Real-Time Simulation of Parameter-Dependent Fluid Flows through Deep Learning-Based Reduced Order Models

Abstract

Simulating fluid flows in different virtual scenarios is of key importance in engineering applications. However, high-fidelity, full-order models relying, e.g., on the finite element method, are unaffordable whenever fluid flows must be simulated in almost real-time. Reduced order models (ROMs) relying, e.g., on proper orthogonal decomposition (POD) provide reliable approximations to parameter-dependent fluid dynamics problems in rapid times. However, they might require expensive hyper-reduction strategies for handling parameterized nonlinear terms, and enriched reduced spaces (or Petrov–Galerkin projections) if a mixed velocity–pressure formulation is considered, possibly hampering the evaluation of reliable solutions in real-time. Dealing with fluid–structure interactions entails even greater difficulties. The proposed deep learning (DL)-based ROMs overcome all these limitations by learning, in a nonintrusive way, both the nonlinear trial manifold and the reduced dynamics. To do so, they rely on deep neural networks, after performing a former dimensionality reduction through POD, enhancing their training times substantially. The resulting POD-DL-ROMs are shown to provide accurate results in almost real-time for the flow around a cylinder benchmark, the fluid–structure interaction between an elastic beam attached to a fixed, rigid block and a laminar incompressible flow, and the blood flow in a cerebral aneurysm.

Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models

“…The reference domain represents a 2D pipe containing a circular obstacle with radius r = 0.05 centered in x obs = (0.2, 0.2), i.e., Ω = (0, 2.2) × (0, 0.41)\ Br (0.2, 0.2) (see Figure 11 for reference); this is a well-known benchmark test case already addressed in [34,30]. The domain's boundary is We assign no-slip boundary conditions on Γ 1 , while a parabolic inflow profile h(x, t; µ) = 4U (t, µ)x 2 (0.41 − x 2 ) 0.41 2 , 0 , with U (t; µ) = µ sin(πt/8), ( 25)…”

“…With the same spirit, POD-DL-ROMs [30] enable a more efficient training stage and the use of much larger FOM dimensions, without affecting network complexity, thanks to a prior dimensionality reduction of FOM snapshots through randomized POD (rPOD) [31], and a multi-fidelity pretraining stage, where different models (exploiting, e.g., coarser discretizations or simplified physical models) can be combined to iteratively initialize network parameters. This latter strategy has proven to be effective for instance in the real-time approximation of cardiac electrophysiology problems [32,33] and problems in fluid dynamics [34].…”

A Direct-Forcing Immersed Boundary Method for Incompressible Flows Based on Physics-Informed Neural Network

“…Over the past few years, machine learning (ML) has permeated into various research areas of fluid mechanics [1], e.g., reduced-order modeling [2,3], wake-type clustering and classification [4][5][6][7], development of turbulence closure model [8][9][10], flow optimization and active control [11][12][13][14][15][16], to name a few. The successful application of ML to these areas relies on the availability of large-scale data, which are obtained from CFD simulations or experimental observations.…”

A Galerkin/POD Reduced-Order Model from Eigenfunctions of Non-Converged Time Evolution Solutions in a Convection Problem

“…Reduced-order models based on POD usually operate in two phases: (1) an off-line phase, where proper bases for the problem unknowns are computed from snapshots, and (2) an on-line phase, where the original partial differential equations are projected over the aforementioned bases. The POD technique has been used extensively to produce reduced-order models for fluid flows [12,[14][15][16][17][18][19][20]22,[41][42][43] or convective flows [8,13,44,45]. Regarding the off-line phase, there are different types of snapshots that can be considered: solutions at different values of the bifurcation parameter [44], unconverged numerical solutions for a unique value of R [25,46], or other variants of the method [47][48][49].…”