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Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations

arXiv 1703.10230 ↗First posted 29 Mar 2017stat.ML cs.NA math.APcited by 72

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Abstract (arXiv)

We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where: (1) all we observe are noisy data on black-box initial conditions, and (2) we are interested in quantifying the uncertainty associated with such noisy data in our solutions to time-dependent partial differential equations. Our method circumvents the need for spatial discretization of the differential operators by proper placement of Gaussian process priors. This is an attempt to construct structured and data-efficient learning machines, which are explicitly informed by the underlying physics that possibly generated the observed data. The effectiveness of the proposed approach is demonstrated through several benchmark problems involving linear and nonlinear time-dependent operators. In all examples, we are able to recover accurate approximations of the latent solutions, and consistently propagate uncertainty, even in cases involving very long time integration.

Key facts. Cited by 72 papers in the map (data through 14 Oct 2026). First posted on arXiv 29 Mar 2017.

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Citing papers per year: 2021: 15; 2022: 18; 2023: 15; 2024: 7; 2025: 12; 2026: 5.

72 papers.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  2. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  3. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  4. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 35
  5. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  6. 2022On the influence of over-parameterization in manifold based surrogates and deep neural operatorscited by 29
  7. 2022On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networkscited by 23
  8. 2022Scalable algorithms for physics-informed neural and graph networkscited by 21
  9. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  10. 2022A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problemscited by 20
  11. 2022Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equationscited by 20
  12. 2022Neural Implicit Representations for Physical Parameter Inference from a Single Videocited by 19
38 more
  1. 2021Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networkscited by 18
  2. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  3. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  4. 2021Bayesian Numerical Methods for Nonlinear Partial Differential Equationscited by 15
  5. 2023Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Priorcited by 15
  6. 2023A spectrum of physics-informed Gaussian processes for regression in engineeringcited by 14
  7. 2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
  8. 2021Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equationscited by 13
  9. 2022MultiAuto-DeepONet: A Multi-resolution Autoencoder DeepONet for Nonlinear Dimension Reduction, Uncertainty Quantification and Operator Learning of Forward and Inverse Stochastic Problemscited by 13
  10. 2024Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methodscited by 11
  11. 2022Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priorscited by 9
  12. 2022Learning "best" kernels from data in Gaussian process regression. With application to aerodynamicscited by 9
  13. 2024Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learningcited by 9
  14. 2022D-CIPHER: Discovery of Closed-form Partial Differential Equationscited by 7
  15. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 7
  16. 2022Physics Informed Neural Network for Dynamic Stress Predictioncited by 6
  17. 2023Bayesian polynomial neural networks and polynomial neural ordinary differential equationscited by 6
  18. 2024Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methodscited by 5
  19. 2023Physics-informed Gaussian process model for Euler-Bernoulli beam elementscited by 4
  20. 2024Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulationcited by 4
  21. 2025Physics-Informed Machine Learning in Biomedical Science and Engineeringcited by 4
  22. 2021A Temporal Kernel Approach for Deep Learning with Continuous-time Informationcited by 3
  23. 2021Learning Free-Surface Flow with Physics-Informed Neural Networkscited by 3
  24. 2022PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equationscited by 3
  25. 2023On the Integration of Physics-Based Machine Learning with Hierarchical Bayesian Modeling Techniquescited by 3
  26. 2023EPINN-NSE: Enhanced Physics-Informed Neural Networks for Solving Navier-Stokes Equationscited by 3
  27. 2023Exact Inference for Continuous-Time Gaussian Process Dynamicscited by 3
  28. 2023Tasks Makyth Models: Machine Learning Assisted Surrogates for Tipping Pointscited by 3
  29. 2025The Vanishing Gradient Problem for Stiff Neural Differential Equationscited by 3
  30. 2025Walsh-Hadamard Neural Operators for Solving PDEs with Discontinuous Coefficientscited by 3
  31. 2021Learning ODEs via Diffeomorphisms for Fast and Robust Integrationcited by 2
  32. 2022Incorporating Sum Constraints into Multitask Gaussian Processescited by 2
  33. 2022High Precision Differentiation Techniques for Data-Driven Solution of Nonlinear PDEs by Physics-Informed Neural Networkscited by 2
  34. 2023Decoding Mean Field Games from Population and Environment Observations By Gaussian Processescited by 2
  35. 2024Scaling up Probabilistic PDE Simulators with Structured Volumetric Informationcited by 2
  36. 2025Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problemscited by 2
  37. 2025Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learningcited by 2
  38. 2021Dominant motion identification of multi-particle system using deep learning from videocited by 1

50 most cited of 72 All 72 (JSON)