Turbulent flows are chaotic and multi-scale dynamical systems, which have large numbers of degrees of freedom. Turbulent flows, however, can be modeled with a smaller number of degrees of freedom when using an appropriate coordinate system, which is the goal of dimensionality reduction via nonlinear autoencoders. Autoencoders are expressive tools, but they are difficult to interpret. This article aims to propose a method to aid the interpretability of autoencoders. First, we introduce the decoder decomposition, a post-processing method to connect the latent variables to the coherent structures of flows. Second, we apply the decoder decomposition to analyze the latent space of synthetic data of a two-dimensional unsteady wake past a cylinder. We find that the dimension of latent space has a significant impact on the interpretability of autoencoders. We identify the physical and spurious latent variables. Third, we apply the decoder decomposition to the latent space of wind-tunnel experimental data of a three-dimensional turbulent wake past a bluff body. We show that the reconstruction error is a function of both the latent space dimension and the decoder size, which are correlated. Finally, we apply the decoder decomposition to rank and select latent variables based on the coherent structures that they represent. This is useful to filter unwanted or spurious latent variables or to pinpoint specific coherent structures of interest. The ability to rank and select latent variables will help users design and interpret nonlinear autoencoders.

Crystallographic distortions in the alternating aluminium and silicon tetrahedral framework of sodalite (Na8Al6Si6O24Cl2), and beryllium and silicon in helvine (Mn8Be6Si6O24S2), (framework designated SOD) are described in terms of a set of condensed normal mode amplitudes and phases derived from an ideal tetrahedron of a theoretical aristotype phase. For a sodalite-structured hettotype phase in space group $P{\overline 4} 3n$, these normal modes transform as the irreducible representations A1, E(α) and T1(z) of point-group ${\overline 4} 3m$, where to a good approximation A1 acts as a pure breathing mode, E(α) as a polyhedral distortive mode and T1(z) as a rigid unit rotation about the unique ${\overline 4}$ axis of the T-site under consideration. Parameterisation of the mode amplitudes in terms of low-order polynomials as a function of thermodynamic variable permits the crystal structure of sodalite-structured phases to be accurately interpolated at intermediate values of the thermodynamic variable. Published data for the high-temperature behaviour of sodalite have been re-analysed in terms of mode amplitudes which accurately reproduce the temperature dependence of the bond lengths, bond angles and the Al–O–Si inter-polyhedral angle. Full expressions for these derived structural parameters in terms of mode amplitudes and the lattice parameter are tabulated and agree with experimental results to within one estimated standard deviation of the experimental parameter. The potential for mode decomposition in lower symmetry SOD framework crystal structures is illustrated by deriving an aristotype structure for tugtupite (Na8Al2Be2Si8O24Cl2) at room temperature in space group $I{\overline 4}$.