Barron-Wiener-Laguerre models
Rahul Manavalan, Filip Tronarp
TL;DR
The paper addresses causal operator learning for time-series and systems identification with uncertainty quantification. It proposes a hybrid approach that combines Laguerre-parameterized stable linear dynamics (Wiener-Laguerre) with probabilistic Barron-type nonlinear readouts, enabling posterior predictive uncertainty. The authors formulate Bayesian estimation for the Barron components, derive predictive distributions, and validate the framework on synthetic systems and nonlinear oscillators. This work bridges classical system identification with modern measure-based function approximation to yield a structured, uncertainty-aware time-series modeling paradigm with practical implications for reliable operator learning.
Abstract
We propose a probabilistic extension of Wiener-Laguerre models for causal operator learning. Classical Wiener-Laguerre models parameterize stable linear dynamics using orthonormal Laguerre bases and apply a static nonlinear map to the resulting features. While structurally efficient and interpretable, they provide only deterministic point estimates. We reinterpret the nonlinear component through the lens of Barron function approximation, viewing two-layer networks, random Fourier features, and extreme learning machines as discretizations of integral representations over parameter measures. This perspective naturally admits Bayesian inference on the nonlinear map and yields posterior predictive uncertainty. By combining Laguerre-parameterized causal dynamics with probabilistic Barron-type nonlinear approximators, we obtain a structured yet expressive class of causal operators equipped with uncertainty quantification. The resulting framework bridges classical system identification and modern measure-based function approximation, providing a principled approach to time-series modeling and nonlinear systems identification.