On quadrature rules for solving Partial Differential Equations using Neural Networks

Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may arise in these applications and propose several alternatives to overcome them, namely: Monte Carlo methods, adaptive integration, polynomial approximations of the Neural Network output, and the inclusion of regularization terms in the loss. We also discuss the advantages and limitations of each proposed numerical integration scheme. We advocate the use of Monte Carlo methods for high dimensions (above 3 or 4), and adaptive integration or polynomial approximations for low dimensions (3 or below). The use of regularization terms is a mathematically elegant alternative that is valid for any spatial dimension; however, it requires certain regularity assumptions on the solution and complex mathematical analysis when dealing with sophisticated Neural Networks

Deep learning enhanced principal component analysis for structural health monitoring

This paper proposes a Deep Learning Enhanced Principal Component Analysis (PCA) approach for outlier detection to assess the structural condition of bridges. We employ partially explainable autoencoder architecture to replicate and enhance the data compression and reconstruction ability of PCA. The particularity of the method lies in the addition of residual connections to account for nonlinearities. We apply the proposed method to monitoring data obtained from two bridges under real operation conditions and compare the results before and after adding the residual connections. Results show that the addition of residual connections enhances the outlier detection ability of the network, allowing to detect lighter damages

Bridge damage identification under varying environmental and operational conditions combining Deep Learning and numerical simulations

This work proposes a novel supervised learning approach to identify damage in operating bridge structures. We propose a method to introduce the effect of environmental and operational conditions into the synthetic damage scenarios employed for training a Deep Neural Network, which is applicable to large-scale complex structures. We apply a clustering technique based on Gaussian Mixtures to effectively select Q representative measurements from a long-term monitoring dataset. We employ these measurements as the target response to solve various Finite Element Model Updating problems before generating different damage scenarios. The synthetic and experimental measurements feed two Deep Neural Networks that assess the structural health condition in terms of damage severity and location. We demonstrate the applicability of the proposed method with a real full-scale case study: the Infante Dom Henrique bridge in Porto. A comparative study reveals that neglecting different environmental and operational conditions during training detracts the damage identification task. By contrast, our method provides successful results during a synthetic validation

Supervised Deep Learning with Finite Element simulations for damage identification in bridges

This work proposes a supervised Deep Learning approach for damage identification in bridge structures. We employ a hybrid methodology that incorporates Finite Element simulations to enrich the training phase of a Deep Neural Network with synthetic damage scenarios. The neural network is based on autoencoders and its particular architecture allows to activate or deactivate nonlinear connections under need. The methodology intends to contribute to the progress towards the applicability of Structural Health Monitoring practices in full-scale bridge structures. The ultimate goal is to estimate the location and severity of damage from measurements of the dynamic response of the structure. The damages we seek to detect correspond to material degradations that affect wide areas of the structure by reducing its stiffness properties. Our method allows a feasible adaptation to large systems with complex parametrizations and structural particularities. We investigate the performance of the proposed method on two full-scale instrumented bridges, obtaining adequate results for the testing datasets even in presence of measurement uncertainty. Besides, the method successfully predicts the damage condition for two real damage scenarios of increasing severity available in one of the bridges.This work has received funding from the European’s Union Horizon 2020 research and innovation program under the grant agreement No 769373 (FORESEE project). Authors would like to acknowledge the Basque Government funding within the ELKARTEK programme (SIGZE project (KK-2021/00095)). This work was financially supported by: Base Funding - UIDB/04708/ 2020 of the CONSTRUCT - Instituto de I&D em Estruturas e Construções - funded by national funds through the FCT/MCTES (PIDDAC). David Pardo has received funding from: the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS); the European Regional Development Fund (ERDF) through the Interreg V-A Spain-France-Andorra program POCTEFA 2014-2020 Project PIXIL (EFA362/19); the Spanish Ministry of Science and Innovation projects with references PID2019-108111RB-I00 (FEDER/AEI) and PDC2021-121093-I00, the ‘‘BCAM Severo Ochoa’’ accreditation of excellence (SEV-2017-0718); and the Basque Government through the BERC 2018-2021 program, the three Elkartek projects 3KIA (KK-2020/00049), EXPERTIA (KK-2021/00048), and SIGZE (KK-2021/00095), and the Consolidated Research Group MATHMODE (IT1294-19) given by the Department of Education

Supervised Deep Learning with Finite Element simulations for damage identification in bridges

This work proposes a supervised Deep Learning approach for damage identification in bridge structures. We employ a hybrid methodology that incorporates Finite Element simulations to enrich the training phase of a Deep Neural Network with synthetic damage scenarios. The neural network is based on autoencoders and its particular architecture allows to activate or deactivate nonlinear connections under need. The methodology intends to contribute to the progress towards the applicability of Structural Health Monitoring practices in full-scale bridge structures. The ultimate goal is to estimate the location and severity of damage from measurements of the dynamic response of the structure. The damages we seek to detect correspond to material degradations that affect wide areas of the structure by reducing its stiffness properties. Our method allows a feasible adaptation to large systems with complex parametrizations and structural particularities. We investigate the performance of the proposed method on two full-scale instrumented bridges, obtaining adequate results for the testing datasets even in presence of measurement uncertainty. Besides, the method successfully predicts the damage condition for two real damage scenarios of increasing severity available in one of the bridges