# Machine Learning of Linear Differential Equations using Gaussian Processes

arXiv: https://arxiv.org/abs/1701.02440 · HTML: https://mlsanity.com/paper/1701.02440/

## Key facts

- Cited by 140 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2017.

## Built on by (most cited first, 30 of 140)

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- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Learning Vortex Dynamics for Fluid Inference and Prediction](https://mlsanity.com/paper/2301.11494/) (2023)
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- [BEAGLE: Forensics of Deep Learning Backdoor Attack for Better Defense](https://mlsanity.com/paper/2301.06241/) (2023)
- [The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation](https://mlsanity.com/paper/2103.11488/) (2021)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate](https://mlsanity.com/paper/2102.02617/) (2021)
- [Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2110.11847/) (2021)
- [MultiAuto-DeepONet: A Multi-resolution Autoencoder DeepONet for Nonlinear Dimension Reduction, Uncertainty Quantification and Operator Learning of Forward and Inverse Stochastic Problems](https://mlsanity.com/paper/2204.03193/) (2022)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)
- [Deep-OSG: Deep Learning of Operators in Semigroup](https://mlsanity.com/paper/2302.03358/) (2023)
- [PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations](https://mlsanity.com/paper/2412.05994/) (2024)

## Abstract (arXiv)

This work leverages recent advances in probabilistic machine learning to discover conservation laws expressed by parametric linear equations. Such equations involve, but are not limited to, ordinary and partial differential, integro-differential, and fractional order operators. Here, Gaussian process priors are modified according to the particular form of such operators and are employed to infer parameters of the linear equations from scarce and possibly noisy observations. Such observations may come from experiments or "black-box" computer simulations.