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Zeitschriftenartikel zum Thema „PINNs“

Autor: Grafiati

Veröffentlicht am 30. Juni 2021

Zuletzt aktualisiert am 29. Juli 2025

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1

Khalid, Salman, Muhammad Haris Yazdani, Muhammad Muzammil Azad, Muhammad Umar Elahi, Izaz Raouf, and Heung Soo Kim. "Advancements in Physics-Informed Neural Networks for Laminated Composites: A Comprehensive Review." Mathematics 13, no. 1 (2024): 17. https://doi.org/10.3390/math13010017.

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Physics-Informed Neural Networks (PINNs) integrate physics principles with machine learning, offering innovative solutions for complex modeling challenges. Laminated composites, characterized by their anisotropic behavior, multi-layered structures, and intricate interlayer interactions, pose significant challenges for traditional computational methods. PINNs address these issues by embedding governing physical laws directly into neural network architectures, enabling efficient and accurate modeling. This review provides a comprehensive overview of PINNs applied to laminated composites, highlig

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Faroughi, Salah A., Ramin Soltanmohammadi, Pingki Datta, Seyed Kourosh Mahjour, and Shirko Faroughi. "Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media." Mathematics 12, no. 1 (2023): 63. http://dx.doi.org/10.3390/math12010063.

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Simulating solute transport in heterogeneous porous media poses computational challenges due to the high-resolution meshing required for traditional solvers. To overcome these challenges, this study explores a mesh-free method based on deep learning to accelerate solute transport simulation. We employ Physics-informed Neural Networks (PiNN) with a periodic activation function to solve solute transport problems in both homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that rely on large training datasets, PiNNs use stron

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Kim, Jaeseung, and Hwijae Son. "Causality-Aware Training of Physics-Informed Neural Networks for Solving Inverse Problems." Mathematics 13, no. 7 (2025): 1057. https://doi.org/10.3390/math13071057.

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Inverse Physics-Informed Neural Networks (inverse PINNs) offer a robust framework for solving inverse problems governed by partial differential equations (PDEs), particularly in scenarios with limited or noisy data. However, conventional inverse PINNs do not explicitly incorporate causality, which hinders their ability to capture the sequential dependencies inherent in physical systems. This study introduces Causal Inverse PINNs (CI-PINNs), a novel framework that integrates directional causality constraints across both temporal and spatial domains. Our approach leverages customized loss functi

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Feng, Zhi-Ying, Xiang-Hua Meng, and Xiao-Ge Xu. "The data-driven localized wave solutions of KdV-type equations via physics-informed neural networks with a priori information." AIMS Mathematics 9, no. 11 (2024): 33263–85. http://dx.doi.org/10.3934/math.20241587.

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<p>In the application of physics-informed neural networks (PINNs) for solutions of partial differential equations, the optimizer may fall into a bad local optimal solution during the training of the network. In this case, the shape of the desired solution may deviate from that of the real solution. To address this problem, we have combined the priori information and knowledge transfer with PINNs. The physics-informed neural networks with a priori information (pr-PINNs) were introduced here, which allow the optimizer to converge to a better solution, improve the training accuracy, and red

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Li, Zhenyu. "A Review of Physics-Informed Neural Networks." Applied and Computational Engineering 133, no. 1 (2025): 165–73. https://doi.org/10.54254/2755-2721/2025.20636.

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This article presents Physics-Informed Neural Networks (PINNs), which integrate physical laws into neural network training to model complex systems governed by partial differential equations (PDEs). PINNs enhance data efficiency, allowing for accurate predictions with less training data, and have applications in fields such as biomedical engineering, geophysics, and material science. Despite their advantages, PINNs face challenges like learning high-frequency components and computational overhead. Proposed solutions include causality constraints and improved boundary condition handling. A nume

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Chen, Yanlai, Yajie Ji, Akil Narayan, and Zhenli Xu. "TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNs." Computer Methods in Applied Mechanics and Engineering 430 (October 2024): 117198. http://dx.doi.org/10.1016/j.cma.2024.117198.

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Ma, Shaojuan, Baolan Li, Hufei Li, and Hui Xiao. "PINNs Method for Sloving the Probability Response of the Stochastic Linear System with Fractional Gaussian Noise." Journal of Physics: Conference Series 3004, no. 1 (2025): 012016. https://doi.org/10.1088/1742-6596/3004/1/012016.

