# Gaussian process learning of nonlinear dynamics

arXiv: https://arxiv.org/abs/2312.12193 · HTML: https://mlsanity.com/paper/2312.12193/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 48 references that are in the map.
- First posted on arXiv 19 Dec 2023.

## Builds on (most cited first, 21 of 21)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective](https://mlsanity.com/paper/1810.03440/) (2018)
- [Reduced operator inference for nonlinear partial differential equations](https://mlsanity.com/paper/2102.00083/) (2021)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Bayesian operator inference for data-driven reduced-order modeling](https://mlsanity.com/paper/2204.10829/) (2022)
- [Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes](https://mlsanity.com/paper/2009.07444/) (2020)
- [Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior](https://mlsanity.com/paper/2305.09017/) (2023)
- [Operator inference with roll outs for learning reduced models from scarce and low-quality data](https://mlsanity.com/paper/2212.01418/) (2022)
- [Multi-fidelity classification using Gaussian processes: accelerating the prediction of large-scale computational models](https://mlsanity.com/paper/1905.03406/) (2019)
- [Model Order Reduction based on Runge-Kutta Neural Network](https://mlsanity.com/paper/2103.13805/) (2021)
- [Uncertainty quantification for nonlinear solid mechanics using reduced order models with Gaussian process regression](https://mlsanity.com/paper/2302.08216/) (2023)
- [Learning orbital dynamics of binary black hole systems from gravitational wave measurements](https://mlsanity.com/paper/2102.12695/) (2021)
- [Uncertainty quantification of a three-dimensional in-stent restenosis model with surrogate modelling](https://mlsanity.com/paper/2111.06173/) (2021)
- [Deep Kernel Learning of Dynamical Models from High-Dimensional Noisy Data](https://mlsanity.com/paper/2208.12975/) (2022)

## Built on by (most cited first, 5 of 5)

- [Learning Governing Equations of Unobserved States in Dynamical Systems](https://mlsanity.com/paper/2404.18572/) (2024)
- [A parametric framework for kernel-based dynamic mode decomposition using deep learning](https://mlsanity.com/paper/2409.16817/) (2024)
- [Towards scientific machine learning for granular material simulations -- challenges and opportunities](https://mlsanity.com/paper/2504.08766/) (2025)
- [A unified perspective of Gaussian process approximation for differential equations](https://mlsanity.com/paper/2607.06292/) (2026)
- [Gaussian process learning with flow map refinement for parameter estimation in dynamical systems](https://mlsanity.com/paper/2608.22324/) (2026)

## Abstract (arXiv)

One of the pivotal tasks in scientific machine learning is to represent underlying dynamical systems from time series data. Many methods for such dynamics learning explicitly require the derivatives of state data, which are not directly available and can be approximated conventionally by finite differences. However, the discrete approximations of time derivatives may result in poor estimations when state data are scarce and/or corrupted by noise, thus compromising the predictiveness of the learned dynamical models. To overcome this technical hurdle, we propose a new method that learns nonlinear dynamics through a Bayesian inference of characterizing model parameters. This method leverages a Gaussian process representation of states, and constructs a likelihood function using the correlation between state data and their derivatives, yet prevents explicit evaluations of time derivatives. Through a Bayesian scheme, a probabilistic estimate of the model parameters is given by the posterior distribution, and thus a quantification is facilitated for uncertainties from noisy state data and the learning process. Specifically, we will discuss the applicability of the proposed method to several typical scenarios for dynamical systems: identification and estimation with an affine parametrization, nonlinear parametric approximation without prior knowledge, and general parameter estimation for a given dynamical system.