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1
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 increase model accuracy with less total network parameters for noiseless models.
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Wang, Jing, Yubo Li, Anping Wu, et al. "Multi-Step Physics-Informed Deep Operator Neural Network for Directly Solving Partial Differential Equations." Applied Sciences 14, no. 13 (2024): 5490. http://dx.doi.org/10.3390/app14135490.
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This paper establishes a method for solving partial differential equations using a multi-step physics-informed deep operator neural network. The network is trained by embedding physics-informed constraints. Different from traditional neural networks for solving partial differential equations, the proposed method uses a deep neural operator network to indirectly construct the mapping relationship between the variable functions and solution functions. This approach makes full use of the hidden information between the variable functions and independent variables. The process whereby the model captures incredibly complex and highly nonlinear relationships is simplified, thereby making network learning easier and enhancing the extraction of information about the independent variables in partial differential systems. In terms of solving partial differential equations, we verify that the multi-step physics-informed deep operator neural network markedly improves the solution accuracy compared with a traditional physics-informed deep neural operator network, especially when the problem involves complex physical phenomena with large gradient changes.
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Hofmann, Tobias, Jacob Hamar, Marcel Rogge, Christoph Zoerr, Simon Erhard, and Jan Philipp Schmidt. "Physics-Informed Neural Networks for State of Health Estimation in Lithium-Ion Batteries." Journal of The Electrochemical Society 170, no. 9 (2023): 090524. http://dx.doi.org/10.1149/1945-7111/acf0ef.
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One of the most challenging tasks of modern battery management systems is the accurate state of health estimation. While physico-chemical models are accurate, they have high computational cost. Neural networks lack physical interpretability but are efficient. Physics-informed neural networks tackle the aforementioned shortcomings by combining the efficiency of neural networks with the accuracy of physico-chemical models. A physics-informed neural network is developed and evaluated against three different datasets: A pseudo-two-dimensional Newman model generates data at various state of health points. This dataset is fused with experimental data from laboratory measurements and vehicle field data to train a neural network in which it exploits correlation from internal modeled states to the measurable state of health. The resulting physics-informed neural network performs best with the synthetic dataset and achieves a root mean squared error below 2% at estimating the state of health. The root mean squared error stays within 3% for laboratory test data, with the lowest error observed for constant current discharge samples. The physics-informed neural network outperforms several other purely data-driven methods and proves its advantage. The inclusion of physico-chemical information from simulation increases accuracy and further enables broader application ranges.
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Karakonstantis, Xenofon, Diego Caviedes-Nozal, Antoine Richard, and Efren Fernandez-Grande. "Room impulse response reconstruction with physics-informed deep learning." Journal of the Acoustical Society of America 155, no. 2 (2024): 1048–59. http://dx.doi.org/10.1121/10.0024750.
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A method is presented for estimating and reconstructing the sound field within a room using physics-informed neural networks. By incorporating a limited set of experimental room impulse responses as training data, this approach combines neural network processing capabilities with the underlying physics of sound propagation, as articulated by the wave equation. The network's ability to estimate particle velocity and intensity, in addition to sound pressure, demonstrates its capacity to represent the flow of acoustic energy and completely characterise the sound field with only a few measurements. Additionally, an investigation into the potential of this network as a tool for improving acoustic simulations is conducted. This is due to its proficiency in offering grid-free sound field mappings with minimal inference time. Furthermore, a study is carried out which encompasses comparative analyses against current approaches for sound field reconstruction. Specifically, the proposed approach is evaluated against both data-driven techniques and elementary wave-based regression methods. The results demonstrate that the physics-informed neural network stands out when reconstructing the early part of the room impulse response, while simultaneously allowing for complete sound field characterisation in the time domain.
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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 numerical experiment demonstrates the effectiveness of PINNs in solving the one-dimensional heat conduction equation, showcasing enhanced model stability and accuracy. Overall, PINNs represent a significant advancement in merging machine learning with physics.
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Pu, Ruilong, and Xinlong Feng. "Physics-Informed Neural Networks for Solving Coupled Stokes–Darcy Equation." Entropy 24, no. 8 (2022): 1106. http://dx.doi.org/10.3390/e24081106.
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In this paper, a grid-free deep learning method based on a physics-informed neural network is proposed for solving coupled Stokes–Darcy equations with Bever–Joseph–Saffman interface conditions. This method has the advantage of avoiding grid generation and can greatly reduce the amount of computation when solving complex problems. Although original physical neural network algorithms have been used to solve many differential equations, we find that the direct use of physical neural networks to solve coupled Stokes–Darcy equations does not provide accurate solutions in some cases, such as rigid terms due to small parameters and interface discontinuity problems. In order to improve the approximation ability of a physics-informed neural network, we propose a loss-function-weighted function strategy, a parallel network structure strategy, and a local adaptive activation function strategy. In addition, the physical information neural network with an added strategy provides inspiration for solving other more complicated problems of multi-physical field coupling. Finally, the effectiveness of the proposed strategy is verified by numerical experiments.
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Kenzhebek, Y., T. S. Imankulov, and D. Zh Akhmed-Zaki. "PREDICTION OF OIL PRODUCTION USING PHYSICS-INFORMED NEURAL NETWORKS." BULLETIN Series of Physics & Mathematical Sciences 76, no. 4 (2021): 45–50. http://dx.doi.org/10.51889/2021-4.1728-7901.06.
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In recent years, modern information technologies have been actively used in various industries. The oil industry is no exception, since high-performance computing technologies, artificial intelligence algorithms, methods of collecting, processing and storing information are actively used to solve the problems of increasing oil recovery. Deep learning has made remarkable strides in a variety of applications, but its use for solving partial differential equations has only recently emerged. In particular, you can replace traditional numerical methods with a neural network that approximates the solution to a partial differential equation. Physically Informed Neural Networks (PINNs) embed partial differential equations into the neural network loss function using automatic differentiation. A numerical algorithm and PINN have been developed for solving the one-dimensional pressure equation from the Buckley-Leverett mathematical model. The results of numerical solution and prediction of the PINN neural network for solving the pressure equation are obtained.
