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

Identifying the kernels that govern memory and nonlocal effects in integro-differential equations (IDEs) is often an ill-posed inverse problem. Traditional methods frequently rely on problem-specific analytical derivations or restrictive assumptions. This paper proposes a data-driven framework that embeds a neural network directly into a differentiable IDE solver. By minimizing the mismatch between predicted and observed spatiotemporal data, the network learns the unknown kernel. To ensure the learned kernels are physically meaningful, the authors employ two Kolmogorov-Arnold Network (KAN) parameterizations: a soft-constrained Chebyshev KAN (Cheb-KAN) and a hard-constrained Monotone-Convex KAN (MC-KAN). The latter uses Bernstein polynomials to enforce positivity, monotonic decrease, and convexity by construction.

The researchers evaluated their framework on three benchmark problems: a 1D Volterra equation, a 1D viscoelastic wave PIDE, and a 2D nonlocal reaction-diffusion equation. For the 1D problems, both the soft-constrained and hard-constrained methods successfully recovered the correct kernel functional forms with comparable accuracy. However, in the more challenging 2D nonlocal problem characterized by sparse and noisy observations, the hard-constrained MC-KAN consistently outperformed the Cheb-KAN. This indicates that embedding physical constraints directly into the network architecture provides superior robustness for multidimensional kernel discovery.

This work provides a flexible, general-purpose tool for scientific machine learning that does not require case-specific adjoint derivations. By producing interpretable, closed-form expressions for kernels via symbolic regression, the framework bridges the gap between black-box neural network models and the need for physically interpretable scientific models. The success of the MC-KAN approach highlights the importance of architectural design in solving ill-posed inverse problems in physical sciences.