Neural network closures for nonlinear model order reduction

San, Omer; Maulik, Romit

doi:10.1007/s10444-018-9590-z

Cited by 134 publications

(93 citation statements)

References 86 publications

(96 reference statements)

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“…We note that data‐driven closure modeling for non‐ROM settings is an extremely active research area (see, eg, the works of Duraisamy et al and Ling et al). We also note that there are other DD‐ROM closure models . We emphasize, however, that these DD‐ROM closure models are different from our DDC‐ROM in the following respects.…”

Section: Introductionmentioning

confidence: 81%

“…We also note that there are other DD-ROM closure models. [30][31][32][33][34][35][36][37] We emphasize, however, that these DD-ROM closure models are different from our DDC-ROM in the following respects.…”

Section: Introductionmentioning

confidence: 94%

“…"> 1.They do not use spatial filtering (as in large‐eddy simulation (LES)) to isolate the Correction term. 2.They generally do closure modeling for both linear and nonlinear terms …”

Section: Introductionmentioning

confidence: 99%

“…They generally do closure modeling for both linear and nonlinear terms. [35][36][37] 3. They only use the ROM projection to define the Correction term.…”

Section: Introductionmentioning

confidence: 99%

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Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Mohebujjaman

Rebholz

Iliescu

2018

Numerical Methods in Fluids

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Summary We have recently proposed a data‐driven correction reduced‐order model (DDC‐ROM) framework for the numerical simulation of fluid flows, which can be formally written as follows. The new DDC‐ROM was constructed by using reduced‐order model spatial filtering and data‐driven reduced‐order model closure modeling (for the Correction term) and was successfully tested in the numerical simulation of a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. In this paper, we propose a physically constrained DDC‐ROM (CDDC‐ROM), which aims at improving the physical accuracy of the DDC‐ROM. The new physical constraints require that the CDDC‐ROM operators satisfy the same type of physical laws (ie, the Correction term's linear component should be dissipative, and the Correction term's nonlinear component should conserve energy) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data‐driven modeling step, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDC‐ROM and standard DDC‐ROM for a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. To this end, we consider a reproductive regime as well as a predictive (ie, cross‐validation) regime in which we use as little as 50% of the original training data. The numerical investigation clearly shows that the new CDDC‐ROM is significantly more accurate than the DDC‐ROM in both regimes.

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Section: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 94%

“…"> 1.They do not use spatial filtering (as in large‐eddy simulation (LES)) to isolate the Correction term. 2.They generally do closure modeling for both linear and nonlinear terms …”

Section: Introductionmentioning

confidence: 99%

“…They generally do closure modeling for both linear and nonlinear terms. [35][36][37] 3. They only use the ROM projection to define the Correction term.…”

Section: Introductionmentioning

confidence: 99%

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Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Mohebujjaman

Rebholz

Iliescu

2018

Numerical Methods in Fluids

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“…The POD is complemented by a Galerkin projection (GP) in which the dynamics of the system is modeled. Recently, there is a significant effort to use machine learning algorithms in model order reduction of nonlinear fluid flows [6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Data-driven recovery of hidden physics in reduced order modeling of fluid flows

et al. 2020

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In this article, we introduce a modular hybrid analysis and modeling (HAM) approach to account for hidden physics in reduced order modeling (ROM) of parameterized systems relevant to fluid dynamics. The hybrid ROM framework is based on using the first principles to model the known physics in conjunction with utilizing the data-driven machine learning tools to model remaining residual that is hidden in data. This framework employs proper orthogonal decomposition as a compression tool to construct orthonormal bases and Galerkin projection (GP) as a model to built the dynamical core of the system. Our proposed methodology hence compensates structural or epistemic uncertainties in models and utilizes the observed data snapshots to compute true modal coefficients spanned by these bases. The GP model is then corrected at every time step with a data-driven rectification using a long short-term memory (LSTM) neural network architecture to incorporate hidden physics. A Grassmannian manifold approach is also adapted for interpolating basis functions to unseen parametric conditions. The control parameter governing the system's behavior is thus implicitly considered through true modal coefficients as input features to the LSTM network. The effectiveness of the HAM approach is discussed through illustrative examples that are generated synthetically to take hidden physics into account. Our approach thus provides insights addressing a fundamental limitation of the physics-based models when the governing equations are incomplete to represent underlying physical processes.

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Deep Learning‐Based Reduced Order Models for Cardiac Electrophysiology

Fresca,

Dedè,

Manzoni

2024

Big Data Analysis and Artificial Intelligence for Medical Sciences

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Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration. More importantly, these models feature coherent structures that propagate over time, such as wavefronts. These systems can hardly be reduced to lower dimensional problems by conventional reduced order models (ROMs) such as, e.g., the reduced basis method. This is primarily due to the low regularity of the solution manifold (with respect to the problem parameters), as well as to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To overcome this difficulty, in this paper we propose a new, nonlinear approach relying on deep learning (DL) algorithms-such as deep feedforward neural networks and convolutional autoencodersto obtain accurate and efficient ROMs, whose dimensionality matches the number of system parameters. We show that the proposed DL-ROM framework can efficiently provide solutions to parametrized electrophysiology problems, thus enabling multi-scenario analysis in pathological cases. We investigate four challenging test cases in cardiac electrophysiology, thus demonstrating that DL-ROM outperforms classical projection-based ROMs.

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Neural network closures for nonlinear model order reduction

Cited by 134 publications

References 86 publications

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Data-driven recovery of hidden physics in reduced order modeling of fluid flows

Deep Learning‐Based Reduced Order Models for Cardiac Electrophysiology

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