The Polynomial Mirror:An Architecture-Agnostic Algebraic Framework for Interpreting and Enhancing Neural Networks

We propose the Polynomial Mirror, a theoretical framework to algebraically approximate any trained neural network using only polynomial functions. By replacing activation functions and affine transformations with polynomial approximations, we construct a symbolic representation of the original network. Relying on classical results in approximation theory, this framework invites rigorous exploration of neural networks from a purely algebraic and interpretable perspective.

Beyond interpretability, this framework also enables neuron-level customization: since each activation function is replaced by a polynomial, the shape of each neuron’s response becomes tunable. This suggests a potential path toward enhancing model performance through fine-grained control of activation behaviors. While this is a predicted consequence, we invite further research into its effectiveness and its relation to prior work on learned activation functions.