Data-driven nonintrusive reduced order modeling for dynamical systems with moving boundaries using Gaussian process regression

Ma, Zewei; Pan, Wenxiao

doi:10.1016/j.cma.2020.113495

Cited by 20 publications

(14 citation statements)

References 32 publications

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“…Furthermore, a GPR is probabilistic and provides a confidence region that quantifies the quality of regression. Such a quantification can either be used to increase the size of the training set [34], to quantify the extrapolation capabilities [26], or to develop an error model [19]. A Lagrange polynomial interpolation is deterministic and does not offer such flexibility.…”

Section: Relation To Previous Workmentioning

confidence: 99%

“…Remark 3 (GPR for vector-valued functions) While training the GPR, we assume that α g (z) has uncorrelated components [19,26]. This certainly introduces some inaccuracies because for most pPDEs, α g (z) has coupled correlated components.…”

Section: Prediction Stepmentioning

confidence: 99%

“…Our MOR technique is no exception to this limitation. Heuristics suggest (see [26]) that the variance of the GP can be used to quantify the extrapolation capabilities, but the success of such a technique-particularly for hyperbolic pPDEs considered in our numerical experiments-is unclear. We include the corners of the parameter domain Z in the parameter samples and avoid extrapolation altogether.…”

Section: Prediction Stepmentioning

confidence: 99%

“…Authors in [4,13], [19] and [22] undertake the first approach and use radial basis functions (RBFs), Gaussian processes (GP) and neural networks, respectively, for regression. Authors in [45,46] opt for the second approach and reduce the Navier-Stokes equations by approximating the evolution operator using RBFs- [47] and [26] provide extensions to problems involving fluid-structure interaction and moving boundaries, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Data-Driven Model Order Reduction for Problems with Parameter-Dependent Jump-Discontinuities

Sarna¹,

Benner²

2021

Preprint

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We propose a data-driven model order reduction (MOR) technique for parametrized partial differential equations that exhibit parameter-dependent jumpdiscontinuities. Such problems have poor-approximability in a linear space and therefore, are challenging for standard MOR techniques. We build upon the methodology of approximating the map between the parameter domain and the expansion coefficients of the reduced basis via regression. The online stage queries the regression model for the expansion coefficients and recovers a reduced approximation for the solution. We propose to apply this technique to a transformed solution that results from composing the solution with a spatial transform. Unlike the (untransformed) solution, it is sufficiently regular along the parameter domain and thus, is well-approximable in a low-dimensional linear space. To recover an approximation for the (untransformed) solution, we propose an online efficient regression-based technique that approximates the inverse of the spatial transform. Our method features a decoupled online and offline stage, and benchmark problems involving hyperbolic and parabolic equations demonstrate its effectiveness.

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Section: Relation To Previous Workmentioning

confidence: 99%

Section: Prediction Stepmentioning

confidence: 99%

Section: Prediction Stepmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Data-Driven Model Order Reduction for Problems with Parameter-Dependent Jump-Discontinuities

Sarna¹,

Benner²

2021

Preprint

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“…The literature related to data-driven reduced-order modeling is vast and spans a great number of engineering and scientific fields ranging from fluid mechanics [6,12,9,8,14] to structural mechanics [13,11,16,1]. In the majority of these works the assumption is plentiful data.…”

Section: Introductionmentioning

confidence: 99%

Optimal criteria and their asymptotic form for data selection in data-driven reduced-order modeling with Gaussian process regression

Sapsis,

Blanchard

2021

Preprint

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We derive criteria for the selection of datapoints used for data-driven reduced-order modeling and other areas of supervised learning based on Gaussian process regression (GPR). While this is a well-studied area in the fields of active learning and optimal experimental design, most criteria in the literature are empirical. Here we introduce an optimality condition for the selection of a new input defined as the minimizer of the distance between the approximated output probability density function (pdf) of the reduced-order model and the exact one. Given that the exact pdf is unknown, we define the selection criterion as the supremum over the unit sphere of the native Hilbert space for the GPR. The resulting selection criterion, however, has a form that is difficult to compute. We combine results from GPR theory and asymptotic analysis to derive a computable form of the defined optimality criterion that is valid in the limit of small predictive variance. The derived asymptotic form of the selection criterion leads to convergence of the GPR model that guarantees a balanced distribution of data resources between probable and large-deviation outputs, resulting in an effective way for sampling towards data-driven reduced-order modeling.

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Nonintrusive reduced‐order modeling approach for parametrized unsteady flow and heat transfer problems

Hua

Sørheim

2022

Numerical Methods in Fluids

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A novel nonintrusive reduced-order modeling approach is proposed for parametrized unsteady flow and heat transfer problems. A set of reduced basis functions are extracted via a proper orthogonal decomposition (POD) method from a collection of high-fidelity numerical solutions (snapshots of spatial distribution at a series of time steps) that are computed for properly chosen parameters (e.g., material property, initial/boundary conditions) using a full-order model (finite-volume/finite-element methods). Here, the time dimension is treated separately from other parameters, reflecting the dynamic features of the flow problems. Dynamic mode decomposition is used to decompose the time-resolved data (POD coefficients) into dynamics modes and reconstruct/predict the dynamic evolution of the flow systems. The POD coefficients at every time step under the condition of a set of parameters are approximated through interpolation with the radial basis function method from those obtained from the snapshots of the chosen parameter space. Hence, the POD coefficients at a set of given time and parameters can be approximated, and the spatial distribution of the solution data can be reconstructed. The current reduced-order model has been applied to the problems governed by a set of parametrized unsteady Navier-Stokes and heat conduction equations. The model capability has been illustrated numerically by three numerical test cases: flow past a cylinder, melt solidification, and dam break. The model prediction capabilities have been evaluated by varying the material properties (viscosity and density) and the heat transfer conditions on some of the boundaries. An error analysis is also carried out to show the model prediction accuracy.

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Data-driven nonintrusive reduced order modeling for dynamical systems with moving boundaries using Gaussian process regression

Cited by 20 publications

References 32 publications

Data-Driven Model Order Reduction for Problems with Parameter-Dependent Jump-Discontinuities

Data-Driven Model Order Reduction for Problems with Parameter-Dependent Jump-Discontinuities

Optimal criteria and their asymptotic form for data selection in data-driven reduced-order modeling with Gaussian process regression

Nonintrusive reduced‐order modeling approach for parametrized unsteady flow and heat transfer problems

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