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Abstract In this paper, the Physics-informed neural networks (PINNs) algorithm is mainly used to solve the time-dependent Fokker-Planck-Kolmogorov (FPK) equation, and eventually obtain the transient probability density function (PDF). Firstly, we derive the FPK equation for a dynamical system driven by multiplicative fractional Gaussian noise (fGn). Secondly, a deep learning method based on PINNs is introduced for resolving the corresponding time-dependent FPK equation. Finally, one examples under fGn excitation condition is discussed to determine the effectiveness and feasibility of PINNs alg

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Ko, Taehwan, Heuisu Kim, Yeoungcheol Shin, et al. "Review of Recent Additive Manufacturing and Welding Research with Application of Physics-Informed Neural Networks." Journal of Welding and Joining 42, no. 4 (2024): 357–65. http://dx.doi.org/10.5781/jwj.2024.42.4.3.

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This review introduces recent research on applying physics-informed neural networks (PINNs) to additive manufacturing and welding. PINNs, which are artificial intelligence models, integrate governing equations containing physical information with artificial neural networks, enabling the modeling of complex physical phenomena at a lower computational cost than traditional numerical models. Although PINNs have been employed in a limited number of studies on welding processes, they have been extensively used in various studies within the field of additive manufacturing. This study reviews the the

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Demir, Kubilay Timur, Kai Logemann, and David S. Greenberg. "Closed-Boundary Reflections of Shallow Water Waves as an Open Challenge for Physics-Informed Neural Networks." Mathematics 12, no. 21 (2024): 3315. http://dx.doi.org/10.3390/math12213315.

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Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs) in fluid dynamics. By using PDE-derived loss functions and auto-differentiation, PINNs can recover solutions without requiring costly simulation data, spatial gridding, or time discretization. However, PINNs often exhibit slow or incomplete convergence, depending on the architecture, optimization algorithms, and complexity of the PDEs. To address these difficulties, a variety of novel and repurposed techniques have been int

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Roh, Dong Min, Minxue He, Zhaojun Bai, et al. "Physics-Informed Neural Networks-Based Salinity Modeling in the Sacramento–San Joaquin Delta of California." Water 15, no. 13 (2023): 2320. http://dx.doi.org/10.3390/w15132320.

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Salinity in estuarine environments has been traditionally simulated using process-based models. More recently, data-driven models including artificial neural networks (ANNs) have been developed for simulating salinity. Compared to process-based models, ANNs yield faster salinity simulations with comparable accuracy. However, ANNs are often purely data-driven and not constrained by physical laws, making it difficult to interpret the causality between input and output data. Physics-informed neural networks (PINNs) are emerging machine-learning models to integrate the benefits of both process-bas

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Tang, Zhuochao, Zhuojia Fu, and Sergiy Reutskiy. "An Extrinsic Approach Based on Physics-Informed Neural Networks for PDEs on Surfaces." Mathematics 10, no. 16 (2022): 2861. http://dx.doi.org/10.3390/math10162861.

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In this paper, we propose an extrinsic approach based on physics-informed neural networks (PINNs) for solving the partial differential equations (PDEs) on surfaces embedded in high dimensional space. PINNs are one of the deep learning-based techniques. Based on the training data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. Using automatic differentiation, the PDEs information could be encoded into NNs and a loss function. To deal with the surface differential operators in the loss function, we combine the e

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Trahan, Corey, Mark Loveland, and Samuel Dent. "Quantum Physics-Informed Neural Networks." Entropy 26, no. 8 (2024): 649. http://dx.doi.org/10.3390/e26080649.

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In this study, the PennyLane quantum device simulator was used to investigate quantum and hybrid, quantum/classical physics-informed neural networks (PINNs) for solutions to both transient and steady-state, 1D and 2D partial differential equations. The comparative expressibility of the purely quantum, hybrid and classical neural networks is discussed, and hybrid configurations are explored. The results show that (1) for some applications, quantum PINNs can obtain comparable accuracy with less neural network parameters than classical PINNs, and (2) adding quantum nodes in classical PINNs can in

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Lee, Jeongsu, Keunhwan Park, and Wonjong Jung. "Physics-Informed Neural Networks for Cantilever Dynamics and Fluid-Induced Excitation." Applied Sciences 14, no. 16 (2024): 7002. http://dx.doi.org/10.3390/app14167002.

