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We study the performance of long short-term memory networks (LSTMs) and neural ordinary differential equations (NODEs) in learning latent-space representations of dynamical equations for an advection-dominated problem given by the viscous Burgers equation. Our formulation is devised in a nonintrusive manner with an equation-free evolution of dynamics in a reduced space with the latter being obtained through a proper orthogonal decomposition. In addition, we leverage the sequential nature of learning for both LSTMs and NODEs to demonstrate their capability for closure in systems which are not completely resolved in the reduced space. We assess our hypothesis for two advection-dominated problems given by the viscous Burgers equation. It is observed that both LSTMs and NODEs are able to reproduce the effects of the absent scales for our test cases more effectively than intrusive dynamics evolution through a Galerkin projection. This result empirically suggests that time-series learning techniques implicitly leverage a memory kernel for coarse-grained system closure as is suggested through the Mori-Zwanzig formalism.

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We describe tests validating progress made toward acceleration and automation of hydrodynamic codes in the regime of developed turbulence by three Deep Learning (DL) Neural Network (NN) schemes trained on Direct Numerical Simulations of turbulence. Even the bare DL solutions, which do not take into account any physics of turbulence explicitly, are impressively good overall when it comes to qualitative description of important features of turbulence. However, the early tests have also uncovered some caveats of the DL approaches. We observe that the static DL scheme, implementing Convolutional GAN and trained on spatial snapshots of turbulence, fails to reproduce intermittency of turbulent fluctuations at small scales and details of the turbulence geometry at large scales. We show that the dynamic NN schemes, namely LAT-NET and Compressed Convolutional LSTM, trained on a temporal sequence of turbulence snapshots are capable to correct for the caveats of the static NN. We suggest a path forward towards improving reproducibility of the large-scale geometry of turbulence with NN.