Barron-Wiener-Laguerre models

TL;DR

This paper introduces a probabilistic extension of Wiener-Laguerre models for causal operator learning, integrating Bayesian inference to provide uncertainty quantification in time-series modeling.

Contribution

It combines Laguerre-parameterized causal dynamics with Barron function approximators, enabling Bayesian inference and uncertainty quantification in classical system identification.

Findings

Provides a structured class of causal operators with uncertainty estimates

Bridges classical system identification and modern measure-based approximation

Enables probabilistic modeling of nonlinear time-series dynamics

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.