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Physics-informed neural networks (PINNs) represent a continuous and differentiable mapping function, approximating solution curves for given differential equations. Recent studies have demonstrated the significant potential of PINNs as an alternative or complementary approach to conventional numerical methods. However, their application in structural dynamics, such as cantilever dynamics and fluid-induced excitations, poses challenges. In particular, limited accuracy and robustness in resolving high-order differential equations, including fourth-order differential equations encountered in stru

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de Cominges Guerra, Ignacio, Wenting Li, and Ren Wang. "A Comprehensive Analysis of PINNs for Power System Transient Stability." Electronics 13, no. 2 (2024): 391. http://dx.doi.org/10.3390/electronics13020391.

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The integration of machine learning in power systems, particularly in stability and dynamics, addresses the challenges brought by the integration of renewable energies and distributed energy resources (DERs). Traditional methods for power system transient stability, involving solving differential equations with computational techniques, face limitations due to their time-consuming and computationally demanding nature. This paper introduces physics-informed Neural Networks (PINNs) as a promising solution for these challenges, especially in scenarios with limited data availability and the need f

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Lawal, Zaharaddeen Karami, Hayati Yassin, Daphne Teck Ching Lai, and Azam Che Idris. "Physics-Informed Neural Network (PINN) Evolution and Beyond: A Systematic Literature Review and Bibliometric Analysis." Big Data and Cognitive Computing 6, no. 4 (2022): 140. http://dx.doi.org/10.3390/bdcc6040140.

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This research aims to study and assess state-of-the-art physics-informed neural networks (PINNs) from different researchers’ perspectives. The PRISMA framework was used for a systematic literature review, and 120 research articles from the computational sciences and engineering domain were specifically classified through a well-defined keyword search in Scopus and Web of Science databases. Through bibliometric analyses, we have identified journal sources with the most publications, authors with high citations, and countries with many publications on PINNs. Some newly improved techniques develo

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WANG Yuduo, CHEN Jiaxin, and LI Biao. "Solving Nonlinear Schrödinger Equations and Parameter Discovery via Extended Mixed-Training Physics-Informed Neural Networks." Acta Physica Sinica 74, no. 16 (2025): 0. https://doi.org/10.7498/aps.74.20250422.

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In recent years, Physics-Informed Neural Networks (PINNs) have provided effcient data-driven methods for solving both forward and inverse problems of partial differential equations (PDEs). However, when addressing complex PDEs, PINNs face significant challenges in computational effciency and accuracy. In this study, we propose the Extended Mixed-Training Physics-Informed Neural Networks (X-MTPINNs), as illustrated in fig. 1., which effectively enhance the capability for solving nonlinear wave problems by integrating the domain decomposition technique of extended Physics-Informed Neural Network

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Du Toit, Jacques Francois, and Ryno Laubscher. "Evaluation of Physics-Informed Neural Network Solution Accuracy and Efficiency for Modeling Aortic Transvalvular Blood Flow." Mathematical and Computational Applications 28, no. 2 (2023): 62. http://dx.doi.org/10.3390/mca28020062.

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Physics-Informed Neural Networks (PINNs) are a new class of machine learning algorithms that are capable of accurately solving complex partial differential equations (PDEs) without training data. By introducing a new methodology for fluid simulation, PINNs provide the opportunity to address challenges that were previously intractable, such as PDE problems that are ill-posed. PINNs can also solve parameterized problems in a parallel manner, which results in favorable scaling of the associated computational cost. The full potential of the application of PINNs to solving fluid dynamics problems i

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Hu, Alice V., and Zbigniew J. Kabala. "Predicting and Reconstructing Aerosol–Cloud–Precipitation Interactions with Physics-Informed Neural Networks." Atmosphere 14, no. 12 (2023): 1798. http://dx.doi.org/10.3390/atmos14121798.

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Interactions between clouds, aerosol, and precipitation are crucial aspects of weather and climate. The simple Koren–Feingold conceptual model is important for providing deeper insight into the complex aerosol–cloud–precipitation system. Recently, artificial neural networks (ANNs) and physics-informed neural networks (PINNs) have been used to study multiple dynamic systems. However, the Koren–Feingold model for aerosol–cloud–precipitation interactions has not yet been studied with either ANNs or PINNs. It is challenging for pure data-driven models, such as ANNs, to accurately predict and recon

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Nair, Tejas, and Merve Gokgol. "Functionality of Physics-Informed Neural Networks and Potential Future Impacts on Artificial Intelligence." Proceedings of London International Conferences, no. 11 (September 9, 2024): 120–24. http://dx.doi.org/10.31039/plic.2024.11.247.

