arXiv subjects

Timon Rabczuk

Publications and source records attributed to Timon Rabczuk.

At least 37 records · Page 2Linked to original sources

arXiv2026

Artificial intelligence for partial differential equations in computational mechanics: A review

In recent years, Artificial intelligence (AI) has become ubiquitous, empowering various fields, especially integrating artificial intelligence and traditional science (AI for Science: Artificial intelligence for science), which has attracted widespread attention. In AI for Science, using artificial intelligence algorithms to solve partial differential equations (AI for PDEs: Artificial intelligence for partial differential equations) has become a focal point in computational mechanics. The core of AI for PDEs is the fusion of data and partial differential equations (PDEs), which can solve almost any PDEs. Therefore, this article provides a comprehensive review of the research on AI for PDEs, summarizing the existing algorithms and theories. The article discusses the applications of AI for PDEs in computational mechanics, including solid mechanics, fluid mechanics, and biomechanics. The existing AI for PDEs algorithms include those based on Physics-Informed Neural Networks (PINNs), Deep Energy Methods (DEM), Operator Learning, and Physics-Informed Neural Operator (PINO). AI for PDEs represents a new method of scientific simulation that provides approximate solutions to specific problems using large amounts of data, then fine-tuning according to specific physical laws, avoiding the need to compute from scratch like traditional algorithms. Thus, AI for PDEs is the prototype for future foundation models in computational mechanics, capable of significantly accelerating traditional numerical algorithms.

eess.SY↗

arXiv2026

Towards Unified AI-Driven Fracture Mechanics: The Extended Deep Energy Method (XDEM)

Physics-Informed Neural Networks (PINNs) have recently emerged as powerful tools for solving partial differential equations (PDEs), with the Deep Energy Method (DEM) proving especially effective in fracture mechanics due to its energy-based formulation. Despite these advances, existing DEM approaches require dense collocation near cracks, face stability challenges, and typically treat discrete and continuous fracture models separately. To overcome these limitations, we introduce the Extended Deep Energy Method (XDEM), a unified deep learning framework that incorporates both displacement discontinuities and crack-tip asymptotics in the discrete setting, while flexibly coupling displacement and phase fields in the continuous setting. This integration enables accurate fracture predictions using uniformly distributed, relatively sparse collocation points. Validation across benchmark problems including stress intensity factor evaluation, straight and kinked crack growth, and complex crack initiation demonstrates that XDEM consistently outperforms standard DEM in accuracy and efficiency. By bridging discrete and phase-field models within a single framework, XDEM establishes a robust foundation for applying AI to fracture mechanics and opens new avenues for predictive modeling in engineering and materials science.

physics.comp-ph↗

arXiv2026

Pretrain Finite Element Method: A Pretraining and Warm-start Framework for PDEs via Physics-Informed Neural Operators

We propose a Pretrained Finite Element Method (PFEM),a physics driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM). PFEM consists of a physics informed pretraining stage and an optional finetuning stage. In the pretraining stage, a neural operator based on the Transolver architecture is trained solely from governing partial differential equations, without relying on labeled solution data. The model operates directly on unstructured point clouds, jointly encoding geometric information, material properties, and boundary conditions, and produces physically consistent initial solutions with extremely high computational efficiency. PDE constraints are enforced through explicit finite element, based differentiation, avoiding the overhead associated with automatic differentiation. In the fine-tuning stage, the pretrained prediction is used as an initial guess for conventional FEM solvers, preserving their accuracy, convergence guarantees, and extrapolation capability while substantially reducing the number of iterations required to reach a prescribed tolerance. PFEM is validated on a broad range of benchmark problems, including linear elasticity and nonlinear hyperelasticity with complex geometries, heterogeneous materials, and arbitrary boundary conditions. Numerical results demonstrate strong generalization in the pretraining stage with relative errors on the order of 1\%, and speedups of up to one order of magnitude in the fine-tuning stage compared to FEM with zero initial guesses.

math.NA↗

arXiv2026

A Computational Model for Flexoelectricity-Driven Contact Electrification

Recent theoretical studies show that nanoscale contact on dielectric substrates can induce flexoelectric polarization large enough to drive electron transfer. This has been supported by experimental evidence, indicating that contact electrification is inherently a coupled electromechanical phenomenon. In this work, we develop a computational model for flexoelectricity-driven contact electrification that integrates finite-deformation flexoelectricity with contact mechanics and physically motivated charge transfer. A tunneling transparency function is introduced to regulate the interfacial channel based on the WKB approximation, capturing the irreversible charge trapping during unloading. Three contact scenarios are investigated with specific hypotheses for charge transfer: unbiased metal-dielectric contact driven by surface polarization, biased contact restricted to carriers of a single polarity, and dielectric-dielectric contact where surface states with finite capacity limit the transferable charge. The model is compared with atomic force microscopy measurements on PMMA and PDAP substrates under both biased and unbiased conditions.For contact between identical dielectric materials, we show that geometric asymmetry in surface curvature is sufficient to induce charge separation, with polarity reversal occurring at a critical surface wavenumber. Three-dimensional simulations on random rough surfaces reproduce the mosaic charge distributions observed experimentally, confirming that contact-induced local strain gradient heterogeneity can generate spatially non-uniform charge patterns without introducing any material inhomogeneity.

physics.app-ph↗

arXiv2026

A phase-field framework for anisotropic viscoelastic-viscoplastic fracture in short fiber-reinforced polymers in hygrothermal environments

This work presents a comprehensive phase-field framework for modeling anisotropic viscoelastic-viscoplastic fracture in short fiber-reinforced polymer (SFRP) composites under hygrothermal environments at finite deformation. The constitutive model employs a multiplicative decomposition of the deformation gradient into viscoelastic and viscoplastic components. An anisotropic phase-field formulation is developed using structural tensors to capture orientation-dependent fracture energy induced by multiple fiber families. Hygrothermal effects are incorporated through moisture-dependent swelling, thermal expansion, and temperature- and moisture-sensitive material parameters within the coupled framework. Numerical investigations demonstrate the framework's capability to capture complex fracture phenomena in SFRPs. Results reveal that fiber orientation fundamentally governs the spatial distribution of crack driving force, with maximum energy accumulation along fiber directions persisting throughout viscous relaxation. The anisotropy parameter controlling directional fracture resistance significantly influences crack path deflection. Hygrothermal degradation substantially reduces both peak load and fracture energy, with moisture absorption and elevated temperature each contributing to decreased mechanical performance. The framework captures the influence of fiber mechanical properties on global load-bearing capacity and crack propagation resistance. This unified computational framework advances the predictive modeling of damage evolution in SFRPs subjected to realistic environmental and mechanical loading conditions.

cs.CE↗

arXiv2026

PENCO: A Physics-Energy-Numerics-Consistent Operator for 3D Phase Field Modeling

Accurate and efficient solutions of spatiotemporal partial differential equations (PDEs), such as phase-field models, are fundamental for understanding interfacial dynamics and microstructural evolution in materials science and fluid mechanics. Neural operators (NOs) have recently emerged as powerful data-driven alternatives to traditional solvers; however, existing architectures often accumulate temporal errors, struggle to generalize over long temporal horizons, and require large training datasets. To overcome these limitations, we propose PENCO (Physics-Energy-Numerics-Consistent Operator), a hybrid operator-learning framework that integrates physical laws with data-driven neural operator methods, using either the Fourier Neural Operator (FNO-4D) or the Multi-Head Neural Operator (MHNO) architecture as the backbone. The formulation introduces an enhanced L^2 Gauss-Lobatto collocation residual around the temporal midpoint that robustly enforces the governing dynamics and significantly improves accuracy, a Fourier-space numerical consistency term that captures the balanced behavior of semi-implicit discretizations, and an energy-dissipation constraint that ensures thermodynamic consistency. Additional low-frequency spectral anchoring and teacher-consistency mechanisms further stabilize learning and suppress long-term error growth. This hybrid design enables PENCO to preserve governing physics while mitigating long-term error growth. Through extensive three-dimensional phase-field benchmarks on periodic cubic domains, covering phase ordering, crystallization, epitaxial growth, and complex pattern formation, PENCO demonstrates superior accuracy, stability, and data efficiency compared to state-of-the-art neural operators, including FNO-4D and MHNO, while maintaining physically consistent evolution. The associated dataset and implementation are available at github.com/MBamdad/PENCO.

physics.comp-ph↗

arXiv2026

Deep Energy Method with Large Language Model assistance: an open-source Streamlit-based platform for solving variational PDEs

Physics-informed neural networks (PINNs) in energy form, also known as the deep energy method (DEM), offer advantages over strong-form PINNs such as lower-order derivatives and fewer hyperparameters, yet dedicated and user-friendly software for energy-form PINNs remains scarce. To address this gap, we present \textbf{LM-DEM} (Large-Model-assisted Deep Energy Method), an open-source, Streamlit-based platform for solving variational partial differential equations (PDEs) in computational mechanics. LM-DEM integrates large language models (LLMs) for geometry modeling: users can generate Gmsh-compatible geometries directly from natural language descriptions or images, significantly reducing the burden of traditional geometry preprocessing. The solution process is driven by the deep energy method, while finite element solutions can be obtained in parallel. The framework supports built-in problems including Poisson, screened Poisson, linear elasticity, and hyperelasticity in two and three dimensions, as well as user-defined energy functionals analogous to the \texttt{UMAT} interface in Abaqus. The source code is available at https://github.com/yizheng-wang/LMDEM, and a web-based version is accessible at https://ai4m.llmdem.com. LM-DEM aims to lower the barrier for practitioners and beginners to adopt energy-form PINNs for variational PDE problems.

math.NA↗

arXiv2026

Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging

This paper proposes a Physics-Informed Kolmogorov-Arnold Network for analyzing elasticity problems in multi-material electronic packaging structures. The method replaces traditional Multi-Layer Perceptrons with Kolmogorov-Arnold Networks within an energy-based Physics-Informed Neural Network framework. By constructing admissible displacement fields satisfying essential boundary conditions and optimizing network parameters through numerical integration, the proposed method effectively handles material property discontinuities. Unlike traditional methods that require domain decomposition and interface constraints for multi-material problems, Kolmogorov-Arnold Networks' trainable B-spline activation functions provide inherent piecewise characteristics. This capability stems from B-splines' local support, which enables effective approximation of discontinuities despite their individual smoothness. Consequently, this approach enables accurate approximation across the entire domain using a single network and simplifying the computational framework. Numerical experiments demonstrate that the proposed method achieves excellent accuracy and robustness in multi-material elasticity problems, validating its practical potential for electronic packaging analysis. Source codes are available at https://github.com/yanpeng-gong/PIKAN-MultiMaterial.

math.NA↗

arXiv2025

Phase-field modeling of multicomponent vesicles in viscoelastic fluid

Multicomponent vesicles suspended in viscoelastic fluids are crucial for understanding a variety of physiological processes. In this work, we develop a continuum surface force (CSF) phase-field model to investigate the hydrodynamics of inextensible multicomponent vesicles in viscoelastic fluid flows with inertial forces. Our model couples a fluid field comprising both Newtonian and Oldroyd-B fluids, a surface concentration field representing the multicomponent distribution on the vesicle membrane, and a phase-field variable governing the membrane evolution. The viscoelasticity effect of extra stress is well incorporated into the full Navier-Stokes equations in the fluid field. The surface concentration field is determined by Cahn-Hilliard equations, while the membrane evolution is governed by a nonlinear advection-diffusion equation. The membrane is coupled to the surrounding fluid through the continuum surface force (CSF) framework. To ensure stable numerical solutions of the highly nonlinear multi-field model, we employ a residual-based variational multiscale (RBVMS) method for the Navier-Stokes equations, a Streamline-Upwind Petrov-Galerkin (SUPG) method for the Oldroyd-B equations, and a standard Galerkin finite element framework for the remaining equations. The system of PDEs is solved using an implicit, monolithic scheme based on the generalized-$α$ time integration method. To enhance spatial accuracy, we employ isogeometric analysis (IGA). We present a series of two-dimensional numerical examples in shear and Poiseuille flows to elucidate the influence of membrane composition and fluid viscoelasticity on the hydrodynamics of multicomponent vesicles.

physics.flu-dyn↗

arXiv2025

A coupled finite element-virtual element method for thermomechanical analysis of electronic packaging structures

This study presents a finite element and virtual element (FE-VE) coupled method for thermomechanical analysis in electronic packaging structures. The approach partitions computational domains strategically, employing FEM for regular geometries to maximize computational efficiency and VEM for complex shapes to enhance geometric flexibility. Interface compatibility is maintained through coincident nodal correspondence, ensuring solution continuity across domain boundaries while reducing meshing complexity and computational overhead. Validation through electronic packaging applications demonstrates reasonable agreement with reference solutions and acceptable convergence characteristics across varying mesh densities. The method effectively captures thermal distributions and stress concentrations in multi-material systems, establishing a practical computational framework for electronic packaging analysis involving complex geometries. Source codes are available at https://github.com/yanpeng-gong/FeVeCoupled-ElectronicPackaging.

math.NA↗

arXiv2025

Physics-informed Machine Learning for Static Friction Modeling in Robotic Manipulators Based on Kolmogorov-Arnold Networks

