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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

With the advantages of fast calculating speed and high precision, the physics-informed neural network method opens up a new approach for numerically solving nonlinear partial differential equations. Based on conserved quantities, we devise…

Exactly Solvable and Integrable Systems · Physics 2022-03-14 Shuning Lin , Yong Chen

In recent years, with the advancements in machine learning and neural networks, algorithms using physics-informed neural networks (PINNs) to solve PDEs have gained widespread applications. While these algorithms are well-suited for a wide…

Numerical Analysis · Mathematics 2026-01-13 Rongxin Lu , Jiwei Jia , Young Ju Lee , Zheng Lu , Chen-Song Zhang

Physics-Informed Neural Networks (PINN) are algorithms from deep learning leveraging physical laws by including partial differential equations together with a respective set of boundary and initial conditions as penalty terms into their…

Machine Learning · Computer Science 2025-09-30 Rafael Bischof , Michael Kraus

Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their…

Soft- and hard-constrained Physics Informed Neural Networks (PINNs) have achieved great success in solving partial differential equations (PDEs). However, these methods still face great challenges when solving the Navier-Stokes equations…

Fluid Dynamics · Physics 2024-11-14 Chuyu Zhou , Tianyu Li , Chenxi Lan , Rongyu Du , Guoguo Xin , Pengyu Nan , Hangzhou Yang , Guoqing Wang , Xun Liu , Wei Li

Inverse problems arise almost everywhere in science and engineering where we need to infer on a quantity from indirect observation. The cases of medical, biomedical, and industrial imaging systems are the typical examples. A very high…

Machine Learning · Computer Science 2025-02-20 Ali Mohammad-Djafari

Physics-informed neural networks (PINNs) provide a promising machine learning framework for solving partial differential equations, but their training often breaks down on challenging problems, sometimes converging to physically incorrect…

Machine Learning · Computer Science 2026-04-28 Sifan Wang , Shawn Koohy , Yiping Lu , Paris Perdikaris

The computation of the seismic wavefield by solving the Helmholtz equation is crucial to many practical applications, e.g., full waveform inversion. Physics-informed neural networks (PINNs) provide functional wavefield solutions represented…

Geophysics · Physics 2024-08-29 Xinquan Huang , Tariq Alkhalifah

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for…

We apply the Physics Informed Neural Network (PINN) to the problem of wildfire fire-front modelling. We use the PINN to solve the level-set equation, which is a partial differential equation that models a fire-front through the…

Physics-informed neural networks (PINNs) have recently emerged as promising data-driven PDE solvers showing encouraging results on various PDEs. However, there is a fundamental limitation of training PINNs to solve multi-dimensional PDEs…

Machine Learning · Computer Science 2023-11-01 Junwoo Cho , Seungtae Nam , Hyunmo Yang , Seok-Bae Yun , Youngjoon Hong , Eunbyung Park

We propose a new semi-analytic physics informed neural network (PINN) to solve singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that offers a promising perspective for finding numerical…

Numerical Analysis · Mathematics 2022-08-22 Gung-Min Gie , Youngjoon Hong , Chang-Yeol Jung

We present the hidden-layer concatenated physics informed neural network (HLConcPINN) method, which combines hidden-layer concatenated feed-forward neural networks, a modified block time marching strategy, and a physics informed approach…

Numerical Analysis · Mathematics 2024-06-11 Yianxia Qian , Yongchao Zhang , Suchuan Dong

We present a Physics-Informed Neural Network (PINN) to simulate the thermochemical evolution of a composite material on a tool undergoing cure in an autoclave. In particular, we solve the governing coupled system of differential equations…

Machine Learning · Computer Science 2021-06-16 Sina Amini Niaki , Ehsan Haghighat , Trevor Campbell , Anoush Poursartip , Reza Vaziri

Physics Informed Neural Networks (PINNs) is a promising application of deep learning. The smooth architecture of a fully connected neural network is appropriate for finding the solutions of PDEs; the corresponding loss function can also be…

Numerical Analysis · Mathematics 2021-12-09 Hwijae Son , Jin Woo Jang , Woo Jin Han , Hyung Ju Hwang

Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs) by directly embedding them into the loss function. Despite their notable success, existing PINNs often exhibit training…

Computational Engineering, Finance, and Science · Computer Science 2026-03-26 Chang Wei , Yuchen Fan , Chin Chun Ooi , Jian Cheng Wong , Heyang Wang , Pao-Hsiung Chiu

Physics-Informed Neural Networks (PINNs) provide a learning-based framework for solving partial differential equations (PDEs) by embedding governing physical laws into neural network training. In practice, however, their performance is…

Machine Learning · Computer Science 2026-01-21 Pancheng Niu , Jun Guo , Qiaolin He , Yongming Chen , Yanchao Shi

This work aims to provide an effective deep learning framework to predict the vector-soliton solutions of the coupled nonlinear equations and their interactions. The method we propose here is a physics-informed neural network (PINN)…

Numerical Analysis · Mathematics 2022-11-23 Shu-Mei Qin , Min Li , Tao Xu , Shao-Qun Dong

Accurate solutions to inverse supersonic compressible flow problems are often required for designing specialized aerospace vehicles. In particular, we consider the problem where we have data available for density gradients from Schlieren…

Numerical Analysis · Mathematics 2022-07-27 Ameya D. Jagtap , Zhiping Mao , Nikolaus Adams , George Em Karniadakis