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Hou, Shubo, Wenchao Wu, and Xiuhong Hao. "Physics-informed neural network for simulating magnetic field of permanent magnet." Journal of Physics: Conference Series 2853, no. 1 (2024): 012018. http://dx.doi.org/10.1088/1742-6596/2853/1/012018.
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Abstract With the rapid development of deep learning, its application in physical field simulation has been widely concerned, and it has begun to lead a new model of meshless simulation. In this paper, research based on physics-informed neural networks is carried out to solve partial differential equations related to the physical laws of electromagnetism. Then the magnetic field simulation is realized. In this method, the governing equation and the boundary conditions containing physical information are embedded into the neural network loss function as constraints, and the backpropagation of neural networks is realized based on automatic differentiation to solve partial differential equations. The high-precision simulation of tile-shaped and rectangular permanent magnet magnetic fields of permanent magnet motors based on physical information neural network is studied, and the error is within 5%. We consider the simulation of magnetic field in two coordinate systems, and realize the joint training of multiple neural networks in multiple sub-domains and different media.
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Pan, Cunliang, Shi Feng, Shengyang Tao, Hongwu Zhang, Yonggang Zheng, and Hongfei Ye. "Physics-Informed Neural Network for Young-Laplace Equation." International Conference on Computational & Experimental Engineering and Sciences 30, no. 1 (2024): 1. http://dx.doi.org/10.32604/icces.2024.011132.
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Yoon, Seunghyun, Yongsung Park, and Woojae Seong. "Improving mode extraction with physics-informed neural network." Journal of the Acoustical Society of America 154, no. 4_supplement (2023): A339—A340. http://dx.doi.org/10.1121/10.0023729.
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This study aims to enhance conventional mode extraction methods in ocean waveguides using a physics-informed neural network (PINN). Mode extraction involves estimating mode wavenumbers and corresponding mode depth functions. The approach considers a scenario with a single frequency source towed at a constant depth and measured from a vertical line array (VLA). Conventional mode extraction methods applied to experimental data face two problems. First, mode shape estimation is limited because the receivers only cover a partial waveguide. Second, the wavenumber spectrum is affected by issues such as Doppler shift and range errors. To address these challenges, we train the PINN with measured data, generating a densely sampled complex pressure field, including the unmeasured region above the VLA. We then apply the same mode extraction methods to both the raw data and the PINN-generated data for comparison. The proposed method is validated using data from the SWellEx-96, demonstrating improved mode extraction performance compared to using raw experimental data directly.
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Satyadharma, Adhika, Heng-Chuan Kan, Ming-Jyh Chern, and Chun-Ying Yu. "Numerical error estimation with physics informed neural network." Computers & Fluids 299 (August 2025): 106700. https://doi.org/10.1016/j.compfluid.2025.106700.
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Schmid, Johannes D., Philipp Bauerschmidt, Caglar Gurbuz, and Steffen Marburg. "Physics-informed neural networks for characterization of structural dynamic boundary conditions." Journal of the Acoustical Society of America 154, no. 4_supplement (2023): A99. http://dx.doi.org/10.1121/10.0022923.
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Structural dynamics simulations are often faced with challenges arising from unknown boundary conditions, leading to considerable prediction uncertainties. Direct measurement of these boundary conditions can be impractical for certain mounting scenarios, such as joints or screw connections. In addition, conventional inverse methods face limitations in integrating measured data and solving inverse problems when the forward model is computationally expensive. In this study, we explore the potential of physics-informed neural networks that incorporate the residual of a partial differential equation into the loss function of a neural network to ensure physically consistent predictions. We train the neural network using noisy boundary displacement data of a structure from a finite element reference solution. The network learns to predict the displacement field within the structure while satisfying the Navier–Lamé equations in the frequency domain. Our results show that physics-informed neural networks accurately predict the displacement field within a three-dimensional structure using only boundary training data. Additionally, differentiating the trained network allows precise characterization of previously unknown boundary conditions and facilitates the assessment of non-measurable quantities, such as the stress tensor.
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Stenkin, Dmitry, and Vladimir Gorbachenko. "Mathematical Modeling on a Physics-Informed Radial Basis Function Network." Mathematics 12, no. 2 (2024): 241. http://dx.doi.org/10.3390/math12020241.
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The article is devoted to approximate methods for solving differential equations. An approach based on neural networks with radial basis functions is presented. Neural network training algorithms adapted to radial basis function networks are proposed, in particular adaptations of the Nesterov and Levenberg-Marquardt algorithms. The effectiveness of the proposed algorithms is demonstrated for solving model problems of function approximation, differential equations, direct and inverse boundary value problems, and modeling processes in piecewise homogeneous media.
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Zhai, Hanfeng, Quan Zhou, and Guohui Hu. "Predicting micro-bubble dynamics with semi-physics-informed deep learning." AIP Advances 12, no. 3 (2022): 035153. http://dx.doi.org/10.1063/5.0079602.
