Spectrally Informed Learning of Fluid Flows

TL;DR

This work proposes a spectrally informed approach to extract low-rank models of fluid flows by leveraging known spectral properties in the learning process and incorporates this knowledge by imposing regularizations on the learned dynamics, which bias the training process toward learning low-frequency structures with corresponding higher power.

Abstract

Accurate and efficient fluid flow models are essential for applications relating to many physical phenomena including geophysical, aerodynamic, and biological systems. While these flows may exhibit rich and multiscale dynamics, in many cases underlying low-rank structures exist which describe the bulk of the motion. These structures tend to be spatially large and temporally slow, and may contain most of the energy in a given flow. The extraction and parsimonious representation of these low-rank dynamics from high-dimensional data is a key challenge. Inspired by the success of physics-informed machine learning methods, we propose a spectrally-informed approach to extract low-rank models of fluid flows by leveraging known spectral properties in the learning process. We incorporate this knowledge by imposing regularizations on the learned dynamics, which bias the training process towards learning low-frequency structures with corresponding higher power. We demonstrate the effectiveness of this method to improve prediction and produce learned models which better match the underlying spectral properties of prototypical fluid flows.

Figures (12)

• Figure 1: We aim to capture low-rank, linear models describing complex systems by exploiting underlying simplicity. We consider a problem where the low-rank state space is sampled to higher dimensional measurements in (a), which can be modeled by the infinite dimensional linear Koopman operator on the space of observables in (b). The blue arrows emphasis the evolution of the underlying dynamical system of interest. In both subfigures the space with larger dimension is placed above.
• Figure 2: We consider a data-driven approach to extracting underlying low-rank linear models that capture the evolution of persistent, energy containing structures in high-dimensional flows. This involves approximating the state space in a so-called latent space and approximating the underlying dynamics with a linear map, implicitly learning both transformations in Figure \ref{['fig:problem setup chart 1']}. In this work we investigate using known spectral properties of the underlying dynamical system to improve the learning of $\bm{\Omega}$.
• Figure 3: Visual description of problem setup, using periodic vortex shedding as a sample system. While the evolution can be described in only three dimensions, DNS data may have $O(1e6)$ or greater dimensions. Our goal is to extract a low-rank linear model which approximates the underlying dynamics using a KAE structure. Isolating the linear dynamics enables the spectral-informing mechanism described in this work.
• Figure 4: The spectral loss is computed as the average difference between the discrete operator spectrum and a prior spectrum, both normalized for comparison. This graphic shows the final operator spectrum in red x's overlaid on the prior spectrum which was determined by taking the spatially averaged temporal power spectrum of sea surface temperature in the Gulf of Mexico for (a) the baseline model and (b) the model with a spectral regularization term. The spectrally-informed model more closely matches the underlying properties, particularly by reducing the power at higher frequencies.
• Figure 5: Comparison of eigenfrequency distribution for 64 models over 25 epochs of training on vortex shedding data for (a) baseline model and (b) low-frequency biased model, and changes from initialization of eigenfrequencies for (c) baseline model and (d) low-frequency biased model. The red region in (a) and (c) highlight the consolidation of low eigenfrequencies to the dominant energy peak during training, the data spectrum is given for comparison ($S(f)$).