We propose a new binary classification model in Machine Learning, called Phase Separation Binary Classifier (PSBC). It consists of a discretization of a nonlinear reaction-diffusion equation (the Allen-Cahn equation), coupled with an ODE . Unlike many feedforward networks that are said to mimic brain or cortical cells functioning, the PSBC is inspired by fluid behavior, namely, on how binary fluids phase separate. Thus, (hyper)parameters have physical meaning, whose effects are carefully studied in several different scenarios : for instance, diffusion introduces interaction among features, whereas reaction plays an active role in classification . PSBC's coefficients are trainable weights, chosen according to a minimization problem using Gradient Descent; optimization relies on a classical Backpropagation Algorithm using weight sharing. Moreover, the model can be seen under the framework of feedforward networks, and is endowed with a nonlinear activation function that is linear in trainable weights but polynomial in other variables  . In view of the model's connection with ODEs and parabolic PDEs, forward propagation amounts to an initial value problem. Thus, stability conditions are established through meshgrid constraints, discrete maximum principles, and, overall, exploiting the concept of Invariant regions , as developed in the work of Chueh, Conway, Smoller, and particularly in the application of their theory to finite-difference methods in the work of Hoff. The PSBC also has interesting model compression properties which are thoroughly discussed. We apply the model to the subset of numbers "0" and "1" of the classical MNIST database, where we are able to discern individuals from both classes with more than 94\\%DIF < accuracy, sometimes using less than 80 variables, a feature that is out of reach of Artificial Neural Networks without weight sharing or feature engineering.\end{abstract} %DIF > accuracy, sometimes using less only about 10\\% of variables.
We propose a new binary classification model called Phase Separation Binary Classifier (PSBC). It consists of a discretization of a nonlinear reaction-diffusion equation coupled with an ODE , and is inspired by fluid behavior, namely, on how binary fluids phase separate. Hence, parameters and hyperparameters have physical meaning, whose effects are carefully studied in several different scenarios . PSBC's coefficients are trainable weights, chosen according to a minimization problem using Gradient Descent; optimization relies on a classical Backpropagation with weight sharing. The model can be seen under the framework of feedforward networks, and is endowed with a nonlinear activation function that is linear in trainable weights but polynomial in other variables , yielding a cost function that is also polynomial . In view of the model's connection with ODEs and parabolic PDEs, forward propagation amounts to an initial value problem. Thus, stability conditions are established using the concept of Invariant regions . Interesting model compression properties are thoroughly discussed. We illustrate the classifier's qualities by applying it to the subset of numbers "0" and "1" of the classical MNIST database, where we are able to discern individuals with more than 94\\%DIF < accuracy, sometimes using less than 80 variables, a feature that is out of reach of Artificial Neural Networks without weight sharing or feature engineering.\end{abstract} %DIF > accuracy, sometimes using less only about 10\\% of variables.
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