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).
arXiv:2606. 28573v1 Announce Type: new Abstract: Modern machine learning models are trained by optimizing high-dimensional non-convex empirical risk functions.
By Andrea Montanari, Kangjie Zhou
arXiv:2501. 18530v3 Announce Type: replace-cross Abstract: We consider a teacher-student model of supervised learning with a fully-trained two-layer neural network whose width $k$ and input dimension $d$ are large and proportional.
By Jean Barbier, Francesco Camilli, Minh-Toan Nguyen, Mauro Pastore, Rudy Skerk
arXiv:2608. 02036v1 Announce Type: new 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.
By Hongyue Jiang, Jianjiang Zhan, Chenzhuo Zhang, Fan Wang
arXiv:2607. 15702v1 Announce Type: cross Abstract: We prove a finite-sample formulation gap for physics-informed learning of nonlinear multiscale elliptic equations.
By Ronald Katende
arXiv:2608. 04882v1 Announce Type: new Abstract: We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture.
By Francesco Camilli, Pierluigi Contucci, Federica Gerace, Emanuele Mingione