Bayesian Modeling and Estimation of Linear Time-Varying Systems using Neural Networks and Gaussian Processes

TL;DR

This paper presents a Bayesian framework utilizing neural networks and Gaussian processes for identifying and estimating linear time-varying systems, effectively quantifying uncertainty and handling system variability.

Contribution

It introduces a unified Bayesian approach modeling the impulse response as a stochastic process, enabling robust inference of LTV systems with uncertainty quantification.

Findings

Successfully infers LTI system properties from a single noisy data pair.

Achieves lower error than classical methods in noise tomography experiments.

Tracks varying LTV impulse responses using structured Gaussian Process priors.

Abstract

The identification of Linear Time-Varying (LTV) systems from input-output data is a fundamental yet challenging ill-posed inverse problem. This work introduces a unified Bayesian framework that models the system's impulse response, $h(t, \tau)$, as a stochastic process. We decompose the response into a posterior mean and a random fluctuation term, a formulation that provides a principled approach for quantifying uncertainty, unifies intrinsic channel variability and epistemic uncertainty through a common posterior representation, and naturally defines a new, useful system class we term Linear Time-Invariant in Expectation (LTIE). To perform inference, we leverage modern machine learning techniques, including Bayesian neural networks and Gaussian Processes, using scalable variational inference. We demonstrate through a series of experiments that our framework can infer the properties of an LTI system from a single noisy input-output pair, including under deliberate additive-noise misspecification, achieve a lower overall error floor than the classical CCF stacking baseline in a simulated ambient noise tomography setting, and track a continuously varying LTV impulse response by using a structured Gaussian Process prior. This work provides a flexible and robust methodology for uncertainty-aware system identification in dynamic environments.