Inferring parameters and reconstruction of two-dimensional turbulent flows with physics-informed neural networks

Inferring parameters and reconstruction of two-dimensional turbulent flows with physics-informed neural networks

Obtaining system parameters and reconstructing the full flow state from limited velocity observations using conventional fluid dynamics solvers can be prohibitively expensive. Here we employ machine learning algorithms to overcome the challenge. As an example, we consider a moderately turbulent fluid flow, excited by a stationary force and described by a two-dimensional Navier-Stokes equation with linear bottom friction. Using dense in time, spatially sparse and probably noisy velocity data, we reconstruct the spatially dense velocity field, infer the pressure and driving force up to a harmonic function and its gradient, respectively, and determine the unknown fluid viscosity and friction coefficient. Both the root-mean-square errors of the reconstructions and their energy spectra are addressed. We study the dependence of these metrics on the degree of sparsity and noise in the velocity measurements. Our approach involves training a physics-informed neural network by minimizing the loss function, which penalizes deviations from the provided data and violations of the governing equations. The suggested technique extracts additional information from velocity measurements, potentially enhancing the capabilities of particle image/tracking velocimetry.

INTRODUCTION

Experimental studies of fluid flows involve measuring their characteristics. The collected data may have low spatio-temporal resolution or be corrupted by measurement noise. Sometimes, the quantity of interest cannot be measured directly, and it has to be reconstructed from other measurements. Various adjoint- and ensemble-variational data assimilation methods address these challenges by combining experimental data with the numerical simulations. The basic idea is to find the flow state that satisfies the governing Navier-Stokes equation and minimizes the deviation from observations. The fundamental downside of such approaches is the computational complexity required to run multiple numerical simulations with various initial conditions and unknown system parameters that are adjusted during optimization. Once they are found, the related flow evolution gives access to the complete system state and at much higher resolution than the original data.

Recently, an alternative approach has been proposed in which the numerical solver is replaced by a physics-informed neural network. The flow estimation is also treated as a minimization problem. The neural network receives coordinates and a moment of time as input and returns flow variables as output. The loss function penalizes deviations from the available data and violations of the governing equations. Let us emphasize that the governing equations are no longer strictly obeyed; rather, their residuals are used as a penalty in the optimization problem. The main advantages of these methods are the relative simplicity of implementation and better suitability for ill-defined problems. In particular, physics-informed neural networks may be used in domains with unknown boundary conditions, whereas numerical simulations without that knowledge are impossible.

A side-by-side comparison of both methodologies was carried out, where it was shown that physics-informed neural networks are generally less accurate for sparse data. However, the situation may change in the future, since physics-informed neural networks are a relatively new technology that is currently being actively improved. Examples of the application of this method to increasingly complex flow configurations can be found in recent reviews.

In this paper, we apply physics-informed neural networks to a specific example of two-dimensional turbulence, which is motivated by experimental studies. We assume that flow measurements reveal the velocity at some locations and time moments. These data can, for example, be obtained by particle image/tracking velocimetry techniques, which track tracing particles over consecutive time frames to determine velocity. The collected data are assumed to be dense in time, but may be sparse in space and contain measurement errors. Density in time means that particle image/tracking velocimetry methods are suitable for measuring instantaneous velocity with reasonable accuracy. Sparsity in space implies that we want to resolve scales that are small compared to the characteristic distance between measurement points. The data assimilation methods we have discussed connect successive moments in time by reproducing the dynamics determined by the Navier-Stokes equation. This allows information about the velocity field from different moments in time to be combined, increasing the accuracy and spatial resolution of possible reconstructions.

Our goal is to enhance the experimental data by increasing their spatial resolution and inferring flow variables and system parameters that were not measured directly. In particular, we aim to reconstruct the spatially dense velocity and pressure fields, establish the driving force, and unknown fluid viscosity and bottom friction coefficient, based only on velocity measurements. Knowledge of these additional quantities will allow us to better describe and characterize the properties of the considered system. The present investigation complements recent studies by exploring a different flow configuration.

The performed analysis shows that sparse (one hundred fifty vectors per image) but accurate velocity measurements are sufficient to reconstruct a dense (six thousand five hundred thirty-six vectors per image) velocity field with a relative root-mean-square error of about zero point two percent and similarly dense pressure and force fields with an accuracy several times worse for typical experimental conditions. The fluid viscosity and the bottom friction are restored with relative errors of several percent. The developed method is robust to small noise (less than or equal to one percent) in the initial velocity data and is even capable of correcting measurement errors based on the fact that the reconstructed quantities should satisfy the Navier-Stokes equation and incompressibility conditions. As the noise level in the initial data increases, the reconstruction accuracy gradually decreases. The changes in reconstruction accuracy with velocity data density are also addressed. Analysis of the energy spectra of velocity fields shows that small scales are the worst reconstructed, especially for noisy measurements.