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On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model

G. Tsialiamanis, C. R. Farrar

TL;DR

A method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest, a neural network.

Abstract

Detection and identification of nonlinearity is a task of high importance for structural dynamics. Detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model herein is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as inputs accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some nonlinear. The statistics of the distributions of the gradients for the different scenarios can be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for "more nonlinear" scenarios.

On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model

TL;DR

A method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest, a neural network.

Abstract

Detection and identification of nonlinearity is a task of high importance for structural dynamics. Detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model herein is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as inputs accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some nonlinear. The statistics of the distributions of the gradients for the different scenarios can be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for "more nonlinear" scenarios.
Paper Structure (7 sections, 7 equations, 4 figures)

This paper contains 7 sections, 7 equations, 4 figures.

Table of Contents

  1. Abstract
  2. Introduction
  3. Machine learning for structural dynamics
  4. A three-storey building dataset
  5. Statistics of the model gradients as a nonlinearity metric
  6. Conclusions
  7. Acknowledgements

Figures (4)

  • Figure 1: Experimental set-up of three-floor and the bumper nonlinearity between the second and third floor shown in the dashed box figueiredo2009structural.
  • Figure 2: Distribution of the gradients of the model for the predictions of the third floor with respect to the accelerations of the previous timestep of the base (blue line), the first floor (orange line) and the second floor (green line). The distributions are shown for two cases, for the baseline state (left) and for state $14$ (right), the most nonlinear case.
  • Figure 3: The values of the metric of equation (\ref{['eq:metrics']}) using the standard deviation as $M$, for the accelerations of the three floors of the structure (left) and the average value of all three floors of the metric for the different states (right).
  • Figure 4: The values of the metric of equation (\ref{['eq:metrics']}) using inverse kurtosis as $M$, for the accelerations of the three floors of the structure (left) and the average value of all three floors of the metric for the different states (right).