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Utilizing physical information to improve the performance of the conventional neural networks is becoming a promising research direction in scientific computing recently. For multiphase flows, it would require significant computational resources for neural network training due to the large gradients near the interface between the two fluids. Based on the idea of the physics-informed neural networks (PINNs), a modified deep learning framework BubbleNet is proposed to overcome this difficulty in the present study. The deep neural network (DNN) with separate sub-nets is adopted to predict physics fields, with the semi-physics-informed part encoding the continuity equation and the pressure Poisson equation [Formula: see text] for supervision and the time discretized normalizer to normalize field data per time step before training. Two bubbly flows, i.e., single bubble flow and multiple bubble flow in a microchannel, are considered to test the algorithm. The conventional computational fluid dynamics software is applied to obtain the training dataset. The traditional DNN and the BubbleNet(s) are utilized to train the neural network and predict the flow fields for the two bubbly flows. Results indicate the BubbleNet frameworks are able to successfully predict the physics fields, and the inclusion of the continuity equation significantly improves the performance of deep NNs. The introduction of the Poisson equation also has slightly positive effects on the prediction results. The results suggest that constructing semi-PINNs by flexibly considering the physical information into neural networks will be helpful in the learning of complex flow problems.
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Karakonstantis, Xenofon, and Efren Fernandez-Grande. "Advancing sound field analysis with physics-informed neural networks." Journal of the Acoustical Society of America 154, no. 4_supplement (2023): A98. http://dx.doi.org/10.1121/10.0022920.
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This work introduces a method that employs physics-informed neural networks to reconstruct sound fields in diverse rooms, including both typical acoustically damped meeting rooms and more spaces of cultural significance, such as concert halls or theatres. The neural network is trained using a limited set of room impulse responses, integrating the expressive capacity of neural networks with the fundamental physics of sound propagation governed by the wave equation. Consequently, the network accurately represents sound fields within an aperture without requiring extensive measurements, regardless of the complexity of the sound field. Notably, our approach extends beyond sound pressure estimation and includes valuable vectorial quantities, such as particle velocity and intensity, resembling classical holography methods. Experimental results confirm the efficacy of the proposed approach, underscoring its reconstruction accuracy and computational efficiency. Moreover, by enabling the acquisition of sound field quantities in the time domain, which were previously challenging to obtain from measurements, our method opens up new frontiers for the analysis and comprehension of sound propagation phenomena in rooms.
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Vo, Van Truong, Samad Noeiaghdam, Denis Sidorov, Aliona Dreglea, and Liguo Wang. "Solving Nonlinear Energy Supply and Demand System Using Physics-Informed Neural Networks." Computation 13, no. 1 (2025): 13. https://doi.org/10.3390/computation13010013.
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Nonlinear differential equations and systems play a crucial role in modeling systems where time-dependent factors exhibit nonlinear characteristics. Due to their nonlinear nature, solving such systems often presents significant difficulties and challenges. In this study, we propose a method utilizing Physics-Informed Neural Networks (PINNs) to solve the nonlinear energy supply–demand (ESD) system. We design a neural network with four outputs, where each output approximates a function that corresponds to one of the unknown functions in the nonlinear system of differential equations describing the four-dimensional ESD problem. The neural network model is then trained, and the parameters are identified and optimized to achieve a more accurate solution. The solutions obtained from the neural network for this problem are equivalent when we compare and evaluate them against the Runge–Kutta numerical method of order 5(4) (RK45). However, the method utilizing neural networks is considered a modern and promising approach, as it effectively exploits the superior computational power of advanced computer systems, especially in solving complex problems. Another advantage is that the neural network model, after being trained, can solve the nonlinear system of differential equations across a continuous domain. In other words, neural networks are not only trained to approximate the solution functions for the nonlinear ESD system but can also represent the complex dynamic relationships between the system’s components. However, this approach requires significant time and computational power due to the need for model training. Furthermore, as this method is evaluated based on experimental results, ensuring the stability and convergence speed of the model poses a significant challenge. The key factors influencing this include the manner in which the neural network architecture is designed, such as the selection of hyperparameters and appropriate optimization functions. This is a critical and highly complex task, requiring experimentation and fine-tuning, which demand substantial expertise and time.
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Kaliuzhniak, Anastasiia, Oleksii Kudi, Yuriy Belokon, and Dmytro Kruglyak. "Developing of neural network computing methods for solving inverse elasticity problems." Eastern-European Journal of Enterprise Technologies 6, no. 7 (132) (2024): 45–52. https://doi.org/10.15587/1729-4061.2024.313795.
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This paper examines the use of neural network methods to solve inverse problems in the mechanics of elastic materials. The aim is to design physics-informed neural networks that can predict the parameters of structural components, and the physical properties of materials based on a specified displacement distribution. A key feature of the specified neural networks is the integration of differential equations and boundary conditions into the loss function calculation. This approach ensures that the error in approximating unknown functions has a direct impact on optimizing the network's weights. As a result, the resulting neural network approximations of unknown functions comply with the differential equations and boundary conditions. To test the capabilities of the designed neural networks, inverse problems involving the bending of plates and beams have been solved, focusing on determining one or two unknown parameters. Comparison of predicted and exact values demonstrates high accuracy of the constructed neural network models, with a relative prediction error of less than 3 % across all cases. Unlike analytical methods for solving inverse problems, the primary advantage of physics-informed neural networks is their flexibility when addressing both linear and nonlinear problems. For instance, the same network architecture can be employed to solve various boundary-value problems without modification. Compared to classical numerical methods, the parallelization capability of neural networks is inherently supported by modern software libraries. Therefore, the application of physics-informed neural networks for solving inverse elasticity problems of plates and beams is effective, as evidenced by the achieved relative errors and the computational robustness of the method. In practice, the proposed solution can be used for relevant calculations during the design of structural elements. The developed software code can also be integrated into automated design systems or computer algebra systems
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Socaciu, Tiberiu, and Paul Pașcu. "Physics-Informed Neural Networks in Pricing Financial Options." BRAIN. Broad Research in Artificial Intelligence and Neuroscience 16, no. 2 (2025): 474. https://doi.org/10.70594/brain/16.2/33.
