Latent Space Subdivision: Stable and Controllable Time Predictions for Fluid Flow

Abstract

We propose an end‐to‐end trained neural network architecture to robustly predict the complex dynamics of fluid flows with high temporal stability. We focus on single‐phase smoke simulations in 2D and 3D based on the incompressible Navier‐Stokes (NS) equations, which are relevant for a wide range of practical problems. To achieve stable predictions for long‐term flow sequences with linear execution times, a convolutional neural network (CNN) is trained for spatial compression in combination with a temporal prediction network that consists of stacked Long Short‐Term Memory (LSTM) layers. Our core contribution is a novel latent space subdivision (LSS) to separate the respective input quantities into individual parts of the encoded latent space domain. As a result, this allows to distinctively alter the encoded quantities without interfering with the remaining latent space values and hence maximizes external control. By selectively overwriting parts of the predicted latent space points, our proposed method is capable to robustly predict long‐term sequences of complex physics problems, like the flow of fluids. In addition, we highlight the benefits of a recurrent training on the latent space creation, which is performed by the spatial compression network. Furthermore, we thoroughly evaluate and discuss several different components of our method.

Enforcing Imprecise Constraints on Generative Adversarial Networks for Emulating Physical Systems

“…This work is complementary to the work presented in a companion paper that studied enforcing statistical constraints [36]. While precise constraints have previously been used as regularization in various fields (e.g., natural language processing [37], lake temperature modeling [38,39], general dynamical systems [25], fluid flow simulations [9,11,12,40], and more specifically in turbulent flow simulations and generation [41][42][43]), the effects, performances, and best practices of imposing imprecise constraints in generative models still need further investigations.…”

“…In recent years, machine learning has been widely adopted in scientific applications, leading to an emerging field referred to as scientific machine learning. Example scientific applications of machine learning include augmenting or constructing data-driven turbulence models [6][7][8], generating realistic animations of flows [9][10][11][12] discovering or solving differential equations [13][14][15][16][17][18][19].…”

“…To this end, the term H(G(Z)) in the loss function (Eq. (10)) is further replaced by log(H(G(Z)) + 1) (11) in our implementation. Since H is a non-negative function by construction, the penalty term log(H(G(Z)) + 1) above is guaranteed to be positive (and thus ensured to be a penalty rather than a reward).…”

Latent space data assimilation by using deep learning

“…This leads to better accuracy. In (Wiewel et al ., 2018; 2020), the authors were mainly interested in computational speed‐up and robust long‐term predictions for fluid flow simulations. These works thus demonstrated the capability of data‐driven approaches for modeling fluid dynamics: they implemented a CAE for spatial compression and stacked long‐short term memory (LSTM) layers to define their surrogate network, that is, a network that performs time propagation.…”

“…A first ingredient of the approach we introduce in the present paper is to replace the (supposedly expensive) time integration of the model with an NN surrogate. Time‐stepping methods based on surrogates have already been explored by several authors (Lu et al ., 2007), (Wiewel et al ., 2018; 2020), (Maulik et al ., 2021), (Vlachas et al ., 2018), (Brajard et al ., 2020), (Pawar and San, 2020). A key question that arises immediately is how to ensure the time stability of the resulting scheme (see (Haber and Ruthotto, 2017), (Haber et al ., 2019) for an investigation within deep learning surrogate models) when the model is repeatedly called to propagate the state over several time steps.…”