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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…

Numerical Analysis · Mathematics 2026-02-10 Yizheng Wang , Cosmin Anitescu , Mohammad Sadegh Eshaghi , Xiaoying Zhuang , Timon Rabczuk , Yinghua Liu

Deep learning and the collocation method are merged and used to solve partial differential equations describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo-Hookean) with…

Machine Learning · Computer Science 2021-11-24 Diab W. Abueidda , Qiyue Lu , Seid Koric

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…

Scientific computing for large deformation of elastic-plastic solids is critical for numerous real-world applications. Classical numerical solvers rely primarily on local discrete linear approximation and are constrained by an inherent…

Machine Learning · Computer Science 2025-06-09 Shilong Tao , Zhe Feng , Haonan Sun , Zhanxing Zhu , Yunhuai Liu

One of the obstacles hindering the scaling-up of the initial successes of machine learning in practical engineering applications is the dependence of the accuracy on the size of the database that "drives" the algorithms. Incorporating the…

Computational Engineering, Finance, and Science · Computer Science 2021-06-09 Wei Li , Martin Z. Bazant , Juner Zhu

Understanding the training dynamics of deep learning models is perhaps a necessary step toward demystifying the effectiveness of these models. In particular, how do data from different classes gradually become separable in their feature…

Machine Learning · Computer Science 2021-10-13 Jiayao Zhang , Hua Wang , Weijie J. Su

The finite element method (FEM) is among the most commonly used numerical methods for solving engineering problems. Due to its computational cost, various ideas have been introduced to reduce computation times, such as domain decomposition,…

Computational Engineering, Finance, and Science · Computer Science 2019-11-07 Andrea Mendizabal , Pablo Márquez-Neila , Stéphane Cotin

This work presents a physics-informed deep learning-based super-resolution framework to enhance the spatio-temporal resolution of the solution of time-dependent partial differential equations (PDE). Prior works on deep learning-based…

Machine Learning · Computer Science 2022-12-09 Rajat Arora , Ankit Shrivastava

Derivation of the probability density evolution provides invaluable insight into the behavior of many stochastic systems and their performance. However, for most real-time applica-tions, numerical determination of the probability density…

Machine Learning · Computer Science 2022-07-06 Seid H. Pourtakdoust , Amir H. Khodabakhsh

In practical structural design and solid mechanics simulations, material properties inherently exhibit random variations within bounded intervals. However, evaluating mechanical responses under continuous material uncertainty remains a…

Machine Learning · Computer Science 2026-03-30 Zhangyong Liang , Huanhuan Gao

We use a space-time discretization based on physics informed deep learning (PIDL) to approximate solutions of a class of rate-dependent strain gradient plasticity models. The differential equation governing the plastic flow, the so-called…

Dynamical Systems · Mathematics 2024-08-14 Ankit Tyagi , Uttam Suman , Mariya Mamajiwala , Debasish Roy

The scientific computation of large deformations in elastic-plastic solids is crucial in various manufacturing applications. Traditional numerical methods exhibit several inherent limitations, prompting Deep Learning (DL) as a promising…

Artificial Intelligence · Computer Science 2026-01-16 Jianheng Tang , Shilong Tao , Zhe Feng , Haonan Sun , Menglu Wang , Zhanxing Zhu , Yunhuai Liu

In the present work, a machine learning based constitutive model for electro-mechanically coupled material behavior at finite deformations is proposed. Using different sets of invariants as inputs, an internal energy density is formulated…

Computational Engineering, Finance, and Science · Computer Science 2022-08-30 Dominik K. Klein , Rogelio Ortigosa , Jesús Martínez-Frutos , Oliver Weeger

We propose a model for rate-independent evolution in elastoplastic materials under external loading, which allows large strains. In the setting of strain-gradient plasticity with multiplicative decomposition of the deformation gradient, we…

Analysis of PDEs · Mathematics 2021-09-01 Martin Kružík , Jiří Zeman

Accurate simulations are essential for engineering applications, and intricate continuum mechanical material models are constructed to achieve this goal. However, the increasing complexity of the material models and geometrical properties…

Computational Engineering, Finance, and Science · Computer Science 2023-11-30 Steffen Kastian , Jannick Kehls , Tim Brepols , Stefanie Reese

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

This work rigorously implements a recent model of large-strain elasto-plastic evolution in single crystals where the plastic flow is driven by the movement of discrete dislocation lines. The model is geometrically and elastically nonlinear,…

Analysis of PDEs · Mathematics 2024-02-27 Filip Rindler

This work presents a novel physics-informed deep learning based super-resolution framework to reconstruct high-resolution deformation fields from low-resolution counterparts, obtained from coarse mesh simulations or experiments. We leverage…

Machine Learning · Computer Science 2022-11-24 Rajat Arora

Finite element modeling is a well-established tool for structural analysis, yet modeling complex structures often requires extensive pre-processing, significant analysis effort, and considerable time. This study addresses this challenge by…

We explore the possibilities of using energy minimization for the numerical modeling of strain localization in solids as a sharp discontinuity in the displacement field. For this purpose, we consider (regularized) strong discontinuity…

Machine Learning · Computer Science 2025-01-14 Omar León , Víctor Rivera , Angel Vázquez-Patiño , Jacinto Ulloa , Esteban Samaniego