Deep Convolutional Architectures for Uncertainty Quantification and Forecast in Inundation Problems

Abstract

<p>Most science and engineering problems are modeled by time-dependent and parametrized nonlinear partial differential equations. Their resolution with traditional computational methods may be too expensive, especially in the context of predictions with uncertainty quantification or optimization, to allow for rapid predictions.  In this talk, we will overview data-driven methods aimed at representing high-fidelity computational models by means of reduced-dimension surrogate ones.  Different approaches will be presented for the uncertainty quantification for reliable predictions and forecasts in inundation problems.</p><p>Particularly, a non-intrusive reduced-order model based on convolutional autoencoders is proposed as a data-driven tool to build an efficient nonlinear reduced-order model for stochastic spatiotemporal large-scale physical problems. The method uses two-level autoencoders to reduce the spatial and temporal dimensions from a set of high-fidelity snapshots collected from an in-house high-fidelity numerical solver of the shallow-water equations. The encoded latent vectors, generated from two compression levels, are then mapped to the input parameters using a regression-based multilayer perceptron. The accuracy of the proposed approach is compared to the linear reduced-order technique-based artificial neural network (POD-ANN) on benchmark tests (the Burgers and Stoker's solutions) and a hypothetical dam-break flow problem over a complex bathymetry river. The numerical results show that the proposed nonlinear framework presents strong predictive abilities to accurately approximate the statistical moments of the outputs for complex stochastic large-scale and time-dependent problems, with low computational cost during the predictive online stage.</p><p>The caveat that remains is the long-term temporal extrapolation for problems marked by sharp gradients and discontinuities. Our study explores forecasting convolutional architectures (LSTM, TCN, and CNN) to obtain accurate solutions for time-steps distant from the training domain, on advection-dominated test cases. A simple convolutional architecture is then proposed and shown to provide accurate results for the forecasts. To evaluate the epistemic uncertainties in the solutions, the methodology of deep ensembles is adopted.</p><p><strong>REFERENCES</strong></p><ul><li>Bhatt, Y. Kumar and A. Soulaïmani. Deep Convolutional Architectures for Extrapolative Forecast in Time-dependent Flow Problems, DOI: 10.48550/arXiv.2209.09651.</li> <li>Abdedou and A. Soulaïmani. Reduced-order modeling for stochastic large-scale and time-dependent problems using deep spatial and temporal convolutional autoencoders. <br>arXiv:2208.03190[physics.flu-dyn].</li> <li>Jacquier, A. Abdedou, V. Delmas and A. Soulaimani. Non-intrusive reduced-order modeling using uncertainty-aware Deep Neural Networks and Proper Orthogonal Decomposition: Application to flood modeling. Journal of Computational Physics. Volume 424, 1 January 2021, 109854.</li> <li>Abdedou and A. Soulaïmani. A non-intrusive reduced-order modeling for uncertainty propagation of time-dependent problems using a B-splines Bézier elements-based method and proper orthogonal decomposition: Application to dam-break flows. Computers & Mathematics with Applications. Volume 102, 15 November 2021, Pages 187-205.</li> <li>Chaudhry and A. Soulaimani. A Comparative Study of Machine Learning Methods for Computational Modeling of the Selective Laser Melting Additive Manufacturing Process. Appl. Sci. 2022, 12(5), 2324; https://doi.org/10.3390/app12052324.</li> <li>Delmas and A. Soulaimani. Parallel high-order resolution of the Shallow-water equations on real large-scale meshes with complex bathymetries. Journal of Computational Physics. Volume 471, 15 December 2022, 111629</li> </ul><p> </p>