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<p dir="ltr"><span>PINNs (Physics-Informed Neural Networks) are neural networks designed to solve Partial Differential Equations (PDEs) by integrating physical knowledge into the learning framework. Constructing a PINN involves defining a neural network to approximate the PDE solution, with the total loss calculated as a combination of the losses associated with the PDE, boundary conditions, initial conditions, and measured data. This concept is employed in practical applications to solve various PDEs, such as the Black-Scholes and Heston equations, which are fundamental in financial option pricing. This approach enables the modelling and pricing of financial options, with the added advantage of parallelising the training process across multiple economic scenarios.</span></p><div><span><br /></span></div>
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Dong, Chenghao. "Solving Differential Equations with Physics-Informed Neural Networks." Theoretical and Natural Science 87, no. 1 (2025): 137–46. https://doi.org/10.54254/2753-8818/2025.20346.
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Solving differential equations is an extensive topic in various fields, such as fluid mechanics and mathematical finance. The recent resurgence in deep neural networks has opened up a brand new track for numerically solving these equations, with the potential to better deal with nonlinear problems and overcome the curse of dimensionality. The Physics-Informed Neural Network (PINN) is one of the fundamental attempts to solve differential equations using deep learning techniques. This paper aims to briefly review the application of PINNs and their variants in solving differential equations through a few simple examples, and to provide practitioners interested in this direction with a quick introduction to the relevant topic
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Zhang, Jianlin, Yake Leng, Hui Dong, Jianuo Yang, and Yinglin Jiao. "Peakons of the shallow-water wave equation implementation with PINNs." Journal of Physics: Conference Series 2964, no. 1 (2025): 012051. https://doi.org/10.1088/1742-6596/2964/1/012051.
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Abstract In this paper, we implement the peakon solution of the CH equation containing highly nonlinear terms using the Physics-Informed Neural Networks, and we define the corresponding neural network models according to different forms of peakon solution expressions. The main implementation is to study the spatio-temporal dynamics behaviour of the peakon solution for the general form of the CH equations. From the experimental results, the Physics-Informed Neural Networks can basically complete the analytical solution prediction of the shallow-water wave equation, and achieve the accurate capture of highly complex nonlinear terms, which proves the successful application of the Physics-Informed Neural Networks method in the peakon solution of the shallow-water wave equation.
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Pannekoucke, Olivier, and Ronan Fablet. "PDE-NetGen 1.0: from symbolic partial differential equation (PDE) representations of physical processes to trainable neural network representations." Geoscientific Model Development 13, no. 7 (2020): 3373–82. http://dx.doi.org/10.5194/gmd-13-3373-2020.
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Abstract. Bridging physics and deep learning is a topical challenge. While deep learning frameworks open avenues in physical science, the design of physically consistent deep neural network architectures is an open issue. In the spirit of physics-informed neural networks (NNs), the PDE-NetGen package provides new means to automatically translate physical equations, given as partial differential equations (PDEs), into neural network architectures. PDE-NetGen combines symbolic calculus and a neural network generator. The latter exploits NN-based implementations of PDE solvers using Keras. With some knowledge of a problem, PDE-NetGen is a plug-and-play tool to generate physics-informed NN architectures. They provide computationally efficient yet compact representations to address a variety of issues, including, among others, adjoint derivation, model calibration, forecasting and data assimilation as well as uncertainty quantification. As an illustration, the workflow is first presented for the 2D diffusion equation, then applied to the data-driven and physics-informed identification of uncertainty dynamics for the Burgers equation.
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Schmid, Johannes. "Physics-informed neural networks for solving the Helmholtz equation." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 267, no. 1 (2023): 265–68. http://dx.doi.org/10.3397/no_2023_0049.
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Discretization-based methods like the finite element method have proven to be effective for solving the Helmholtz equation in computational acoustics. However, it is very challenging to incorporate measured data into the model or infer model input parameters based on observed response data. Machine learning approaches have shown promising potential in data-driven modeling. In practical applications, purely supervised approaches suffer from poor generalization and physical interpretability. Physics-informed neural networks (PINNs) incorporate prior knowledge of the underlying partial differential equation by including the residual into the loss function of an artificial neural network. Training the neural network minimizes the residual of both the differential equation and the boundary conditions and learns a solution that satisfies the corresponding boundary value problem. In this contribution, PINNs are applied to solve the Helmholtz equation within a two-dimensional acoustic duct and mixed boundary conditions are considered. The results show that PINNs are able to solve the Helmholtz equation very accurately and provide a promising data-driven method for physics-based surrogate modeling.
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Yoon, Seunghyun, Yongsung Park, Peter Gerstoft, and Woojae Seong. "Physics-informed neural network for predicting unmeasured ocean acoustic pressure field." Journal of the Acoustical Society of America 154, no. 4_supplement (2023): A97. http://dx.doi.org/10.1121/10.0022916.
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This study employs a physics-informed neural network in an ocean waveguide to predict the unmeasured acoustic pressure field, leveraging partially measured data. The method addresses a scenario where an acoustic source transmits signals across different ranges and is measured by multiple receivers. The acoustic pressure field in ocean waveguides exhibits rapid spatial variations over kilometer-range scales. The fully connected neural networks encounter challenges when approximating high-frequency functions, known as spectral bias. To mitigate this problem, the measured pressure field is transformed into a low-frequency function for training the neural network. We propose two methods sharing the same neural network architecture but utilizing different information. The first method uses a complex value of the pressure field (i.e., both magnitude and phase), while the second method uses only magnitude. We validate the proposed methods using simulations and experimental data from the SWellEx-96 environment. Results demonstrate that the first method exhibits superior performance with sparse data, while the second method works better in real-world scenarios.
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Berrone, Stefano, and Moreno Pintore. "Meshfree Variational-Physics-Informed Neural Networks (MF-VPINN): An Adaptive Training Strategy." Algorithms 17, no. 9 (2024): 415. http://dx.doi.org/10.3390/a17090415.
