Adapting Deep Variational Bayes Filter for Enhanced Confidence Estimation in Finite Element Method Integrated Networks (FEMIN)

TL;DR

The paper tackles the reliability gap in crash simulations that use FEMIN by introducing a probabilistic DVBF that provides uncertainty estimates during online FEMIN runs. By decoding a predictive force from the DVBF's transition and feeding both kinematics and this force into the encoder, the model yields a posterior likelihood from which a mean force is applied to the FEMIN solver and the decoder's std serves as a confidence metric. DVBF outperforms deterministic LSTMs in accuracy and delivers meaningful, calibrated uncertainty (high PICP and interpretable AC), enabling more robust, interpretable FEMIN simulations. The approach achieves competitive computational efficiency, with window-based training enabling scalability to long sequences, and demonstrates strong generalization across BI and TCT crash load cases, highlighting its practical impact for reliable automotive crash analysis.

Abstract

The Finite Element Method (FEM) is a widely used technique for simulating crash scenarios with high accuracy and reliability. To reduce the significant computational costs associated with FEM, the Finite Element Method Integrated Networks (FEMIN) framework integrates neural networks (NNs) with FEM solvers. However, this integration can introduce errors and deviations from full-FEM simulations, highlighting the need for an additional metric to assess prediction confidence, especially when no ground truth data is available. In this study, we adapt the Deep Variational Bayes Filter (DVBF) to the FEMIN framework, incorporating a probabilistic approach to provide qualitative insights into prediction confidence during FEMIN simulations. The adaptation involves using the learned transition model for a predictive decoding step, generating a preliminary force prediction. This predictive force is used alongside the displacement and the velocity data from the FEM solver as input for the encoder model. The decoder reconstructs the likelihood distribution based on the posterior. The mean force of this distribution is applied to the FEM solver, while the predicted standard deviation can be used for uncertainty estimation. Our findings demonstrate that the DVBF outperforms deterministic NN architectures in terms of accuracy. Furthermore, the standard deviation derived from the decoder serves as a valuable qualitative metric for assessing the confidence in FEMIN simulations. This approach enhances the robustness of FEMIN by providing a measure of reliability alongside the simulation results.

Figures (11)

• Figure 1: Principle of FEMIN: A FEM simulation is accelerated by a NN, which replaces a portion of the FEM mesh.
• Figure 2: Flow chart of the modified DVBF for FEMIN. No backpropagation over dashed lines
• Figure 3: Load cases as presented by thel_introducing_2024. The elements depicted in green are in the node set $\mathcal{N}_{\text{replaced}}$, and the gray elements are in $\mathcal{N}_{\text{FEMIN}}$. The boundary nodes $\mathcal{N}_{\text{B}}$ are marked with a red dashed line.
• Figure 4: Comparison of ELBO, NLL, and MSE over epochs for BI load case with $\boldsymbol{\mathbf{p}} = \{\boldsymbol{\mathbf{v}}_{\text{init}}, \boldsymbol{\mathbf{d}}_{\text{impactor}}\}$
• Figure 5: Comparison of ELBO, NLL, and MSE over epochs for BI load case ($\boldsymbol{\mathbf{p}} = \boldsymbol{\mathbf{0}}$)