<p><strong>Key Words</strong>: Uncertainty Quantification, Deep Learning, Space-Time POD, Flood Modeling</p><p><br>While impressive results have been achieved in the well-known fields where Deep Learning allowed for breakthroughs such as computer vision, language modeling, or content generation [1], its impact on different, older fields is still vastly unexplored. In computational fluid dynamics and especially in Flood Modeling, many phenomena are very high-dimensional, and predictions require the use of finite element or volume methods, which can be, while very robust and tested, computational-heavy and may not prove useful in the context of real-time predictions. This led to various attempts at developing Reduced-Order Modeling techniques, both intrusive and non-intrusive. One late relevant addition was a combination of Proper Orthogonal Decomposition with Deep Neural Networks (POD-NN) [2]. Yet, to our knowledge, in this example and more generally in the field, little work has been conducted on quantifying uncertainties through the surrogate model.<br>In this work, we aim at comparing different novel methods addressing uncertainty quantification in reduced-order models, pushing forward the POD-NN concept with ensembles, latent-variable models, as well as encoder-decoder models. These are tested on benchmark problems, and then applied to a real-life application: flooding predictions in the Mille-Iles river in Laval, QC, Canada.<br>For the flood modeling application, our setup involves a set of input parameters resulting from onsite measures. High-fidelity solutions are then generated using our own finite-volume code CuteFlow, which is solving the highly nonlinear Shallow Water Equations. The goal is then to build a non-intrusive surrogate model, that&#8217;s able to <em>know what it know</em>s, and more importantly, <em>know when it doesn&#8217;t</em>, which is still an open research area as far as neural networks are concerned [3].</p><p><br><strong>REFERENCES</strong><br>[1] C. Szegedy, S. Ioffe, V. Vanhoucke, and A. A. Alemi, &#8220;Inception-v4, inception-resnet and the impact of residual connections on learning&#8221;, in Thirty-First AAAI Conference on Artificial Intelligence, 2017.<br>[2] Q. Wang, J. S. Hesthaven, and D. Ray, &#8220;Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem&#8221;, Journal of Computational Physics, vol. 384, pp. 289&#8211;307, May 2019.<br>[3] B. Lakshminarayanan, A. Pritzel, and C. Blundell, &#8220;Simple and scalable predictive uncertainty estimation using deep ensembles&#8221;, in Advances in Neural Information Processing Systems, 2017, pp. 6402&#8211;6413.</p>
Reduced Order Modeling for Parameterized Time-Dependent PDEs using Spatially and Memory Aware Deep Learning