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In this paper, we introduce a Meshfree Variational-Physics-Informed Neural Network. It is a Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions. In order to generate the test space, we exploit an a posteriori error indicator and add test functions only where the error is higher. Four training strategies are proposed and compared. Numerical results show that the accuracy is higher than the one of a Variational-Physics-Informed Neural Network trained with the same number of test functions but defined on a quasi-uniform mesh.
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Bizzi, Arthur, Lucas Nissenbaum, and João M. Pereira. "Neural Conjugate Flows: A Physics-Informed Architecture with Flow Structure." Proceedings of the AAAI Conference on Artificial Intelligence 39, no. 15 (2025): 15576–86. https://doi.org/10.1609/aaai.v39i15.33710.
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We introduce Neural Conjugate Flows (NCF), a class of neural-network architectures equipped with exact flow structure. By leveraging topological conjugation, we prove that these networks are not only naturally isomorphic to a continuous group, but are also universal approximators for flows of ordinary differential equation (ODEs). Furthermore, topological properties of these flows can be enforced by the architecture in an interpretable manner. We demonstrate in numerical experiments how this topological group structure leads to concrete computational gains over other physics informed neural networks in estimating and extrapolating latent dynamics of ODEs, while training up to five times faster than other flow-based architectures.
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Yoon, Seunghyun, Yongsung Park, Peter Gerstoft, and Woojae Seong. "Predicting ocean pressure field with a physics-informed neural network." Journal of the Acoustical Society of America 155, no. 3 (2024): 2037–49. http://dx.doi.org/10.1121/10.0025235.
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Ocean sound pressure field prediction, based on partially measured pressure magnitudes at different range-depths, is presented. Our proposed machine learning strategy employs a trained neural network with range-depth as input and outputs complex acoustic pressure at the location. We utilize a physics-informed neural network (PINN), fitting sampled data while considering the additional information provided by the partial differential equation (PDE) governing the ocean sound pressure field. In vast ocean environments with kilometer-scale ranges, pressure fields exhibit rapidly fluctuating phases, even at frequencies below 100 Hz, posing a challenge for neural networks to converge to accurate solutions. To address this, we utilize the envelope function from the parabolic-equation technique, fundamental in ocean sound propagation modeling. The envelope function shows slower variations across ranges, enabling PINNs to predict sound pressure in an ocean waveguide more effectively. Additional PDE information allows PINNs to capture PDE solutions even with a limited amount of training data, distinguishing them from purely data-driven machine learning approaches that require extensive datasets. Our approach is validated through simulations and using data from the SWellEx-96 experiment.
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Ponomarev, R. Yu, R. R. Ziazev, A. A. Leshchenko, R. R. Migmanov, and M. I. Ivlev. "Flooding system optimization: Advantages of a hybrid approach to developing neural network filtration models." Actual Problems of Oil and Gas 15, no. 4 (2024): 349–63. https://doi.org/10.29222/ipng.2078-5712.2024-15-4.art3.
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Background. Currently, neural networks are increasingly used for processing and forecasting the dynamics of well performance. However, there are a number of limitations in their application for flooding system optimization. Objective. To develop models that can correctly reproduce the impact of the reservoir pressure maintenance system on the operation of production wells. The article considers the problem of modeling the reaction of producing wells to the changes in water injection modes in injection wells using neural network modeling methods. Results. We propose the approaches to the creation and training of physics-informed neural networks for modeling responses in oil production to regime changes in the reservoir pressure maintenance system. The results of testing the training and predictive abilities of PINN (physics-informed neural network) models are presented, and a comparison is made with the results of forecasting on the classical LSTM neural network. Conclusions. With hybrid training of models based on actual data, PINN models make it possible to offset the limitations of classical neural networks.
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Hassanaly, Malik, Peter J. Weddle, Corey R. Randall, Eric J. Dufek, and Kandler Smith. "Rapid Inverse Parameter Inference Using Physics-Informed Neural Networks." ECS Meeting Abstracts MA2024-01, no. 2 (2024): 345. http://dx.doi.org/10.1149/ma2024-012345mtgabs.
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As Li-ion batteries become more essential in today's economy, tools need to be developed to accurately and rapidly diagnose a battery's internal state-of-health. Using a Li-ion battery's (high-rate) voltage response, it is proposed to determine a battery's internal state through Bayesian calibration. However, Bayesian calibration is notoriously slow and requires thousands of model runs. To accelerate parameter inference using Bayesian calibration, a surrogate model is developed to replace the underlying physics-based Li-ion model. Developing a surrogate model for rapid Bayesian calibration analysis is discussed for both the single particle model (SPM) and the pseudo two-dimensional (P2D) model. Surrogate models are constructed using physics-informed neural networks (PINNs) that encode the influence of internal properties on observed voltage responses. In practice, a neural network can be trained by: 1) using simulation results of the physics-based model (i.e., a data-loss approach); 2) using the residuals of the governing equations themselves (i.e., a physics-loss approach); or 3) using a combination of simulation results and governing equation residuals. In the present work, PINNs are developed using a variety of training losses and neural network architectures. In this analysis, it is shown that a PINN surrogate model can be reliably trained with only physics-informed loss. However, using a coupled data-informed and physics-loss approach produced the most accurate PINNs. Figure~\ref{fig:spm_2d} illustrates the absolute relative errors of trained PINN networks using several different training losses and neural network architectures. After determining a consistent training strategy for both the SPM and P2D PINN surrogate models, the PINNs are extended to determine additional internal state-of-health parameters. As more and more parameters were introduced, the PINN training suffered from ``the curse of dimensionality", which was mitigated by using a hierarchical training approach (where a PINN trained with fewer variable model parameters was used to train a PINN with more variable model parameters). Next, the high-dimensionality PINN surrogates are then integrated into Bayesian calibration schemes to identify internal Li-ion battery properties from experimentally measured voltages. Interpreting the high-dimensional parameter posteriors is discussed with respect to model error, parameter prior choices, and experimental errors. Figure 1
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Cai, Zemin, Xiangqi Lin, Tianshu Liu, Fan Wu, Shizhao Wang, and Yun Liu. "Determining pressure from velocity via physics-informed neural network." European Journal of Mechanics - B/Fluids 109 (January 2025): 1–21. http://dx.doi.org/10.1016/j.euromechflu.2024.08.007.
