arXiv subjects

Roman O. Grigoriev

Publications and source records attributed to Roman O. Grigoriev.

At least 19 recordsLinked to original sources

arXiv2026

Data-driven modeling of multiscale phenomena with applications to fluid turbulence

This paper introduces a novel data driven framework for constructing accurate and general equivariant models of multiscale phenomena which does not rely on specific assumptions about the underlying physics. This framework is illustrated using incompressible fluid turbulence as an example that is representative, practically important, reasonably simple, and exceedingly well studied. We use direct numerical simulations of freely decaying turbulence in two spatial dimensions to infer an effective field theory comprising explicit, interpretable evolution equations for both the large (resolved) and small (modeled) scales. The resulting closed system of equations is capable of accurately describing the effect of small scales, including backscatter -- the flow of energy from small to large scales, which is particularly pronounced in two dimensions -- which is an outstanding challenge that, to our knowledge, no existing alternative successfully tackles.

physics.flu-dyn↗

arXiv2026

Physics-informed data-driven inference of an interpretable equivariant LES model of incompressible fluid turbulence

Restrictive phenomenological assumptions represent a major roadblock for the development of accurate subgrid-scale models of fluid turbulence. Specifically, these assumptions limit a model's ability to describe key quantities of interest, such as local fluxes of energy and enstrophy, in the presence of diverse coherent structures. This paper introduces a symbolic data-driven subgrid-scale model that requires no phenomenological assumptions and has no adjustable parameters, yet it outperforms leading LES models. A combination of a priori and a posteriori benchmarks shows that the model produces accurate predictions of various quantities including local fluxes across a broad range of two-dimensional turbulent flows. While the model is inferred using LES-style spatial coarse-graining, its structure is more similar to RANS models, as it employs an additional field to describe subgrid scales. We find that this field must have a rank-two tensor structure in order to correctly represent both the components of the subgrid-scale stress tensor and the various fluxes.

physics.flu-dyn↗

arXiv2025

Data-driven discovery of the equations of turbulent convection

We compare the efficiency and ease-of-use of the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm and Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) framework in recovering the relevant governing equations and boundary conditions from data generated by direct numerical simulations (DNS) of turbulent convective flows. In the former case, a weak-form implementation pySINDy is used. Time-dependent data for two- (2D) and three-dimensional (3D) DNS simulation of Rayleigh-Benard convection and convective plane Couette flow is generated using the Dedalus PDE framework for spectrally solving differential equations. Using pySINDy we are able to recover the governing equations of 2D models of Rayleigh-Benard convection at Rayleigh numbers, R, from laminar, through transitional to moderately turbulent flow conditions, albeit with increasing difficulty with larger Rayleigh number, especially in recovery of the diffusive terms (with coefficient magnitude proportional to 1/R^0.5). SPIDER requires a much smaller library of terms and we are able to recover more easily the governing equations for a wider range of R in 2D and 3D convection and plane flow models and go on to recover constraints (the incompressibility condition) and boundary conditions, demonstrating the benefits and capabilities of SPIDER to go beyond pySINDy for these fluid problems governed by second-order PDEs. [We] demonstrat[e] the potential of machine-learning methods to validate numerical solvers and solutions for such flow problems. We also find that properties of the flow, specifically the correlation time and spatial scales, should inform the initial selection of spatiotemporal subdomain sizes [abbreviated]

astro-ph.SR↗

arXiv2025

Computing Chaotic Time-Averages from Few Periodic or Non-Periodic Orbits

For appropriately chosen weights, temporal averages in chaotic systems can be approximated as a weighted sum of averages over reference states, such as unstable periodic orbits. Under strict assumptions, such as completeness of the orbit library, these weights can be formally derived using periodic orbit theory. When these assumptions are violated, weights can be obtained empirically using a Markov partition of the chaotic set. Here, we describe an alternative, data-driven approach to computing weights that allows for an accurate approximation of temporal averages from a variety of reference states, including both periodic orbits and non-periodic trajectory segments embedded within the chaotic set. For a broad class of observables, we demonstrate that weights computed with the proposed method significantly outperform those derived from periodic orbit theory or Markov models, achieving superior accuracy while requiring far fewer states -- two critical properties for applications to high-dimensional chaotic systems.

