# Learning Nonparametric Volterra Kernels with Gaussian Processes

arXiv: https://arxiv.org/abs/2106.05582 · HTML: https://mlsanity.com/paper/2106.05582/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 36 references that are in the map.
- First posted on arXiv 10 Jun 2021.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Tackling Climate Change with Machine Learning](https://mlsanity.com/paper/1906.05433/) (2019)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Computationally Efficient Bayesian Learning of Gaussian Process State Space Models](https://mlsanity.com/paper/1506.02267/) (2015)
- [A flexible state space model for learning nonlinear dynamical systems](https://mlsanity.com/paper/1603.05486/) (2016)
- [Function-Space Distributions over Kernels](https://mlsanity.com/paper/1910.13565/) (2019)
- [Non-linear process convolutions for multi-output Gaussian processes](https://mlsanity.com/paper/1810.04632/) (2018)
- [Recurrent Gaussian Processes](https://mlsanity.com/paper/1511.06644/) (2015)
- [Black-Box Inference for Non-Linear Latent Force Models](https://mlsanity.com/paper/1906.09199/) (2019)
- [Fast Kernel Approximations for Latent Force Models and Convolved Multiple-Output Gaussian processes](https://mlsanity.com/paper/1805.07460/) (2018)

## Built on by (most cited first, 4 of 4)

- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
- [Shallow and Deep Nonparametric Convolutions for Gaussian Processes](https://mlsanity.com/paper/2206.08972/) (2022)
- [Deep Latent Force Models: ODE-based Process Convolutions for Bayesian Deep Learning](https://mlsanity.com/paper/2311.14828/) (2023)
- [Parametric Nonlinear Volterra Series via Machine Learning: Transonic Aerodynamics](https://mlsanity.com/paper/2410.19514/) (2024)

## Abstract (arXiv)

This paper introduces a method for the nonparametric Bayesian learning of nonlinear operators, through the use of the Volterra series with kernels represented using Gaussian processes (GPs), which we term the nonparametric Volterra kernels model (NVKM). When the input function to the operator is unobserved and has a GP prior, the NVKM constitutes a powerful method for both single and multiple output regression, and can be viewed as a nonlinear and nonparametric latent force model. When the input function is observed, the NVKM can be used to perform Bayesian system identification. We use recent advances in efficient sampling of explicit functions from GPs to map process realisations through the Volterra series without resorting to numerical integration, allowing scalability through doubly stochastic variational inference, and avoiding the need for Gaussian approximations of the output processes. We demonstrate the performance of the model for both multiple output regression and system identification using standard benchmarks.