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Physics-informed neural networks, or PINNs, are indicative of a new approach that involves the use of scientific knowledge, as these programs adhere to laws of physics described by general nonlinear partial differential equations while solving problems that are related to physics. This is accomplished via programming these equations into the loss function, which ensures that the underlying system adheres to these laws. This paper will be discussing how PINNs function and analyze how they make use of physics when solving problems. PINNs can be used to model physical systems and phenomena in the

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Tejas Nair and Merve Gokgol. "Functionality of Physics-Informed Neural Networks and Potential Future Impacts on Artificial Intelligence." London Journal of Interdisciplinary Sciences, no. 4 (February 9, 2025): 65–69. https://doi.org/10.31039/ljis.2025.4.304.

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Physics-informed neural networks, or PINNs, are indicative of a new approach that involves the use of scientific knowledge, as these programs adhere to laws of physics described by general nonlinear partial differential equations while solving problems that are related to physics. This is accomplished via programming these equations into the loss function, which ensures that the underlying system adheres to these laws. This paper will be discussing how PINNs function and analyze how they make use of physics when solving problems. PINNs can be used to model physical systems and phenomena in the

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Malashin, Ivan, Vadim Tynchenko, Andrei Gantimurov, Vladimir Nelyub, and Aleksei Borodulin. "Physics-Informed Neural Networks in Polymers: A Review." Polymers 17, no. 8 (2025): 1108. https://doi.org/10.3390/polym17081108.

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The modeling and simulation of polymer systems present unique challenges due to their intrinsic complexity and multi-scale behavior. Traditional computational methods, while effective, often struggle to balance accuracy with computational efficiency, especially when bridging the atomistic to macroscopic scales. Recently, physics-informed neural networks (PINNs) have emerged as a promising tool that integrates data-driven learning with the governing physical laws of the system. This review discusses the development and application of PINNs in the context of polymer science. It summarizes the re

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Kokhanovskiy, A. Yu, L. M. Dorogin, X. A. Egorova, E. V. Antonov, and D. A. Sinev. "Progress and Perspectives of Physics-Informed Neural Networks for Tribological Applications with Multiphysics Awareness." Reviews on Advanced Materials and Technologies 7, no. 2 (2025): 88–104. https://doi.org/10.17586/2687-0568-2025-7-2-88-104.

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Recent advancements in the field of physics-informed neural networks (PINNs) hold great potential for solving the tribology-related problems, and areas for their applications are systematically reviewed in this article. The tribological applications are viewed as fundamentally dependent on the variety of multiphysics phenomena, which must be taken into account when developing PINNs. Materials data, topology and surface roughness, and analytical tribometry data can be used as multiphysics input for the PINNs specialized in solving friction, lubrication, wear, wetting, heat transfer, structural

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Zhang, Guangtao, Huiyu Yang, Guanyu Pan, Yiting Duan, Fang Zhu, and Yang Chen. "Constrained Self-Adaptive Physics-Informed Neural Networks with ResNet Block-Enhanced Network Architecture." Mathematics 11, no. 5 (2023): 1109. http://dx.doi.org/10.3390/math11051109.

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Physics-informed neural networks (PINNs) have been widely adopted to solve partial differential equations (PDEs), which could be used to simulate physical systems. However, the accuracy of PINNs does not meet the needs of the industry, and severely degrades, especially when the PDE solution has sharp transitions. In this paper, we propose a ResNet block-enhanced network architecture to better capture the transition. Meanwhile, a constrained self-adaptive PINN (cSPINN) scheme is developed to move PINN’s objective to the areas of the physical domain, which are difficult to learn. To demonstrate

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Suhendar, Haris, Muhammad Ridho Pratama, and Michael Setyanto Silambi. "Mesh-Free Solution of 2D Poisson Equation with High Frequency Charge Patterns Using Data-Free Physics Informed Neural Network." Journal of Physics: Conference Series 2866, no. 1 (2024): 012053. http://dx.doi.org/10.1088/1742-6596/2866/1/012053.

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Abstract In this paper, we present a data-free physics-informed neural networks (PINNs) approach for solving two-dimensional (2D) Poisson equation, which is pivotal in fields such as electromagnetics, mechanical engginering, and thermodynamics. Traditional numerical method for solving this equation often require structured mesh generation such as Finite Element Method (FEM), which can be computationally expensive when dealing with high resolution Poisson Equation Solution. To address this challenge, we leverage the capabilities of PINNs capturing pattern of complex system by incorporating phys

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Rao, Shubhanshu, Gaurav Kumar, and Martin Agelin-Chaab. "A Hybrid Framework for Airfoil Optimization: Combining PINNs and Genetic Algorithm (Student Abstract)." Proceedings of the AAAI Conference on Artificial Intelligence 39, no. 28 (2025): 29475–76. https://doi.org/10.1609/aaai.v39i28.35293.

