edu.ucar.opensky::articles:42342 UCAR/NCAR - Library OpenSky Support PO Box 3000 Boulder CO 80307-3000 opensky@ucar.edu http://opensky.ucar.edu/ homepage pointOfContact 2025-12-24T18:07:11.127011 ISO 19115-2 Geographic Information - Metadata - Part 2: Extensions for Imagery and Gridded Data ISO 19115-2:2009(E) https://n2t.net/ark:/85065/d77085qt Bayesian reduced-order deep learning surrogate model for dynamic systems described by partial differential equations 2024-09-01T00:00:00Z publication https://n2t.net/ark:/85065/d77085qt https://doi.org/10.1016/j.cma.2024.117147 Wang, Y. author Zong, Y. author McCreight, James author Hughes, J. author Tartakovsky, A. M. author UCAR/NCAR - Library publisher We propose a reduced-order deep-learning surrogate model for dynamic systems described by time-dependent partial differential equations. This method employs space–time Karhunen–Loève expansions (KLEs) of the state variables and space-dependent KLEs of space-varying parameters to identify the reduced (latent) dimensions. Subsequently, a <a class="topic-link" style="background-color:rgba(0, 0, 0, 0);box-sizing:border-box;color:rgb(31, 31, 31);margin:0px;padding:0px;text-decoration:underline 1px rgb(31, 31, 31);text-underline-offset:1px;word-break:break-word;" href="https://www.sciencedirect.com/topics/chemical-engineering/deep-neural-network" title="Learn more about deep neural network from ScienceDirect&apos;s AI-generated Topic Pages">deep neural network</a> (DNN) is used to map the parameter latent space to the state variable latent space.</div><div class="u-margin-s-bottom" style="-webkit-text-stroke-width:0px;box-sizing:border-box;color:rgb(31, 31, 31);font-family:ElsevierGulliver, Georgia, &quot;Times New Roman&quot;, Times, STIXGeneral, &quot;Cambria Math&quot;, &quot;Lucida Sans Unicode&quot;, &quot;Microsoft Sans Serif&quot;, &quot;Segoe UI Symbol&quot;, &quot;Arial Unicode MS&quot;, serif, sans-serif;font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;margin-bottom:16px !important;margin-left:0px;margin-right:0px;margin-top:0px;orphans:2;padding:0px;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;" id="d1e1441">An approximate Bayesian method is developed for uncertainty quantification (UQ) in the proposed KL-DNN surrogate model. The KL-DNN method is tested for the linear advection–diffusion and nonlinear diffusion equations, and the Bayesian approach for UQ is compared with the deep ensembling (DE) approach, commonly used for quantifying uncertainty in DNN models. It was found that the approximate Bayesian method provides a more informative distribution of the PDE solutions in terms of the coverage of the reference PDE solutions (the percentage of nodes where the reference solution is within the confidence interval predicted by the UQ methods) and log predictive probability. The DE method is found to underestimate uncertainty and introduce bias.</div><div class="u-margin-s-bottom" style="-webkit-text-stroke-width:0px;box-sizing:border-box;color:rgb(31, 31, 31);font-family:ElsevierGulliver, Georgia, &quot;Times New Roman&quot;, Times, STIXGeneral, &quot;Cambria Math&quot;, &quot;Lucida Sans Unicode&quot;, &quot;Microsoft Sans Serif&quot;, &quot;Segoe UI Symbol&quot;, &quot;Arial Unicode MS&quot;, serif, sans-serif;font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;margin-bottom:16px !important;margin-left:0px;margin-right:0px;margin-top:0px;orphans:2;padding:0px;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;" id="d1e1443">For the nonlinear diffusion equation, we compare the KL-DNN method with the Fourier Neural Operator (FNO) method and find that KL-DNN is 10% more accurate and needs less training time than the FNO method. UCAR/NCAR - Library OpenSky Support PO Box 3000 Boulder CO 80307-3000 opensky@ucar.edu http://opensky.ucar.edu/ homepage pointOfContact PDF Text theme Resource Type 2016-01-01T00:00:00Z publication <span style="font-family:Arial;font-size:10pt;font-style:normal;" data-sheets-root="1">Copyright author(s). This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.</span> None eng geoscientificInformation