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) , 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) ; 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 Decomposition (POD) Berkooz et al. (1993) finds the optimal linear subspace that best explains the variation of the state distribution. However, POD is flawed in that it ignores the temporal dependence of state variables. This problem is remedied by the Dynamic Model Decomposition (DMD) Schmid (2010) that finds the optimal linear subspace that best approximates the Koopman operator. However, these data-driven algorithms are irrespective of the nonlinearity in the underlying PDE. Comparatively, data-free methods, such as balanced POD Rowley ( 2005 ... We propose a machine learning approach to identify locally optimal, time reversible, reduced-order fluid dynamic models. We first interpret the linear subspace of fluid velocities as a point on the Grassmannian manifold and study the dependence of reduced trajectories on the choice of subspace. Thanks to the time reversibility, we show that the map from the subspace bases to reduced trajectories is globally differentiable, which allows us to optimize the reduced model via gradient-based Riemannian optimization. We further propose a trajectory-wise discrepancy loss that penalizes the difference between the full-order and the reduced trajectories. To make the optimization tractable, we propose a tensor precomputation scheme to accelerate the back-propagation of gradient information. Figure 1 illustrates the high-level pipeline of our method that fine-tunes the reduced fluid model to minimize the expected trajectory-wise discrepancy loss over the distribution of initial conditions. In essence, our method extends prior optimal reduced bases construction algorithm Berkooz et al. (1993); Schmid (2010) to the nonlinear, idealized fluid dynamic model. As an intrusive approach, our method preserves the desirable property of time reversibility. When compared with POD-type reduced model baseline on a row of idealized fluid simulation benchmarks, our method lowers the discrepancy by 50% -90%. Ū I v+ (v 0 , Ū ) v+ (v 1 , Ū ) v+ (v T , Ū ) L dyn L dyn L dyn

2. RELATED WORK

We review related works on machine learning for solving ODE and PDE, reduced physics models beyond fluid dynamics, and finally learning under hard constraints. Learning for Solving ODE and PDE: To study the complex behavior of dynamic systems, various mathematical models have been proposed for idealized models of fluid, solid, elasto-magnetic fields, etc. However, there are oftentimes subtle discrepancies between these models and real-world observations that are hard to model, in which cases machine learning stands out as an effective approach for acquiring these behaviors from groundtruth data. Chen et al. (2018) propose to learn such dynamics as a general Ordinary Differential Equation (ODE) with the time derivative of state predicted via a neural network. Although this method is applicable to general dynamic systems, it does not reflect the spatial and temporal structures of certain systems, which limits its accuracy, data-efficacy, and scalability to high-dimensional systems such as fluids. Several follow-up works improve the network architecture to reflect additional structures. For example, the inter-dependency



), H 2 -optimizationGugercin  et al. (2006), and modal analysisTaira et al. (2017), identify bases corresponding to the intrinsic property of PDE by analyzing the system transfer matrices in the frequency domain, and are thus independent of data. Unfortunately, these techniques are largely limited to linear systems and their extensions to nonlinear fluid dynamics, such as Yang et al. (2019), are in their infancy.More generally, the construction of reduced fluid models has been formulated as machine learning problems for system identification. The vast majority of prior works generalize the non-intrusive approach and identify the state transfer function via supervised learning in an existing sub-manifold, where the transfer functions are parameterized via radial basis functions Zhang et al. (2016), feedforward networks Hsieh & Tang (1998), recurrent networks Pearlmutter (1989); Wang et al. (2018), etc. More recent approaches jointly learn the state transfer function and identify the sub-manifold via convolutional autoencoder Wu et al. (2021); Hasegawa et al. (2020). Unfortunately, all these non-intrusive learning techniques cannot preserve the time reversible property of idealized fluid, potentially leading to large prediction error or requiring a large dataset.

Figure 1: Given a distribution of initial conditions I, we identify a reduced-order fluid model v+ (v, Ū ) by optimizing the bases Ū that minimize the expected trajectory-wise discrepancy loss L dyn . Our output model v+ (v, Ū ) can perform efficient and as-accurateas-possible fluid trajectory predictions.

