Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees
2026-08-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors show that previous attempts to use neural networks for predicting fields in systems with varying geometry fail because they lead to unstable solutions. They propose a new approach called convex neural energy elements, which ensures the energy functions are convex and depend smoothly on geometry, making the assembled system stable and reliable. Their method combines learned elements that behave like classical finite elements, allowing reuse across different geometries while maintaining accuracy and stability. They prove theoretical error bounds and test their approach on heat conduction and elasticity problems, demonstrating good accuracy and much faster setup times.
neural operatorconvex energyfinite element methodpositive-semidefinite matrixhypernetworkboundary degrees of freedomNewton's methodelliptic PDEplane-strain elasticityregularization nullspace
Authors
Hongyue Jiang, Jianjiang Zhan, Chenzhuo Zhang, Fan Wang
Abstract
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, and Newton converges to spurious minima (247% error) even with 1%-accurate field predictions. We introduce convex neural energy elements: each element exports a scalar energy E(g,U), architecturally convex in its boundary degrees of freedom U and smoothly parameterized by its geometry g, realized as a hypernetwork-generated positive-semidefinite quadratic form (an input-convex correction is reserved for non-quadratic physics). A regularization-nullspace principle--the regularizer's nullspace must contain the physics nullspace--removes an otherwise irreducible bias, and assembled elements inherit the classical guarantee that singular element stiffnesses yield a positive-definite global system. We prove conditional error bounds (energy-to-solution accuracy, element-count scaling, geometry generalization) and verify each experimentally. On heat conduction with elliptic holes, one trained element assembles into 2x2 to 8x8 grids and an L-shaped layout of unseen geometries at 0.6-1.0% relative L2 error, with 175x faster per-geometry setup for boundary-quantity workloads. A second trained element type mixes freely with the first in one monolithic assembly, and a three-dimensional instantiation reaches 0.23% on eight-element assemblies--the guarantees are type- and dimension-agnostic. A plane-strain elasticity element, whose physics nullspace is three-dimensional, lands on the analytically predicted regularization floors. Making the energy the learned object turns neural operators from single-use surrogates into reusable elements that inherit the assembly guarantees of the method they extend.