We consider the problem of learning maximal Lyapunov functions to compute the maximal region of attraction of nonlinear dynamical systems using neural networks. Two obstacles limit existing neural-network approaches: 1) finite-width approximations can fail to preserve the strict Lyapunov inequalities near the equilibrium, where they become arbitrarily weak, and 2) training is necessarily performed on a bounded computational domain for which the classical Lyapunov and Zubov theorems do not directly apply. To address these issues, first we derive Lyapunov and Zubov theorems directly on a prescribed compact computational domain, characterizing the maximal region of attraction within that domain as the solution of a boundary-value PDE. Second, we introduce the Taylor Lyapunov network, an architecture that prescribes the quadratic leading-order term via the system's linearization and represents the higher-order remainder through a structured cubic factorization, guaranteeing that any network in this class is a valid local Lyapunov function by construction. Finally, we prove that Taylor Lyapunov networks are universal approximators of maximal Lyapunov functions on compact domains, and combine these results into an unsupervised, physics-informed training procedure. Numerical experiments on systems of varying dimensions and stability types show that the proposed architecture is efficient at estimating the maximum region of attraction.

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