scholarly journals Reduced Order Modeling for Parameterized Time-Dependent PDEs using Spatially and Memory Aware Deep Learning

2021 ◽  
pp. 101408
Author(s):  
Nikolaj T. Mücke ◽  
Sander M. Bohté ◽  
Cornelis W. Oosterlee
Keyword(s):  
Deep Learning ◽  
Reduced Order Modeling ◽  
Time Dependent ◽  
Reduced Order

scholarly journals A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Stefania Fresca ◽  
Luca Dede’ ◽  
Andrea Manzoni
Keyword(s):  
Deep Learning ◽  
Reduced Order Modeling ◽  
Time Dependent ◽  
Reduced Basis ◽  
Reduced Order ◽  
Nonlinear Approach ◽  
Parameter Values ◽  
Proper Orthogonal ◽  
Convection Dominated ◽  
Very High

AbstractConventional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) may incur in severe limitations when dealing with nonlinear time-dependent parametrized PDEs, as these are strongly anchored to the assumption of modal linear superimposition they are based on. For problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method may yield inefficient reduced order models (ROMs) when very high levels of accuracy are required. To overcome this limitation, in this work, we propose a new nonlinear approach to set ROMs by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of full order model (FOM) solutions obtained for different parameter values. We show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs. Moreover, we assess its accuracy and efficiency on different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.

scholarly journals pyNIROM—A suite of python modules for non-intrusive reduced order modeling of time-dependent problems

2021 ◽  
Vol 10 ◽  
pp. 100129
Author(s):  
Sourav Dutta ◽  
Peter Rivera-Casillas ◽  
Orie M. Cecil ◽  
Matthew W. Farthing
Keyword(s):  
Reduced Order Modeling ◽  
Time Dependent ◽  
Reduced Order ◽  
Time Dependent Problems

Reduced-Order Flood Modeling Using Uncertainty-Aware Deep Neural Networks

2020 ◽  
Author(s):  
Pierre Jacquier ◽  
Azzedine Abdedou ◽  
Azzeddine Soulaïmani
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Uncertainty Quantification ◽  
Surrogate Model ◽  
Deep Neural Networks ◽  
Reduced Order Modeling ◽  
Flood Modeling ◽  
Reduced Order ◽  
Highly Nonlinear ◽  
The Impact

<p><strong>Key Words</strong>: Uncertainty Quantification, Deep Learning, Space-Time POD, Flood Modeling</p><p><br>While impressive results have been achieved in the well-known fields where Deep Learning allowed for breakthroughs such as computer vision, language modeling, or content generation [1], its impact on different, older fields is still vastly unexplored. In computational fluid dynamics and especially in Flood Modeling, many phenomena are very high-dimensional, and predictions require the use of finite element or volume methods, which can be, while very robust and tested, computational-heavy and may not prove useful in the context of real-time predictions. This led to various attempts at developing Reduced-Order Modeling techniques, both intrusive and non-intrusive. One late relevant addition was a combination of Proper Orthogonal Decomposition with Deep Neural Networks (POD-NN) [2]. Yet, to our knowledge, in this example and more generally in the field, little work has been conducted on quantifying uncertainties through the surrogate model.<br>In this work, we aim at comparing different novel methods addressing uncertainty quantification in reduced-order models, pushing forward the POD-NN concept with ensembles, latent-variable models, as well as encoder-decoder models. These are tested on benchmark problems, and then applied to a real-life application: flooding predictions in the Mille-Iles river in Laval, QC, Canada.<br>For the flood modeling application, our setup involves a set of input parameters resulting from onsite measures. High-fidelity solutions are then generated using our own finite-volume code CuteFlow, which is solving the highly nonlinear Shallow Water Equations. The goal is then to build a non-intrusive surrogate model, that’s able to <em>know what it know</em>s, and more importantly, <em>know when it doesn’t</em>, which is still an open research area as far as neural networks are concerned [3].</p><p><br><strong>REFERENCES</strong><br>[1] C. Szegedy, S. Ioffe, V. Vanhoucke, and A. A. Alemi, “Inception-v4, inception-resnet and the impact of residual connections on learning”, in Thirty-First AAAI Conference on Artificial Intelligence, 2017.<br>[2] Q. Wang, J. S. Hesthaven, and D. Ray, “Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem”, Journal of Computational Physics, vol. 384, pp. 289–307, May 2019.<br>[3] B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simple and scalable predictive uncertainty estimation using deep ensembles”, in Advances in Neural Information Processing Systems, 2017, pp. 6402–6413.</p>

scholarly journals Application of deep learning method to Reynolds stress models of channel flow based on reduced-order modeling of DNS data

2018 ◽  
Vol 31 (1) ◽  
pp. 58-65 ◽  
Author(s):  
Zhen Zhang ◽  
Xu-dong Song ◽  
Shu-ran Ye ◽  
Yi-wei Wang ◽  
Chen-guang Huang ◽  
...  
Keyword(s):  
Deep Learning ◽  
Reynolds Stress ◽  
Channel Flow ◽  
Reduced Order Modeling ◽  
Learning Method ◽  
Reduced Order ◽  
Stress Models
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