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

• URL: http://arxiv.org/abs/2607.11110v1
• Date: Mon, 13 Jul 2026 05:40:37 GMT
• Title: Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov--Arnold Networks
• Abstract summary: Existing identification methods often rely on problem-specific analytical derivations, specialized observation requirements, or restrictive assumptions about the kernel.<n>We propose a differentiable-solver-based framework for discovering memory and nonlocal kernels directly fromtemporal observations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 problem-specific 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 differentiable-solver-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 Bernstein-polynomial-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 solution-reconstruction 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.

Related papers

• Controlling False Discovery in Arbitrarily Structured Hypothesis Spaces via Reproducing Kernels [47.16593717985932]
  We introduce a framework that unifies continuous domains, graphs, and hierarchies under a single algorithm through kernel choice alone.<n>We provide two decision rules which we prove to control the False Discovery Rate (FDR)<n>We validate our method on two sources: spatial locations derived from high-dimensional real-world datasets, and a differential gene expression task utilizing protein-protein interaction graphs.
  arXiv  Detail & Related papers  (2026-05-17T17:42:56Z)
• Discovery of interaction and diffusion kernels in particle-to-mean-field multi-agent systems [41.99844472131922]
  We propose a data-driven framework to learn interaction kernels in multi-agent systems.<n>Our approach aims at identifying the functional form of nonlocal interaction and diffusion terms directly from trajectory data.
  arXiv  Detail & Related papers  (2026-03-16T21:19:25Z)
• Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristics [3.2116198597240846]
  We present a unified theoretical and computational framework for constructing reproducing kernels tailored to transport equations.<n>These eigenfunctions satisfy a transport-type partial differential equation (PDE) that we invert.<n>We show that the associated kernel eigenfunctions (Mercer modes) converge in L2 to true Koopman eigenfunctions when the latter lie in the RKHS.
  arXiv  Detail & Related papers  (2026-03-06T20:46:20Z)
• KROM: Kernelized Reduced Order Modeling [3.988493458010939]
  KROM formulates PDE solution as a minimum-norm (Gaussian-process) recovery problem in an RKHS.<n>A central ingredient is an empirical kernel constructed from a snapshot library of PDE solutions.
  arXiv  Detail & Related papers  (2026-02-27T22:52:22Z)
• Toward Efficient Kernel-Based Solvers for Nonlinear PDEs [19.975293084297014]
  We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs)<n>In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, our approach eliminates these operators from the kernel.<n>We model the solution using a standard kernel form and differentiate the interpolant to compute the derivatives.
  arXiv  Detail & Related papers  (2024-10-15T01:00:43Z)
• Exact dynamics of quantum dissipative $XX$ models: Wannier-Stark localization in the fragmented operator space [49.1574468325115]
  We find an exceptional point at a critical dissipation strength that separates oscillating and non-oscillating decay. We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace.
  arXiv  Detail & Related papers  (2024-05-27T16:11:39Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Score-based Diffusion Models in Function Space [137.70916238028306]
  Diffusion models have recently emerged as a powerful framework for generative modeling.<n>This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space.<n>We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Learning Discretized Neural Networks under Ricci Flow [48.47315844022283]
  We study Discretized Neural Networks (DNNs) composed of low-precision weights and activations.<n>DNNs suffer from either infinite or zero gradients due to the non-differentiable discrete function during training.
  arXiv  Detail & Related papers  (2023-02-07T10:51:53Z)
• Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles [1.776746672434207]
  We introduce a nonparametric algorithm to learn interaction kernels of mean-field equations for 1st-order systems of interacting particles. By at least squares with regularization, the algorithm learns the kernel on data-adaptive hypothesis spaces efficiently.
  arXiv  Detail & Related papers  (2020-10-29T15:37:17Z)
• Deterministic error bounds for kernel-based learning techniques under bounded noise [0.0]
  We consider the problem of reconstructing a function from a finite set of noise-corrupted samples. Two kernel algorithms are analyzed, namely kernel ridge regression and $varepsilon$-support vector regression.
  arXiv  Detail & Related papers  (2020-08-10T10:16:00Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.