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Fasel U, Kutz JN, Brunton BW, Brunton SL. Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control. Proc Math Phys Eng Sci 2022; 478:20210904. [PMID: 35450025 PMCID: PMC9006119 DOI: 10.1098/rspa.2021.0904] [Citation(s) in RCA: 11] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Accepted: 03/10/2022] [Indexed: 12/17/2022] Open
Abstract
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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Affiliation(s)
- U Fasel
- Department of Mechanical Engineering, University of Washington, Seattle, WA, USA
| | - J N Kutz
- Department of Applied Mathematics, University of Washington, Seattle, WA, USA
| | - B W Brunton
- Department of Biology, University of Washington, Seattle, WA, USA
| | - S L Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA, USA
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Callaham JL, Loiseau JC, Rigas G, Brunton SL. Nonlinear stochastic modelling with Langevin regression. Proc Math Phys Eng Sci 2021; 477:20210092. [PMID: 35153564 PMCID: PMC8299553 DOI: 10.1098/rspa.2021.0092] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2021] [Accepted: 05/04/2021] [Indexed: 12/18/2022] Open
Abstract
Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker-Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.
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Affiliation(s)
- J. L. Callaham
- Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
| | - J.-C. Loiseau
- Laboratoire DynFluid, Arts et Mètiers ParisTech, 75013 Paris, France
| | - G. Rigas
- Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
| | - S. L. Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
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Fonzi N, Brunton SL, Fasel U. Data-driven nonlinear aeroelastic models of morphing wings for control. Proc Math Phys Eng Sci 2020; 476:20200079. [PMID: 32831607 DOI: 10.1098/rspa.2020.0079] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2020] [Accepted: 06/16/2020] [Indexed: 01/05/2023] Open
Abstract
Accurate and efficient aeroelastic models are critically important for enabling the optimization and control of highly flexible aerospace structures, which are expected to become pervasive in future transportation and energy systems. Advanced materials and morphing wing technologies are resulting in next-generation aeroelastic systems that are characterized by highly coupled and nonlinear interactions between the aerodynamic and structural dynamics. In this work, we leverage emerging data-driven modelling techniques to develop highly accurate and tractable reduced-order aeroelastic models that are valid over a wide range of operating conditions and are suitable for control. In particular, we develop two extensions to the recent dynamic mode decomposition with control (DMDc) algorithm to make it suitable for flexible aeroelastic systems: (1) we introduce a formulation to handle algebraic equations, and (2) we develop an interpolation scheme to smoothly connect several linear DMDc models developed in different operating regimes. Thus, the innovation lies in accurately modelling the nonlinearities of the coupled aerostructural dynamics over multiple operating regimes, not restricting the validity of the model to a narrow region around a linearization point. We demonstrate this approach on a high-fidelity, three-dimensional numerical model of an airborne wind energy system, although the methods are generally applicable to any highly coupled aeroelastic system or dynamical system operating over multiple operating regimes. Our proposed modelling framework results in real-time prediction of nonlinear unsteady aeroelastic responses of flexible aerospace structures, and we demonstrate the enhanced model performance for model predictive control. Thus, the proposed architecture may help enable the widespread adoption of next-generation morphing wing technologies.
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Affiliation(s)
- N Fonzi
- CMASLab, ETH Zurich, 8092 Zurich, Switzerland
| | - S L Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
| | - U Fasel
- CMASLab, ETH Zurich, 8092 Zurich, Switzerland
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Mangan NM, Askham T, Brunton SL, Kutz JN, Proctor JL. Model selection for hybrid dynamical systems via sparse regression. Proc Math Phys Eng Sci 2019; 475:20180534. [PMID: 31007544 PMCID: PMC6451978 DOI: 10.1098/rspa.2018.0534] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2018] [Accepted: 01/25/2019] [Indexed: 12/14/2022] Open
Abstract
Hybrid systems are traditionally difficult to identify and analyse using classical dynamical systems theory. Moreover, recently developed model identification methodologies largely focus on identifying a single set of governing equations solely from measurement data. In this article, we develop a new methodology, Hybrid-Sparse Identification of Nonlinear Dynamics, which identifies separate nonlinear dynamical regimes, employs information theory to manage uncertainty and characterizes switching behaviour. Specifically, we use the nonlinear geometry of data collected from a complex system to construct a set of coordinates based on measurement data and augmented variables. Clustering the data in these measurement-based coordinates enables the identification of nonlinear hybrid systems. This methodology broadly empowers nonlinear system identification without constraining the data locally in time and has direct connections to hybrid systems theory. We demonstrate the success of this method on numerical examples including a mass–spring hopping model and an infectious disease model. Characterizing complex systems that switch between dynamic behaviours is integral to overcoming modern challenges such as eradication of infectious diseases, the design of efficient legged robots and the protection of cyber infrastructures.
