Gaussian Process Hydrodynamics (2209.10707v3)

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.

• The paper presents a novel method that leverages divergence-free GP priors conditioned on vorticity to approximate turbulent fluid flows.
• It employs a Lagrangian particle-based framework with physics-informed additive kernels, bypassing direct pressure calculations while ensuring incompressibility.
• The method delivers robust uncertainty estimates, aligns with known turbulence power laws, and reduces computational costs in fluid dynamics simulations.

Overview of Gaussian Process Hydrodynamics

The paper introduces a novel method termed Gaussian Process Hydrodynamics (GPH) for approximating solutions to the Navier-Stokes (NS) and Euler equations. Hinging on the principles of Gaussian Processes (GPs), GPH constitutes a Lagrangian particle-based framework akin to Smoothed Particle Hydrodynamics (SPH). Nonetheless, it diverges from SPH by representing discrete information about the flow rather than particles of matter. At the heart of GPH is the conditioning of a divergence-free GP prior on vorticity at particle locations, incorporating known physics through physics-informed additive kernels. The consequent model enables a statistical analysis of turbulence, eschewing the need to solve pressure equations and offering a coherent probabilistic interpretation.

Methodology and Design

The paper outlines the construction of GPH, beginning with the representation of the velocity field as a GP conditioned on vorticity information tethered to particles advected by the flow. The GP is defined using a matrix-valued kernel that inherently respects incompressibility and fluid-structure boundary conditions. This divergence-free kernel is achieved through the differentiation of a scalar kernel, defined on the toroidal domain to account for periodic boundary conditions.

A significant element within GPH is mode decomposition informed by the Richardson cascade, wherein the flow is decomposed into modes acting at separate scales. This decomposition is realized through additive kernels that capture multiscale processes and energy transfer, crucial for qualitatively and quantitatively understanding turbulence.

Numerical Results and Implications

The numerical results featured in the paper illustrate the efficacy of GPH in aligning with known power laws, including the well-documented two-thirds law of turbulence. The method shows robustness in coarse-graining turbulence and provides uncertainty estimates, offering insights into the accuracy and stability of the solution. GPH's variational framework enables the addition or removal of particles based on uncertainty estimates, and its probabilistic foundation facilitates seamless integration into uncertainty quantification pipelines.

Theoretical and Practical Implications

Theoretically, GPH serves as a bridge between numerical approximation and statistical inference, reinforcing the synergy between the two domains in resolving PDEs from partial information. It offers a fresh perspective, casting turbulence as information loss and quantifying it through a source term error measure.

Practically, the proposed method affords a significant reduction in computational resources compared to traditional methods. By bypassing the explicit computation of pressure and leveraging physics-informed kernels, GPH heralds a computationally efficient paradigm adaptable to diverse fluid dynamics scenarios. The capability for real-time data assimilation further positions GPH as a powerful tool for predictive modeling and control in fluid dynamics.

Future Directions

The paper lays fertile ground for future research avenues, including extensions to three-dimensional turbulence and the incorporation of complex boundary conditions into the kernel design. An exploration into the integration of state-of-the-art sparse GP methodologies for kernel matrix inversion could further enhance scalability. Moreover, formal comparative studies with existing ANN-based methods could substantiate the advantages and limitations of GPH in broader contexts, while ongoing developments in kernel flows might provide novel strategies to infer appropriate kernels when the underlying physics is partially understood.

In summary, Gaussian Process Hydrodynamics offers a pioneering and sophisticated framework for solving complex nonlinear PDEs in fluid dynamics, characterized by its adherence to physical laws and its adaptability to evolving information paradigms.