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Achieving optimal design is a crucial aspect of any design process for safe and efficient operation. Such tasks typically require numerous simulations over many iterations, which can become computationally expensive. This paper proposes a novel method that combines Physics-informed Neural Networks (PINNs) with a Genetic Algorithm to optimize the parameters of an airfoil that aims to achieve favourable aerodynamic conditions. Traditional solvers are computationally expensive for performing such tasks, but using PINNs can significantly reduce this while keeping accuracy high. The proposed approa

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Kim, Seunggoo, Donwoo Lee, and Seungjae Lee. "Performance Improvement of Seismic Response Prediction Using the LSTM-PINN Hybrid Method." Biomimetics 10, no. 8 (2025): 490. https://doi.org/10.3390/biomimetics10080490.

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Accurate and rapid prediction of structural responses to seismic loading is critical for ensuring structural safety. Recently, there has been active research focusing on the application of deep learning techniques, including Physics-Informed Neural Networks (PINNs) and Long Short-Term Memory (LSTM) networks, to predict the dynamic behavior of structures. While these methods have shown promise, each comes with distinct limitations. PINNs offer physical consistency but struggle with capturing long-term temporal dependencies in nonlinear systems, while LSTMs excel in learning sequential data but

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Farea, Amer, Olli Yli-Harja, and Frank Emmert-Streib. "Understanding Physics-Informed Neural Networks: Techniques, Applications, Trends, and Challenges." AI 5, no. 3 (2024): 1534–57. http://dx.doi.org/10.3390/ai5030074.

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Physics-informed neural networks (PINNs) represent a significant advancement at the intersection of machine learning and physical sciences, offering a powerful framework for solving complex problems governed by physical laws. This survey provides a comprehensive review of the current state of research on PINNs, highlighting their unique methodologies, applications, challenges, and future directions. We begin by introducing the fundamental concepts underlying neural networks and the motivation for integrating physics-based constraints. We then explore various PINN architectures and techniques f

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Liu, Yuhao, Junjie Hou, Ping Wei, Jie Jin, and Renjie Zhang. "Research and Application of ROM Based on Res-PINNs Neural Network in Fluid System." Symmetry 17, no. 2 (2025): 163. https://doi.org/10.3390/sym17020163.

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In the design of fluid systems, rapid iteration and simulation verification are essential, and reduced-order modeling techniques can significantly improve computational efficiency and accuracy. However, traditional Physics-Informed Neural Networks (PINNs) often face challenges such as vanishing or exploding gradients when learning flow field characteristics, limiting their ability to capture complex fluid dynamics. This study presents an enhanced reduced-order model (ROM): Physics-Informed Neural Networks based on Residual Networks (Res-PINNs). By integrating a Residual Network (ResNet) module

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Zhang, Yong, Huanhe Dong, Jiuyun Sun, Zhen Wang, Yong Fang, and Yuan Kong. "The New Simulation of Quasiperiodic Wave, Periodic Wave, and Soliton Solutions of the KdV-mKdV Equation via a Deep Learning Method." Computational Intelligence and Neuroscience 2021 (November 26, 2021): 1–9. http://dx.doi.org/10.1155/2021/8548482.

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How to solve the numerical solution of nonlinear partial differential equations efficiently and conveniently has always been a difficult and meaningful problem. In this paper, the data-driven quasiperiodic wave, periodic wave, and soliton solutions of the KdV-mKdV equation are simulated by the multilayer physics-informed neural networks (PINNs) and compared with the exact solution obtained by the generalized Jacobi elliptic function method. Firstly, the different types of solitary wave solutions are used as initial data to train the PINNs. At the same time, the different PINNs are applied to l

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Sarma, Antareep Kumar, Sumanta Roy, Chandrasekhar Annavarapu, Pratanu Roy, and Shriram Jagannathan. "Interface PINNs (I-PINNs): A physics-informed neural networks framework for interface problems." Computer Methods in Applied Mechanics and Engineering 429 (September 2024): 117135. http://dx.doi.org/10.1016/j.cma.2024.117135.

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Ortiz Ortiz, Rubén Darío, Oscar Martínez Núñez, and Ana Magnolia Marín Ramírez. "Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks." Mathematics 12, no. 21 (2024): 3430. http://dx.doi.org/10.3390/math12213430.

