Error Approximation and Bias Correction in Dynamic Problems using a Recurrent Neural Network/Finite Element Hybrid Model

TL;DR

The results show that the proposed hybrid model is capable of approximating model discrepancies to a high degree of accuracy and accordingly correct low-fidelity models.

Abstract

This work proposes a hybrid modeling framework based on recurrent neural networks (RNNs) and the finite element (FE) method to approximate model discrepancies in time dependent, multi-fidelity problems, and use the trained hybrid models to perform bias correction of the low-fidelity models. The hybrid model uses FE basis functions as a spatial basis and RNNs for the approximation of the time dependencies of the FE basis' degrees of freedom. The training data sets consist of sparse, non-uniformly sampled snapshots of the discrepancy function, pre-computed from trajectory data of low- and high-fidelity dynamic FE models. To account for data sparsity and prevent overfitting, data upsampling and local weighting factors are employed, to instigate a trade-off between physically conforming model behavior and neural network regression. The proposed hybrid modeling methodology is showcased in three highly non-trivial engineering test-cases, all featuring transient FE models, namely, heat diffusion out of a heat sink, eddy-currents in a quadrupole magnet, and sound wave propagation in a cavity. The results show that the proposed hybrid model is capable of approximating model discrepancies to a high degree of accuracy and accordingly correct low-fidelity models.

Figures (12)

• Figure 1: Hybrid model architecture. The comprises $N_p$ units and a linear output layer and approximates the coefficients of the discrepancy function. The latter are subsequently combined with the shape functions in the resulting hybrid model. The vectors $\mathbf{c}_{t}$ and $\mathbf{h}_t$, $t = t_k,...,t_{k+N_p-1}$, denote the cell states and hidden states of the units. The 's inputs and outputs are highlighted in purple.
• Figure 2: Overview of the data exchange in the hybrid model's training process. Details regarding loss computation and pre-processing of the training data $\hat{\delta}_{t_k}$ are given in Sections \ref{['sec:hybrid_model']}, \ref{['sec:upsampling']}, and \ref{['sec:non_dimensionalisation']}.
• Figure 3: The high-fidelity solution $\mathbf{x}^{\text{hifi}}_{t_k}$ is defined on $T_{\text{hifi}}$. The intermediary time steps of the low-fidelity model are on the time axis. $N_{I_1}=3$ denotes the number of intermediary steps for the interval $I_1$.
• Figure 4: Left: Schematic of a heat sink cross-section, where $\Omega_{\text{con}}$ is the thermally conductive domain and $\Omega_{\text{air}}$ the non-conductive domain. Non-homogeneous Dirichlet boundary condition are applied at the heat sink base at $\partial \Omega_{\text{nd}} = \partial \Omega \cap \partial \Omega_{\text{con}}$. Right: Plot of the thermal conductivity of the material along the middle fin, where $\kappa_{\text{hifi}}$ is the conductivity with defects and $\kappa_{\text{lofi}}$ without. The conductivity in $\Omega_{\text{air}}$ is $\kappa = 0.5 W\left(mK\right)^{-1}$ for both cases.
• Figure 5: Spatially integrated discrepancy function $\|\delta_{t}\|_{2}$ for $t_0=0 \,s$ until $t_{100}=2\,s$. The reference and the hybrid model with and without artificial upsampling are depicted. The training data is indicated at the respective time steps with $\mathbf{\times}$.