Deep Convolutional Architectures for Uncertainty Quantification and Forecast in Inundation Problems

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.

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.

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.

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