mlsanityData to 7 Oct 2026

Inferring solutions of differential equations using noisy multi-fidelity data

arXiv 1607.04805 ↗First posted 16 Jul 2016cs.LGcited by 38

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Abstract (arXiv)

For more than two centuries, solutions of differential equations have been obtained either analytically or numerically based on typically well-behaved forcing and boundary conditions for well-posed problems. We are changing this paradigm in a fundamental way by establishing an interface between probabilistic machine learning and differential equations. We develop data-driven algorithms for general linear equations using Gaussian process priors tailored to the corresponding integro-differential operators. The only observables are scarce noisy multi-fidelity data for the forcing and solution that are not required to reside on the domain boundary. The resulting predictive posterior distributions quantify uncertainty and naturally lead to adaptive solution refinement via active learning. This general framework circumvents the tyranny of numerical discretization as well as the consistency and stability issues of time-integration, and is scalable to high-dimensions.

Key facts. Cited by 38 papers in the map (data through 7 Oct 2026). First posted on arXiv 16 Jul 2016.

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Citing papers per year: 2021: 8; 2022: 13; 2023: 7; 2024: 6; 2025: 4.

38 papers.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  2. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  3. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 74
  4. 2021Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantitiescited by 31
  5. 2021Computational Graph Completioncited by 28
  6. 2022Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approachcited by 25
  7. 2023Multi-fidelity reduced-order surrogate modelingcited by 18
  8. 2021The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimationcited by 14
  9. 2023A DeepONet multi-fidelity approach for residual learning in reduced order modelingcited by 14
  10. 2021Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy datacited by 13
  11. 2023A spectrum of physics-informed Gaussian processes for regression in engineeringcited by 13
  12. 2022Multi-fidelity wavelet neural operator with application to uncertainty quantificationcited by 12
26 more
  1. 2021Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEscited by 10
  2. 2024GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applicationscited by 10
  3. 2022Physics Informed Neural Network for Dynamic Stress Predictioncited by 6
  4. 2023Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftingscited by 6
  5. 2022Physically Consistent Learning of Conservative Lagrangian Systems with Gaussian Processescited by 5
  6. 2022A Deep Learning Approach for Predicting Two-dimensional Soil Consolidation Using Physics-Informed Neural Networks (PINN)cited by 4
  7. 2022Multi-fidelity Monte Carlo: a pseudo-marginal approachcited by 4
  8. 2023Physics-informed Gaussian process model for Euler-Bernoulli beam elementscited by 4
  9. 2024Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulationcited by 4
  10. 2021Learning Free-Surface Flow with Physics-Informed Neural Networkscited by 3
  11. 2022PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equationscited by 3
  12. 2022Quantum Extremal Learningcited by 3
  13. 2021Physics-Coupled Spatio-Temporal Active Learning for Dynamical Systemscited by 2
  14. 2022Modelling of physical systems with a Hopf bifurcation using mechanistic models and machine learningcited by 2
  15. 2022A hybrid data driven-physics constrained Gaussian process regression framework with deep kernel for uncertainty quantificationcited by 1
  16. 2024Uncertainty-aware multi-fidelity surrogate modeling with noisy datacited by 1
  17. 2025HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEscited by 1
  18. 2022Domain-aware Control-oriented Neural Models for Autonomous Underwater Vehiclescited by 0
  19. 2022Use of BNNM for interference wave solutions of the gBS-like equation and comparison with PINNscited by 0
  20. 2023Bayesian Reasoning for Physics Informed Neural Networkscited by 0
  21. 2024PINN-BO: A Black-box Optimization Algorithm using Physics-Informed Neural Networkscited by 0
  22. 2024Reconstructing Blood Flow in Data-Poor Regimes: A Vasculature Network Kernel for Gaussian Process Regressioncited by 0
  23. 2024Open-Source High-Speed Flight Surrogate Modeling Frameworkcited by 0
  24. 2025A general physics-constrained method for the modelling of equation's closure terms with sparse datacited by 0
  25. 2025On the performance of multi-fidelity and reduced-dimensional neural emulators for inference of physiological boundary conditionscited by 0
  26. 2025Physics-Informed Reward Machinescited by 0