finite-element quantity recovery

This work addresses ill-posed inverse problems governed by partial differential equations (PDEs). We propose a physics-informed, data-driven iterative regularization method that integrates physical modeling with deep learning. Specifically, we construct a graph structure via finite-element discretization, embed the forward operator into a graph neural network (GNN) framework, and design a physically interpretable GNN-based regularizer. During iteration, coefficient reconstruction and the regularization prior are jointly optimized. Unlike conventional Tikhonov or total variation regularization, our approach achieves robustness, generality, and interpretability without requiring strong prior assumptions. Experiments demonstrate significantly improved reconstruction accuracy over classical methods under highly ill-conditioned settings and low signal-to-noise ratios. The method establishes a novel paradigm for synergistic data–physics integration in solving PDE-constrained inverse problems.

This work proposes implicit Finite Operator Learning (iFOL), a physics-informed operator learning framework for multiphysics problems governed by coupled partial differential equations on arbitrary domains, which operates without requiring ground-truth labeled data. Built upon the finite element weighted residual formulation, iFOL establishes a resolution-independent mapping from input parameters to the solution space and provides a unified treatment of complex geometries and multiphysics coupling. Implemented within the Folax system on the JAX platform, the approach integrates FNO, DeepONet, and iFOL, leveraging finite element residuals to construct physics-constrained loss functions. Experiments demonstrate that iFOL achieves high efficiency on complex geometries in two- and three-dimensional nonlinear thermo-mechanical coupling and industrial casting scenarios, while FNO excels in accuracy on regular domains; furthermore, a single-network end-to-end training strategy significantly outperforms baseline methods.

This work addresses the challenge of escalating computational cost and degraded accuracy in FEONet when applied to large-scale problems due to the increasing number of elements. Inspired by the local sparsity inherent in finite element methods, the authors propose a sparse neural network architecture that establishes a direct mapping from parameters to solutions, enabling efficient solution of parametric partial differential equations without requiring training data. By incorporating locally sparse connectivity and a data-free training mechanism, the method substantially reduces both computational and memory overhead while offering theoretical guarantees on approximation accuracy and stability. Numerical experiments demonstrate that the proposed approach achieves computational efficiency significantly higher than that of the original FEONet, while maintaining comparable high accuracy and robustness.

Traditional finite element methods (FEM) suffer from strong mesh dependency and high computational cost for multiscale complex problems, while existing neural operators exhibit poor reusability and prohibitive training overhead. To address these limitations, this work proposes the Neural Operator Element Method (NOEM): the first framework embedding reusable neural operators into the FEM paradigm. NOEM constructs physics-consistent neural operator elements over complex subdomains and seamlessly couples them with standard FEM via the variational principle. It employs deep operator networks to model subdomain mappings, supporting nonlinearity, multiscale behavior, and arbitrary geometries. Experiments demonstrate that NOEM substantially reduces degrees of freedom and computational cost while preserving optimal convergence rates, scalability, and cross-problem transferability. By unifying data-driven learning with physics-based discretization, NOEM achieves a favorable balance among accuracy, efficiency, and generalization—outperforming conventional FEM and state-of-the-art neural operators in both theoretical rigor and practical applicability.

To address the insufficient physical consistency in surrogate modeling of quasi-static stress fields in solid mechanics, this paper proposes the Physics-encoded Fourier Neural Operator (PeFNO). Grounded in stress potential theory, PeFNO intrinsically enforces the divergence-free constraint—required by mechanical equilibrium—by parameterizing a stress potential function and analytically deriving the stress field therefrom, thereby embedding physical laws directly into the network architecture rather than relying on soft penalty terms in the loss function. Evaluated on heterogeneous polycrystalline material under uniaxial tension, PeFNO reduces the equilibrium error in predicted stress fields by 42% on average compared to baseline models including Physics-guided FNO (PgFNO) and Physics-informed FNO (PiFNO), achieving both high accuracy and strict adherence to conservation laws. This work establishes the first end-to-end, stress-potential-driven physics encoding framework, introducing a new paradigm for interpretable and verifiable mechanical surrogate modeling.

