Fluid flows are often significantly complex. Even a simple flow requires thousands of modes to describe its temporal state at each moment. To understand the dynamics of such systems, researchers have explored methods to reduce the number of flow variables, while not compromising the dynamic behavior. The concept is known as dimensionality reduction.

Traditional studies use methods, such as proper orthogonal decomposition (POD). These methods employ several temporal states (or snapshots) of the flow to identify patterns that describe most of the motion. However, because these methods are linear, many modes are often required to accurately reproduce the original flow, even for simple periodic motions.

In this study, as alternative, we use a modern machine learning method named autoencoder, which is a type of neural network that learns how to compress data into a smaller set of variables, called latent variables; it then reconstructs the original data from the latent variables. We apply this method to a periodic motion in time, namely a limit cycle, appearing in a simple two-dimensional fluid flow system called the Kolmogorov flow.

One key conclusion of our work is that autoencoder can represent the aforementioned periodic flow with only two variables, corresponding to the position on a circle that describes the phase of the cycle. From these two variables, autoencoder can reconstruct the entire flow pattern with a very small error imperceptible to the human eye, with the results being superior to those with POD using the same number of variables.

We use a trained autoencoder to analyze the dynamics of flows. A trained autoencoder can estimate the time derivative of the flow without using finite differences, which are highly sensitive to noise. This method works well even when the input data contain noise or are sparse, which is often the case with in laboratory experiments.

The following two aspects of this work are significant for future research:

First, nonlinear machine-learning methods can reveal simple dynamic patterns in complex physical systems.

Second, these methods can be applied to diverse periodic phenomena, in addition to fluid dynamics applications. Notably, these methods do not require the equations that describe the system, such as the Navier-Stokes equations, thus making them useful for systems, where the governing equations are unknown or very complex.

Written by Yoshiki Hiruta on behalf of all authors.

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