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Liu, Xue, Wei Cheng, Ji Xing, et al. "Physics-informed Neural Network for system identification of rotors." IFAC-PapersOnLine 58, no. 15 (2024): 307–12. http://dx.doi.org/10.1016/j.ifacol.2024.08.546.
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Shi, Yan, and Michael Beer. "Physics-informed neural network classification framework for reliability analysis." Expert Systems with Applications 258 (December 2024): 125207. http://dx.doi.org/10.1016/j.eswa.2024.125207.
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Costa, Erbet Almeida, Carine Menezes Rebello, Vinícius Viena Santana, and Idelfonso B. R. Nogueira. "Physics-informed neural network uncertainty assessment through Bayesian inference." IFAC-PapersOnLine 58, no. 14 (2024): 652–57. http://dx.doi.org/10.1016/j.ifacol.2024.08.411.
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Hassanaly, Malik, Peter J. Weddle, Kandler Smith, Subhayan De, Alireza Doostan, and Ryan King. "Physics-Informed Neural Network Modeling of Li-Ion Batteries." ECS Meeting Abstracts MA2022-02, no. 3 (2022): 174. http://dx.doi.org/10.1149/ma2022-023174mtgabs.
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Li-ion batteries (LIB) are a promising solution to enable the storage of intermittent energy sources due to their high energy density. However, LIBs are known to significantly degrade after about 1000 charge-discharge cycles. LIBs degrade following different degradation modes and at a rate that depends on the operating conditions (e.g., external temperature, load). To plan the installation of batteries, appropriate understanding and prediction capabilities of their lifecycle is needed. In particular, the LIB degradation model needs to be transferable to variable operating conditions throughout the LIB lifetime. To this end, degradation models of individual LIB battery properties are sought to allow for sufficient granularity in the degradation model. High-fidelity numerical models of LIBs such as the pseudo-two-dimensional (P2D) model have been shown to accurately represent the charge-discharge-cycle of an LIB if the physical parameters used in the model are accurately estimated. Given observations of battery charge-discharged cycles, the objective is to use the P2D model to infer the values of all the battery properties, throughout the battery life. To prevent overfitting and account for the sparse data availability, the overarching objective is to enable Bayesian calibration to solve the inverse problem. Given the number of physical parameters, and the number of cycles to simulate, adjusting parameters directly via P2D forward runs is computationally intractable. This work describes the development of a surrogate model that would replace numerical integration of the P2D equations to significantly reduce the cost of the forward runs. To capture parameter dependencies, a physics-informed neural network (PINN) is developed as a surrogate substitute for the P2D model. The inverse modeling approach is illustrated in the Figure (top). The PINN is advantageous as it needs little to no observational data, which avoids offsetting the reduced inference computational cost with an increased training data generation burden. However, PINNs are notoriously difficult to train in stiff dynamical systems such as the P2D equations. Here, we discuss the specific training procedure that is adopted to efficiently cover parameter space, handle model stiffness, enforce initial, boundary conditions, and treat variables of different magnitudes. Furthermore, a verification procedure akin to ones used in computational fluid dynamics is implemented to ensure that the right governing equations are implemented. An emphasis is placed on verifying the governing equation even in presence of numerical errors. The training procedure and loss convergence are described to highlight training instabilities encountered. In addition, the training cost is evaluated and put in perspective of the forward integration of the P2D equations. Through ablation studies, we discuss what model components are the most critical to appropriately capture P2D solutions. The trained PINN is validated against numerical solutions of the P2D model (sample results are shown in Figure, bottom). In particular, it is assessed whether the PINN can replicate numerical solutions for parameter values not represented in the training data which is key in ensuring that the surrogate can be used for parameter calibration. Figure 1
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Guler Bayazit, Nilgun. "Physics informed neural network consisting of two decoupled stages." Engineering Science and Technology, an International Journal 46 (October 2023): 101489. http://dx.doi.org/10.1016/j.jestch.2023.101489.
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Sha, Yanliang, Jun Lan, Yida Li, and Quan Chen. "A Physics-Informed Recurrent Neural Network for RRAM Modeling." Electronics 12, no. 13 (2023): 2906. http://dx.doi.org/10.3390/electronics12132906.
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Extracting behavioral models of RRAM devices is challenging due to their unique “memory” behaviors and rapid developments, for which well-established modeling frameworks and systematic parameter extraction processes are not available. In this work, we propose a physics-informed recurrent neural network (PiRNN) methodology to generate behavioral models of RRAM devices from practical measurement/simulation data. The proposed framework can faithfully capture the evolution of internal state and its impacts on the output. A series of modifications informed by the RRAM device physics are proposed to enhance the modeling capabilities. The integration strategy of Verilog-A equivalent circuits, is also developed for compatibility with existing general-purpose circuit simulators. The Verilog-A model can be easily adopted into the SPICE-type simulator for the circuit design with a variable step that differs from the training process. Numerical experiments with real RRAM devices data demonstrate the feasibility and advantages of the proposed methodology.
