发表机构
University of Utah（犹他大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究从稀疏有噪观测中发现积分 - 微分方程的记忆和非局部核这一逆问题，提出基于可微求解器框架，用约束柯尔莫哥洛夫 - 阿诺德网络参数化并施加物理约束，经训练和符号回归得到表示，实验表明构造约束比软惩罚更具鲁棒性。
AI 中文摘要
从稀疏且有噪声的观测中发现控制积分 - 微分方程（IDE）的记忆或非局部核是一个不适定的逆问题。现有识别方法有局限。本文提出基于可微求解器的框架，直接从时空观测中发现记忆和非局部核。在求解器中，未知核用约束柯尔莫哥洛夫 - 阿诺德网络（KAN）参数化，通过两种方法施加物理约束。训练后用符号回归获得可解释的闭式表示。在多个基准测试上评估，结果表明在从稀疏且有噪声的观测中进行多维核发现时，通过构造强制施加物理形状约束比软惩罚更具鲁棒性。
英文摘要
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.
