Hamiltonian Neural Networks Based Networks for Learning

The problem of learning represents a gateway to understanding intelligence in brains and machines. Many researchers believe that supervised learning will become a key technology for extracting information from the flood of data around us. The supervised learning techniques, i.e. learning from examples, can be seen as an implementation of the mappings y = F(x), relying on the fitting of given data pairs {xk ,yk}. The key point is that the fitting should be predictive and uncover an underlying physical law, which is then used in a predictive or anticipatory way. A great number of models implementing the supervised learning techniques have been proposed in literature. Artificial Neural Networks (ANN), Radial Basis Functions (RBF), Support Vector Machines (SVM) and Fuzzy Logic based models (ANFIS) should be here mentioned. Support Vector Machines, distinctive tools for data classification, are the product of statistical learning theory. Recently, a new learning algorithm named Regularized Least Squares Classification (RLSC) has been proposed. The RLSC concept relyies on multivariate function approximation with regularization theory as a natural framework for solving ill-posed problems of approximation. It is worth noting that SVM and RBF models can be regarded as special cases in the framework of approximation and regularization theory. On the other hand, the Hamiltonian Neural Networks (HNN) based orthogonal filters can be regarded as a natural implementation of the regularization technique. Using Hamiltonian Neural Networks based spectrum analysis, recognition, and memorization, gives rise to mapping implementations with skew-symmetric and symmetric kernels. The purpose of this chapter is to present how very large scale networks for learning can be designed by using HNN-based orthogonal filters and, specifically, by using 8dimensional (octonionic) modules. The unique feature of HNN is the fact that they can exist as either algorithms or physically implementable devices. In this chapter we mainly concentrate on algorithmic description of HNN-based networks. Moreover, since the structures of HNN can be based on the family of Hurwitz-Radon matrices, we present here how to design large scale nonlinear mappings by using neural networks with weight matrices determined by Hurwitz-Radon matrices. Hence, this chapter consists of the following issues: Fundamentals of HNN Family of Hurwitz-Radon matrices RLSC basics O pe n A cc es s D at ab as e w w w .in te ch w eb .o rg