POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition

Fresca, Stefania; Manzoni, Andrea

doi:10.1016/j.cma.2021.114181

Cited by 163 publications

(99 citation statements)

References 28 publications

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“…Note that the idea of coupling the proper orthogonal decomposition with machine learning approaches was already investigated, even for time-parameter dependent systems (parametrized PDEs), see e.g. [10]. However, utilizing the shifted POD variant is, up to the authors knowledge, a novelty.…”

Section: Podiann and Spodiann Frameworkmentioning

confidence: 99%

“…While sPOD exhibits great promise in approximation of data originating from transport-dominated systems via just a few spatial modes in the respective co-moving frames, it practically disallows for utilization of the standard projection-based approaches to the reduced order model construction [2]. In order to allow for practical application of sPOD in MOR, we propose to replace the projection framework by an interpolation between the discrete values {η r } ℓ r=1 utilizing an artificial neural network (ANN), an idea usable for parametrized PDEs and similar to the one presented in [9] and in [10].…”

Section: Introductionmentioning

confidence: 99%

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Shifted Proper Orthogonal Decomposition and Artificial Neural Networks for Time-Continuous Reduced Order Models of Transport-Dominated Systems

Kovárnová¹,

Krah²,

Reiß³

et al. 2022

Topical Problems of Fluid Mechanics 2022

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Transport-dominated systems are pervasive in both industrial and scientific applications. However, they provide a challenge for common mode-based model order reduction (MOR) approaches, as they often require a large number of linear modes to obtain a sufficiently accurate reduced order model (ROM). In this work, we utilize the shifted proper orthogonal decomposition (sPOD), a methodology tailored for MOR of transport-dominated systems, and combine it with an interpolation based on artificial neural networks (ANN) to obtain a time-continuous ROM usable in engineering practice. The resulting MOR framework is purely data-driven, i.e., it does not require any information on the full order model (FOM) structure, which extends its applicability. On the other hand, compared to the standard projection-based approaches to MOR, the dimensionality reduction utilizing sPOD and ANN is significantly more computationally expensive since it requires a solution of high-dimensional optimization problems.

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Section: Podiann and Spodiann Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Shifted Proper Orthogonal Decomposition and Artificial Neural Networks for Time-Continuous Reduced Order Models of Transport-Dominated Systems

Kovárnová¹,

Krah²,

Reiß³

et al. 2022

Topical Problems of Fluid Mechanics 2022

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“…This work extends the POD-DL-ROM framework [30] in two directions: first, it replaces the CAE architecture of POD-DL-ROM with a long short-term memory (LSTM) based autoencoder [35,36], in order to better take into account time evolution when dealing with nonlinear unsteady parametrized PDEs (µ-POD-LSTM-ROM); second, it aims at performing extrapolation forward in time (compared to the training time window) of the PDE solution, for unseen values of the input parameters -a task often missed by traditional projection-based ROMs. Our final goal is to predict the PDE solution on a larger time domain (T in , T end ) than the one, (0, T ), used for the ROM training -here 0 ≤ T in ≤ T end and T end > T .…”

Section: Introductionmentioning

confidence: 97%

“…DL-ROMs outperform POD-based ROMs such as the reduced basis method -regarding both numerical accuracy and computational efficiency at testing stage. With the same spirit, POD-DL-ROMs [30] enable a more efficient training stage and the use of much larger FOM dimensions, without affecting network complexity, thanks to a prior dimensionality reduction of FOM snapshots through randomized POD (rPOD) [31], and a multi-fidelity pretraining stage, where different models (exploiting, e.g., coarser discretizations or simplified physical models) can be combined to iteratively initialize network parameters. This latter strategy has proven to be effective for instance in the real-time approximation of cardiac electrophysiology problems [32,33] and problems in fluid dynamics [34].…”

Section: Introductionmentioning

confidence: 99%

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Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models

Fatone¹,

Fresca²,

Manzoni³

2022

Preprint

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Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs -built, e.g., exclusively through proper orthogonal decomposition (POD) -when applied to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can achieve extreme efficiency in the training stage and faster than real-time performances at testing, thanks to a prior dimensionality reduction through POD and a DL-based prediction framework. Nonetheless, they share with conventional ROMs poor performances regarding time extrapolation tasks. This work aims at taking a further step towards the use of DL algorithms for the efficient numerical approximation of parametrized PDEs by introducing the µt-POD-LSTM-ROM framework. This novel technique extends the POD-DL-ROM framework by adding a two-fold architecture taking advantage of long short-term memory (LSTM) cells, ultimately allowing long-term prediction of complex systems' evolution, with respect to the training window, for unseen input parameter values. Numerical results show that this recurrent architecture enables the extrapolation for time windows up to 15 times larger than the training time domain, and achieves better testing time performances with respect to the already lightning-fast POD-DL-ROMs.

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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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POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition

Cited by 163 publications

References 28 publications

Shifted Proper Orthogonal Decomposition and Artificial Neural Networks for Time-Continuous Reduced Order Models of Transport-Dominated Systems

Shifted Proper Orthogonal Decomposition and Artificial Neural Networks for Time-Continuous Reduced Order Models of Transport-Dominated Systems

Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models

Deep Learning‐Based Reduced Order Models for Cardiac Electrophysiology

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