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

arXiv: https://arxiv.org/abs/2507.12878 · HTML: https://mlsanity.com/paper/2507.12878/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 41 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 9 of 9)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [Flipout: Efficient Pseudo-Independent Weight Perturbations on Mini-Batches](https://mlsanity.com/paper/1803.04386/) (2018)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy data](https://mlsanity.com/paper/2103.03385/) (2021)
- [Sparse Bayesian Deep Learning for Dynamic System Identification](https://mlsanity.com/paper/2107.12910/) (2021)
- [System Identification via Meta-Learning in Linear Time-Varying Environments](https://mlsanity.com/paper/2010.14664/) (2020)
- [Safe Time-Varying Optimization based on Gaussian Processes with Spatio-Temporal Kernel](https://mlsanity.com/paper/2409.18000/) (2024)
- [Revisiting the Equivalence of Bayesian Neural Networks and Gaussian Processes: On the Importance of Learning Activations](https://mlsanity.com/paper/2410.15777/) (2024)

## Abstract (arXiv)

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, τ), 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.