toggle visibility Search & Display Options

Select All    Deselect All
 |   | 
Details
   print
  Records Links
Author (up) Aylwin, R.; Henriquez, F.; Schwab, C. doi  openurl
  Title ReLU Neural Network Galerkin BEM Type
  Year 2023 Publication Journal of Scientific Computing Abbreviated Journal J. Sci. Comput.  
  Volume 95 Issue 2 Pages 41  
  Keywords BOUNDARY INTEGRAL METHOD; ELEMENT METHODS; P-VERSION; CONVERGENCE; FRAMEWORK; EQUATIONS; OPERATORS; SCREEN  
  Abstract We introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation-based method, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built by using NNs that, in turn, employ the so-called Rectified Linear Units (ReLU) as the underlying activation function. The ReLU-NNs used to approximate the solutions to the BIEs depend nonlinearly on the parameters characterizing the NNs themselves. Consequently, the computation of a numerical solution to a BIE by means of ReLU-NNs boils down to a fine tuning of these parameters, in network training. We argue that ReLU-NNs of fixed depth and with a variable width allow us to recover well-known approximation rate results for the standard Galerkin Boundary Element Method (BEM). This observation hinges on existing well-known properties concerning the regularity of the solution of the BIEs on Lipschitz, polygonal boundaries, i.e. accounting for the effect of corner singularities, and the expressive power of ReLU-NNs over different classes of functions. We prove that shallow ReLU-NNs, i.e. networks having a fixed, moderate depth but with increasing width, can achieve optimal order algebraic convergence rates. We propose novel loss functions for NN training which are obtained using computable, local residual a posteriori error estimators with ReLU-NNs for the numerical approximation of BIEs. We find that weighted residual estimators, which are reliable without further assumptions on the quasi-uniformity of the underlying mesh, can be employed for the construction of computationally efficient loss functions for ReLU-NN training. The proposed framework allows us to leverage on state-of-the-art computational deep learning technologies such as TENSORFLOW and TPUs for the numerical solution of BIEs using ReLU-NNs. Exploratory numerical experiments validate our theoretical findings and indicate the viability of the proposed ReLU-NN Galerkin BEM approach.  
  Address  
  Corporate Author Thesis  
  Publisher Place of Publication Editor  
  Language Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 0885-7474 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000954568900001 Approved  
  Call Number UAI @ alexi.delcanto @ Serial 1769  
Permanent link to this record
 

 
Author (up) Aylwin, R.; Jerez-Hanckes, C.; Schwab, C.; Zech, J. doi  openurl
  Title Multilevel Domain Uncertainty Quantification in Computational Electromagnetics Type
  Year 2023 Publication Mathematical Models and Methods in Applied Sciences Abbreviated Journal Math. Models Methods Appl. Sci.  
  Volume 33 Issue 04 Pages 877-921  
  Keywords  
  Abstract We continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), 8:301--341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping the physical domains to a nominal polygonal domain with piecewise smooth maps. The regularity of the pullback solutions on the nominal domain is characterized in piecewise Sobolev spaces. We prove error convergence rates and optimize the algorithmic steering of parameters for edge-element discretizations in the nominal domain combined with: (a) multilevel Monte Carlo sampling, and (b) multilevel, sparse-grid quadrature for computing the expectation of the solutions with respect to uncertain domain ensembles. In addition, we analyze sparse-grid interpolation to compute surrogates of the domain-to-solution mappings. All calculations are performed on the polyhedral nominal domain, which enables the use of standard simplicial finite element meshes. We provide a rigorous fully discrete error analysis and show, in all cases, that dimension-independent algebraic convergence is achieved. For the multilevel sparse-grid quadrature methods, we prove higher order convergence rates which are free from the so-called curse of dimensionality, i.e. independent of the number of parameters used to parametrize the admissible shapes. Numerical experiments confirm our theoretical results and verify the superiority of the sparse-grid methods.  
  Address  
  Corporate Author Thesis  
  Publisher Place of Publication Editor  
  Language Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 0218-2025 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000961040400001 Approved  
  Call Number UAI @ alexi.delcanto @ Serial 1699  
Permanent link to this record
 

 
Author (up) Aylwin, R.; Jerez-Hanckes, C.; Schwab, C.; Zech, J. doi  openurl
  Title Domain Uncertainty Quantification in Computational Electromagnetics Type
  Year 2020 Publication Siam-Asa Journal On Uncertainty Quantification Abbreviated Journal SIAM-ASA J. Uncertain. Quantif.  
  Volume 8 Issue 1 Pages 301-341  
  Keywords computational electromagnetics; uncertainty quantification; finite elements; shape holomorphy; sparse grid quadrature; Bayesian inverse problems  
  Abstract We study the numerical approximation of time-harmonic, electromagnetic fields inside a lossy cavity of uncertain geometry. Key assumptions are a possibly high-dimensional parametrization of the uncertain geometry along with a suitable transformation to a fixed, nominal domain. This uncertainty parametrization results in families of countably parametric, Maxwell-like cavity problems that are posed in a single domain, with inhomogeneous coefficients that possess finite, possibly low spatial regularity, but exhibit holomorphic parametric dependence in the differential operator. Our computational scheme is composed of a sparse grid interpolation in the high-dimensional parameter domain and an Hcurl -conforming edge element discretization of the parametric problem in the nominal domain. As a stepping-stone in the analysis, we derive a novel Strang-type lemma for Maxwell-like problems in the nominal domain, which is of independent interest. Moreover, we accommodate arbitrary small Sobolev regularity of the electric field and also cover uncertain isotropic constitutive or material laws. The shape holomorphy and edge-element consistency error analysis for the nominal problem are shown to imply convergence rates for multilevel Monte Carlo and for quasi-Monte Carlo integration, as well as sparse grid approximations, in uncertainty quantification for computational electromagnetics. They also imply expression rate estimates for deep ReLU networks of shape-to-solution maps in this setting. Finally, our computational experiments confirm the presented theoretical results.  
  Address [Aylwin, Ruben] Pontificia Univ Catolica Chile, Sch Engn, Santiago 7820436, Chile, Email: rdaylwin@uc.cl;  
  Corporate Author Thesis  
  Publisher Siam Publications Place of Publication Editor  
  Language English Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 2166-2525 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000551383300011 Approved  
  Call Number UAI @ eduardo.moreno @ Serial 1207  
Permanent link to this record
Select All    Deselect All
 |   | 
Details
   print

Save Citations:
Export Records: