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Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical…

Numerical Analysis · Mathematics 2022-03-17 Vidhi Zala , Robert M. Kirby , Akil Narayan

Piecewise affine neural networks (PANNs) provide a principled geometric perspective on neural network expressivity by characterizing the input--output map as a continuous piecewise affine (CPA) function whose complexity is governed by the…

Machine Learning · Computer Science 2026-05-13 Yi Wei , Xuan Qi , Furao Shen , Jian Zhao , Vittorio Murino , Cigdem Beyan

We introduce a comprehensive data-driven framework aimed at enhancing the modeling of physical systems, employing inference techniques and machine learning enhancements. As a demonstrative application, we pursue the modeling of cathodic…

Computational Physics · Physics 2025-02-04 Christian Jacobsen , Jiayuan Dong , Mehdi Khalloufi , Xun Huan , Karthik Duraisamy , Maryam Akram , Wanjiao Liu

We present polynomial-augmented neural networks (PANNs), a novel machine learning architecture that combines deep neural networks (DNNs) with a polynomial approximant. PANNs combine the strengths of DNNs (flexibility and efficiency in…

Machine Learning · Computer Science 2025-02-25 Madison Cooley , Shandian Zhe , Robert M. Kirby , Varun Shankar

The modeling of large deformation fracture mechanics has been a challenging problem regarding the accuracy of numerical methods and their ability to deal with considerable changes in deformations of meshes where having the presence of…

Numerical Analysis · Computer Science 2019-03-21 Hai D. Huynh , Phuong Tran , Xiaoying Zhuang , H. Nguyen-Xuan

This paper presents a comprehensive computational framework for investigating thermo-elastic fracture in transversely isotropic materials, where classical linear elasticity fails to predict physically realistic behavior near stress…

Numerical Analysis · Mathematics 2025-10-08 Saugata Ghosh , Dambaru Bhatta , S. M. Mallikarjunaiah

Graph neural networks (GNNs) naturally align with sparse operators and unstructured discretizations, making them a promising paradigm for physics-informed machine learning in computational mechanics. Motivated by discrete physics losses and…

Machine Learning · Computer Science 2026-02-10 Jianchuan Yang , Xi Chen , Jidong Zhao

Physics-informed neural networks (PINNs) have emerged as a promising numerical method based on deep learning for modeling boundary value problems, showcasing promising results in various fields. In this work, we use PINNs to discretize…

Computational Physics · Physics 2024-06-10 Michel Nohra , Steven Dufour

The analysis of parametric and non-parametric uncertainties of very large dynamical systems requires the construction of a stochastic model of said system. Linear approaches relying on random matrix theory and principal componant analysis…

Machine Learning · Statistics 2023-02-02 Hamza Boukraichi , Nissrine Akkari , Fabien Casenave , David Ryckelynck

Numerical modeling errors are unavoidable in finite element analysis. The presence of model errors inherently reflects both model accuracy and uncertainty. To date there have been few methods for explicitly quantifying errors at points of…

Machine Learning · Computer Science 2024-11-19 Bozhou Zhuang , Sashank Rana , Brandon Jones , Danny Smyl

This work presents a physics-informed neural network (PINN) based framework to model the strain-rate and temperature dependence of the deformation fields in elastic-viscoplastic solids. To avoid unbalanced back-propagated gradients during…

Materials Science · Physics 2022-11-24 Rajat Arora , Pratik Kakkar , Biswadip Dey , Amit Chakraborty

The Kohn-Sham equation is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanosciences. In this paper, we study the adaptive finite element…

Numerical Analysis · Mathematics 2013-11-22 Huajie Chen , Xiaoying Dai , Xingao Gong , Lianhua He , Aihui Zhou

As inelastic structures are ubiquitous in many engineering fields, a central task in computational mechanics is to develop accurate, robust and efficient tools for their analysis. Motivated by the poor performances exhibited by standard…

Numerical Analysis · Mathematics 2018-09-21 Nicola A. Nodargi

Machine learning techniques are increasingly used to predict material behavior in scientific applications and offer a significant advantage over conventional numerical methods. In this work, an Artificial Neural Network (ANN) model is used…

Computational Engineering, Finance, and Science · Computer Science 2022-09-08 Olivier Pantalé , Pierre Tize Mha , Amèvi Tongne

We present an elastic simulator for domains defined as evolving implicit functions, which is efficient, robust, and differentiable with respect to both shape and material. This simulator is motivated by applications in 3D reconstruction: it…

Graphics · Computer Science 2025-04-09 Gilles Daviet , Tianchang Shen , Nicholas Sharp , David I. W. Levin

FeFETs hold strong potential for advancing memory and logic technologies, but their inherent randomness arising from both operational cycling and fabrication variability poses significant challenges for accurate and reliable modeling.…

Machine Learning · Computer Science 2025-08-06 Tasnia Nobi Afee , Jack Hutchins , Md Mazharul Islam , Thomas Kampfe , Ahmedullah Aziz

Emulator embedded neural networks, which are a type of physics informed neural network, leverage multi-fidelity data sources for efficient design exploration of aerospace engineering systems. Multiple realizations of the neural network…

Machine Learning · Computer Science 2023-09-14 Atticus Beachy , Harok Bae , Jose Camberos , Ramana Grandhi

Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering…

Computational Engineering, Finance, and Science · Computer Science 2022-10-05 Shahed Rezaei , Ali Harandi , Ahmad Moeineddin , Bai-Xiang Xu , Stefanie Reese

Fiber-reinforced soft biological tissues are typically modeled as hyperelastic, anisotropic, and nearly incompressible materials. To enforce incompressibility a multiplicative split of the deformation gradient into a volumetric and an…

Numerical Analysis · Mathematics 2022-04-13 Elias Karabelas , Matthias A. F. Gsell , Gundolf Haase , Gernot Plank , Christoph M. Augustin

In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a…

Numerical Analysis · Mathematics 2020-08-21 Tingting Hao , Manman Ma , Xuejun Xu