摘要
We present a Gaussian process(GP)approach,called Gaussian process hydrodynamics(GPH)for approximating the solution to the Euler and Navier-Stokes(NS)equations.Similar to smoothed particle hydrodynamics(SPH),GPH is a Lagrangian particle-based approach that involves the tracking of a finite number of particles transported by a 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 priorξon the velocity field and conditioning it on the vorticity at the particle locations.Known physics(e.g.,the Richardson cascade and velocityincrement power laws)is incorporated into the GP prior by using physics-informed additive kernels.This is equivalent to expressingξas a sum of independent GPsξl,which we call modes,acting at different scales(each modeξlself-activates to represent the formation of eddies at the corresponding scales).This approach enables a quantitative analysis of the Richardson cascade through the analysis of the activation of these modes,and enables us to analyze coarse-grain turbulence statistically rather than deterministically.Because GPH is formulated by using the vorticity equations,it does not require solving a pressure equation.By enforcing incompressibility and fluid-structure boundary conditions through the selection of a kernel,GPH requires significantly fewer particles than SPH.Because GPH has a natural probabilistic interpretation,the numerical results come with uncertainty estimates,enabling their incorporation into an uncertainty quantification(UQ)pipeline and adding/removing particles(quanta of information)in an adapted manner.The proposed approach is suitable for analysis because it inherits the complexity of state-of-the-art solvers for dense kernel matrices and results in 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
基金
supported by the Air Force Office of Scientific Research under the MURI award number FA9550-20-1-0358(Machine Learning and Physics-Based Modeling and Simulation)
by the Department of Energy under the award number DE-SC0023163(SEA-CROGS:Scalable,Efficient,and Accelerated Causal Reasoning Operators,Graphs and Spikes for Earth and Embedded Systems)。
