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Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based ROMs

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MDLW03 - Deep learning and partial differential equations

Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs – built, e.g., through proper orthogonal decomposition (POD) – when applied to nonlinear time-dependent parametrized PDEs. Although extremely efficient at testing time, when evaluating the PDE solution for any new testing-parameter instance, DL-ROMs require an expensive training stage. To avoid this latter, a prior dimensionality reduction through POD , and a multi-fidelity pretraining stage, are introduced, yielding the POD -DL-ROM framework, which allows to solve time-dependent PDEs even faster than in real-time. A further step has led us to introduce LSTM networks instead than convolutional autoencoders, ultimately obtaining POD -LSTM-ROMs that better grasp the time evolution of the PDE system, to enhance predictions for new testing-parameter instances. Equipped with LST Ms, POD -LSTM-ROMs also allow us to perform extrapolation of the PDE solution forward in time, that is, on a (much) larger time domain than the one used to train the ROM , for unseen values of the input parameters – a task often missed by traditional projection-based ROMs. To this aim, we train a POD -LSTM-ROM on snapshots acquired on a given time interval, and then approximate the solution on a much longer time window, taking advantage of a LSTM architecture in a different form besides the POD -LSTM-ROM introduced before. Building predictions for future times based on the past, these coupled architectures mimic the behavior of common numerical solvers for dynamical systems. We assess the performance of the proposed framework on several examples, ranging from low-dimensional, nonperiodic systems to applications in structural mechanics dealing with micro electromechanical systems, obtaining faster than real-time simulations that are able to preserve a remarkable accuracy across the entire time domain considered during training.

This talk is part of the Isaac Newton Institute Seminar Series series.

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