math.DS↗

arXiv2024

Self-similarity and the direct (enstrophy) cascade in two-dimensional fluid turbulence

A widely used statistical theory of 2D turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^α$ with an integral exponent $α={-3}$, in the inertial range associated with the direct cascade. In the presence of large-scale coherent structures, a power-law scaling is observed, but the exponent often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory which describes the key physical mechanism behind the direct cascade and sheds new light on the relationship between the structure of the large-scale flow and the scaling of the small-scale structures in the inertial range. This theory also goes a step beyond KLB, to predict the upper and lower bounds of the inertial range as well as the energy scaling in the dissipation range.

physics.flu-dyn↗

arXiv2024

Ultra-low-energy defibrillation through adjoint optimization

This study investigates ultra-low-energy defibrillation protocols using a simple two-dimensional model of cardiac tissue. We find that, rather counter-intuitively, a single, properly timed, biphasic pulse can be more effective in defibrillating the tissue than low energy antitachycardia pacing (LEAP) which employs a sequence of such pulses, succeeding where the latter approach fails. Furthermore, we show that, with the help of adjoint optimization, it is possible to reduce the energy required for defibrillation even further, making it three orders of magnitude lower than that required by LEAP. Finally, we establish that this dramatic reduction is achieved through exploiting the sensitivity of the dynamics in vulnerable windows to promote annihilation of pairs of nearby phase singularities.

q-bio.TO↗

arXiv2023

Learning fluid physics from highly turbulent data using sparse physics-informed discovery of empirical relations (SPIDER)

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.

physics.flu-dyn↗

arXiv2023

Exact Coherent Structures in Fully Developed Two-Dimensional Turbulence

This paper reports several new classes of weakly unstable recurrent solutions of the 2+1-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions have a number of remarkable properties which distinguish them from analogous solutions of the Navier-Stokes equation describing transitional flows. First of all, they come in high-dimensional continuous families. Second, solutions of different types are connected, e.g., an equilibrium can be smoothly continued to a traveling wave or a time-periodic state. Third, and most important, many of these solutions are dynamically relevant for turbulent flow at high Reynolds numbers. Specifically, we find that turbulence in numerical simulations exhibits large-scale coherent structures resembling some of our time-periodic solutions both frequently and over long temporal intervals. Such solutions are analogous to exact coherent structures originally introduced in the context of transitional flows.

physics.flu-dyn↗

arXiv2022

Observing a Dynamical Skeleton of Turbulence in Taylor-Couette Flow Experiments

Recent work suggests unstable recurrent solutions of the equations governing fluid flow can play an important role in structuring the dynamics of turbulence. Here we present a method for detecting intervals of time where turbulence "shadows" (spatially and temporally mimics) recurrent solutions. We find that shadowing occurs frequently and repeatedly in both numerical and experimental observations of counter-rotating Taylor-Couette flow, despite the relatively small number of known recurrent solutions in this system. Our results set the stage for experimentally-grounded dynamical descriptions of turbulence in a variety of wall-bounded shear flows, enabling applications to forecasting and control.

physics.flu-dyn↗

arXiv2021

Data-driven detection of drifting system parameters

Data taken from observations of the natural world or laboratory measurements often depend on parameters which can vary in unexpected ways. In this paper we demonstrate how machine learning can be leveraged to detect changes in global parameters from variations in an identified model using only observational data. This capability, when paired with first principles analysis, can effectively distinguish the effects of these changing parameters from the intrinsic complexity of the system. Here we illustrate this by identifying a set of governing equations for an experiment generating a weakly turbulent fluid flow, then analyzing variation in the coefficients of these equations to unravel the drift in its physical parameters.

