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

arXiv: https://arxiv.org/abs/2404.15618 · HTML: https://mlsanity.com/paper/2404.15618/

## 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 (most cited first, 23 of 23)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs](https://mlsanity.com/paper/2208.01565/) (2022)
- [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Gaussian Process Regression constrained by Boundary Value Problems](https://mlsanity.com/paper/2012.11857/) (2020)
- [Multi-fidelity wavelet neural operator with application to uncertainty quantification](https://mlsanity.com/paper/2208.05606/) (2022)
- [A foundational neural operator that continuously learns without forgetting](https://mlsanity.com/paper/2310.18885/) (2023)
- [Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems](https://mlsanity.com/paper/2306.12047/) (2023)
- [Generative adversarial wavelet neural operator: Application to fault detection and isolation of multivariate time series data](https://mlsanity.com/paper/2401.04004/) (2024)

## Built on by (most cited first, 4 of 4)

- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation](https://mlsanity.com/paper/2512.09165/) (2025)
- [Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics](https://mlsanity.com/paper/2409.10972/) (2024)
- [From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systems](https://mlsanity.com/paper/2506.15906/) (2025)

## 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.