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In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers’ equation, such as shock wave formation and the smoothing effects of viscosity, and sh

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Tkachov, Yurii, and Oleh Murashko. "Physics-Informed Neural Networks in Aerospace: A Structured Taxonomy with Literature Review." Challenges and Issues of Modern Science 4, no. 1 (2025): 4–28. https://doi.org/10.15421/cims.4.313.

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Purpose. This study aims to develop a structured four-tier taxonomy that systematically organizes aerospace engineering tasks suitable for the application of Physics-Informed Neural Networks (PINNs), while validating this classification through a literature review and identifying opportunities for future research. Design / Method / Approach. The methodology involves grouping tasks into four distinct tiers—Physical Modeling, Dynamic Analysis, Functional Assessment, and System-Level Assessment—based on their physical, operational, and systemic characteristics. This framework is subsequently popu

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Duñabeitia, Miren K., Susana Hormilla, Isabel Salcedo, and Jose I. Peña. "Ectomycorrhizae synthesized between Pinus radiata and eight fungi associated with Pinns spp." Mycologia 88, no. 6 (1996): 897–908. http://dx.doi.org/10.1080/00275514.1996.12026730.

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Helali, Saloua, Shadiah Albalawi, and Nizar Bel Hadj Ali. "Harnessing Physics-Informed Neural Networks for Performance Monitoring in SWRO Desalination." Water 17, no. 3 (2025): 297. https://doi.org/10.3390/w17030297.

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Seawater Reverse Osmosis (SWRO) desalination is a critical technology for addressing global water scarcity, yet its performance can be hindered by complex process dynamics and operational inefficiencies. This study investigates the revolutionary potential of Physics-Informed Neural Networks (PINNs) for modeling SWRO desalination processes. PINNs are subsets of machine learning algorithms that incorporate physical information to help provide physically meaningful neural network models. The proposed approach is here demonstrated using operating data collected over several months in a Seawater RO

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Bandai, Toshiyuki, and Teamrat A. Ghezzehei. "Forward and inverse modeling of water flow in unsaturated soils with discontinuous hydraulic conductivities using physics-informed neural networks with domain decomposition." Hydrology and Earth System Sciences 26, no. 16 (2022): 4469–95. http://dx.doi.org/10.5194/hess-26-4469-2022.

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Abstract. Modeling water flow in unsaturated soils is vital for describing various hydrological and ecological phenomena. Soil water dynamics is described by well-established physical laws (Richardson–Richards equation – RRE). Solving the RRE is difficult due to the inherent nonlinearity of the processes, and various numerical methods have been proposed to solve the issue. However, applying the methods to practical situations is very challenging because they require well-defined initial and boundary conditions. Recent advances in machine learning and the growing availability of soil moisture d

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Serkin, Leonid, and Tatyana L. Belyaeva. "Physics-Informed Neural Networks for Higher-Order Nonlinear Schrödinger Equations: Soliton Dynamics in External Potentials." Mathematics 13, no. 11 (2025): 1882. https://doi.org/10.3390/math13111882.

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This review summarizes the application of physics-informed neural networks (PINNs) for solving higher-order nonlinear partial differential equations belonging to the nonlinear Schrödinger equation (NLSE) hierarchy, including models with external potentials. We analyze recent studies in which PINNs have been employed to solve NLSE-type evolution equations up to the fifth order, demonstrating their ability to obtain one- and two-soliton solutions, as well as other solitary waves with high accuracy. To provide benchmark solutions for training PINNs, we employ analytical methods such as the noniso

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Roy Sarkar, Dibakar, Chandrasekhar Annavarapu, and Pratanu Roy. "Adaptive Interface-PINNs (AdaI-PINNs) for inverse problems: Determining material properties for heterogeneous systems." Finite Elements in Analysis and Design 249 (July 2025): 104373. https://doi.org/10.1016/j.finel.2025.104373.

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Brumand-Poor, Faras, Florian Barlog, Nils Plückhahn, Matteo Thebelt, Niklas Bauer, and Katharina Schmitz. "Physics-Informed Neural Networks for the Reynolds Equation with Transient Cavitation Modeling." Lubricants 12, no. 11 (2024): 365. http://dx.doi.org/10.3390/lubricants12110365.

