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Machine learning offers an intriguing alternative to first-principle analysis for discovering new physics from experimental data. However, to date, purely data-driven methods have only proven successful in uncovering physical laws describing simple, low-dimensional systems with low levels of noise. Here we demonstrate that combining a data-driven methodology with some general physical principles enables discovery of a quantitatively accurate model of a non-equilibrium spatially extended system from high-dimensional data that is both noisy and incomplete. We illustrate this using an experimental weakly turbulent fluid flow where only the velocity field is accessible. We also show that this hybrid approach allows reconstruction of the inaccessible variables – the pressure and forcing field driving the flow. Reinbold et al. propose a physics-informed data-driven approach that successfully discovers a dynamical model using high-dimensional, noisy and incomplete experimental data describing a weakly turbulent fluid flow. This approach is relevant to other non-equilibrium spatially-extended systems.

We show how a complete mathematical model of a physical process can be learned directly from data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality and symmetry, a weak formulation of differential equations and sparse regression. To illustrate this, we extract a complete system of governing equations of fluid dynamics – the Navier–Stokes equation, the continuity equation and the boundary conditions – as well as the pressure-Poisson and energy equations, from numerical data describing a highly turbulent channel flow in three dimensions. Whether they represent known or unknown physics, relations discovered using this approach take the familiar form of partial differential equations, which are easily interpretable and readily provide information about the relative importance of different physical effects. The proposed approach offers insight into the quality of the data, serving as a useful diagnostic tool. It is also remarkably robust, yielding accurate results for very high noise levels, and should thus be well suited for analysis of experimental data.

We show how a complete mathematical description of a complicated physical phenomenon can be learned from observational data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality, and symmetry, a weak formulation of differential equations, and sparse regression. To illustrate this, we extract a system of governing equations describing flows of incompressible Newtonian fluids -- the Navier-Stokes equation, the continuity equation, and the boundary conditions -- from numerical data describing a highly turbulent channel flow in three dimensions. These relations have the familiar form of partial differential equations, which are easily interpretable and readily provide information about the relative importance of different physical effects as well as insight into the quality of the data, serving as a useful diagnostic tool. The approach described here is remarkably robust, yielding accurate results for very high noise levels, and should thus be well-suited to experimental data.

Machine learning offers an intriguing alternative to first-principles analysis for discovering new physics from experimental data. However, to date, purely data-driven methods have only proven successful in uncovering physical laws describing simple, low-dimensional systems with low levels of noise. Here we demonstrate that combining a data-driven methodology with some general physical principles enables discovery of a quantitatively accurate model of a non-equilibrium spatially-extended system from high-dimensional data that is both noisy and incomplete. We illustrate this using an experimental weakly turbulent fluid flow where only the velocity field is accessible. We also show that this hybrid approach allows reconstruction of the inaccessible variables -- the pressure and forcing field driving the flow.

Sparse regression has recently emerged as an attractive approach for discovering models of spatiotemporally complex dynamics directly from data. In many instances, such models are in the form of nonlinear partial differential equations (PDEs); hence sparse regression typically requires evaluation of various partial derivatives. However, accurate evaluation of derivatives, especially of high order, is infeasible when the data are noisy, which has a dramatic negative effect on the result of regression. We present a novel and rather general approach that addresses this difficulty by using a weak formulation of the problem. For instance, it allows accurate reconstruction of PDEs involving high-order derivatives, such as the Kuramoto-Sivashinsky equation, from data with a considerable amount of noise. The flexibility of our approach also allows reconstruction of PDE models that involve latent variables which cannot be measured directly with acceptable accuracy. This is illustrated by reconstructing a model for a weakly turbulent flow in a thin fluid layer, where neither the forcing nor the pressure field is known.