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Liu, Chen-Xu, Xinghao Wang, Weiming Liu, Yi-Fan Yang, Gui-Lan Yu, and Zhanli Liu. "A physics-informed neural network for Kresling origami structures." International Journal of Mechanical Sciences 269 (May 2024): 109080. http://dx.doi.org/10.1016/j.ijmecsci.2024.109080.
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Tarkhov, Dmitriy, Tatiana Lazovskaya, and Galina Malykhina. "Constructing Physics-Informed Neural Networks with Architecture Based on Analytical Modification of Numerical Methods by Solving the Problem of Modelling Processes in a Chemical Reactor." Sensors 23, no. 2 (2023): 663. http://dx.doi.org/10.3390/s23020663.
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A novel type of neural network with an architecture based on physics is proposed. The network structure builds on a body of analytical modifications of classical numerical methods. A feature of the constructed neural networks is defining parameters of the governing equations as trainable parameters. Constructing the network is carried out in three stages. In the first step, a neural network solution to an equation corresponding to a numerical scheme is constructed. It allows for forming an initial low-fidelity neural network solution to the original problem. At the second stage, the network with physics-based architecture (PBA) is further trained to solve the differential equation by minimising the loss function, as is typical in works devoted to physics-informed neural networks (PINNs). In the third stage, the physics-informed neural network with architecture based on physics (PBA-PINN) is trained on high-fidelity sensor data, parameters are identified, or another task of interest is solved. This approach makes it possible to solve insufficiently studied PINN problems: selecting neural network architecture and successfully initialising network weights corresponding to the problem being solved that ensure rapid convergence to the loss function minimum. It is advisable to use the devised PBA-PINNs in the problems of surrogate modelling and modelling real objects with multi-fidelity data. The effectiveness of the approach proposed is demonstrated using the problem of modelling processes in a chemical reactor. Experiments show that subsequent retraining of the initial low-fidelity PBA model based on a few high-accuracy data leads to the achievement of relatively high accuracy.
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Silva, Roberto Mamud Guedes da, Helio dos Santos Migon, and Antônio José da Silva Neto. "Parameter estimation in the pollutant dispersion problem with Physics-Informed Neural Networks." Ciência e Natura 45, esp. 3 (2023): e74615. http://dx.doi.org/10.5902/2179460x74615.
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In this work, the inverse problem of parameter estimation in the advection-dispersion-reaction equation, modelling the pollutant dispersion in a river, is studied with a Neural Network approach. In the direct problem, the dispersion, velocity and reaction parameters are known and then the initial and boundary value problem is solved by classical numerical methods, where it is used as input dataset for the inverse problem and formulation. In the inverse problem, we know the dispersion and the velocity parameters and also the information about the pollutant concentration from the synthetic experimental data, and then the aim is to estimate the reaction parameter in the advection-dispersion-reaction equation. This inverse problem is solved by an usual Artificial Neural Network (ANN) and by a Physics-Informed Neural Network (PINN), which is a special type of neural networks that includes in its formulation the physical laws that describe the phenomena involved. Numerical experiments with both the ANN and PINN are presented, demonstrating the feasibility of the approach considered.
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Liu, Zhixiang, Yuanji Chen, Ge Song, Wei Song, and Jingxiang Xu. "Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics." Mathematics 11, no. 19 (2023): 4147. http://dx.doi.org/10.3390/math11194147.
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Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.
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Oluwasakin, Ebenezer O., and Abdul Q. M. Khaliq. "Optimizing Physics-Informed Neural Network in Dynamic System Simulation and Learning of Parameters." Algorithms 16, no. 12 (2023): 547. http://dx.doi.org/10.3390/a16120547.
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Artificial neural networks have changed many fields by giving scientists a strong way to model complex phenomena. They are also becoming increasingly useful for solving various difficult scientific problems. Still, people keep trying to find faster and more accurate ways to simulate dynamic systems. This research explores the transformative capabilities of physics-informed neural networks, a specialized subset of artificial neural networks, in modeling complex dynamical systems with enhanced speed and accuracy. These networks incorporate known physical laws into the learning process, ensuring predictions remain consistent with fundamental principles, which is crucial when dealing with scientific phenomena. This study focuses on optimizing the application of this specialized network for simultaneous system dynamics simulations and learning time-varying parameters, particularly when the number of unknowns in the system matches the number of undetermined parameters. Additionally, we explore scenarios with a mismatch between parameters and equations, optimizing network architecture to enhance convergence speed, computational efficiency, and accuracy in learning the time-varying parameter. Our approach enhances the algorithm’s performance and accuracy, ensuring optimal use of computational resources and yielding more precise results. Extensive experiments are conducted on four different dynamical systems: first-order irreversible chain reactions, biomass transfer, the Brusselsator model, and the Lotka-Volterra model, using synthetically generated data to validate our approach. Additionally, we apply our method to the susceptible-infected-recovered model, utilizing real-world COVID-19 data to learn the time-varying parameters of the pandemic’s spread. A comprehensive comparison between the performance of our approach and fully connected deep neural networks is presented, evaluating both accuracy and computational efficiency in parameter identification and system dynamics capture. The results demonstrate that the physics-informed neural networks outperform fully connected deep neural networks in performance, especially with increased network depth, making them ideal for real-time complex system modeling. This underscores the physics-informed neural network’s effectiveness in scientific modeling in scenarios with balanced unknowns and parameters. Furthermore, it provides a fast, accurate, and efficient alternative for analyzing dynamic systems.
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Li, Jianfeng, Liangying Zhou, Jingwei Sun, and Guangzhong Sun. "Physically plausible and conservative solutions to Navier-Stokes equations using Physics-Informed CNNs." JUSTC 53 (2023): 1. http://dx.doi.org/10.52396/justc-2022-0174.
