# Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics

arXiv: https://arxiv.org/abs/2409.10972 · HTML: https://mlsanity.com/paper/2409.10972/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 36 references that are in the map.
- First posted on arXiv 17 Sep 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)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [The Promises and Pitfalls of Deep Kernel Learning](https://mlsanity.com/paper/2102.12108/) (2021)
- [Pathwise Conditioning of Gaussian Processes](https://mlsanity.com/paper/2011.04026/) (2020)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models](https://mlsanity.com/paper/2108.12956/) (2021)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent](https://mlsanity.com/paper/2306.11589/) (2023)
- [A representer theorem for deep kernel learning](https://mlsanity.com/paper/1709.10441/) (2017)
- [A Modern Retrospective on Probabilistic Numerics](https://mlsanity.com/paper/1901.04457/) (2019)
- [Probabilistic selection of inducing points in sparse Gaussian processes](https://mlsanity.com/paper/2010.09370/) (2020)
- [Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems](https://mlsanity.com/paper/2304.12541/) (2023)
- [Generative adversarial wavelet neural operator: Application to fault detection and isolation of multivariate time series data](https://mlsanity.com/paper/2401.04004/) (2024)
- [Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations](https://mlsanity.com/paper/2404.15618/) (2024)
- [Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification](https://mlsanity.com/paper/2408.00681/) (2024)
- [MD-NOMAD: Mixture density nonlinear manifold decoder for emulating stochastic differential equations and uncertainty propagation](https://mlsanity.com/paper/2404.15731/) (2024)

## Built on by (most cited first, 3 of 3)

- [GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Models](https://mlsanity.com/paper/2608.21652/) (2026)
- [From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systems](https://mlsanity.com/paper/2506.15906/) (2025)
- [Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems](https://mlsanity.com/paper/2508.09623/) (2025)

## Abstract (arXiv)

The growing demand for accurate, efficient, and scalable solutions in computational mechanics highlights the need for advanced operator learning algorithms that can efficiently handle large datasets while providing reliable uncertainty quantification. This paper introduces a novel Gaussian Process (GP) based neural operator for solving parametric differential equations. The approach proposed leverages the expressive capability of deterministic neural operators and the uncertainty awareness of conventional GP. In particular, we propose a ``neural operator-embedded kernel'' wherein the GP kernel is formulated in the latent space learned using a neural operator. Further, we exploit a stochastic dual descent (SDD) algorithm for simultaneously training the neural operator parameters and the GP hyperparameters. Our approach addresses the (a) resolution dependence and (b) cubic complexity of traditional GP models, allowing for input-resolution independence and scalability in high-dimensional and non-linear parametric systems, such as those encountered in computational mechanics. We apply our method to a range of non-linear parametric partial differential equations (PDEs) and demonstrate its superiority in both computational efficiency and accuracy compared to standard GP models and wavelet neural operators. Our experimental results highlight the efficacy of this framework in solving complex PDEs while maintaining robustness in uncertainty estimation, positioning it as a scalable and reliable operator-learning algorithm for computational mechanics.