Antonio López Montes

arXiv2026

Neural Architectures as Functional Priors in Physics-Informed Control Problems

In this work we investigate the role of neural architectures as implicit functional priors in control problems governed by ordinary differential equations. Rather than focusing on highly complex problems, our objective is to investigate architecture-dependent effects in controlled dynamical systems within the simplest physically interpretable settings possible. In particular, we study a controlled linear RLC electrical circuit and a nonlinear Duffing-type dynamical system. Both systems are analyzed first through classical optimal-control formulations and later through PINN-based approaches. We compare different combinations of multilayer perceptrons (MLPs) and Fourier-based KAN-like architectures, and analyze their influence on the resulting controls. The numerical experiments suggest that different architectural choices systematically generate qualitatively distinct controls, even under identical governing equations, loss functionals, initial and target states, training parameters and physical constraints. Significant differences appear in the spectral structure, smoothness, energy distribution, and phase-space behavior of the learned solutions. A central observation of this work is the emergence of a functional specialization phenomenon when the neural architectures are allowed sufficient freedom to shape the structure of the learned controls. More specifically, in the systems considered here, Fourier-based architectures tend to produce trajectories with richer oscillatory content, whereas smoother low-frequency-biased architectures tend to generate more regular and energetically efficient controls. This suggests that different functional components of the control problem may be handled more efficiently by different neural architectures, leading to an implicit specialization between state representation and control generation.

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arXiv2026

A Unified Benchmark of Physics-Informed Neural Networks and Kolmogorov-Arnold Networks for Ordinary and Partial Differential Equations

Physics-Informed Neural Networks (PINNs) have emerged as a powerful mesh-free framework for solving ordinary and partial differential equations by embedding the governing physical laws directly into the loss function. However, their classical formulation relies on multilayer perceptrons (MLPs), whose fixed activation functions and global approximation biases limit performance in problems with oscillatory behavior, multiscale dynamics, or sharp gradients. In parallel, Kolmogorov-Arnold Networks (KANs) have been introduced as a functionally adaptive architecture based on learnable univariate transformations along each edge, providing richer local approximations and improved expressivity. This work presents a systematic and controlled comparison between standard MLP-based PINNs and their KAN-based counterparts, Physics-Informed Kolmogorov-Arnold Networks (PIKANs), using identical physics-informed formulations and matched parameter budgets to isolate the architectural effect. Both models are evaluated across a representative collection of ODEs and PDEs, including cases with known analytical solutions that allow direct assessment of gradient reconstruction accuracy. The results show that PIKANs consistently achieve more accurate solutions, converge in fewer iterations, and yield superior gradient estimates, highlighting their advantage for physics-informed learning. These findings underline the potential of KAN-based architectures as a next-generation approach for scientific machine learning and provide rigorous evidence to guide model selection in differential equation solving.

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