mlsanityData to 14 Oct 2026

Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations

arXiv 2404.15618 ↗First posted 24 Apr 2024stat.ML cs.LGcited by 4

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Abstract (arXiv)

The study of neural operators has paved the way for the development of efficient approaches for solving partial differential equations (PDEs) compared with traditional methods. However, most of the existing neural operators lack the capability to provide uncertainty measures for their predictions, a crucial aspect, especially in data-driven scenarios with limited available data. In this work, we propose a novel Neural Operator-induced Gaussian Process (NOGaP), which exploits the probabilistic characteristics of Gaussian Processes (GPs) while leveraging the learning prowess of operator learning. The proposed framework leads to improved prediction accuracy and offers a quantifiable measure of uncertainty. The proposed framework is extensively evaluated through experiments on various PDE examples, including Burger's equation, Darcy flow, non-homogeneous Poisson, and wave-advection equations. Furthermore, a comparative study with state-of-the-art operator learning algorithms is presented to highlight the advantages of NOGaP. The results demonstrate superior accuracy and expected uncertainty characteristics, suggesting the promising potential of the proposed framework.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 23 of its 40 references that are in the map. First posted on arXiv 24 Apr 2024.

Builds on

23 of 40 references.

References · 23

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  4. 2018GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Accelerationcited by 609
  5. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
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  7. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
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  12. 2020A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challengescited by 52
11 more
  1. 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 44
  2. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 41
  3. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  4. 2022Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEscited by 25
  5. 2022Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problemscited by 21
  6. 2018Algorithmic Linearly Constrained Gaussian Processescited by 20
  7. 2020Gaussian Process Regression constrained by Boundary Value Problemscited by 15
  8. 2022Multi-fidelity wavelet neural operator with application to uncertainty quantificationcited by 12
  9. 2023A foundational neural operator that continuously learns without forgettingcited by 11
  10. 2023Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problemscited by 9
  11. 2024Generative adversarial wavelet neural operator: Application to fault detection and isolation of multivariate time series datacited by 6

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Cited by year

Citing papers per year: 2024: 2; 2025: 2.

4 papers.

  1. 2024Linearization Turns Neural Operators into Function-Valued Gaussian Processescited by 12
  2. 2025Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximationcited by 6
  3. 2024Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanicscited by 3
  4. 2025From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systemscited by 0