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MDLW03 - Deep learning and partial differential equations

Conventional reduced order models (ROMs) anchored to the assumption of modal linear superimposition, such as proper orthogonal decomposition (POD), may reveal inefficient when dealing with nonlinear time-dependent parametrized PDEs, especially for problems featuring coherent structures propagating over time. To enhance ROM efficiency, we propose a nonlinear approach to set ROMs by exploiting deep learning (DL) algorithms, such as convolutional neural networks. In the resulting DL-ROM, both the nonlinear trial manifold and the nonlinear reduced dynamics are learned in a non-intrusive way by relying on DL algorithms trained on a set of full order model (FOM) snapshots, obtained for different parameter values. Performing then a former dimensionality reduction on FOM snapshots through POD enables, when dealing with large-scale FOMs, to speedup training times, and decrease the network complexity, substantially. Accuracy and efficiency of the resulting POD -DL-ROM technique are assessed on different parametrized PDE problems in cardiac electrophysiology, computational mechanics and fluid dynamics, possibly accounting for fluid-structure interaction (FSI) effects, where new queries to the POD -DL-ROM can be computed in real-time. S. Fresca, A. Manzoni 2022. POD -DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition. Computer Methods in Applied Mechanics and Engineering, 388, 114181. S. Fresca, A. Manzoni L. Dedè, A. Quarteroni 2021. POD enhanced deep learning-based reduced order models for the real-timesimulation of cardiac electrophysiology in the left atrium. Frontiers in Physiology, 12, 1431. S. Fresca, A. Manzoni 2021. Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models. Fluids, 6(7), 259. S. Fresca, A. Manzoni, L. Dedè 2021. A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. Journal of Scientific Computing, 87(2):1-36. S. Fresca, A. Manzoni, L. Dedè, A. Quarteroni 2020. Deep learning-based reduced order models in cardiac electrophysiology. PLOS ONE , 15(10):1-32.  

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