# Physics-informed, boundary-constrained Gaussian process regression for the reconstruction of fluid flow fields

arXiv: https://arxiv.org/abs/2507.17582 · HTML: https://mlsanity.com/paper/2507.17582/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 54 references that are in the map.
- First posted on arXiv 23 Jul 2025.

## Builds on (most cited first, 15 of 15)

- [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)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions](https://mlsanity.com/paper/2212.07564/) (2022)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Gaussian Process Regression constrained by Boundary Value Problems](https://mlsanity.com/paper/2012.11857/) (2020)
- [Inferring turbulent velocity and temperature fields and their statistics from Lagrangian velocity measurements using physics-informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2407.15727/) (2024)
- [Know Your Boundaries: Constraining Gaussian Processes by Variational Harmonic Features](https://mlsanity.com/paper/1904.05207/) (2019)
- [Learning "best" kernels from data in Gaussian process regression. With application to aerodynamics](https://mlsanity.com/paper/2206.02563/) (2022)
- [Gaussian Process Hydrodynamics](https://mlsanity.com/paper/2209.10707/) (2022)

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

- [A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianity](https://mlsanity.com/paper/2511.21281/) (2025)

## Abstract (arXiv)

Gaussian process regression techniques have been used in fluid mechanics for the reconstruction of flow fields from a reduction-of-dimension perspective. A main ingredient in this setting is the construction of adapted covariance functions, or kernels, to obtain such estimates. In this paper, we present a general method for constraining a prescribed Gaussian process on an arbitrary compact set. The kernel of the pre-defined process must be at least continuous and may include other information about the studied phenomenon. This general boundary-constraining framework can be implemented with high flexibility for a broad range of engineering applications. From this, we derive physics-informed kernels for simulating two-dimensional velocity fields of an incompressible (divergence-free) flow around aerodynamic profiles. These kernels allow to define Gaussian process priors satisfying the incompressibility condition and the prescribed boundary conditions along the profile in a continuous manner. We describe an adapted numerical method for the boundary-constraining procedure parameterized by a measure on the compact set. The relevance of the methodology and performances are illustrated by numerical simulations of flows around a cylinder and a NACA 0412 airfoil profile, for which no observation at the boundary is needed at all.