Friction modeling plays a crucial role in achieving high-precision motion control in robotic operating systems. Traditional static friction models (such as the Stribeck model) are widely used due to their simple forms; however, they typically require predefined functional assumptions, which poses significant challenges when dealing with unknown functional structures. To address this issue, this paper proposes a physics-inspired machine learning approach based on the Kolmogorov Arnold Network (KAN) for static friction modeling of robotic joints. The method integrates spline activation functions with a symbolic regression mechanism, enabling model simplification and physical expression extraction through pruning and attribute scoring, while maintaining both high prediction accuracy and interpretability. We first validate the method's capability to accurately identify key parameters under known functional models, and further demonstrate its robustness and generalization ability under conditions with unknown functional structures and noisy data. Experiments conducted on both synthetic data and real friction data collected from a six-degree-of-freedom industrial manipulator show that the proposed method achieves a coefficient of determination greater than 0.95 across various tasks and successfully extracts concise and physically meaningful friction expressions. This study provides a new perspective for interpretable and data-driven robotic friction modeling with promising engineering applicability.

cs.RO↗

arXiv2025

Multi-Head Neural Operator for Modelling Interfacial Dynamics

Interfacial dynamics underlie a wide range of phenomena, including phase transitions, microstructure coarsening, pattern formation, and thin-film growth, and are typically described by stiff, time-dependent nonlinear partial differential equations (PDEs). Traditional numerical methods, including finite difference, finite element, and spectral techniques, often become computationally prohibitive when dealing with high-dimensional problems or systems with multiple scales. Neural operators (NOs), a class of deep learning models, have emerged as a promising alternative by learning mappings between function spaces and efficiently approximating solution operators. In this work, we introduce the Multi-Head Neural Operator (MHNO), an extended neural operator framework specifically designed to address the temporal challenges associated with solving time-dependent PDEs. Unlike existing neural operators, which either struggle with error accumulation or require substantial computational resources for high-dimensional tensor representations, MHNO employs a novel architecture with time-step-specific projection operators and explicit temporal connections inspired by message-passing mechanisms. This design allows MHNO to predict all time steps after a single forward pass, while effectively capturing long-term dependencies and avoiding parameter overgrowth. We apply MHNO to solve various phase field equations, including antiphase boundary motion, spinodal decomposition, pattern formation, atomic scale modeling, and molecular beam epitaxy growth model, and compare its performance with existing NO-based methods. Our results show that MHNO achieves superior accuracy, scalability, and efficiency, demonstrating its potential as a next-generation computational tool for phase field modeling. The code and data supporting this work is publicly available at https://github.com/eshaghi-ms/MHNO.

physics.comp-ph↗

arXiv2025

A Pretraining-Finetuning Computational Framework for Material Homogenization

Homogenization is a fundamental tool for studying multiscale physical phenomena. Traditional numerical homogenization methods, heavily reliant on finite element analysis, demand significant computational resources, especially for complex geometries, materials, and high-resolution problems. To address these challenges, we propose PreFine-Homo, a novel numerical homogenization framework comprising two phases: pretraining and fine-tuning. In the pretraining phase, a Fourier Neural Operator (FNO) is trained on large datasets to learn the mapping from input geometries and material properties to displacement fields. In the fine-tuning phase, the pretrained predictions serve as initial solutions for iterative algorithms, drastically reducing the number of iterations needed for convergence. The pretraining phase of PreFine-Homo delivers homogenization results up to 1000 times faster than conventional methods, while the fine-tuning phase further enhances accuracy. Moreover, the fine-tuning phase grants PreFine-Homo unlimited generalization capabilities, enabling continuous learning and improvement as data availability increases. We validate PreFine-Homo by predicting the effective elastic tensor for 3D periodic materials, specifically Triply Periodic Minimal Surfaces (TPMS). The results demonstrate that PreFine-Homo achieves high precision, exceptional efficiency, robust learning capabilities, and strong extrapolation ability, establishing it as a powerful tool for multiscale homogenization tasks.

cs.CE↗

arXiv2025

Transfer Learning in Physics-Informed Neural Networks: Full Fine-Tuning, Lightweight Fine-Tuning, and Low-Rank Adaptation

AI for PDEs has garnered significant attention, particularly Physics-Informed Neural Networks (PINNs). However, PINNs are typically limited to solving specific problems, and any changes in problem conditions necessitate retraining. Therefore, we explore the generalization capability of transfer learning in the strong and energy form of PINNs across different boundary conditions, materials, and geometries. The transfer learning methods we employ include full finetuning, lightweight finetuning, and Low-Rank Adaptation (LoRA). The results demonstrate that full finetuning and LoRA can significantly improve convergence speed while providing a slight enhancement in accuracy.

cs.LG↗

arXiv2025

Energy-based physics-informed neural network for frictionless contact problems under large deformation

