Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees

This paper addresses the limitations of the neural-operator element method by proposing convex neural energy elements that enable a reusable library of geometry-parameterized element types. The study identifies that traditional field-predicting neural operators can lead to indefinite Hessians, causing convergence issues in optimization tasks. To remedy this, the authors introduce a methodology where each element generates a scalar energy that is architecturally convex in its boundary degrees of freedom and smoothly parameterized by geometry. Through a regularization-nullspace principle, biases are mitigated, and the classical guarantee of a positive-definite global system is preserved upon assembly. Extensive experiments demonstrate the approach's efficacy, achieving relative L2 errors as low as 0.6–1.0% for complex geometries while significantly enhancing computational efficiency. A promising finding is that the assembly of multiple trained element types can coexist within a single framework, thereby extending the application's versatility across various dimensions and types—proving the generalized stability and error guarantees critical for practical implementations of neural operators in engineering applications.