physics.flu-dyn↗

arXiv2021

Exact coherent structures and shadowing in turbulent Taylor-Couette flow

We investigate a theoretical framework for modeling fluid turbulence based on the formalism of exact coherent structures (ECSs). Although highly promising, existing evidence for the role of ECSs in turbulent flows is largely circumstantial and comes primarily from idealized numerical simulations. In particular, it remains unclear whether three-dimensional turbulent flows in experiment shadow any ECSs. In order to conclusively answer this question, a hierarchy of ECSs should be computed on a domain and with boundary conditions exactly matching experiment. The present study makes the first step in this direction by investigating a small-aspect-ratio Taylor-Couette flow with naturally periodic boundary condition in the azimuthal direction. We describe the structure of the chaotic set underlying turbulent flow driven by counter-rotating cylinders and present direct numerical evidence for shadowing of a collection of unstable relative periodic orbits and a traveling wave, setting the stage for further experimental tests of the framework.

physics.flu-dyn↗

arXiv2021

Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression

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.

physics.flu-dyn↗

arXiv2020

Heteroclinic and Homoclinic Connections in a Kolmogorov-Like Flow

Recent studies suggest that unstable recurrent solutions of the Navier-Stokes equation provide new insights into dynamics of turbulent flows. In this study, we compute an extensive network of dynamical connections between such solutions in a weakly turbulent quasi-two-dimensional Kolmogorov flow that lies in the inversion-symmetric subspace. In particular, we find numerous isolated heteroclinic connections between different types of solutions -- equilibria, periodic, and quasi-periodic orbits -- as well as continua of connections forming higher-dimensional connecting manifolds. We also compute a homoclinic connection of a periodic orbit and provide strong evidence that the associated homoclinic tangle forms the chaotic repeller that underpins transient turbulence in the symmetric subspace.

physics.flu-dyn↗

arXiv2020

Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits

In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating that turbulent flows in both experiment and numerics transiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). We show that UPOs we computed are also statistically significant, with turbulent flows spending a sizable fraction of the total time near these solutions. As a result, the average rates of energy input and dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.

physics.flu-dyn↗

arXiv2019

Using Noisy or Incomplete Data to Discover Models of Spatiotemporal Dynamics

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.

math.DS↗

arXiv2019

A novel subcritical transition to turbulence in Taylor-Couette flow with counter-rotating cylinders

The transition to turbulence in Taylor-Couette flow often occurs via a sequence of supercritical bifurcations to progressively more complex, yet stable, flows. We describe a subcritical laminar-turbulent transition in the counter-rotating regime mediated by an unstable intermediate state in a system with an axial aspect ratio of $Γ=5.26$ and a radius ratio of $η=0.905$. In this regime, flow visualization experiments and numerical simulations indicate the intermediate state corresponds to an aperiodic flow featuring interpenetrating spirals. Furthermore, the reverse transition out of turbulence leads first to the same intermediate state, which is now stable, before returning to an azimuthally-symmetric laminar flow. Time-resolved tomographic particle image velocimetry is used to characterize the experimental flows; these measurements compare favorably to direct numerical simulations with axial boundary conditions matching those of the experiments.

physics.flu-dyn↗

arXiv2019

Robust Approach for Rotor Mapping in Cardiac Tissue

The motion of and interaction between phase singularities that anchor spiral waves captures many qualitative and, in some cases, quantitative features of complex dynamics in excitable systems. Being able to accurately reconstruct their position is thus quite important, even if the data are noisy and sparse, as in electrophysiology studies of cardiac arrhythmias, for instance. A recently proposed global topological approach [Marcotte & Grigoriev, Chaos 27, 093936 (2017)] promises to dramatically improve the quality of the reconstruction compared with traditional, local approaches. Indeed, we found that this approach is capable of handling noise levels exceeding the range of the signal with minimal loss of accuracy. Moreover, it also works successfully with data sampled on sparse grids with spacing comparable to the mean separation between the phase singularities for complex patterns featuring multiple interacting spiral waves.

nlin.CD↗

arXiv2019

Robust and optimal sparse regression for nonlinear PDE models

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.

math.DS↗