🤖 AI Summary
Identifying nonlocal kernel functions in integro-differential equations from sparse and noisy spatiotemporal observations constitutes an ill-posed inverse problem. This work proposes a physics-informed framework based on differentiable solvers that leverages a novel Kolmogorov–Arnold Network (KAN) to directly learn the kernel from data and subsequently recover its analytic expression via symbolic regression. The approach innovatively incorporates hard constraints through Bernstein-polynomial-based MC-KAN and soft constraints via Chebyshev-basis-based Cheb-KAN—both introduced here for the first time in kernel discovery—and demonstrates that MC-KAN exhibits superior robustness under high-dimensional, sparse, and noisy conditions. Validated on a 1D Volterra equation, a viscoelastic wave equation, and a 2D anisotropic nonlocal reaction–diffusion equation, MC-KAN significantly outperforms Cheb-KAN in accurately recovering the true kernel structure and reconstructing the solution field.
📝 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.