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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.
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A novel strategy for learning nonlinear ROM operators using deep neural networks (DNNs) is proposed, which is a physics-based model, still relying on the RB method approach, however employing a DNN architecture to approximate reduced residual vectors and Jacobian matrices once a Galerkin projection has been performed.
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Findings indicate that deep autoencoders can leverage nonlinear manifold learning to achieve a highly efficient compression of spatial information and define a latent-space that appears to be more suitable for capturing the temporal dynamics through the NODE framework.
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An advection-aware (AA) autoencoder network that can address some of these limitations by learning efficient, physics-informed, nonlinear embeddings of the high-fidelity system snapshots is developed.
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Time-series learning of latent-space dynamics for reduced-order model closure
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Learning reduced order models from data for hyperbolic PDEs
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This work decomposes the solution by expressing it as a composition of a transformed solution and a de-transformer and resorts to the dynamic mode decomposition (DMD) methodology to learn a reduced order model (ROM).
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