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:2605. 24651v2 Announce Type: replace-cross Abstract: We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $\varphi$-finite element method ($\varphi$-FEM).
By Bokai Zhu, Yizheng Wang, Qinghui Zhang, Timon Rabczuk
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:2608. 05437v1 Announce Type: cross Abstract: Supervised training of finite-element (FE) surrogate models requires reference solutions, and each reference solution is obtained by solving the system that the surrogate is intended to replace.
By Ruifeng Cao (The University of Manchester), Xidan Song (Wuhan University)
The paper introduces a differentiable finite element framework that discovers hyperelastic constitutive laws from limited experimental data, such as boundary-only displacement measurements and global reaction forces. By embedding the nonlinear finite element equilibrium problem into the learning loop, the method evaluates candidate strain‑energy densities through the deformation fields they produce, enforcing mechanical equilibrium as a constraint. The constitutive response is modeled with Hyperelastic Neural Networks, a structure‑preserving class that guarantees physical admissibility, including residual energy and stress‑free conditions, frame indifference, isotropic symmetry, polyconvexity, coercivity, and controlled volumetric growth. Numerical experiments in two and three dimensions show accurate recovery of hyperelastic isotropic responses, robustness to noise, and generalization across geometries, loading, and boundary conditions.
By Francesco Regazzoni
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:2607. 20339v1 Announce Type: new Abstract: Constitutive modeling under uncertainty remains a central challenge for reliable mechanics simulations, particularly when the available stress-deformation data are sparse, noisy, or heterogeneous.
By Somesh Pratap Singh, Govinda Anantha Padmanabha, Jingye Tan, Steven Yang, Reese E. Jones, D. Thomas Seidl, Nikolaos Bouklas
arXiv:2608. 08322v1 Announce Type: cross Abstract: Physics-informed neural networks applied to the level-set formulation of interface advection commonly augment the residual and initial-condition losses with an eikonal regulariser, penalising the deviation of $\|\nabla\phi\|$ from unity.
By Muhammad Akbar Khan
arXiv:2607. 27000v1 Announce Type: cross Abstract: Optimization in non-convex neural network models is strongly influenced by the geometry of the solution space: sparse, isolated, point-like clusters are typically algorithmically inaccessible, whereas wide and flat regions can be found efficiently despite being relatively rare.
By Enrico M. Malatesta, Alessandra Passalacqua, Riccardo Zecchina
Constitutive modeling under uncertainty remains a central challenge for reliable mechanics simulations, particularly when the available stress-deformation data are sparse, noisy, or heterogeneous. We propose interval and fuzzy physics-augmented neural networks (iPANNs and fPANNs) for uncertainty-aware hyperelastic constitutive modeling.
arXiv:2607. 22004v1 Announce Type: new Abstract: Energy natural gradient descent (ENGD) aligns parameter updates with the curvature of an underlying function-space energy, but existing formulations assume an unconstrained Euclidean parameter domain.
By Zhangyong Liang, Huanhuan Gao
arXiv:2412.04596v3 Announce Type: replace
Abstract: We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The appro...
By Mats G. Larson, Carl Lundholm, Anna Persson