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Partial differential equations (PDEs) are an essential computational kernel in physics and engineering. With the advance of deep learning, physics-informed neural networks (PINNs), as a mesh-free method, have shown great potential for fast…

Machine Learning · Computer Science 2023-06-19 Junjun Yan , Xinhai Chen , Zhichao Wang , Enqiang Zhoui , Jie Liu

In this work, we present the physics-informed neural network (PINN) model applied particularly to dynamic problems in solid mechanics. We focus on forward and inverse problems. Particularly, we show how a PINN model can be used efficiently…

Neural and Evolutionary Computing · Computer Science 2025-12-16 Vijay Kag , Venkatesh Gopinath

The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial…

Machine Learning · Computer Science 2024-03-28 Vikas Dwivedi , Nishant Parashar , Balaji Srinivasan

Combining machine learning with physics is a trending approach for discovering unknown dynamics, and one of the most intensively studied frameworks is the physics-informed neural network (PINN). However, PINN often fails to optimize the…

Machine Learning · Computer Science 2023-11-29 Yuichi Kajiura , Jorge Espin , Dong Zhang

Physics-Informed Neural Networks (PINNs) present a transformative approach for smart grid modeling by integrating physical laws directly into learning frameworks, addressing critical challenges of data scarcity and physical consistency in…

Machine Learning · Computer Science 2025-09-01 Julen Cestero , Carmine Delle Femine , Kenji S. Muro , Marco Quartulli , Marcello Restelli

Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…

Numerical Analysis · Mathematics 2022-08-29 Antonio Tadeu Azevedo Gomes , Larissa Miguez da Silva , Frederic Valentin

The integration of physics-based knowledge with machine learning models is increasingly shaping the monitoring, diagnostics, and prognostics of electrical transformers. In this two-part series, the first paper introduced the foundations of…

Machine Learning · Computer Science 2025-12-30 Jose I. Aizpurua

Accurate modeling of closure terms is a critical challenge in engineering and scientific research, particularly when data is sparse (scarse or incomplete), making widely applicable models difficult to develop. This study proposes a novel…

Machine Learning · Computer Science 2025-05-09 Tian Chen , Shengping Liu , Li Liu , Heng Yong

I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the…

Computational Physics · Physics 2024-03-04 Hubert Baty

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

Recent works have explored the potential of machine learning as data-driven turbulence closures for RANS and LES techniques. Beyond these advances, the high expressivity and agility of physics-informed neural networks (PINNs) make them…

Machine Learning · Computer Science 2021-03-08 Didier Lucor , Atul Agrawal , Anne Sergent

Physics-informed neural networks (PINNs) are an influential method of solving differential equations and estimating their parameters given data. However, since they make use of neural networks, they provide only a point estimate of…

Machine Learning · Computer Science 2024-05-15 Lena Podina , Mahdi Torabi Rad , Mohammad Kohandel

In this paper, we review the new method Physics-Informed Neural Networks (PINNs) that has become the main pillar in scientific machine learning, we present recent practical extensions, and provide a specific example in data-driven discovery…

Machine Learning · Computer Science 2024-09-02 Maziar Raissi , Paris Perdikaris , Nazanin Ahmadi , George Em Karniadakis

The utilization of Deep Neural Networks (DNNs) in physical science and engineering applications has gained traction due to their capacity to learn intricate functions. While large datasets are crucial for training DNN models in fields like…

Machine Learning · Computer Science 2025-08-05 Vamsi Sai Krishna Malineni , Suresh Rajendran

As an emerging technology in deep learning, physics-informed neural networks (PINNs) have been widely used to solve various partial differential equations (PDEs) in engineering. However, PDEs based on practical considerations contain…

Machine Learning · Computer Science 2021-11-11 Yuhao Huang

Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In…

Analysis of PDEs · Mathematics 2019-09-04 Dongkun Zhang , Lu Lu , Ling Guo , George Em Karniadakis

We present pseudo-differential enhanced physics-informed neural networks (PINNs), an extension of gradient enhancement but in Fourier space. Gradient enhancement of PINNs dictates that the PDE residual is taken to a higher differential…

Machine Learning · Computer Science 2026-05-06 Andrew Gracyk

To plan and optimize energy storage demands that account for Li-ion battery aging dynamics, techniques need to be developed to diagnose battery internal states accurately and rapidly. This study seeks to reduce the computational resources…

We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the…

Numerical Analysis · Mathematics 2024-04-02 Simin Shekarpaz , Fanhai Zeng , George Karniadakis

In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…

Optimization and Control · Mathematics 2024-09-06 Michael Hintermüller , Denis Korolev