Physics-constrained, low-dimensional models for magnetohydrodynamics: First-principles and data-driven approaches

Kaptanoglu, Alan A.; Morgan, Kyle; Hansen, Chris; Brunton, Steven L.

doi:10.1103/physreve.104.015206

Cited by 70 publications

(27 citation statements)

References 107 publications

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“…These models may be ordinary differential equations (ODEs) [16] or partial differential equations (PDEs) [17,29]. SINDy has been applied to a number of challenging model discovery problems, including for reduced-order models of fluid dynamics [30][31][32][33][34][35] and plasma dynamics [36][37][38], turbulence closures [39][40][41], mesoscale ocean closures [42], nonlinear optics [43], computational chemistry [44] and numerical integration schemes [45]. SINDy has been widely adopted, in part, because it is highly extensible.…”

Section: Introductionmentioning

confidence: 99%

Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control

et al. 2022

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Sparse model identification enables the discovery of nonlinear dynamical systems purely from data; however, this approach is sensitive to noise, especially in the low-data limit. In this work, we leverage the statistical approach of bootstrap aggregating (bagging) to robustify the sparse identification of the nonlinear dynamics (SINDy) algorithm. First, an ensemble of SINDy models is identified from subsets of limited and noisy data. The aggregate model statistics are then used to produce inclusion probabilities of the candidate functions, which enables uncertainty quantification and probabilistic forecasts. We apply this ensemble-SINDy (E-SINDy) algorithm to several synthetic and real-world datasets and demonstrate substantial improvements to the accuracy and robustness of model discovery from extremely noisy and limited data. For example, E-SINDy uncovers partial differential equations models from data with more than twice as much measurement noise as has been previously reported. Similarly, E-SINDy learns the Lotka Volterra dynamics from remarkably limited data of yearly lynx and hare pelts collected from 1900 to 1920. E-SINDy is computationally efficient, with similar scaling as standard SINDy. Finally, we show that ensemble statistics from E-SINDy can be exploited for active learning and improved model predictive control.

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Section: Introductionmentioning

confidence: 99%

Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control

et al. 2022

Self Cite

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“…In particular [6] used a robust method termed SR3 (sparse relaxed regularized regression) to extract the sparse, nonzero coefficients of the dynamical model. Since its introduction, SINDy has been applied to a wide range of systems, including for reduced-order models of fluid dynamics [7]- [14] and plasma dynamics [15], [16], turbulence closures [17]- [19], nonlinear optics [20], numerical integration schemes [21], discrepancy modeling [22], [23], boundary value problems [24], multiscale dynamics [25], identifying dynamics on Poincare maps [26], [27], tensor formulations [28], and systems with stochastic dynamics [29], [30]. It can also be used to jointly discovery coordinates and dynamics simultaneously [31], [32].…”

Section: Introductionmentioning

confidence: 99%

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

Delahunt

Kutz

2022

IEEE Access

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We consider the data-driven discovery of governing equations from time-series data in the limit of high noise. The algorithms developed describe an extensive toolkit of methods for circumventing the deleterious effects of noise in the context of the sparse identification of nonlinear dynamics (SINDy) framework. We offer two primary contributions, both focused on noisy data acquired from a system ẋ = f (x). First, we propose, for use in high-noise settings, an extensive toolkit of critically enabling extensions for the SINDy regression method, to progressively cull functionals from an over-complete library and yield a set of sparse equations that regress to the derivate ẋ. This toolkit includes: (regression step) weight timepoints based on estimated noise, use ensembles to estimate coefficients, and regress using FFTs; (culling step) leverage linear dependence of functionals, and restore and protect culled functionals based on Figures of Merit (FoMs). In a novel Assessment step, we define FoMs that compare model predictions to the original time-series (i.e., x(t) rather than ẋ(t)). These innovations can extract sparse governing equations and coefficients from high-noise time-series data (e.g., 300% added noise). For example, it discovers the correct sparse libraries in the Lorenz system, with median coefficient estimate errors equal to 1%−3% (for 50% noise), 6%−8% (for 100% noise); and 23%−25% (for 300% noise). The enabling modules in the toolkit are combined into a single method, but the individual modules can be tactically applied in other equation discovery methods (SINDy or not) to improve results on high-noise data. Second, we propose a technique, applicable to any model discovery method based on ẋ = f (x), to assess the accuracy of a discovered model in the context of non-unique solutions due to noisy data. Currently, this non-uniqueness can obscure a discovered model's accuracy and thus a discovery method's effectiveness. We describe a technique that uses linear dependencies among functionals to transform a discovered model into an equivalent form that is closest to the true model, enabling more accurate assessment of a discovered model's accuracy.

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“…Moreover, neural networks have also been utilized to learn the optimal map between filtered and unfiltered variables in the approximate deconvolution framework for SGS modeling [26,27]. Apart from SGS closure modeling, machine learning (ML) and in particular DL is being increasingly applied for different problems in fluid mechanics, like superresolution of turbulent flows [28,29], Reynolds-Average Navier-Stokes (RANS) closure modeling [30][31][32], and reduced-order modeling [33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

Frame invariant neural network closures for Kraichnan turbulence

Pawar,

San,

Rasheed

et al. 2022

Preprint

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Numerical simulations of geophysical and atmospheric flows have to rely on parameterizations of subgrid scale processes due to their limited spatial resolution. Despite substantial progress in developing parameterization (or closure) models for subgrid scale (SGS) processes using physical insights and mathematical approximations, they remain imperfect and can lead to inaccurate predictions. In recent years, machine learning has been successful in extracting complex patterns from high-resolution spatio-temporal data, leading to improved parameterization models, and ultimately better coarse grid prediction. However, the inability to satisfy known physics and poor generalization hinders the application of these models for real-world problems. In this work, we propose a frame invariant closure approach to improve the accuracy and generalizability of deep learning-based subgrid scale closure models by embedding physical symmetries directly into the structure of the neural network. Specifically, we utilized specialized layers within the convolutional neural network in such a way that desired constraints are theoretically guaranteed without the need for any regularization terms. We demonstrate our framework for a two-dimensional decaying turbulence test case mostly characterized by the forward enstrophy cascade. We show that our frame invariant SGS model (i) accurately predicts the subgrid scale source term, (ii) respects the physical symmetries such as translation, Galilean, and rotation invariance, and (iii) is numerically stable when implemented in coarse-grid simulation with generalization to different initial conditions and Reynolds number. This work builds a bridge between extensive physics-based theories and data-driven modeling paradigms, and thus represents a promising step towards the development of physically consistent data-driven turbulence closure models.

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Physics-constrained, low-dimensional models for magnetohydrodynamics: First-principles and data-driven approaches

Cited by 70 publications

References 107 publications

Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control

Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

Frame invariant neural network closures for Kraichnan turbulence

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