mlsanityData to 14 Oct 2026

Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations

arXiv 2208.12515 ↗First posted 26 Aug 2022cs.LG cs.SC stat.MLcited by 14

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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.

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

20 of 75 references.

References · 20

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2018GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Accelerationcited by 609
  3. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 140
  4. 2013Structure Discovery in Nonparametric Regression through Compositional Kernel Searchcited by 88
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  6. 2017Linearly constrained Gaussian processescited by 42
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  8. 2015Modeling and interpolation of the ambient magnetic field by Gaussian processescited by 28
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  11. 2020Calibrated Adaptive Probabilistic ODE Solverscited by 21
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8 more
  1. 2018Algorithmic Linearly Constrained Gaussian Processescited by 20
  2. 2017Scaling up the Automatic Statistician: Scalable Structure Discovery using Gaussian Processescited by 19
  3. 2020Bayesian ODE Solvers: The Maximum A Posteriori Estimatecited by 19
  4. 2021Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernelscited by 19
  5. 2020Linearly Constrained Gaussian Processes with Boundary Conditionscited by 14
  6. 2022On boundary conditions parametrized by analytic functionscited by 4
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Citing papers per year: 2022: 2; 2023: 2; 2024: 5; 2025: 3; 2026: 2.

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  1. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  2. 2022A Kernel Approach for PDE Discovery and Operator Learningcited by 7
  3. 2024Transition Constrained Bayesian Optimization via Markov Decision Processescited by 7
  4. 2024Physics-Informed Variational State-Space Gaussian Processescited by 7
  5. 2026Conditioning Gaussian Processes on Almost Anythingcited by 4
  6. 2024Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equationscited by 2
  7. 2025Gaussian Process Regression for Inverse Problems in Linear PDEscited by 2
  8. 2025Physics-Informed Diffusion Models for Unsupervised Anomaly Detection in Multivariate Time Seriescited by 2
  9. 2023PDE-constrained Gaussian process surrogate modeling with uncertain data locationscited by 1
  10. 2024Future Aware Safe Active Learning of Time Varying Systems using Gaussian Processescited by 1
  11. 2023Bayesian Reasoning for Physics Informed Neural Networkscited by 0
  12. 2024Physics-informed Gaussian Processes as Linear Model Predictive Controllercited by 0
2 more
  1. 2025Comparing EPGP Surrogates and Finite Elements Under Degree-of-Freedom Paritycited by 0
  2. 2026Fast Reconstruction of Exact Maxwell Dynamics from Sparse Datacited by 0