# From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systems

arXiv: https://arxiv.org/abs/2506.15906 · HTML: https://mlsanity.com/paper/2506.15906/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 37 references that are in the map.
- First posted on arXiv 18 Jun 2025.

## Builds on (most cited first, 22 of 22)

- [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)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets](https://mlsanity.com/paper/1406.7343/) (2014)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [LNO: Laplace Neural Operator for Solving Differential Equations](https://mlsanity.com/paper/2303.10528/) (2023)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs](https://mlsanity.com/paper/2208.01565/) (2022)
- [Stochastic Gradient Descent for Gaussian Processes Done Right](https://mlsanity.com/paper/2310.20581/) (2023)
- [Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learning](https://mlsanity.com/paper/1608.05639/) (2016)
- [Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architectures](https://mlsanity.com/paper/2501.00016/) (2024)
- [A foundational neural operator that continuously learns without forgetting](https://mlsanity.com/paper/2310.18885/) (2023)
- [Learning bias corrections for climate models using deep neural operators](https://mlsanity.com/paper/2302.03173/) (2023)
- [Randomized prior wavelet neural operator for uncertainty quantification](https://mlsanity.com/paper/2302.01051/) (2023)
- [Actually Sparse Variational Gaussian Processes](https://mlsanity.com/paper/2304.05091/) (2023)
- [Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations](https://mlsanity.com/paper/2404.15618/) (2024)
- [Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics](https://mlsanity.com/paper/2409.10972/) (2024)
- [Derivation of Output Correlation Inferences for Multi-Output (aka Multi-Task) Gaussian Process](https://mlsanity.com/paper/2501.07964/) (2025)

## Abstract (arXiv)

Operator learning offers a powerful paradigm for solving parametric partial differential equations (PDEs), but scaling probabilistic neural operators such as the recently proposed Gaussian Processes Operators (GPOs) to high-dimensional, data-intensive regimes remains a significant challenge. In this work, we introduce a novel, scalable GPO, which capitalizes on sparsity, locality, and structural information through judicious kernel design. Addressing the fundamental limitation of cubic computational complexity, our method leverages nearest-neighbor-based local kernel approximations in the spatial domain, sparse kernel approximation in the parameter space, and structured Kronecker factorizations to enable tractable inference on large-scale datasets and high-dimensional input. While local approximations often introduce accuracy trade-offs due to limited kernel interactions, we overcome this by embedding operator-aware kernel structures and employing expressive, task-informed mean functions derived from neural operator architectures. Through extensive evaluations on a broad class of nonlinear PDEs - including Navier-Stokes, wave advection, Darcy flow, and Burgers' equations - we demonstrate that our framework consistently achieves high accuracy across varying discretization scales. These results underscore the potential of our approach to bridge the gap between scalability and fidelity in GPO, offering a compelling foundation for uncertainty-aware modeling in complex physical systems.