches typically reformulate the problem as a Volterra integral equation for the unknown kernel and use problem-specific information such as boundary flux measurements [12] or integral overdetermination constraints [13] . The resulting equations are then solved using analytical techniques, including eigenfunction expansions [14], Laplace transforms [15], resol

FAQs

What problem does the document address?

It addresses how to discover the memory or nonlocal kernel of an integro-differential equation from sparse and noisy spatiotemporal observations, formulated as an ill-posed inverse problem.

How are the unknown kernels parameterized in the proposed approach?

The unknown kernel is represented with a constrained Kolmogorov–Arnold Network (KAN) parameterization inside a differentiable solver framework.

What is the difference between MC-KAN and Cheb-KAN?

MC-KAN enforces physical shape constraints through coefficient constraints (positivity, monotonic decrease, convexity) constructed by a Bernstein-polynomial design, while Cheb-KAN encourages the same properties using soft penalty terms.

Jul 16, 2026

Upload by Quinn