Traditional Finite Element Analysis (FEA) has long relied on a "compute-assemble-solve" paradigm that requires the manual derivation of weak forms and the implementation of rigid, element-centric assembly loops. As computational mechanics advances toward coupled multiphysics, non-local constraints, and data-driven constitutive laws, this legacy architecture often becomes a significant bottleneck, forcing a trade-off between physical expressivity and numerical performance. In this work, we present an energy-centric FEM framework that fundamentally decouples the definition of physics from discretization by representing systems through a single, monolithic global potential energy functional. By leveraging automatic differentiation (AD) and a novel sparse-assembly strategy based on distance-2 graph coloring, we circumvent the "Assembly Trap" characteristic of traditional GPU-accelerated solvers. Our approach eliminates manual linearization and kernel-level rigidity, allowing the compiler to generate exact, machine-precision tangents for complex potentials, including deep neural constitutive models (NCM). We demonstrate that by operating directly on the global functional, we can realize both sparse stiffness matrix and matrix-free Jacobian-Vector Products. Both the sparse assembly of stiffness matrix and matrix-free JVPs scale linearly up to 10 million degrees of freedom (DOFs) on a single GPU. Beyond tackling classical complex problems like contact resolution, cohesive fracture, and multiphysics, our proposed framework seamlessly integrates neural networks, from quadrature-level Neural Constitutive models to domain-level Neural Operator Element Method (NOEM), while implicitly coupling AI- and physics-driven regions via variational summation. We believe this framework turns the FEM into a differentiable tool for discovery, making it easier to describe and solve complex problems. It paves a path toward a system where accurate simulations and machine learning work together in a single, scalable and numerically reliable framework.