LEARNING REDUCED FLUID DYNAMICS

Abstract

Predicting the state evolution of ultra high-dimensional, time-reversible fluid dynamic systems is a crucial but computationally expensive task. Model reduction has been proven an effective method to reduce computational costs by learning a low-dimensional state embedding. However, existing reduced models are irrespective of either the time reversible property or the nonlinear dynamics, leading to sub-optimal performance. We propose a model-based approach to identify locally optimal, model-reduced, time reversible, nonlinear fluid dynamic systems. Our main idea is to use stochastic Riemann optimization to obtain a high-quality reduced fluid model by minimizing the expected trajectory-wise model reduction error over a given distribution of initial conditions. To this end, our method formulates the reduced fluid dynamics as an invertible state transfer function parameterized by the reduced subspace. We further show that the reduced trajectories are differentiable with respect to the subspace bases over the entire Grassmannian manifold, under proper choices of timestep sizes and numerical integrators. Finally, we propose a loss function measuring the trajectory-wise discrepancy between the original and reduced models. By tensor precomputation, we show that gradient information of such loss function can be evaluated efficiently over a long trajectory without time-integrating the high-dimensional dynamic system. Through evaluations on a row of simulation benchmarks, we show that our method reduces the discrepancy by 50% -90% over conventional reduced models.

1. INTRODUCTION

High-dimensional Partial Differential Equations (PDE), especially fluid dynamic systems, find vast applications in the field of scientific computation Moin & Mahesh (1998) ; Alfonsi (2009) Bridson (2015) , to name a few. A fundamental task of all these applications lies in the efficient prediction of numerical solutions over a long horizon. In design prototyping, for example, a designer needs to quickly preview the fluid flow surrounding an aerial vehicle in order to refine its form factor. In a game engine, a fluid simulator needs to achieve real-time performance to provide interactive special effects for players. Although abundant numerical tools Petrila & Trif (2004) ; Demkowicz et al. (1989) have been developed over the past decades with improved efficacy, their algorithmic complexity is still challenging the limits of current computational resources. As a parallel effort, the idealized, incompressible, inviscid Eulerian fluid should be time reversible and energy preserving Duponcheel et al. (2008) , and dedicated numerical schemes are proposed to faithfully preserve these properties in a discrete setting Rowley & Marsden (2002); Pavlov et al. (2011) . This implies that the initial condition of a trajectory can be recovered from any state thereafter and the discrete total energy is a constant throughout the predicted trajectory. Although idealized fluid models are not pursued in applications, their accurate prediction is an important criterion of reliable numerical schemes. Since their proposal Berkooz et al. (1993); Rowley (2005) , model reduction has been quickly established as one of the most effective approaches that can significantly reduce the PDE prediction cost. By restricting the state variables to low-dimensional linear and nonlinear sub-manifolds, the dimension of associated dynamic system can be reduced by orders of magnitude. Over the years, several data-driven and data-free approaches have been proposed to identify sub-manifolds that can capture the complex dynamic behaviors of fluids. The earliest data-driven method of Proper Orthogonal 1



, PDEconstrained optimization Biegler et al. (2003); Herzog & Kunisch (2010), design prototyping Baysal & Eleshaky (1992); Zang & Green (1999), fluidic devices design Du et al. (2020); Li et al. (2022), and digital entertainment Bridson & Batty (2010);