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Gaining insight into tribological systems is crucial for optimizing efficiency and prolonging operational lifespans in technical systems. Experimental investigations are time-consuming and costly, especially for reciprocating seals in fluid power systems. Elastohydrodynamic lubrication (EHL) simulations offer an alternative but demand significant computational resources. Physics-informed neural networks (PINNs) provide a promising solution using physics-based approaches to solve partial differential equations. While PINNs have successfully modeled hydrodynamics with stationary cavitation, they

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Song, Chao, Tariq Alkhalifah, and Umair Bin Waheed. "A versatile framework to solve the Helmholtz equation using physics-informed neural networks." Geophysical Journal International 228, no. 3 (2021): 1750–62. http://dx.doi.org/10.1093/gji/ggab434.

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SUMMARY Solving the wave equation to obtain wavefield solutions is an essential step in illuminating the subsurface using seismic imaging and waveform inversion methods. Here, we utilize a recently introduced machine-learning based framework called physics-informed neural networks (PINNs) to solve the frequency-domain wave equation, which is also referred to as the Helmholtz equation, for isotropic and anisotropic media. Like functions, PINNs are formed by using a fully connected neural network (NN) to provide the wavefield solution at spatial points in the domain of interest, in which the coo

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Kang, Namgyu, Byeonghyeon Lee, Youngjoon Hong, Seok-Bae Yun, and Eunbyung Park. "PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 7 (2023): 8186–94. http://dx.doi.org/10.1609/aaai.v37i7.25988.

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With the increases in computational power and advances in machine learning, data-driven learning-based methods have gained significant attention in solving PDEs. Physics-informed neural networks (PINNs) have recently emerged and succeeded in various forward and inverse PDE problems thanks to their excellent properties, such as flexibility, mesh-free solutions, and unsupervised training. However, their slower convergence speed and relatively inaccurate solutions often limit their broader applicability in many science and engineering domains. This paper proposes a new kind of data-driven PDEs so

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Kaewnuratchadasorn, Chawit, Jiaji Wang, and Chul‐Woo Kim. "Physics‐informed neural operator solver and super‐resolution for solid mechanics." Computer-Aided Civil and Infrastructure Engineering, July 11, 2024. http://dx.doi.org/10.1111/mice.13292.

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AbstractPhysics‐Informed Neural Networks (PINNs) have solved numerous mechanics problems by training to minimize the loss functions of governing partial differential equations (PDEs). Despite successful development of PINNs in various systems, computational efficiency and fidelity prediction have remained profound challenges. To fill such gaps, this study proposed a Physics‐Informed Neural Operator Solver (PINOS) to achieve accurate and fast simulations without any required data set. The training of PINOS adopts a weak form based on the principle of least work for static simulations and a stor

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Sun, Jiuyun, Huanhe Dong, and Yong Fang. "Physical informed memory networks for solving PDEs: Implementation and Applications." Communications in Theoretical Physics, January 3, 2024. http://dx.doi.org/10.1088/1572-9494/ad1a0e.

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Abstract With the advent of physics informed neural networks (PINNs), deep learning has gained interest for solving nonlinear partial differential equations (PDEs) in recent years. In this paper, physics informed memory networks (PIMNs) are proposed as a new approach to solve PDEs by using physical laws and dynamic behavior of PDEs. Unlike the fully connected structure of the PINNs, the PIMNs construct the long-term dependence of the dynamics behavior with the help of the long short-term memory network (LSTM). Meanwhile, the PDEs residuals are approximated using difference schemes in the form

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Deguchi, Shota, and Mitsuteru Asai. "Dynamic and norm-based weights to normalize imbalance in back-propagated gradients of physics-informed neural networks." Journal of Physics Communications, July 4, 2023. http://dx.doi.org/10.1088/2399-6528/ace416.

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Abstract Physics-Informed Neural Networks (PINNs) have been a promising machine learning model for evaluating various physical problems. Despite their success in solving many types of partial differential equations (PDEs), some problems have been found to be difficult to learn, implying that the baseline PINNs is biased towards learning the governing PDEs while relatively neglecting given initial or boundary conditions. In this work, we propose Dynamically Normalized Physics-Informed Neural Networks (DN-PINNs), a method to train PINNs while evenly distributing multiple back-propagated gradient

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Cao, Zhen, Kai Liu, Kun Luo, Sifan Wang, Liang Jiang, and Jianren Fan. "Surrogate modeling of multi-dimensional premixed and non-premixed combustion using pseudo-time stepping physics-informed neural networks." Physics of Fluids 36, no. 11 (2024). http://dx.doi.org/10.1063/5.0235674.

