# Learning unknown ODE models with Gaussian processes

arXiv: https://arxiv.org/abs/1803.04303 · HTML: https://mlsanity.com/paper/1803.04303/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2018.

## Built on by (most cited first, 30 of 54)

- [Neural graphical modelling in continuous-time: consistency guarantees and algorithms](https://mlsanity.com/paper/2105.02522/) (2021)
- [Continuous-Time Model-Based Reinforcement Learning](https://mlsanity.com/paper/2102.04764/) (2021)
- [Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series](https://mlsanity.com/paper/2301.11308/) (2023)
- [SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations](https://mlsanity.com/paper/2502.02472/) (2025)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [Amortized Reparametrization: Efficient and Scalable Variational Inference for Latent SDEs](https://mlsanity.com/paper/2312.10550/) (2023)
- [Variational multiple shooting for Bayesian ODEs with Gaussian processes](https://mlsanity.com/paper/2106.10905/) (2021)
- [Scalable Inference in SDEs by Direct Matching of the Fokker-Planck-Kolmogorov Equation](https://mlsanity.com/paper/2110.15739/) (2021)
- [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)
- [Zero-shot Imputation with Foundation Inference Models for Dynamical Systems](https://mlsanity.com/paper/2402.07594/) (2024)
- [Continual Learning from Demonstration of Robotics Skills](https://mlsanity.com/paper/2202.06843/) (2022)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [Sparse Variational Student-t Processes](https://mlsanity.com/paper/2312.05568/) (2023)
- [Free-Form Variational Inference for Gaussian Process State-Space Models](https://mlsanity.com/paper/2302.09921/) (2023)
- [Bayesian Gaussian Process ODEs via Double Normalizing Flows](https://mlsanity.com/paper/2309.09222/) (2023)
- [Learning Governing Equations of Unobserved States in Dynamical Systems](https://mlsanity.com/paper/2404.18572/) (2024)
- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Uncertainty and Structure in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2305.13290/) (2023)
- [Learning Interacting Dynamical Systems with Latent Gaussian Process ODEs](https://mlsanity.com/paper/2205.11894/) (2022)
- [Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernels](https://mlsanity.com/paper/2310.05387/) (2023)
- [Neural Stochastic Flows: Solver-Free Modelling and Inference for SDE Solutions](https://mlsanity.com/paper/2510.25769/) (2025)
- [Learning nonparametric ordinary differential equations from noisy data](https://mlsanity.com/paper/2206.15215/) (2022)
- [Symbolic Regression on Sparse and Noisy Data with Gaussian Processes](https://mlsanity.com/paper/2309.11076/) (2023)
- [Machine Learning for Predicting Chaotic Systems](https://mlsanity.com/paper/2407.20158/) (2024)
- [Learning particle swarming models from data with Gaussian processes](https://mlsanity.com/paper/2106.02735/) (2021)
- [The Normalized Cross Density Functional: A Framework to Quantify Statistical Dependence for Random Processes](https://mlsanity.com/paper/2212.04631/) (2022)
- [Exact Inference for Continuous-Time Gaussian Process Dynamics](https://mlsanity.com/paper/2309.02351/) (2023)
- [System-Aware Neural ODE Processes for Few-Shot Bayesian Optimization](https://mlsanity.com/paper/2406.02352/) (2024)
- [Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching](https://mlsanity.com/paper/2409.01416/) (2024)
- [Foundation Inference Models for Ordinary Differential Equations](https://mlsanity.com/paper/2602.08733/) (2026)

## Abstract (arXiv)

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model's capabilities to infer dynamics from sparse data and to simulate the system forward into future.