Architecturally constrained neural energy representations for variationally well-posed higher-order continua

Abstract

Neural constitutive models have recently emerged as powerful tools for data-driven continuum mechanics, yet their integration into higher-order theories remains largely empirical and often lacks structural guarantees. In this work, we develop a rigorous variational framework for invariant neural representations of stored energy densities in strain-gradient continuum models. The energy is expressed as a neural mapping of isotropic invariants of the infinitesimal strain tensor and its gradient, thereby enforcing objectivity and isotropy at the representation level. We prove thermodynamic admissibility through potential structure, establish coercivity in $H^2(\mathrm{\Omega })$ , and demonstrate weak lower semicontinuity and existence of minimizers via the direct method of the calculus of variations. A detailed Legendre–Hadamard analysis yields explicit sufficient conditions for preservation of strong ellipticity expressed directly in invariant space, clarifying how neural parametrizations influence stability and strain localization. We further discuss the relation of these conditions to convexity, rank-one convexity, and quasiconvexity and outline a polyconvex neural extension suitable for finite-strain settings. The proposed framework embeds neural energy representations into a functional-analytic structure consistent with generalized continuum mechanics, thereby providing a mathematically controlled alternative to black-box constitutive learning in higher-order elasticity.