Numerical methods for contact mechanics are of great importance in engineering applications, enabling the prediction and analysis of complex surface interactions under various conditions. In this work, we propose an energy-based physics-informed neural network (PINNs) framework for solving frictionless contact problems under large deformation. Inspired by microscopic Lennard-Jones potential, a surface contact energy is used to describe the contact phenomena. To ensure the robustness of the proposed PINN framework, relaxation, gradual loading and output scaling techniques are introduced. In the numerical examples, the well-known Hertz contact benchmark problem is conducted, demonstrating the effectiveness and robustness of the proposed PINNs framework. Moreover, challenging contact problems with the consideration of geometrical and material nonlinearities are tested. It has been shown that the proposed PINNs framework provides a reliable and powerful tool for nonlinear contact mechanics. More importantly, the proposed PINNs framework exhibits competitive computational efficiency to the commercial FEM software when dealing with those complex contact problems. The codes used in this manuscript are available at https://github.com/JinshuaiBai/energy_PINN_Contact.(The code will be available after acceptance)

cs.CE↗

arXiv2024

Applications of Scientific Machine Learning for the Analysis of Functionally Graded Porous Beams

This study investigates different Scientific Machine Learning (SciML) approaches for the analysis of functionally graded (FG) porous beams and compares them under a new framework. The beam material properties are assumed to vary as an arbitrary continuous function. The methods consider the output of a neural network/operator as an approximation to the displacement fields and derive the equations governing beam behavior based on the continuum formulation. The methods are implemented in the framework and formulated by three approaches: (a) the vector approach leads to a Physics-Informed Neural Network (PINN), (b) the energy approach brings about the Deep Energy Method (DEM), and (c) the data-driven approach, which results in a class of Neural Operator methods. Finally, a neural operator has been trained to predict the response of the porous beam with functionally graded material under any porosity distribution pattern and any arbitrary traction condition. The results are validated with analytical and numerical reference solutions. The data and code accompanying this manuscript will be publicly available at https://github.com/eshaghi-ms/DeepNetBeam.

cs.LG↗

arXiv2024

Robust and Feature-Preserving Offset Meshing

We introduce a novel offset meshing approach that can robustly handle a 3D surface mesh with an arbitrary geometry and topology configurations, while nicely capturing the sharp features on the original input for both inward and outward offsets. Compared to the existing approaches focusing on constant-radius offset, to the best of our knowledge, we propose the first-ever solution for mitered offset that can well preserve sharp features. Our method is designed based on several core principals: 1) explicitly generating the offset vertices and triangles with feature-capturing energy and constraints; 2) prioritizing the generation of the offset geometry before establishing its connectivity, 3) employing exact algorithms in critical pipeline steps for robustness, balancing the use of floating-point computations for efficiency, 4) applying various conservative speed up strategies including early reject non-contributing computations to the final output. Our approach further uniquely supports variable offset distances on input surface elements, offering a wider range practical applications compared to conventional methods. We have evaluated our method on a subset of Thinkgi10K, containing models with diverse topological and geometric complexities created by practitioners in various fields. Our results demonstrate the superiority of our approach over current state-of-the-art methods in terms of element count, feature preservation, and non-uniform offset distances of the resulting offset mesh surfaces, marking a significant advancement in the field.

cs.GR↗

arXiv2024

A first-principles study on the physical properties of two-dimensional Nb3Cl8, Nb3Br8 and Nb3I8

In a recent advance, Nb3Cl8 two-dimensional crystals with a kagome lattice and electronic topological flat bands has been experimentally fabricated (Nano Lett. 2022, 22, 4596). In this work motivated by the aforementioned progress, we conduct first-principles calculations to explore the structural, phonon dispersion relations, single-layer exfoliation energies and mechanical features of the Nb3X8 (X=Cl, Br, I) nanosheets. Acquired phonon dispersion relations reveal the dynamical stability of the Nb3X8 (X=Cl, Br, I) monolayers. In order to isolate single-layer crystals from bulk counterparts, we predicted exfoliation energies of 0.24, 0.27 and 0.28 J/m2, for the Nb3Cl8, Nb3Br8 and Nb3I8 monolayers, respectively, which are noticeably lower than that of the graphene. We found that the Nb3X8 monolayers are relatively strong nanosheets with isotropic elasticity and anisotropic tensile strength. It is moreover shown that by increasing the atomic weight of halogen atoms in the Nb3X8 nanosheets, mechanical characteristics decline. Presented results provide a useful vision about the key physical properties of novel 2D systems of Nb3X8 (X=Cl, Br, I).

cond-mat.mes-hall↗