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Artificial Intelligence and Machine learning have been widely used in variousfields of mathematical computing, physical modeling, computational science,communication science, and stochastic analysis. Approaches based on DeepArtificial Neural Networks (DANN) are very popular in our days. Depending onthe learning task, the exact form of DANNs is determined via their multi-layerarchitecture, activation functions and the so-called loss function. However,for a majority of deep learning approaches based on DANNs, the kernel structureof neural signal processing remains the same, where the node response isencoded as a linear superposition of neural activity, while the non-linearityis triggered by the activation functions. In the current paper, we suggest toanalyze the neural signal processing in DANNs from the point of view ofhomogeneous chaos theory as known from polynomial chaos expansion (PCE). Fromthe PCE perspective, the (linear) response on each node of a DANN could be seenas a $1^{st}$ degree multi-variate polynomial of single neurons from theprevious layer, i.e. linear weighted sum of monomials. From this point of view,the conventional DANN structure relies implicitly (but erroneously) on aGaussian distribution of neural signals. Additionally, this view revels that bydesign DANNs do not necessarily fulfill any orthogonality or orthonormalitycondition for a majority of data-driven applications. Therefore, the prevailinghandling of neural signals in DANNs could lead to redundant representation asany neural signal could contain some partial information from other neuralsignals. To tackle that challenge, we suggest to employ the data-drivengeneralization of PCE theory known as arbitrary polynomial chaos (aPC) toconstruct a corresponding multi-variate orthonormal representations on eachnode of a DANN to obtain Deep arbitrary polynomial chaos neural networks.

The Deep Arbitrary Polynomial Chaos Neural Network or how Deep Artificial Neural Networks could benefit from Data-Driven Homogeneous Chaos Theory