Our paper titled Neural-network learning of SPOD latent dynamics was just accepted for publication in Journal of Computational Physics. Thanks to all authors: Andrea Lario (SISSA, Italy), Romit Maulik (Argonne National Laboratory, USA), Oliver Schmidt (University of California San Diego, USA), and Gianluigi Rozza (SISSA, Italy).

Plain-language description

Deep learning strategies are gaining traction as a tool for surrogate modeling (also referred to as emulation or non-intrusive reduced order modeling) of dynamical systems. Data associated to dynamical systems usually have both a space and a time component. For instance, one can think of the evolution in time of the temperature at various locations worldwide. The various locations where the temperature is provided constitute the space component of the data, while the time snapshots constitute the time component. A common case in computational physics is when we have access to values of a given quantity at S spatial locations (or grid points), for N time snapshots. In this case, the dimension of the problem is S x N, where the spatial component S might be extremely large. The key in surrogate modeling or emulation is to identify a suitable representation of the data that reduces the high dimensionality of S to a more computationally manageable number, that we denote with Sr << S. The new reduced space of dimension Sr, often referred to as the latent space, is where we seek to learn the time evolution of the system. The learnt reduced space can then be used to reconstruct the original high-dimensional space by inverting the data compression. Emulation techniques are particularly useful when fast predictions are required, e.g., in the context of digital twins.


We aim to reconstruct the latent space dynamics of high dimensional, quasi-stationary systems using model order reduction via the spectral proper orthogonal decomposition (SPOD). The proposed method is based on three fundamental steps: in the first, once that the mean flow field has been subtracted from the realizations (also referred to as snapshots), we compress the data from a high-dimensional representation to a lower dimensional one by constructing the SPOD latent space; in the second, we build the time-dependent coefficients by projecting the snapshots containing the fluctuations onto the SPOD basis and we learn their evolution in time with the aid of recurrent neural networks; in the third, we reconstruct the high-dimensional data from the learnt lower-dimensional representation. The proposed method is demonstrated on two different test cases, namely, a compressible jet flow, and a geophysical problem known as the Madden-Julian Oscillation. An extensive comparison between SPOD and the equivalent POD-based counterpart is provided and differences between the two approaches are highlighted. The numerical results suggest that the proposed model is able to provide low rank predictions of complex statistically stationary data and to provide insights into the evolution of phenomena characterized by specific range of frequencies. The comparison between POD and SPOD surrogate strategies highlights the need for further work on the characterization of the interplay of error between data reduction techniques and neural network forecasts.