# Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2208.12515 · HTML: https://mlsanity.com/paper/2208.12515/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 75 references that are in the map.
- First posted on arXiv 26 Aug 2022.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Structure Discovery in Nonparametric Regression through Compositional Kernel Search](https://mlsanity.com/paper/1302.4922/) (2013)
- [Manifold Gaussian Processes for Regression](https://mlsanity.com/paper/1402.5876/) (2014)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [Generative Modeling of Turbulence](https://mlsanity.com/paper/2112.02548/) (2021)
- [Probabilistic ODE Solvers with Runge-Kutta Means](https://mlsanity.com/paper/1406.2582/) (2014)
- [Calibrated Adaptive Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.08202/) (2020)
- [A Probabilistic State Space Model for Joint Inference from Differential Equations and Data](https://mlsanity.com/paper/2103.10153/) (2021)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Scaling up the Automatic Statistician: Scalable Structure Discovery using Gaussian Processes](https://mlsanity.com/paper/1706.02524/) (2017)
- [Bayesian ODE Solvers: The Maximum A Posteriori Estimate](https://mlsanity.com/paper/2004.00623/) (2020)
- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Linearly Constrained Gaussian Processes with Boundary Conditions](https://mlsanity.com/paper/2002.00818/) (2020)
- [On boundary conditions parametrized by analytic functions](https://mlsanity.com/paper/2205.03185/) (2022)
- [Designing Robust Biotechnological Processes Regarding Variabilities using Multi-Objective Optimization Applied to a Biopharmaceutical Seed Train Design](https://mlsanity.com/paper/2205.03261/) (2022)
- [Adjoint-aided inference of Gaussian process driven differential equations](https://mlsanity.com/paper/2202.04589/) (2022)

## Built on by (most cited first, 14 of 14)

- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [A Kernel Approach for PDE Discovery and Operator Learning](https://mlsanity.com/paper/2210.08140/) (2022)
- [Transition Constrained Bayesian Optimization via Markov Decision Processes](https://mlsanity.com/paper/2402.08406/) (2024)
- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Conditioning Gaussian Processes on Almost Anything](https://mlsanity.com/paper/2605.21041/) (2026)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [Gaussian Process Regression for Inverse Problems in Linear PDEs](https://mlsanity.com/paper/2502.04276/) (2025)
- [Physics-Informed Diffusion Models for Unsupervised Anomaly Detection in Multivariate Time Series](https://mlsanity.com/paper/2508.11528/) (2025)
- [PDE-constrained Gaussian process surrogate modeling with uncertain data locations](https://mlsanity.com/paper/2305.11586/) (2023)
- [Future Aware Safe Active Learning of Time Varying Systems using Gaussian Processes](https://mlsanity.com/paper/2405.10581/) (2024)
- [Bayesian Reasoning for Physics Informed Neural Networks](https://mlsanity.com/paper/2308.13222/) (2023)
- [Physics-informed Gaussian Processes as Linear Model Predictive Controller](https://mlsanity.com/paper/2412.04502/) (2024)
- [Comparing EPGP Surrogates and Finite Elements Under Degree-of-Freedom Parity](https://mlsanity.com/paper/2511.04518/) (2025)
- [Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data](https://mlsanity.com/paper/2605.20514/) (2026)

## Abstract (arXiv)

Data in many applications follows systems of Ordinary Differential Equations (ODEs). This paper presents a novel algorithmic and symbolic construction for covariance functions of Gaussian Processes (GPs) with realizations strictly following a system of linear homogeneous ODEs with constant coefficients, which we call LODE-GPs. Introducing this strong inductive bias into a GP improves modelling of such data. Using smith normal form algorithms, a symbolic technique, we overcome two current restrictions in the state of the art: (1) the need for certain uniqueness conditions in the set of solutions, typically assumed in classical ODE solvers and their probabilistic counterparts, and (2) the restriction to controllable systems, typically assumed when encoding differential equations in covariance functions. We show the effectiveness of LODE-GPs in a number of experiments, for example learning physically interpretable parameters by maximizing the likelihood.