# Tensor Decomposition Through Neural Architectures
## 1 Introduction  
Machine learning techniques are designed for extracting the operator (or model) that transforms input data into output data, according to an existing, but unavailable, subtenant physics. Sometimes, they are expected to perform the clustering of unlabeled or the classification of labeled data; the former is called unsupervised (for instance, k-means clustering [1, 2]), the latter supervised (e.g., SVM [3], Decision Trees (DTs) [4] or its random forest counterpart [5], and neural networks [6], among many other choices). At other times, the output is a continuous variable or field, and the associated procedures are the so-called regression techniques. Many types of data are processed by regressions, such as scalars or vectors, tabulated data, images, graphs, curves, and time-series, and different regression technologies were designed to efficiently operate on them. Thus, regressions based on the use of regularized polynomials [7], Gaussian processes, Support Vector Regression, Decision Trees, and random forest, to cite but a few, are widely considered in many applications [8]. When the amount of data is large enough, neural networks are preferred [6], with a diversity of architectures to accommodate the diversity of input and output data formats. Thus, the simplest architecture consists of the so-called multi-layer perceptron, which links the input data to the output layer of the neural network, using a series of consecutive layers,each constituted of a number of neurons. Available input/output data are employed to train the network, that is, computing the weights associated to the neurons' connections and the neurons' bias. For treating more complex data structures, images or graphs, specific neural architectures have been designed with the aim of ensuring invariance properties. Thus, convolution is employed when treating images (Convolutional Neural Networks, CNNs [9]), extended to non-regular data structures (graphs) in the so-called Graph Neural Networks, GNNs [10]. On the other hand, time series, widely encountered when monitoring engineering systems, imply learning stable and causal models. A panoply of techniques exist to model time series. Some techniques are used for predicting the system state evolution, for instance, rNN or LSTM [11; 12], while others are used to learn the operator itself, for instance, ResNet or NeuralODE [13]. The introduction of existing knowledge enormously facilitates the learning process. When the partial differential equation is assumed to be known, Physics-Informed Neural Networks (PINNs) represent an appealing choice, with the existing data enriching the a priori existing knowledge (the PDE) [14]. When a prior model is not available, first principles can be taken into consideration. Thus, Thermodynamics-Informed Neural Networks (TINNs), also known as Structure-Preserving Neural Networks (SPNNs), become an excellent choice for learning within the right thermodynamical setting [15; 16; 17; 18; 19; 20; 21; 22; 23]. Engineering applications produce very heavy solutions from simulations, or capture extremely rich images that then need to be processed. Sometimes, these solutions or images depend on a series of parameters, represented by scalars, vectors, or tensors. An appealing choice for manipulating images relies on the use of convolution neural networks, which successfully performed in many engineering problems [24]. In order to alleviate the computational cost in manipulating images or parametric solutions, expressed as matrices or tensors, tensor decomposition can be considered. For instance, singular value decomposition (SVD), closely related to Principal Component Analysis (PCA), enables researchers to extract the most significant modes able to optimally approximate a matrix [25]. On the other hand, a procedure for solving parametric partial differential equations (pPDEs) was proposed in our earlier works to obtain the tensor decomposition of the solution, with the tensor dimensions concerning space, time, and the different parameters involved in the pPDEs.