  This work begins by establishing a mathematical formalization between
different geometrical interpretations of Neural Networks, providing a first
contribution. From this starting point, a new interpretation is explored, using
the idea of implicit vector fields moving data as particles in a flow. A new
architecture, Vector Fields Neural Networks(VFNN), is proposed based on this
interpretation, with the vector field becoming explicit. A specific
implementation of the VFNN using Euler's method to solve ordinary differential
equations (ODEs) and gaussian vector fields is tested. The first experiments
present visual results remarking the important features of the new architecture
and providing another contribution with the geometrically interpretable
regularization of model parameters. Then, the new architecture is evaluated for
different hyperparameters and inputs, with the objective of evaluating the
influence on model performance, computational time, and complexity. The VFNN
model is compared against the known basic models Naive Bayes, Feed Forward
Neural Networks, and Support Vector Machines(SVM), showing comparable, or
better, results for different datasets. Finally, the conclusion provides many
new questions and ideas for improvement of the model that can be used to
increase model performance.
