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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 effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly…

Computational Physics · Physics 2021-11-03 Guofei Pang , Lu Lu , George Em Karniadakis

The physics-informed neural network (PINN) is effective in solving the partial differential equation (PDE) by capturing the physics constraints as a part of the training loss function through the Automatic Differentiation (AD). This study…

Computational Physics · Physics 2022-02-17 Zixue Xiang , Wei Peng , Weien Zhou , Wen Yao

Physics-informed neural networks (PINNs) are an emerging technique to solve partial differential equations (PDEs). In this work, we propose a simple but effective PINN approach for the phase-field model of ferroelectric microstructure…

Materials Science · Physics 2024-09-06 Lan Shang , Sizheng Zheng , Jin Wang , Jie Wang

Partial differential equations (PDEs) provide a mathematical foundation for simulating and understanding intricate behaviors in both physical sciences and engineering. With the growing capabilities of deep learning, data$-$driven approaches…

Machine Learning · Computer Science 2025-10-14 Narayan S Iyer , Bivas Bhaumik , Ram S Iyer , Satyasaran Changdar

Physics-Informed Neural Networks (PINNs) are a novel computational approach for solving partial differential equations (PDEs) with noisy and sparse initial and boundary data. Although, efficient quantification of epistemic and aleatoric…

Machine Learning · Computer Science 2025-05-02 Júlia Vicens Figueres , Juliette Vanderhaeghen , Federica Bragone , Kateryna Morozovska , Khemraj Shukla

Physics-informed neural networks (PINNs) have recently emerged as a prominent paradigm for solving partial differential equations (PDEs), yet their training strategies remain underexplored. While hard prioritization methods inspired by…

Machine Learning · Computer Science 2025-12-22 Zhaoqian Gao , Min Yanga

Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional…

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

Physics-informed neural networks (PINNs) have recently emerged as a promising way to compute the solutions of partial differential equations (PDEs) using deep neural networks. However, despite their significant success in various fields, it…

Numerical Analysis · Mathematics 2024-07-15 Seungchan Ko , Sang Hyeon Park

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their…

Machine Learning · Computer Science 2026-03-17 Aleksander Krasowski , René P. Klausen , Aycan Celik , Sebastian Lapuschkin , Wojciech Samek , Jonas Naujoks

Physics-Informed Neural Network (PINN) is a novel multi-task learning framework useful for solving physical problems modeled using differential equations (DEs) by integrating the knowledge of physics and known constraints into the…

Machine Learning · Computer Science 2024-09-18 Shivprasad Kathane , Shyamprasad Karagadde

Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for solving partial differential equations (PDEs). However, training PINNs from scratch is often computationally intensive and time-consuming. To address this problem,…

Numerical Analysis · Mathematics 2024-10-21 Sidi Wu

In the realm of computational fluid dynamics, traditional numerical methods, which heavily rely on discretization, typically necessitate the formulation of partial differential equations (PDEs) in conservative form to accurately capture…

A method for solving elasticity problems based on separable physics-informed neural networks (SPINN) in conjunction with the deep energy method (DEM) is presented. Numerical experiments have been carried out for a number of problems showing…

Numerical Analysis · Mathematics 2024-01-25 Vasiliy A. Es'kin , Danil V. Davydov , Julia V. Gur'eva , Alexey O. Malkhanov , Mikhail E. Smorkalov

In this work, a physics-informed neural networks (PINNs) based algorithm is used for simulation of nonlinear 1D and 2D Burgers' type models. This scheme relies on a neural network built to approximate the problem solution and use a trial…

Machine Learning · Computer Science 2025-06-17 Ajeet Singh , Ram Jiwari , Vikram , Ujjwal Saini

Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to…

Numerical Analysis · Mathematics 2023-05-23 Victorita Dolean , Alexander Heinlein , Siddhartha Mishra , Ben Moseley

We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm and Benjamin-Ono equations. The stability of solutions…

Numerical Analysis · Mathematics 2022-05-19 Genming Bai , Ujjwal Koley , Siddhartha Mishra , Roberto Molinaro

Physics-informed neural networks (PINNs) were recently proposed in [1] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution while a PDE-induced NN is coupled to the solution NN,…

Computational Physics · Physics 2019-10-22 Xiaoli Chen , Jinqiao Duan , George Em Karniadakis

Deep neural networks (DNNs), especially physics-informed neural networks (PINNs), have recently become a new popular method for solving forward and inverse problems governed by partial differential equations (PDEs). However, these methods…

Machine Learning · Computer Science 2023-10-26 Wenbo Cao , Weiwei Zhang