# Gaussian Process Hydrodynamics

arXiv: https://arxiv.org/abs/2209.10707 · HTML: https://mlsanity.com/paper/2209.10707/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 50 references that are in the map.
- First posted on arXiv 21 Sep 2022.

## Builds on (most cited first, 13 of 13)

- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Sparse Cholesky factorization by Kullback-Leibler minimization](https://mlsanity.com/paper/2004.14455/) (2020)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity](https://mlsanity.com/paper/1706.02205/) (2017)
- [Learning dynamical systems from data: a simple cross-validation perspective](https://mlsanity.com/paper/2007.05074/) (2020)
- [Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation](https://mlsanity.com/paper/2005.11375/) (2020)
- [Do ideas have shape? Idea registration as the continuous limit of artificial neural networks](https://mlsanity.com/paper/2008.03920/) (2020)
- [Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games](https://mlsanity.com/paper/1503.03467/) (2015)
- [Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods](https://mlsanity.com/paper/2103.10935/) (2021)
- [Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series](https://mlsanity.com/paper/2111.13037/) (2021)
- [Learning "best" kernels from data in Gaussian process regression. With application to aerodynamics](https://mlsanity.com/paper/2206.02563/) (2022)
- [Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients](https://mlsanity.com/paper/1606.07686/) (2016)

## Built on by (most cited first, 4 of 4)

- [Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows](https://mlsanity.com/paper/2602.15472/) (2026)
- [Physics-informed, boundary-constrained Gaussian process regression for the reconstruction of fluid flow fields](https://mlsanity.com/paper/2507.17582/) (2025)
- [A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianity](https://mlsanity.com/paper/2511.21281/) (2025)
- [Adaptive multi-resolution Gaussian processes: Scalable exact inference with naturally data-sparse covariance matrices](https://mlsanity.com/paper/2609.30348/) (2026)

## Abstract (arXiv)

We present a Gaussian Process (GP) approach (Gaussian Process Hydrodynamics, GPH) for approximating the solution of the Euler and Navier-Stokes equations. As in Smoothed Particle Hydrodynamics (SPH), GPH is a Lagrangian particle-based approach involving the tracking of a finite number of particles transported by the flow. However, these particles do not represent mollified particles of matter but carry discrete/partial information about the continuous flow. Closure is achieved by placing a divergence-free GP prior xi on the velocity field and conditioning on vorticity at particle locations. Known physics (e.g., the Richardson cascade and velocity-increments power laws) is incorporated into the GP prior through physics-informed additive kernels. This approach allows us to coarse-grain turbulence in a statistical manner rather than a deterministic one. By enforcing incompressibility and fluid/structure boundary conditions through the selection of the kernel, GPH requires much fewer particles than SPH. Since GPH has a natural probabilistic interpretation, numerical results come with uncertainty estimates enabling their incorporation into a UQ pipeline and the adding/removing of particles (quantas of information) in an adapted manner. The proposed approach is amenable to analysis, it inherits the complexity of state-of-the-art solvers for dense kernel matrices, and it leads to a natural definition of turbulence as information loss. Numerical experiments support the importance of selecting physics-informed kernels and illustrate the major impact of such kernels on accuracy and stability. Since the proposed approach has a Bayesian interpretation, it naturally enables data assimilation and making predictions and estimations based on mixing simulation data with experimental data.