# Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry

arXiv: https://arxiv.org/abs/2509.02617 · HTML: https://mlsanity.com/paper/2509.02617/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 61 references that are in the map.
- First posted on arXiv 1 Sep 2025.

## Builds on (most cited first, 12 of 12)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [A machine learning framework for data driven acceleration of computations of differential equations](https://mlsanity.com/paper/1807.09519/) (2018)
- [Physics-enhanced deep surrogates for partial differential equations](https://mlsanity.com/paper/2111.05841/) (2021)
- [Operator Learning with Gaussian Processes](https://mlsanity.com/paper/2409.04538/) (2024)
- [PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs](https://mlsanity.com/paper/2405.08558/) (2024)

## Built on by (most cited first, 1 of 1)

- [Bayesian Nonlinear PDE Inference via Gaussian Process Collocation with Application to the Richards Equation](https://mlsanity.com/paper/2510.23550/) (2025)

## Abstract (arXiv)

Parametric partial differential equations (PDEs) serve as fundamental mathematical tools for modeling complex physical phenomena, yet repeated high-fidelity numerical simulations across parameter spaces remain computationally prohibitive. In this work, we propose a physical law-corrected prior Gaussian process (LC-prior GP) for efficient surrogate modeling of parametric PDEs. The proposed method employs proper orthogonal decomposition (POD) to represent high-dimensional discrete solutions in a low-dimensional modal coefficient space, significantly reducing the computational cost of kernel optimization compared with standard GP approaches in full-order spaces. The governing physical laws are further incorporated to construct a law-corrected prior to overcome the limitation of existing physics-informed GP methods that rely on linear operator invariance, which enables applications to nonlinear and multi-coupled PDE systems without kernel redesign. Furthermore, the radial basis function-finite difference (RBF-FD) method is adopted for generating training data, allowing flexible handling of irregular spatial domains. The resulting differentiation matrices are independent of solution fields, enabling efficient optimization in the physical correction stage without repeated assembly. The proposed framework is validated through extensive numerical experiments, including nonlinear multi-parameter systems and scenarios involving multi-coupled physical variables defined on different two-dimensional irregular domains to highlight the accuracy and efficiency compared with baseline approaches.