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Affiliation(s)
- N M Mangan
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
| | - T Askham
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
| | - S L Brunton
- Institute for Disease Modeling, Bellevue, WA 98005, USA
| | - J N Kutz
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
| | - J L Proctor
- Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
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Kaiser E, Kutz JN, Brunton SL. Sparse identification of nonlinear dynamics for model predictive control in the low-data limit. Proc Math Phys Eng Sci 2018; 474:20180335. [PMID: 30839858 PMCID: PMC6283900 DOI: 10.1098/rspa.2018.0335] [Citation(s) in RCA: 59] [Impact Index Per Article: 9.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2018] [Accepted: 10/11/2018] [Indexed: 02/07/2023] Open
Abstract
Data-driven discovery of dynamics via machine learning is pushing the frontiers of modelling and control efforts, providing a tremendous opportunity to extend the reach of model predictive control (MPC). However, many leading methods in machine learning, such as neural networks (NN), require large volumes of training data, may not be interpretable, do not easily include known constraints and symmetries, and may not generalize beyond the attractor where models are trained. These factors limit their use for the online identification of a model in the low-data limit, for example following an abrupt change to the system dynamics. In this work, we extend the recent sparse identification of nonlinear dynamics (SINDY) modelling procedure to include the effects of actuation and demonstrate the ability of these models to enhance the performance of MPC, based on limited, noisy data. SINDY models are parsimonious, identifying the fewest terms in the model needed to explain the data, making them interpretable and generalizable. We show that the resulting SINDY-MPC framework has higher performance, requires significantly less data, and is more computationally efficient and robust to noise than NN models, making it viable for online training and execution in response to rapid system changes. SINDY-MPC also shows improved performance over linear data-driven models, although linear models may provide a stopgap until enough data is available for SINDY. SINDY-MPC is demonstrated on a variety of dynamical systems with different challenges, including the chaotic Lorenz system, a simple model for flight control of an F8 aircraft, and an HIV model incorporating drug treatment.
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Affiliation(s)
- E. Kaiser
- Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195
| | - J. N. Kutz
- Department of Applied Mathematics, University of Washington, Seattle, WA, 98195
| | - S. L. Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195
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Mangan NM, Kutz JN, Brunton SL, Proctor JL. Model selection for dynamical systems via sparse regression and information criteria. Proc Math Phys Eng Sci 2017; 473:20170009. [PMID: 28878554 PMCID: PMC5582175 DOI: 10.1098/rspa.2017.0009] [Citation(s) in RCA: 48] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2017] [Accepted: 07/26/2017] [Indexed: 11/13/2022] Open
Abstract
We develop an algorithm for model selection which allows for the consideration of a combinatorially large number of candidate models governing a dynamical system. The innovation circumvents a disadvantage of standard model selection which typically limits the number of candidate models considered due to the intractability of computing information criteria. Using a recently developed sparse identification of nonlinear dynamics algorithm, the sub-selection of candidate models near the Pareto frontier allows feasible computation of Akaike information criteria (AIC) or Bayes information criteria scores for the remaining candidate models. The information criteria hierarchically ranks the most informative models, enabling the automatic and principled selection of the model with the strongest support in relation to the time-series data. Specifically, we show that AIC scores place each candidate model in the strong support, weak support or no support category. The method correctly recovers several canonical dynamical systems, including a susceptible-exposed-infectious-recovered disease model, Burgers’ equation and the Lorenz equations, identifying the correct dynamical system as the only candidate model with strong support.
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Affiliation(s)
- N M Mangan
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.,Institute for Disease Modeling, Bellevue, WA 98005, USA
| | - J N Kutz
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
| | - S L Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
| | - J L Proctor
- Institute for Disease Modeling, Bellevue, WA 98005, USA
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Abstract
BACKGROUND A case of avascular necrosis (AN) of the navicular bone, in a 24-year-old woman with Type 1 diabetes with peripheral neuropathy, in the absence of any history of direct trauma is presented. The clinical and radiological features at presentation suggested an evolving Charcot arthropathy (CA), but subsequent serial X-rays clearly confirmed AN. CONCLUSIONS Swelling and foot deformity in association with long-standing diabetic peripheral neuropathy is suggestive of CA, although AN, a less common condition, may show the same clinical features. It is therefore important to undertake further confirmatory radiological investigations if there is any doubt about the diagnosis.
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Affiliation(s)
- Y P Samarasinghe
- Beta Cell Diabetes Centre, Chelsea and Westminster Hospital NHS Trust, London, UK.
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