Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problems

Abstract

The increasingly wide use of deep machine learning techniques in computational mechanics has significantly accelerated simulations of problems that were considered unapproachable just a few years ago. However, in critical applications such as Digital Twins for engineering or medicine, fast responses are not enough; reliable results must also be provided. In certain cases, traditional deterministic methods may not be optimal as they do not provide a measure of confidence in their predictions or results, especially in inverse problems where the solution may not be unique or the initial data may not be entirely reliable due to the presence of noise, for instance. Classic deep neural networks also lack a clear measure to quantify the uncertainty of their predictions. In this work, we present a variational graph neural network (VGNN) architecture that integrates variational layers into its architecture to model the probability distribution of weights. Unlike computationally expensive full Bayesian networks, our approach strategically introduces variational layers exclusively in the decoder, allowing us to estimate cognitive uncertainty and statistical uncertainty at a relatively lower cost. In this work, we validate the proposed methodology in two cases of solid mechanics: the identification of the value of the elastic modulus with nonlinear distribution in a 2D elastic problem and the location and quantification of the loads applied to a 3D hyperelastic beam, in both cases using only the displacement field of each test as input data. The results show that the model not only recovers the physical parameters with high precision, but also provides confidence intervals consistent with the physics of the problem, as well as being able to locate the position of the applied load and estimate its value, giving a confidence interval for that experiment.

Figures (17)

• Figure 1: A sketch of the Variational Graph Neural Network Architecture.
• Figure 2: 2D Plate with a nonlinear distribution of the elastic modulus under plane stress.
• Figure 3: Three examples of experiments used in neural network training. The first row represents the actual distribution of the Elastic Module, the next two rows represent the corresponding displacements --horizontal and vertical, respectively-- for the behavior of the material represented in the initial row.
• Figure 4: Relative RMSE error for training and test data for the problem of a 2D plate subjected to plane stress with nonlinear elastic modulus distributions.
• Figure 5: Test example of the plate subjected to flat stress. The upper left shows the distribution of the actual elastic modulus (Ground Truth), while the upper right shows the average prediction of that field. The bottom row shows the lower and upper limits of that field, on the left and right, respectively.