An advanced hybrid deep adversarial autoencoder for parameterized nonlinear fluid flow modelling
Introduction
Numerical simulations of nonlinear fluid flows are used to describe the complex evolution of many physical processes. Accurate simulations of nonlinear fluid flows are of great importance to many fields such as atmosphere, flood and ocean modelling, which systems often exhibit rich flow dynamics in both space and time [1], [2]. The numerical simulations have benefited from the availability of high-resolution spatio-temporal data due to recent advances in measurement techniques [3], which make the studies of more complex flows possible. However, the computational cost involved in solving complex problems is intensive which still precludes the development in these areas. In order to address the issue of the high computational cost, this paper proposes a hybrid deep adversarial autoencoder to solve fluid problems in an efficient manner.
Recent advances in machine learning technologies are increasingly of interest for efficiently simulating flows in dynamical systems [4], [5], [6]. Machine learning has demonstrated its potential capability in fluid flow applications [7], such as feature extraction [8], [9], [10], dimensionality reduction [5], [11] and superresolution [12]. For example, Kim et al. [13] adopted a convolutional neural network (CNN) to extract flow features and reconstruct fluid fields. Gonzalez and Balajewicz [11] developed a deep convolutional recurrent autoencoder for nonlinear model reduction, which identified the low-dimensional feature dynamics by the high-dimensional datasets and modelled flow dynamics on the underlying manifold space. Xie et al. [12] proposed a temporally coherent generative model to produce the super-resolution fluid flows. Despite the fact that these research efforts demonstrated how to successfully reconstruct flow dynamics, it is noted that they commonly do not take into account the temporal and spatial evolution of inputs or parameters, which is crucial for realistic dynamical systems [14].
Most recently, it has been shown that Generative Adversarial Network (GAN) is an efficient approach for reconstructing the spatial distribution of tracers and flow fields [15]. GAN introduced by Goodfellow et al. [16], has emerged as having a leading role in recreating the distributions of complex data [12], [17]. The key feature of GAN is the adversarial strategy consisting of two modules. GAN defines a learning generative network by transforming a latent variable into a state variable using nonlinear functions. Then GAN drives the learning process by discriminating the observed data from the generated data in a discriminator network. Because of the special adversarial architecture, GAN has demonstrated great capability in producing high-resolution samples, mostly in images, e.g., image synthesis [18], semantic image editing [19], style transfer [20] etc.
For efficient GAN training and accurate spatio-temporal fluid flow prediction, variational autoencoder (VAE) [21] has been introduced to GAN in this study. VAE has been widely used in various research areas, such as text generation [22], facial attribute prediction [23], image generation [24], [25], graph generation [26], music synthesis [27], and speech emotion classification [28], etc. The hybrid deep adversarial autoencoder (VAE-GAN) developed here takes advantage of both GAN and VAE. GAN allows for training on large datasets and is fast to yield visually high-resolution images, but the flexible architecture is easy to result in the model collapse problem and generate unreal results [29]. The VAE is attractive for achieving better log-likelihoods than GAN [30], [31], this suggests a hybrid VAE-GAN to better represent all the training data and discourage model-collapse problem in GAN [29].
The hybrid VAE-GAN developed here is a robust and efficient numerical tool for accurate prediction of parameterized nonlinear fluid flows. The advantages of the hybrid VAE-GAN include:
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The proposed method exploits spatial features by use of convolutional neural networks. It will be advantageous over the traditional reduced order models (ROMs) based on proper orthogonal decomposition (POD) [32], [33], Galerkin regression [34], [35], cluster-based reduced order model (CROM) [36], [37] and dynamic mode decomposition (DMD) [38], which are used for model reduction of dynamical systems by linear combinations of the original snapshots. In convolutional neural networks, the high-dimensional datasets are compressed into the low-dimensional representations by nonlinearity functions in a convolutional encoder. In this way, the predictive fluid flows containing both high nonlinearity and chaotic nature can be represented by a convolutional decoder.
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The low-dimensional representations, several orders of magnitude smaller than the dimensional size of the original datasets, are applied into the adversarial network for representation learning and parameter optimization, thus accelerating the computation process.
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With the trained hybrid VAE-GAN, for given unseen inputs, the spatio-temporal features can be automatically extracted in the encoder. Consequently, accurate predictive nonlinear fluid fields can be further obtained in the decoder in an efficient manner.
