Interpolation-Based Model Order Reduction for Polynomial Systems

Benner, Peter; Goyal, Pawan

doi:10.1137/19m1259171

Cited by 30 publications

(38 citation statements)

References 37 publications

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“…whereÂ( ) ∈ R r×r andĤ( ) ∈ R r×r 2 with r ≪ N. If the FOM is available in explicit form, for example, in matrix-vector form, then intrusive MOR techniques can be applied, such as POD 37,38 and interpolation-based methods. 39 Assuming an explicit form of the FOM is available, the ROM can be constructed with the projection matrix V ∈ R N×r so that w(t, ) ≈ Vŵ(t, ), for all t ≥ 0 and ∈  obtained by POD. Then, reduced-order matrices of the system ( 7) can be computed as follows:Â…”

Section: Operator Inference Approach To Learning Parameterized Reduced-operatorsmentioning

confidence: 99%

Learning reduced‐order dynamics for parametrized shallow water equations from data

Yıldız

Goyal

Benner

et al. 2021

Numerical Methods in Fluids

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This paper discusses a non-intrusive data-driven model order reduction method that learns low-dimensional dynamical models for a parametrized shallow water equation. We consider the shallow water equation in non-traditional form (NTSWE). We focus on learning low-dimensional models in a non-intrusive way. That means, we assume not to have access to a discretized form of the NTSWE in any form. Instead, we have snapshots that can be obtained using a black-box solver. Consequently, we aim at learning reduced-order models only from the snapshots. Precisely, a reduced-order model is learnt by solving an appropriate least-squares optimization problem in a low-dimensional subspace. Furthermore, we discuss computational challenges that particularly arise from the optimization problem being ill-conditioned. Moreover, we extend the non-intrusive model order reduction framework to a parametric case, where we make use of the parameter dependency at the level of the partial differential equation. We illustrate the efficiency of the proposed non-intrusive method to construct reduced-order models for NTSWE and compare it with an intrusive method (proper orthogonal decomposition). We furthermore discuss the predictive capabilities of both models outside the range of the training data. K E Y W O R D Sdata-driven modeling, model order reduction, operator inference, scientific machine learning, shallow water equation INTRODUCTIONShallow water equations (SWE) are a popular set of hyperbolic partial differential equations (PDEs) with the capability of describing geophysical wave phenomena, for example, the Kelvin and Rossby waves in the atmosphere and the oceans. They are frequently used in geophysical flow prediction, 1 investigation of baroclinic instability, 2,3 and planetary flows. 4 In this paper, we study a model order reduction (MOR) technique for SWE. MOR techniques allow us to construct low-dimensional models or reduced-order models (ROMs) for a large-scale dynamical system. We refer to the books 5,6 for an overview of the available techniques. These ROMs are computationally efficient and accurate, and are worthy when a full order model (FOM) needs to be simulated multiple-times for different parameter settings. Additionally, ROMs are even more valuable in the case of SWE, when the interest lies in simulating the model for a very long time horizon. MOR problems for SWE have been intensively studied in the literature, see, for example, References [7-12].

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Section: Operator Inference Approach To Learning Parameterized Reduced-operatorsmentioning

confidence: 99%

Learning reduced‐order dynamics for parametrized shallow water equations from data

Yıldız

Goyal

Benner

et al. 2021

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

show abstract

“…, X Mt , to which we apply operator inference as described in Section 2.4. We use a first-order forward difference scheme to approximate the time derivative as in (8). Additionally, for benchmarking purposes, we also construct a reduced model ( 6) via the intrusive process described in Section 2.3.…”

Section: Stability Of Inferred Modelsmentioning

confidence: 99%

“…First, there is system identification that originated in the systems and control community [32]. Antoulas and collaborators introduced the Loewner approach [3,35,5], which has been extended from linear time-invariant systems to parameterized [25], switched [23], structured [53], delayed [52], bilinear [6], quadratic bilinear [22], and polynomial [8] systems as well as to learning from time-domain data [41,28]. There is also dynamic mode decomposition (DMD) [51,47,59,30] that best-fits linear operators to state trajectories in L 2 norm.…”

Section: Introductionmentioning

confidence: 99%

Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference

Sawant¹,

Kramer²,

Peherstorfer³

2021

Preprint

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Operator inference learns low-dimensional dynamical-system models with polynomial nonlinear terms from trajectories of high-dimensional physical systems (non-intrusive model reduction). This work focuses on the large class of physical systems that can be well described by models with quadratic nonlinear terms and proposes a regularizer for operator inference that induces a stability bias onto quadratic models. The proposed regularizer is physics informed in the sense that it penalizes quadratic terms with large norms and so explicitly leverages the quadratic model form that is given by the underlying physics. This means that the proposed approach judiciously learns from data and physical insights combined, rather than from either data or physics alone. Additionally, a formulation of operator inference is proposed that enforces model constraints for preserving structure such as symmetry and definiteness in the linear terms. Numerical results demonstrate that models learned with operator inference and the proposed regularizer and structure preservation are accurate and stable even in cases where using no regularization or Tikhonov regularization leads to models that are unstable.

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“…On the other hand methods developed for linear systems overcome such reliance on training data establishing rigor regarding error bounds, structure preservation and even ROM error optimally (see [12], [13], [14]). Some of these methods such as balanced truncation or moment matching methods with their advantages are being recently transferred for systems with weak/structural nonlinearity such as bilinear [15], quadratic [16] and polynomial [17] systems. Since developing global optimal reduction methods for general nonlinear systems can be claimed to be an intractable task.…”

Section: Introductionmentioning

confidence: 99%

Towards Real-Time Monitoring and Control of Water Networks

Elkhashap¹,

Rüschen²,

Abel³

2022

Preprint

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Water networks are used in numerous applications, all of which have the essential task of real-time monitoring and control of water states. A framework for the generation of efficient models of water networks suitable for real-time monitoring and control purposes is proposed. The proposed models preserve the distributed parameter character of the network local elements. Hence, the spatial resolution of the property under consideration is recovered. The realtime feasibility of the network model is ensured by means of reduced order modeling of the models constituting components. A novel model order reduction procedure that preserves the model parametric dependency is introduced. The proposed concept is evaluated with the water temperature as the property under consideration. The formulated model is applied for the prediction of the water temperature within an experimental test bench of a 60 meter exemplary circulation network at the company VIEGA. A reduced order model (ROM) with a 50 mm spatial resolution, i.e. 1200 discretization points, is constructed and utilized as the identification model for a single path of the test bench. Afterwards, the ROM is evaluated in a Hardware in the Loop experiment for the prediction of the downstream temperature showing high prediction accuracy with mean relative error below 3.5%. The ROM single step computation time stayed below 2 msec highlighting the realtime potential of the method. Moreover, full network model validation experiments featuring both diffusion and transport dominated parts were conducted. The network model could predict the temperature evolution, flow rate, and pressure accurately at the different paths of the network with mean relative errors below 4%, 2%, and 2%, respectively.

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Interpolation-Based Model Order Reduction for Polynomial Systems

Cited by 30 publications

References 37 publications

Learning reduced‐order dynamics for parametrized shallow water equations from data

Learning reduced‐order dynamics for parametrized shallow water equations from data

Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference

Towards Real-Time Monitoring and Control of Water Networks

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