Neural networks are increasingly used in finance, and complex systems. One of the relevant characteristics of financial systems, and other complex systems, is the intricate dependency structure between variables. Such a dependency structure is a high-order network of relations and it is very important to be able to represent it within the models’ architecture. Traditional neural networks have not been designed to capture such complexity and its dynamics. We propose to investigate the design and testing of a novel deep learning neural network architecture, termed Homological Neural Network (HNN) with a higher-order graphical structure that can adjust dynamically to better model the multivariate dynamical complexity of data with practical applications in finance and more.