As a byproduct of the AI revolution, computational solid and structural mechanics is rapidly evolving, not only through the adoption of new concepts such as physics-informed neural networks and data-driven constitutive laws, but also through the widespread availability of novel computing hardware, particularly GPUs and TPUs alongside traditional CPUs. This shift raises a fundamental question: do methods originally designed for CPU-based computing remain optimal in this new heterogeneous hardware landscape? We recently introduced a new class of solvers, inspired by advances in data-driven computational mechanics, termed Phase Space Iterative Solvers (PSIs)[1]. These methods recast the solution of a discretized mechanics problem as the simultaneous enforcement of two conditions in an abstract phase space: membership in a physically admissible set (satisfying equilibrium with external forces) and in a materially admissible set (satisfying the constitutive law). The solution is obtained through successive projections between these two sets until convergence; a global projection onto the equilibrium set, which involves solving a banded linear system, and a local projection onto the constitutive set, performed elementwise and thus trivially parallelizable. In this talk, we explore hybrid CPU–GPU implementations of PSIs, assigning each projection to the hardware best suited to its computational characteristics, and demonstrate how such a division of labor can be leveraged for efficient and scalable solid mechanics simulations.