# Operator Learning with Gaussian Processes

arXiv: https://arxiv.org/abs/2409.04538 · HTML: https://mlsanity.com/paper/2409.04538/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 76 references that are in the map.
- First posted on arXiv 6 Sep 2024.

## Builds on (most cited first, 30 of 33)

- [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)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [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)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Deep Gaussian Processes for Multi-fidelity Modeling](https://mlsanity.com/paper/1903.07320/) (2019)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (2021)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Multi-Fidelity Cost-Aware Bayesian Optimization](https://mlsanity.com/paper/2211.02732/) (2022)
- [Kernel Interpolation for Scalable Online Gaussian Processes](https://mlsanity.com/paper/2103.01454/) (2021)
- [GP+: A Python Library for Kernel-based learning via Gaussian Processes](https://mlsanity.com/paper/2312.07694/) (2023)
- [Do ideas have shape? Idea registration as the continuous limit of artificial neural networks](https://mlsanity.com/paper/2008.03920/) (2020)
- [Data Fusion with Latent Map Gaussian Processes](https://mlsanity.com/paper/2112.02206/) (2021)
- [Improving Output Uncertainty Estimation and Generalization in Deep Learning via Neural Network Gaussian Processes](https://mlsanity.com/paper/1707.05922/) (2017)

## Built on by (most cited first, 18 of 18)

- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Polynomial Chaos Expansion for Operator Learning](https://mlsanity.com/paper/2508.20886/) (2025)
- [Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis](https://mlsanity.com/paper/2503.01036/) (2025)
- [Operator Learning at Machine Precision](https://mlsanity.com/paper/2511.19980/) (2025)
- [Verification and Validation for Trustworthy Scientific Machine Learning](https://mlsanity.com/paper/2502.15496/) (2025)
- [Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows](https://mlsanity.com/paper/2602.15472/) (2026)
- [Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2512.17884/) (2025)
- [Thompson Sampling in Function Spaces via Neural Operators](https://mlsanity.com/paper/2506.21894/) (2025)
- [Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry](https://mlsanity.com/paper/2509.02617/) (2025)
- [Physics-based deep kernel learning for parameter estimation in high dimensional PDEs](https://mlsanity.com/paper/2509.14054/) (2025)
- [Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning](https://mlsanity.com/paper/2606.17513/) (2026)
- [Kernel-based Operator Learning: Error Analysis, Budget Allocation, and a Physics-Informed Extension](https://mlsanity.com/paper/2607.06287/) (2026)
- [Flow Battery Manifold Design with Heterogeneous Inputs Through Generative Adversarial Neural Networks](https://mlsanity.com/paper/2508.08863/) (2025)
- [Scalable multitask Gaussian processes for complex mechanical systems with functional covariates](https://mlsanity.com/paper/2602.20640/) (2026)
- [Scalable Gaussian process modeling of parametrized spatio-temporal fields](https://mlsanity.com/paper/2603.00290/) (2026)
- [Learning solution operators of PDEs with sparse approximation methods](https://mlsanity.com/paper/2606.06046/) (2026)
- [Neural means and kernel corrections for operator learning](https://mlsanity.com/paper/2609.00389/) (2026)
- [Gaussian-process surrogate indicators for residual-based adaptive GMsFEM](https://mlsanity.com/paper/2609.26843/) (2026)

## Abstract (arXiv)

Operator learning focuses on approximating mappings G^dagger:U rightarrowV between infinite-dimensional spaces of functions, such as u: Omegaurightarrowℝ and v: Omegavrightarrowℝ. This makes it particularly suitable for solving parametric nonlinear partial differential equations (PDEs). While most machine learning methods for operator learning rely on variants of deep neural networks (NNs), recent studies have shown that Gaussian Processes (GPs) are also competitive while offering interpretability and theoretical guarantees. In this paper, we introduce a hybrid GP/NN-based framework for operator learning that leverages the strengths of both methods. Instead of approximating the function-valued operator G^dagger, we use a GP to approximate its associated real-valued bilinear form widetildeG^dagger: U×V^*rightarrowℝ. This bilinear form is defined by widetildeG^dagger(u,varphi) := [varphi,G^dagger(u)], which allows us to recover the operator G^dagger through G^dagger(u)(y)=widetildeG^dagger(u,δy). The GP mean function can be zero or parameterized by a neural operator and for each setting we develop a robust training mechanism based on maximum likelihood estimation (MLE) that can optionally leverage the physics involved. Numerical benchmarks show that (1) it improves the performance of a base neural operator by using it as the mean function of a GP, and (2) it enables zero-shot data-driven models for accurate predictions without prior training. Our framework also handles multi-output operators where G^dagger:U rightarrowprods=1SVs, and benefits from computational speed-ups via product kernel structures and Kronecker product matrix representations.