Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problems

2026-03-31

AI Summary

This work addresses the challenge of reliably quantifying prediction uncertainty in computational mechanics inverse problems, where traditional methods and standard deep learning approaches often fail—particularly under noisy data or non-unique solutions. The authors propose a variational graph neural network (VGNN) that innovatively embeds a variational layer solely within the decoder, enabling joint modeling of epistemic and aleatoric uncertainties at a reduced computational cost. Taking displacement fields as input, the method achieves high-fidelity inverse identification of material parameters and loading conditions while producing physically consistent confidence intervals. Demonstrated on two-dimensional nonlinear elastic modulus identification and three-dimensional hyperelastic beam load localization tasks, the model matches the accuracy of full Bayesian approaches while offering significantly improved computational efficiency.

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.

Problem

• uncertainty quantification
• inverse problems
• computational mechanics
• deep learning
• Digital Twins

Innovation

• Variational Graph Neural Network
• Uncertainty Quantification
• Inverse Problems
• Epistemic Uncertainty
• Computational Mechanics