This paper investigates how models of spatiotemporal dynamics in the form of nonlinear partial differential equations can be identified directly from noisy data using a combination of sparse regression and weak formulation. Using the 4th-order Kuramoto-Sivashinsky equation for illustration, we show how this approach can be optimized in the limits of low and high noise, achieving accuracy that is orders of magnitude better than what existing techniques allow. In particular, we derive the scaling relation between the accuracy of the model, the parameters of the weak formulation, and the properties of the data, such as its spatial and temporal resolution and the level of noise.

In spatially extended systems, it is common to find latent variables that are hard, or even impossible, to measure with acceptable precision, but are crucially important for the proper description of the dynamics. This substantially complicates construction of an accurate model for such systems using data-driven approaches. The present paper illustrates how physical constraints can be employed to overcome this limitation using the example of a weakly turbulent quasi-two-dimensional Kolmogorov flow driven by a steady Lorenz force with an unknown spatial profile. Specifically, the terms involving latent variables in the partial differential equations governing the dynamics can be eliminated at the expense of raising the order of that equation. We show that local polynomial interpolation combined with symbolic regression can handle sparse data on grids that are representative of typical experimental measurement techniques such as particle image velocimetry. However, we also find that the reconstructed model is sensitive to measurement noise and trace this sensitivity to the presence of high order spatial and/or temporal derivatives.

Partial differential equations (PDEs) provide macroscopic descriptions of systems in numerous fields, such as physics, biology, and chemistry. Additionally, with increasingly vast amounts of data becoming available with the advancement of technology, machine learning is now offering an alternative to traditional model construction (eg. from first principles). This alternative is particularly attractive for systems that are too complex for derivation from first principles to be tractable, or worse, where no first principles are known at all. The research presented in this thesis advances the current state of data-driven PDE modeling. The fundamental approach involves converting a candidate PDE into a system of algebraic equations linear in model parameters via a carefully designed evaluation procedure, and then using sparse regression to narrow down to the model that best fits the data. Existing regression methods, when applied to PDEs, rely on linear systems that are sensitive to noise corruption in the observations. However, regression methods can quickly explore the fitness of many different model options, and so are the default choice for systems that could be described by a large set of potential models.Several data-driven approaches to model discovery have been designed recently, but most have been shown to be inadequate for application to high-dimensional data described by PDEs. Some approaches (like training a neural network to ’learn’ the dynamics, from which the model can be backed out of the network parameters) are less susceptible to noise corruption, but are currently ill-suited to finding the best model out of a large set of candidates, since it does not directly accomodate changes to the model structure. On the other hand, symbolic regression methods of the type mentioned above can discern between different potential models efficiently. They do well for low-dimensional systems described by ordinary differential equations (ODEs), but have yet to see successful application to PDEs for experimental data. The main problem is that constructing the linear system prior to performing the regression has traditionally required locally evaluating derivatives of discrete and noisy data. Derivatives (especially of higher order) computed from discrete and noisy data is notoriously inaccurate and has hindered many attempts at PDE model identification.Besides relying on local (inaccurate) derivative information, existing regression methods rarely constrain the library of candidate model terms. However, it is found here that utilizing knowledge of relevant physics and symmetry can ensure that the candidate terms are both interpretable and not un-physical. The main breakthrough of this thesis is developing an alternative approach to building the linear system fed to the regression algorithm. By considering the weak form of the candidate model, the derivatives on many model terms can be moved onto a weight function whose derivatives are analytically known. The linear system is then filled with integral values that are far less sensitive to noise corruption. Furthermore, the weight function can be carefully designed to remove dependence on certain latent variables from the weak form, which enables identification of PDEs that would otherwise be impossible to work with due to the missing information. These latent variables can later be reconstructed using the available data, domain knowledge, and (crucially) the model identified using the weak formulation. The preceding approach was tested on a number of synthetic examples, and then applied to experimental turbulent fluid flow data to obtain a 2D model consistent over a range of driving values. In summary, the research presented here develops and validates a methodology for data-driven discovery of PDEs that is robust to noise and latent variables, and it demonstrates the ability to do so on real-world data for the first time.