This paper introduces a polyconvex physics‑augmented neural network (PANN) that uses learned paired directional features to model the anisotropic hyperelastic response of a representative volume element. The network’s energy functions are constructed with convex, componentwise nondecreasing cores (ICNN or ICKAN) and satisfy objectivity, stress‑free reference, two‑dimensional polyconvexity, and prescribed growth conditions. Experiments on multicavity and single‑cavity periodic cells show that the constrained PANN achieves sub‑percent stress and displacement errors, while an unconstrained model loses rank‑one convexity and under‑predicts compression stresses beyond the training data.
By S. Ares de Parga, A. Cornejo, J. A. Hern\'andez, R. Rossi
arXiv:2608. 09707v1 Announce Type: cross Abstract: Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research.
By Yu Liu, Jan Kronqvist, Fabricio Oliveira
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
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).
The paper introduces the "lift" technique for training input‑convex neural networks, replacing the traditional non‑negative weight constraint enforced by projected gradient descent or a softplus map. By adding a learnable slack variable and an unconstrained network that processes a permutation‑invariant batch summary, the lift couples batch‑dependent latent weights to the gradient, increasing update variance and enabling faster escape from the softplus shoulder. Experiments show that when the softplus method stalls at the shoulder, the lift achieves tighter fits and reconstructs targets roughly three times faster, while both methods agree when the shoulder is rarely reached.
By Ali Siahkoohi
The paper introduces Shallow Universal Polynomial Networks (SUPNs), a single‑hidden‑layer architecture that replaces standard activations with learnable multivariate polynomials, thereby increasing expressivity per unit. SUPNs achieve universal approximation by either widening the network or increasing polynomial complexity, and the authors prove convergence rates and quasi‑optimal parameter choices. Empirical tests across a wide range of smooth, nonsmooth, and discontinuous functions in up to ten dimensions show that SUPNs often outperform deep neural networks and Kolmogorov‑Arnold networks in accuracy and variability while requiring fewer parameters and lower training cost.
By Zachary Morrow, Michael Penwarden, Brian Chen, Aurya Javeed, Akil Narayan, John D. Jakeman
arXiv:2607. 20378v1 Announce Type: new Abstract: Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability.
By Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee, Aida Pakniyat, Dara Rahmati, Saeid Gorgin
arXiv:2608. 09523v1 Announce Type: new Abstract: Deep neural network (DNN) training with stochastic gradient descent (SGD) and its variants achieves strong empirical performance, yet classical optimization theory does not fully explain this success.
By Binchuan Qi
The paper introduces a variational, label‑free physics‑informed graph neural network (PI‑GNN) that models heterogeneous solid mechanics by embedding material heterogeneity into the discretization rather than the neural network’s trial field. The PI‑GNN operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy without penalty terms or interface weights, yielding a discrete energy equivalent to the finite element Ritz functional. Across small‑strain elasticity and finite‑strain Neo‑Hookean hyperelasticity in 2D and 3D, the method achieves von Mises errors below 3.58 % over a wide stiffness‑contrast range, outperforming strong‑form PINNs and reducing displacement errors significantly.
By Aashay Rajan Yadav, Amiya Prakash Das, Ratna Kumar Annabattula
arXiv:2605.04722v2 Announce Type: replace-cross
Abstract: Input convex neural networks (ICNNs) are used to learn convex objectives whose minimizers define decisions, making efficient and reliable opt...
By Kang Liu, Jianchen Hu, Wei Peng
arXiv:2605.17692v2 Announce Type: replace
Abstract: We show that the training problem of a deep linear neural network under the squared loss admits an exact convex reformulation in a lifted space ove...
By Karthik Prakhya, Alp Yurtsever
arXiv:2607. 02003v1 Announce Type: cross Abstract: Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics.
By Matej Benko, Pierre Bousquet, Iwona Chlebicka, B{\l}a\.zej Miasojedow