2026 Jiang Convex Neural Energy Finite Elements

Extending the neural-operator element method from individually trained, fixed-geometry neural elements to a library of reusable, geometry-parameterized element types fails structurally: a field-predicting operator trained by value regression induces an energy whose assembled Hessian is indefinite, resulting in significant error even with accurate field predictions. This study introduces convex neural energy elements, which export a scalar energy E(g, U), convex in its boundary degrees of freedom U and smoothly parameterized by its geometry g. A hypernetwork-generated positive-semidefinite quadratic form, along with a regularization-nullspace principle, alleviates bias and ensures positive-definite global stiffness in assembled elements. Conditional error bounds are proven and verified experimentally, demonstrating effectiveness in heat conduction applications with various geometries, achieving 0.6–1.0% relative L2 error and significantly faster setup times for per-geometry workloads. Additional element types integrate seamlessly, maintaining performance across dimensions, with analyses confirming that the learned energy transforms neural operators into reusable elements inheriting assembly guarantees. This work bridges a gap in efficient, reliable finite-element assembly in computational mechanics.