Gaussian Process Hydrodynamics

TL;DR

This paper introduces Gaussian Process Hydrodynamics (GPH), a probabilistic, particle-based method for simulating fluid flows that incorporates physics-informed kernels, enabling efficient turbulence modeling, uncertainty quantification, and data assimilation.

Contribution

The paper presents a novel GP-based approach for fluid dynamics that reduces particle count, integrates physics-informed kernels, and provides uncertainty estimates, advancing turbulence modeling and simulation accuracy.

Findings

GPH requires fewer particles than SPH for accurate simulations.

Physics-informed kernels significantly improve accuracy and stability.

GPH naturally supports uncertainty quantification and data assimilation.

Abstract

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.