Data to 7 Oct 2026

Machine Learning of Linear Differential Equations using Gaussian Processes

arXiv 1701.02440 ↗First posted 10 Jan 2017cs.LG cs.NA math.NAcited by 139

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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.

Key facts. Cited by 139 papers in the map (data through 7 Oct 2026). First posted on arXiv 10 Jan 2017.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 23; 2022: 33; 2023: 29; 2024: 13; 2025: 28; 2026: 13.

139 papers.

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2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
2021PySINDy: A comprehensive Python package for robust sparse system identificationcited by 88
2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
2021Encoding physics to learn reaction-diffusion processescited by 61
2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 41
2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 40
2021Learning Functional Priors and Posteriors from Data and Physicscited by 37
2022Delta-PINNs: physics-informed neural networks on complex geometriescited by 28
2022Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approachcited by 25
2021Latent Map Gaussian Processes for Mixed Variable Metamodelingcited by 23
2022AutoIP: A United Framework to Integrate Physics into Gaussian Processescited by 20

38 more

2022Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEscited by 19
2023Benchmarking sparse system identification with low-dimensional chaoscited by 19
2022Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation datacited by 18
2023Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarkscited by 18
2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
2023Learning Vortex Dynamics for Fluid Inference and Predictioncited by 17
2024Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantificationcited by 17
2023BEAGLE: Forensics of Deep Learning Backdoor Attack for Better Defensecited by 15
2021The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimationcited by 14
2022Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equationscited by 14
2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
2021Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equationscited by 13
2022MultiAuto-DeepONet: A Multi-resolution Autoencoder DeepONet for Nonlinear Dimension Reduction, Uncertainty Quantification and Operator Learning of Forward and Inverse Stochastic Problemscited by 13
2023A spectrum of physics-informed Gaussian processes for regression in engineeringcited by 13
2021Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Datacited by 12
2024PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representationscited by 11
2021TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problemscited by 10
2022Bayesian Physics Informed Neural Networks for Data Assimilation and Spatio-Temporal Modelling of Wildfirescited by 10
2021A Review of Physics-based Machine Learning in Civil Engineeringcited by 9
2022Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priorscited by 9
2023Deep-OSG: Deep Learning of Operators in Semigroupcited by 9
2023Gaussian processes for Bayesian inverse problems associated with linear partial differential equationscited by 9
2024Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learningcited by 9
2021Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operatorscited by 8
2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
2022Algorithmic Differentiation for Automated Modeling of Machine Learned Force Fieldscited by 7
2022A Kernel Approach for PDE Discovery and Operator Learningcited by 7
2024Toward Efficient Kernel-Based Solvers for Nonlinear PDEscited by 7
2022Physics Informed Neural Network for Dynamic Stress Predictioncited by 6
2023Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discoverycited by 6
2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 6
2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
2023Stochastic stiffness identification and response estimation of Timoshenko beams via physics-informed Gaussian processescited by 5
2023Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernelscited by 5
2023Gaussian process learning of nonlinear dynamicscited by 5
2021Deep learning neural networks for the third-order nonlinear Schrodinger equation: Solitons, breathers, and rogue wavescited by 4
2022Modeling unknown dynamical systems with hidden parameterscited by 4

50 most cited of 139 All 139 (JSON)