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Physics-informed neural networks (PINNs) have emerged as a promising alternative to conventional computational fluid dynamics (CFD) approaches for solving and modeling multi-dimensional flow fields. They offer instant inference speed and cost-effectiveness without the need for training datasets. However, compared to common data-driven methods, purely learning the physical constraints of partial differential equations and boundary conditions is much more challenging and prone to convergence issues leading to incorrect local optima. This training robustness issue significantly increases the diff

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Li, Zhihui, Francesco Montomoli, and Sanjiv Sharma. "Investigation of Compressor Cascade Flow Using Physics-Informed Neural Networks with Adaptive Learning Strategy." AIAA Journal, February 29, 2024, 1–11. http://dx.doi.org/10.2514/1.j063562.

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In this study, we utilize the emerging physics-informed neural networks (PINNs) approach for the first time to predict the flowfield of a compressor cascade. Different from conventional training methods, a new adaptive learning strategy that mitigates gradient imbalance through incorporating adaptive weights in conjunction with a dynamically adjusting learning rate is used during the training process to improve the convergence of PINNs. The performance of PINNs is assessed here by solving both the forward and inverse problems. In the forward problem, by encapsulating the physical relations amo

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Fang Ze, Pan YongQuan, Dai Dong, and Zhang JunBo. "Physics-informed neural networks based on source term decoupled and its application in discharge plasma simulation." Acta Physica Sinica, 2024, 0. http://dx.doi.org/10.7498/aps.73.20240343.

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In recent years, the artificial intelligence computing paradigm represented by physics-informed neural networks (PINNs) has received great attention in the field of plasma numerical simulation. However, the plasma chemical system considered in related research is relatively simplified, and there is still a gap in the research on solving the more complex multi-particle low-temperature fluid model based on PINNs. In more complex chemical systems, the coupling relationships between particle densities as well as between particle densities and mean electron energy become more intricate. Therefore,

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Rodriguez-Torrado, Ruben, Pablo Ruiz, Luis Cueto-Felgueroso, et al. "Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem." Scientific Reports 12, no. 1 (2022). http://dx.doi.org/10.1038/s41598-022-11058-2.

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AbstractPhysics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum

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Moschou, Sofia P., Elliot Hicks, Rishi Parekh, Dhruv Mathew, Shoumik Majumdar, and Nektarios Vlahakis. "Physics-Informed Neural Networks for modeling astrophysical shocks." Machine Learning: Science and Technology, August 16, 2023. http://dx.doi.org/10.1088/2632-2153/acf116.

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Abstract Physics-Informed Neural Networks (PINNs) are machine learning models that integrate data-based learning with partial differential equations (PDEs). In this work, for the first time we extend PINNs to model the numerically challenging case of astrophysical shock waves in the presence of a stellar gravitational field. Notably, PINNs suffer from competing losses during gradient descent that can lead to poor performance especially in physical setups involving multiple scales, which is the case for shocks in the gravitationally stratified solar atmosphere. We applied PINNs in three differe

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Biswas, Saykat Kumar, and N. K. Anand. "Three-dimensional laminar flow using physics informed deep neural networks." Physics of Fluids 35, no. 12 (2023). http://dx.doi.org/10.1063/5.0180834.

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Physics informed neural networks (PINNs) have demonstrated their effectiveness in solving partial differential equations (PDEs). By incorporating the governing equations and boundary conditions directly into the neural network architecture with the help of automatic differentiation, PINNs can approximate the solution of a system of PDEs with good accuracy. Here, an application of PINNs in solving three-dimensional (3D) Navier–Stokes equations for laminar, steady, and incompressible flow is presented. Notably, our approach involves deploying PINNs using feed-forward deep neural networks (DNNs)

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Suarez, Juan Esteban, and Michael Hecht. "Polynomial Differentiation Decreases the Training Time Complexity of Physics-Informed Neural Networks and Strengthens their Approximation Power." Machine Learning: Science and Technology, September 13, 2023. http://dx.doi.org/10.1088/2632-2153/acf97a.

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Abstract We present novel approximates of variational losses, being applicable for the training of physics-informed neural networks (PINNs). The formulations reflect classic Sobolev space theory for partial differential equations and their weak formulations. The loss approximates rest on polynomial differentiation realised by an extension of classic Gauss-Legendre cubatures, we term Sobolev cubatures, and serve as a replacement of automatic differentiation. We prove the training time complexity of the resulting Sobolev-PINNs with polynomial differentiation to be less than required by PINNs rel

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