Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networks

Mücke, Nikolaj Takata; Christiansen, Lasse Hjuler; Engsig-Karup, Allan Peter; Jørgensen, John Bagterp

doi:10.1109/cdc40024.2019.9029284

“…The linear nature of the POD operator (or equivalent ones) imposes accuracy limitations, as it constrains the dynamics to evolve in a linear subspace, instead of the original manifold. This limitation has been addressed efficiently in [12,13,14,15,16], where neural networks or convolutional auto-encoders are employed to learn the reduced order manifold and LSTMs or equivalent techniques to propagate the respective dynamic phenomena forward in time.…”

Section: Introductionmentioning

confidence: 99%

A Physics-based Reduced Order Model with Machine Learning Boosted Hyper-Reduction.

Konstantinos,

Najera-Flores,

Martinez

et al. 2022

Proposed for Presentation at the IMAC Conference in ,

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Physics-based Reduced Order Models (ROMs) tend to rely on projection-based reduction. This family of approaches utilizes a series of responses of the full-order model to assemble a suitable basis, subsequently employed to formulate a set of equivalent, low-order equations through projection. However, in a nonlinear setting, physics-based ROMs require an additional approximation to circumvent the bottleneck of projecting and evaluating the nonlinear contributions on the reduced space. This scheme is termed hyper-reduction and enables substantial computational time reduction. The aforementioned hyper-reduction scheme implies a trade-off, relying on a necessary sacrifice on the accuracy of the nonlinear terms' mapping to achieve rapid or even real-time evaluations of the ROM framework. Since time is essential, especially for digital twins representations in structural health monitoring applications, the hyper-reduction approximation serves as both a blessing and a curse. Our work scrutinizes the possibility of exploiting machine learning (ML) tools in place of hyper-reduction to derive more accurate surrogates of the nonlinear mapping. By retaining the POD-based reduction and introducing the machine learning boosted surrogate(s) directly on the reduced coordinates, we aim to substitute the projection and update process of the nonlinear terms when integrating forward in time on the low-order dimension. Our approach explores a proof of concept case study based on a Nonlinear Autoregressive neural network with eXogenous Inputs (NARX-NN), trying to potentially derive a superior physics-based ROM in terms of efficiency, suitable for (near) real-time evaluations. The proposed ML-boosted ROM (N3-pROM) is validated in a multi-degree of freedom shear frame under ground-motion excitation featuring hysteretic nonlinearities.

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“…Various methods to circumvent that problem, such as the discrete empirical interpolation methods [123], already exist. In recent years, there are also studies exploring approximating F l,δt with neural networks [108,119].…”

Section: Linear Dimensionality Reductionmentioning

confidence: 99%

Deep Learning for Real-Time Inverse Problems and Data Assimilation with Uncertainty Quantification for Digital Twins

Mücke

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We develop novel approaches to enhance the functionality and efficiency of digital twins through deep learning techniques. Digital twins, sophisticated virtual models of physical systems, serve as dynamic counterparts, mirroring real-world entities' behavior and performance. Their primary role includes real-time monitoring, flaw detection, automated control, and proactive maintenance. To perform such tasks it is necessary to use various methods from scientific computing to combine data with physics models. Furthermore, to assess the reliability of a digital twin, it is important to not only make predictions, but also quantify the uncertainty of estimations and predictions. In this regard, Bayesian methods serve as a natural framework to pose the problems. However, the complexity and real-time operation requirements pose significant computational challenges. The thesis explores the integration of deep learning in scientific computing for enabling digital twins. The focus is on overcoming the computational hurdles in real-time data assimilation and inverse problem-solving through surrogate modeling. Deep learning, with its capability to handle high-dimensional functions in both deterministic and stochastic settings is shown to be a viable solution to these challenges. This work investigates several key areas, beginning with a brief introduction to digital twins, data assimilation, and inverse problems. It proceeds to elaborate on the role of Bayesian inversion problems in static and dynamic contexts, highlighting the importance of uncertainty quantification. The focus is on the challenges of real-time computation and the potential of Bayesian methods, despite their computational intensity. The thesis investigates deep learning architectures like dense and residual neural networks, convolutional neural networks, recurrent neural networks and transformers. The core contributions are presented in the four main chapters of the thesis: 1) Reduced Order Modeling for Parameterized Time-Dependent Partial Differential Equations: Introducing a deep learning framework for solving parameterized PDEs using spatially and memory-aware neural networks, demonstrated on linear advection and incompressible Navier-Stokes equations. 2) Markov Chain Generative Adversarial Neural Networks: Enhancing Bayesian inversion for static problems with sparse observations using a unique algorithm combining MCMC and GANs, demonstrated on a Darcy flow problem and pipe flow leakage detection. 3) Probabilistic Digital Twin for Leak Localization: Developing a probabilistic framework using supervised Wasserstein Autoencoders for leak localization in water distribution networks, tested on several network models. 4) The Deep Latent Space Particle Filter: Proposing a real-time nonlinear data assimilation method for dynamic PDEs with uncertainty quantification, using transformer-based dimensionality reduction and time-stepping in particle filters, demonstrated on various dynamic systems. This thesis contributes significantly to digital twins, addressing computational limitations and opening new avenues for practical applications in various industries.

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A Physics-Based Reduced Order Model with Machine Learning-Boosted Hyper-Reduction

Vlachas

¹

,

Najera-Flores²,

Martinez³

et al. 2012

Topics in Modal Analysis &Amp; Parameter Identification, Volume 8

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Physics-based Reduced Order Models (ROMs) tend to rely on projection-based reduction. This family of approaches utilizes a series of responses of the full-order model to assemble a suitable basis, subsequently employed to formulate a set of equivalent, low-order equations through projection. However, in a nonlinear setting, physics-based ROMs require an additional approximation to circumvent the bottleneck of projecting and evaluating the nonlinear contributions on the reduced space. This scheme is termed hyper-reduction and enables substantial computational time reduction. The aforementioned hyper-reduction scheme implies a trade-off, relying on a necessary sacrifice on the accuracy of the nonlinear terms' mapping to achieve rapid or even real-time evaluations of the ROM framework. Since time is essential, especially for digital twins representations in structural health monitoring applications, the hyper-reduction approximation serves as both a blessing and a curse. Our work scrutinizes the possibility of exploiting machine learning (ML) tools in place of hyper-reduction to derive more accurate surrogates of the nonlinear mapping. By retaining the POD-based reduction and introducing the machine learning boosted surrogate(s) directly on the reduced coordinates, we aim to substitute the projection and update process of the nonlinear terms when integrating forward in time on the low-order dimension. Our approach explores a proof of concept case study based on a Nonlinear Autoregressive neural network with eXogenous Inputs (NARX-NN), trying to potentially derive a superior physics-based ROM in terms of efficiency, suitable for (near) real-time evaluations. The proposed ML-boosted ROM (N3-pROM) is validated in a multi-degree of freedom shear frame under ground-motion excitation featuring hysteretic nonlinearities.

show abstract

Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networks

Cited by 5 publications

References 17 publications

A Physics-based Reduced Order Model with Machine Learning Boosted Hyper-Reduction.

A Physics-based Reduced Order Model with Machine Learning Boosted Hyper-Reduction.

Deep Learning for Real-Time Inverse Problems and Data Assimilation with Uncertainty Quantification for Digital Twins

A Physics-Based Reduced Order Model with Machine Learning-Boosted Hyper-Reduction

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