  Polynomial expansions are important in the analysis of neural network
nonlinearities. They have been applied thereto addressing well-known
difficulties in verification, explainability, and security. Existing approaches
span classical Taylor and Chebyshev methods, asymptotics, and many numerical
approaches. We find that while these individually have useful properties such
as exact error formulas, adjustable domain, and robustness to undefined
derivatives, there are no approaches that provide a consistent method yielding
an expansion with all these properties. To address this, we develop an
analytically modified integral transform expansion (AMITE), a novel expansion
via integral transforms modified using derived criteria for convergence. We
show the general expansion and then demonstrate application for two popular
activation functions, hyperbolic tangent and rectified linear units. Compared
with existing expansions (i.e., Chebyshev, Taylor, and numerical) employed to
this end, AMITE is the first to provide six previously mutually exclusive
desired expansion properties such as exact formulas for the coefficients and
exact expansion errors (Table II). We demonstrate the effectiveness of AMITE in
two case studies. First, a multivariate polynomial form is efficiently
extracted from a single hidden layer black-box Multi-Layer Perceptron (MLP) to
facilitate equivalence testing from noisy stimulus-response pairs. Second, a
variety of Feed-Forward Neural Network (FFNN) architectures having between 3
and 7 layers are range bounded using Taylor models improved by the AMITE
polynomials and error formulas. AMITE presents a new dimension of expansion
methods suitable for analysis/approximation of nonlinearities in neural
networks, opening new directions and opportunities for the theoretical analysis
and systematic testing of neural networks.