This work addresses the challenge of scaling traditional finite element surrogate models, which rely on costly reference solutions for supervised training. The authors propose a novel unsupervised training approach that eliminates the need for reference solutions by rigorously establishing, for the first time, an exact equivalence between discrete potential energy and stiffness-norm error. Leveraging this relationship, they design a gradient-consistent training mechanism that integrates discrete energy functionals, stiffness-weighted error analysis, and a JEPA (Joint-Embedding Predictive Architecture) framework. Validation across synthetic benchmarks and 16 experimental cases demonstrates that the resulting energy gap effectively controls displacement error. Furthermore, the study reveals that Euclidean error is ill-suited as a primary evaluation metric and delineates the method’s applicability boundaries.

This work addresses the challenge of efficiently and accurately estimating quantities of interest (QoI) in multi-query linear problems, where conventional approaches suffer from high computational costs and strong dependence on load configurations. The authors propose a novel reduced-order modeling paradigm based on the adjoint problem, shifting the focus of model reduction from the primal to the adjoint equation for the first time. By introducing a parameterized kernel function to replace the full external load, the method constructs a load-independent surrogate model. Demonstrated on Poisson’s equation and plane-stress elasticity problems, the approach achieves rapid convergence and significantly outperforms traditional primal-based reduction strategies. It enables high-fidelity QoI estimation while supporting fast multi-scenario evaluation and virtual chart generation, thereby greatly enhancing the generality and efficiency of early-stage design optimization.

This work addresses the instability and low integration efficiency of the virtual element method in large-deformation nonlinear problems by proposing a stabilized approach that combines scaled boundary parameterization with reduced integration. By performing a Taylor expansion of constitutive quantities about the cross-sectional centroid, the weak form is analytically integrated, requiring only a single integration point per cross-section. This strategy drastically reduces the number of integration points while effectively handling hyperelastic anisotropic and elastoplastic large-deformation scenarios. Numerical experiments demonstrate that the method accurately captures structural responses and inelastic behavior across various materials and loading conditions, exhibiting particularly superior performance when physical elements closely resemble their reference configurations.

This work addresses the high computational cost of traditional finite element methods in problems involving local nonlinearities, fine-scale features, or long-time dynamics, as well as the limited geometric flexibility of existing hybrid approaches. To overcome these challenges, the authors propose a non-overlapping Schwarz alternating coupling framework that efficiently integrates finite elements with neural operators. Information exchange between subdomains is achieved through Neumann–Dirichlet interface conditions, eliminating overlapping regions to avoid redundant computation. The framework employs Point-DeepONet to directly handle unstructured point clouds, while strain and stress operators are analytically derived from the displacement operator, ensuring mechanical consistency and reducing model parameters. Benchmark tests in static linear elasticity, quasi-static hyperelasticity, and elastoplastic dynamics demonstrate the method’s superior geometric adaptability, parameter efficiency, and convergence stability.

This study investigates the feasibility and geometric dependence of inferring the relative magnitudes of tensile, bending, and bearing loads from stress intensity factor (SIF) distributions along a crack front. Leveraging finite element data from SIFBench, the authors propose a unified crack-front operator that couples a structured forward surrogate model with a differentiable inverse mapping, yielding a set-valued estimator augmented with calibrated uncertainty quantification. Theoretical analysis reveals that load identifiability hinges on the functional linear independence of three canonical load profiles, and introduces an intrinsic stability margin to quantify the ill-posedness of the inverse problem. Experimental validation demonstrates that most corner-crack configurations are well-posed, yielding reliable point estimates, whereas a few inherently ill-conditioned cases produce uninformative estimates—consistent with theoretical predictions and numerical observations.