cs.LGMar 31, 2026

Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problems

Authors: David Gonzalez, Alba Muixi, Beatriz Moya, Elias Cueto

Organizations: Keysight-UZ Chair of the Spanish National Strategy on AI. Aragon Institute of Engineering Research (I3A), Universidad de Zaragoza, Spain · Laboratori de Càlcul Numèric (LaCàn), Universitat Politècnica de Catalunya - BarcelonaTech (UPC), Spain · Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Barcelona, Spain · PIMM Lab. Arts et Métiers Institute of Technology, Paris, France

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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 & tables

Figure 1: Example of a 2D finite elements mesh (left) and the equivalent self-informed graph (right). This graph is the one used in the example in Section 4.1 .
Figure 2: A sketch of the Variational Graph Neural Network Architecture.
Figure 3: 2D Plate with a nonlinear distribution of the elastic modulus under plane stress.
Figure 4: 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.
HyperparameterValue
nBatch22
Latent space dimension2525 neurons
Message Passing steps ( mm )55
σ1\sigma_{1}1exp⁡(1)\frac{1}{\exp(1)}
σ2\sigma_{2}1exp⁡(2)\frac{1}{\exp(2)}
Activation functionSwishLayer
Table 1: Most important hyperparameters used in example 4.1
Figure 5: 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 6: A first test example of the plate subjected to plane stress. (a) Distribution of the actual elastic modulus (Ground Truth), (b) Average prediction of that field, (c) Lower limit of that field, (d) Upper limit of that field.
Figure 7: Second test example of the plate subjected to plane stress. (a) Distribution of the actual elastic modulus (Ground Truth), (b) Average prediction of that field, (c) Lower limit of that field, (d) Upper limit of that field.
Figure 8: Third example test of the plate subjected to plane stress. (a) Distribution of the actual elastic modulus (Ground Truth), (b) Average prediction of that field, (c) Lower limit of that field, (d) Upper limit of that field.
Figure 9: Spatial distribution of the absolute error ∣y−μ∣|y-\mu| (left), alongside the spatial distribution of the uncertainty spreads (right) for first, second and third 2D plate examples, respectively.
Figure 10: Example of a loaded mesh associated with the hyperelastic beam problem.
Figure 11: Schematic procedure for training the 3D hyperelastic beam problem.
HyperparameterValue
nBatch22
Latent space dimension1212 neurons
Message Passing steps ( mm )44
σ1\sigma_{1}1exp⁡(1)\frac{1}{\exp(1)}
σ2\sigma_{2}1exp⁡(2)\frac{1}{\exp(2)}
Activation functionSwishLayer
Table 2: Most important hyperparameters used in Example 4.2
Figure 12: Relative RMSE error for training and test data for the 3D Hyperelastic Beam problem.
ExampleRef. LoadLower Bd.MeanUpper Bd.Spread SizeRel. Error
1st1^{st}−12857-12857−12824-12824−12834-12834−12814-1281420.3820.381.5E−31.5E-3
2nd2^{nd}−60714-60714−60710-60710−60719-60719−60701-6070117.7417.742.9E−42.9E-4
3rd3^{rd}−11428-11428−11427-11427−11436-11436−11418-1141817.9417.941.5E−31.5E-3
Table 3: Load values in loaded nodes in example 4.2
Figure 13: First test example of the 3D hyperelastic beam subjected to bending. (a) Reference distribution of the applied load, (b) Average prediction of the load value and its location. (c) and (d) Lower and upper limits of the load value distributions, respectively.
Figure 14: Second test example of the 3D hyperelastic beam subjected to bending. (a) Reference distribution of the applied load, (b) Average prediction of the load value and its location. (c) and (d) Lower and upper limits of the load value distributions, respectively.
Figure 15: Third test example of the 3D hyperelastic beam subjected to bending. (a) Reference distribution of the applied load, (b) Average prediction of the load value and its location. (c) and (d) Lower and upper limits of the load value distributions, respectively.
Figure 16: Relationship between load distance to Dirichlet boundary and variance on loaded nodes. Colored points are related to absolute value of applied load.
Figure 17: Comparative of geometries of trained beam and the new shorter one.
Figure 18: Relative RMSE error for test data for the new (shorter) 3D hyperelastic Beam problem.
Figure 19: First test example of the new hyperelastic beam subjected to bending. (a) Distribution of the applied load (Ground Truth), (b) average prediction of the load value and its location. (c) and (d) lower and upper limits of the load value distributions, respectively.
Figure 20: Second test example of the new hyperelastic beam subjected to bending. (a) Distribution of the applied load (Ground Truth), (b) average prediction of the load value and its location. (c) and (d) lower and upper limits of the load value distributions, respectively.
Figure 21: Third test example of the new hyperelastic beam subjected to bending. (a) Distribution of the applied load (Ground Truth), (b) average prediction of the load value and its location. (c) and (d) lower and upper limits of the load value distributions, respectively.

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