# Tensor Decomposition Through Neural Architectures
## 3 Numerical Results
### A 2D Problem with a Field as Input  
This section addresses the 2D heat transfer problem  
\[-\
abla\cdot(K(\mathbf{x})\
abla u(\mathbf{x}))=Q(\mathbf{x}), \tag{12}\]  
defined in \(\mathbf{x}\in\Omega=[-0.5,0.5]\times[-0.5,0.5]\), with the source \(Q(\mathbf{x})=1\) in \([-0.2,0.2]\times[-0.1,0.1]\), vanishing outside. The temperature is enforced to zero on the domain boundary, i.e., \(u(\mathbf{x}\in\partial\Omega)=0\). Here, the conductivity is assumed to be evolving in \(\Omega\), and in our simulations, we assumed it to be described by  
\[K(\mathbf{x})=1+100\bigg{(}sin\bigg{(}\frac{\xi\pi x}{0.5}\bigg{)}\bigg{)}^{2}, \tag{13}\]  
with \(\xi\) being a uniformly distributed random variable taking values in the interval defined by \(\xi=0\) and \(\xi=10\). To generate the data set, 1000 values of \(\xi\) were selected, \(\{\xi_{1},\xi_{2},\ldots\}\), leading to 1000 conductivity distributions \(K^{i}(\mathbf{x})\), and 1000 associated solutions \(u^{i}(\mathbf{x})\), \(u^{i}(\mathbf{x})\equiv u(\mathbf{x};\xi_{i})\). One of these solutions is depicted in Figure 5 along with the associated conductivity distribution. In order to described the input, that is, the conductivity, its space decomposition was performed by applying the SVD on its matrix representation. Thus, each one of the generated  
Figure 4: Singular value prediction on the training and test sets. Figure 5: Solution of the 2D steady-state heat transfer problem for a particular choice of the conductivity distribution. conductivities, represented in a discrete manner by \(\mathbf{K}^{i}\), \(i=1,\ldots,1000\), with \(\mathbf{K}^{i}_{rs}=K^{i}(x_{r},y_{s})\), is decomposed again by applying the SVD, which, after truncation, results in  
\[\mathbf{K}^{i}\approx\mathbb{V}^{i}_{K}\mathbb{D}^{i}_{K}\mathbb{W}^{i^{T}}_{K}, \tag{14}\]  
and, similarly, for the solutions themselves, expressed from \(\mathbf{U}^{i}\), with components \(\mathbf{U}^{i}_{rs}=u^{i}(x_{r},y_{s})\),  
\[\mathbf{U}^{i}\approx\mathbb{V}^{i}\mathbb{D}^{i}\mathbb{W}^{i^{T}}. \tag{15}\]  
Now, the goal is to train a surrogate \(\mathcal{F}(\cdot)\) relating both reduced bases, that is,  
\[[\mathbb{V},\mathbb{W}]=\mathcal{F}(\mathbb{V}_{K},\mathbb{W}_{K}) \tag{16}\]  
The surrogate model architecture of \(\mathcal{F}\) is given in Appendix A, Figure 11. The procedure performance is illustrated in Figures 6 and 7 for a sample in the training and test sets, respectively. The average relative error is \(\epsilon=0.83\%\) in the training set and \(\epsilon=0.77\%\) in the test set. The slightly lower error in the test set can be explained by the use of the dropout layers in the neural network architecture, which prevents overfitting. Finally, the decomposition of the output field was correlated to the decomposition of the input field by predicting the singular values of the output. For that purpose, a surrogate model \(\mathcal{S}\) was trained to perform  
\[\mathbb{D}=\mathcal{S}(\mathbb{V}_{K},\mathbb{D}_{K},\mathbb{W}_{K}) \tag{17}\]  
Since the singular values \(\mathbb{D}\) decay in amplitude quickly and their amplitude changes a great deal, a different model \(\mathcal{S}\) was built for each value of \(\mathbb{D}\), to avoid a common normal  
Figure 6: Predicted reduced basis vectors for the 2D steady-state heat transfer problem, for a solution in the training set. ization in which the convergence using typical losses such as the mean square error of the mean absolute error may be difficult. The surrogate model architecture of \(\mathcal{S}\) is given in Appendix A, Figure A2. The results for the eigenvalues \(\mathbb{D}_{1}\) and \(\mathbb{D}_{2}\) are illustrated in Figure 8a,b, respectively. A combination of dense and convolution neural layers are used in the surrogate model. The results demonstrate the ability to predict the output singular values, as a function of the input parameters.