This is the first time that the hybrid VAE-GAN is adopted to address parameterized nonlinear fluid flow problems. It is expected to make a breakthrough in predicting accurate nonlinear fluid flows with the high-speed computation.
The remainder of this paper is organized as follows. Section 2 presents the governing equation for nonlinear fluid flows in dynamic systems. Methodologies of VAE and GAN are briefly introduced in Sections 3 Variational autoencoder for model reduction, 4 Generative adversarial network, respectively. The hybrid VAE-GAN for parameterized nonlinear fluid fields is described in detail in Section 5. Section 6 demonstrates the performance of the hybrid VAE-GAN using a flow past a cylinder and water column collapse as test cases. Finally in Section 7, conclusions are presented.
Section snippets
Governing equations
In the computational fluid dynamic models, nonlinear fluid flows are usually simulated by solving nonlinear partial differential equations, such as Navier–Stokes equations. In general, a parameterized partial differential equation for a spatio-temporal fluid flow problem can be written as: where denotes a nonlinear partial differential operator, is the state variable vector (for example, pressure, density, velocity, etc.), represents the spatial
Variational autoencoder for model reduction
VAE was introduced by Kingma and Welling [21], which combines Bayesian inference with deep learning. The VAE is a generative model which aims to produce the desired local variable from the underlying latent variable (as shown in Fig. 1). Mathematically, let be a prior distribution of , and the probability of the local state variable be modelled by where is the conditional distribution of the local state variable given , which is modelled by
Generative adversarial network
The GAN is generally implemented with a minimax game in a system of two players [39]. One player is responsible for generating the new samples from a random dataset , while another player aims to discriminate the real samples from the generated samples . The former player is called the generator and the latter is the discriminator .
In the discriminator, if the real samples are accepted while if the generated samples () are rejected. Unlike the
Hybrid deep adversarial autoencoder for nonlinear fluid flow modelling
In this paper, for nonlinear fluid flow modelling, a hybrid deep learning fluid model based on deep adversarial autoencoder (VAE-GAN) is proposed by combining a VAE and a GAN.
In spatio-temporal fluid flow simulations, the state variable vector represents the fluid flow distribution as and the state variable vector in Eq. (11) can be rewritten as: where is the state variable
Numerical examples
In this section, two examples, flow past a cylinder and water column collapse, are used to illustrate the capabilities of the hybrid VAE-GAN in resolving nonlinear fluid flow problem governed by the Navier–Stokes equations. The input–output datasets in both examples were obtained by running the unstructured mesh finite element fluid model () [44] (referred as the original high fidelity model). In Table 1, the module architecture of the encoder, decoder and discriminator used for two
Conclusions
In this work, a hybrid VAE-GAN method has been, for the first time, used to predict nonlinear fluid flows in varying parameterized space. For any given input parameters , the hybrid VAE-GAN is capable of predicting accurately dynamic nonlinear fluid flows and remains highly efficient in simulations since it combines both advantages of VAE and GAN.
The performance of the hybrid VAE-GAN has been demonstrated by a flow past a cylinder test case and a second case of water column collapse. To
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The Authors acknowledge the support of: China Scholarship Council (No. 201806270238) and funding from the EPSRC (MAGIC) (EP/N010221/1) and INHALE (EP/T003189/1), and the Royal Society (IEC/ NS- FC/170563) in the UK, as well as the joint KAUST-Imperial research project (EACPR_P83206).
References (47)
- et al.
Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks
J. Comput. Phys.
(2019) - et al.
Data-driven modelling of nonlinear spatio-temporal fluid flows using a deep convolutional generative adversarial network
Comput. Methods Appl. Mech. Engrg.
(2020) - et al.
A domain decomposition non-intrusive reduced order model for turbulent flows
Comput. & Fluids
(2019) - et al.
Sparsity enabled cluster reduced-order models for control
J. Comput. Phys.
(2018) - et al.
Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations
Comput. Methods Appl. Mech. Engrg.
(2001) - et al.
Dam break flow solution using artificial neural network
Ocean Eng.
(2017) - et al.
Physics-informed autoencoders for Lyapunov-stable fluid flow prediction
(2019) - et al.
Rapid spatio-temporal flood prediction and uncertainty quantification using a deep learning method
J. Hydrol.
(2019) - et al.
Modal analysis of fluid flows: Applications and outlook
(2019) - et al.
Deep learning for universal linear embeddings of nonlinear dynamics
Nature Commun.
(2018)