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Gaussian process regression + deep neural network autoencoder for probabilistic surrogate modeling in nonlinear mechanics of solids

Saurabh Deshpande, Hussein Rappel, Mark Hobbs, Stéphane P. A. Bordas, Jakub Lengiewicz

TL;DR

An innovative approach that combines autoencoder deep neural networks with the probabilistic regression capabilities of Gaussian processes is introduced, which is computationally efficient as well as accurate in predicting non-linear deformations of solid bodies subjected to external forces.

Abstract

Many real-world applications demand accurate and fast predictions, as well as reliable uncertainty estimates. However, quantifying uncertainty on high-dimensional predictions is still a severely under-investigated problem, especially when input-output relationships are non-linear. To handle this problem, the present work introduces an innovative approach that combines autoencoder deep neural networks with the probabilistic regression capabilities of Gaussian processes. The autoencoder provides a low-dimensional representation of the solution space, while the Gaussian process is a Bayesian method that provides a probabilistic mapping between the low-dimensional inputs and outputs. We validate the proposed framework for its application to surrogate modeling of non-linear finite element simulations. Our findings highlight that the proposed framework is computationally efficient as well as accurate in predicting non-linear deformations of solid bodies subjected to external forces, all the while providing insightful uncertainty assessments.

Gaussian process regression + deep neural network autoencoder for probabilistic surrogate modeling in nonlinear mechanics of solids

TL;DR

An innovative approach that combines autoencoder deep neural networks with the probabilistic regression capabilities of Gaussian processes is introduced, which is computationally efficient as well as accurate in predicting non-linear deformations of solid bodies subjected to external forces.

Abstract

Many real-world applications demand accurate and fast predictions, as well as reliable uncertainty estimates. However, quantifying uncertainty on high-dimensional predictions is still a severely under-investigated problem, especially when input-output relationships are non-linear. To handle this problem, the present work introduces an innovative approach that combines autoencoder deep neural networks with the probabilistic regression capabilities of Gaussian processes. The autoencoder provides a low-dimensional representation of the solution space, while the Gaussian process is a Bayesian method that provides a probabilistic mapping between the low-dimensional inputs and outputs. We validate the proposed framework for its application to surrogate modeling of non-linear finite element simulations. Our findings highlight that the proposed framework is computationally efficient as well as accurate in predicting non-linear deformations of solid bodies subjected to external forces, all the while providing insightful uncertainty assessments.
Paper Structure (16 sections, 25 equations, 13 figures, 3 tables)

This paper contains 16 sections, 25 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Obtaining latent representations with autoencoder neural networks
  4. Gaussian Process Regression in the latent space
  5. Projecting latent GP predictions to the full field space using decoder
  6. Finite element formulation for non-linear deformations of solid bodies
  7. Results
  8. Dataset generation information
  9. Implementation details
  10. Autoencoder networks: Architectures & training details
  11. Gaussian process details
  12. Sources of prediction errors
  13. Performance study of the GP+autoencoder framework
  14. GP performance in the latent space
  15. Uncertainty for missing data region
  16. ...and 1 more sections

Figures (13)

  • Figure 1: Two-stage training. (a) First, the autoencoder neural network is trained to compress full field displacement data to corresponding latent space representations. (b) Afterwards, the GP model is trained to provide the probabilistic force-displacement mapping in the latent space.
  • Figure 2: For an unseen input force $\mathbf{f}^*$, the GP is used to predict the latent displacement distribution (i.e., our uncertainty about the displacements is described by a probability distribution). Subsequently, these latent displacements are projected to the full-field displacement distribution by using the decoder component of the autoencoder network.
  • Figure 3: Schematics of examples considered in this work. (a) 2D beam discretised with quad elements is subjected to point loads on the nodes lying on red line; this example is adapted from DESHPANDE2022115307. (b) 3D liver is subjected to body forces.
  • Figure 4: CNN autoencoder architecture for used for the 2D beam case. Input is provided to the network in the structured mesh format, the number at the top of the convolution layer output indicates the number of channels in the respective layer. Input is applied series of convolution and pooling/upsampling layers until the original dimension is retrieved. In the bottleneck level, fully connected layers (dense layer) are applied and the number the top of their output represents the number of units present in it.
  • Figure 5: Fully connected autoencoder architecture used for the liver case. Skip connections are indicated with the (+) sign. The number of units for each dense layer output is indicated at the bottom. The latent representation dimension is denoted as $L$, set to $L = 16$ for the 3D liver case.
  • ...and 8 more figures