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Physics-informed Neural Network (PINN) is an emerging approach for efficiently solving partial differential equations (PDEs) using neural networks. Physics-informed Convolutional Neural Network (PICNN), a variant of PINN enhanced by convolutional neural networks (CNNs), has achieved better results on a series of PDEs since the parameter-sharing property of CNNs is effective to learn spatial dependencies. However, applying existing PICNN-based methods to solve Navier-Stokes equations can generate oscillating predictions, which are inconsistent with the laws of physics and the conservation properties. To address this issue, we propose a novel method that combines PICNN with the finite volume method to obtain physically plausible and conservative solutions to Navier-Stokes equations. We derive the second-order upwind difference scheme of Navier-Stokes equations using the finite volume method. Then we use the derived scheme to calculate the partial derivatives and construct the physics-informed loss function. The proposed method is assessed by experiments on steady-state Navier-Stokes equations under different scenarios, including convective heat transfer, lid-driven cavity flow, etc. The experimental results demonstrate that our method can effectively improve the plausibility and the accuracy of the predicted solutions from PICNN.
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Hooshyar, Saman, and Arash Elahi. "Sequencing Initial Conditions in Physics-Informed Neural Networks." Journal of Chemistry and Environment 3, no. 1 (2024): 98–108. http://dx.doi.org/10.56946/jce.v3i1.345.
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The scientific machine learning (SciML) field has introduced a new class of models called physics-informed neural networks (PINNs). These models incorporate domain-specific knowledge as soft constraints on a loss function and use machine learning techniques to train the model. Although PINN models have shown promising results for simple problems, they are prone to failure when moderate level of complexities are added to the problems. We demonstrate that the existing baseline models, in particular PINN and evolutionary sampling (Evo), are unable to capture the solution to differential equations with convection, reaction, and diffusion operators when the imposed initial condition is non-trivial. We then propose a promising solution to address these types of failure modes. This approach involves coupling Curriculum learning with the baseline models, where the network first trains on PDEs with simple initial conditions and is progressively exposed to more complex initial conditions. Our results show that we can reduce the error by 1 – 2 orders of magnitude with our proposed method compared to regular PINN and Evo.
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Olivieri, Marco, Mirco Pezzoli, Fabio Antonacci, and Augusto Sarti. "A Physics-Informed Neural Network Approach for Nearfield Acoustic Holography." Sensors 21, no. 23 (2021): 7834. http://dx.doi.org/10.3390/s21237834.
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In this manuscript, we describe a novel methodology for nearfield acoustic holography (NAH). The proposed technique is based on convolutional neural networks, with autoencoder architecture, to reconstruct the pressure and velocity fields on the surface of the vibrating structure using the sampled pressure soundfield on the holographic plane as input. The loss function used for training the network is based on a combination of two components. The first component is the error in the reconstructed velocity. The second component is the error between the sound pressure on the holographic plane and its estimate obtained from forward propagating the pressure and velocity fields on the structure through the Kirchhoff–Helmholtz integral; thus, bringing some knowledge about the physics of the process under study into the estimation algorithm. Due to the explicit presence of the Kirchhoff–Helmholtz integral in the loss function, we name the proposed technique the Kirchhoff–Helmholtz-based convolutional neural network, KHCNN. KHCNN has been tested on two large datasets of rectangular plates and violin shells. Results show that it attains very good accuracy, with a gain in the NMSE of the estimated velocity field that can top 10 dB, with respect to state-of-the-art techniques. The same trend is observed if the normalized cross correlation is used as a metric.
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Yonekura, Kazuo. "A Short Note on Physics-Guided GAN to Learn Physical Models without Gradients." Algorithms 17, no. 7 (2024): 279. http://dx.doi.org/10.3390/a17070279.
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This study briefly describes the concept of guided training of deep neural networks (DNNs) to learn physically reasonable solutions. The proposed method does not need the gradients of the physical equations, although the conventional physics-informed models need the gradients. DNNs are widely used to predict phenomena in physics and mechanics. One of the issues with DNNs is that their output does not always satisfy physical equations. One approach to consider with physical equations is adding a residual of the equations into the loss function; this is called physics-informed neural network (PINN). One feature of PINNs is that the physical equations and corresponding residuals must be implemented as part of a neural network model. In addition, the residual does not always converge to a small value. The proposed model is a physics-guided generative adversarial network (PG-GAN) that uses a GAN architecture, in which physical equations are used to judge whether the neural network’s output is consistent with physics. The proposed method was applied to a simple problem to assess its potential usability.
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Duarte, D. H. G., P. D. S. de Lima, and J. M. de Araújo. "Outlier-resistant physics-informed neural network." Physical Review E 111, no. 2 (2025). https://doi.org/10.1103/physreve.111.l023302.
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Manikkan, Sreehari, and Balaji Srinivasan. "Transfer physics informed neural network: a new framework for distributed physics informed neural networks via parameter sharing." Engineering with Computers, July 19, 2022. http://dx.doi.org/10.1007/s00366-022-01703-9.
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Liu Jin-Pin, Wang Bing-Zhong, Chen Chuan-Sheng, and Wang Ren. "Inverse design of microwave waveguide devices based on deep physics-informed neural networks." Acta Physica Sinica, 2023, 0. http://dx.doi.org/10.7498/aps.72.20230031.
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Using physics-informed neural networks to solve physical inverse problems is becoming a trend.However,it is difficult to solve the scheme that only introduces physical knowledge through the loss function.Constructing a reasonable loss function to make the results converge becomes a challenge.To address the challenge of physics-informed neural network models for inverse design of electromagnetic devices,a deep physics-informed neural network is introduced by using pattern matching method.The physical equations have been integrated into the network structure when the network is constructed.This feature makes the deep physics-informed neural network have a more concise loss function and higher computational efficiency when solving physical inverse problems.In addition,the training parameters of deep physics-informed neural networks are physically meaningful compared to traditional physicsinformed neural networks.Users can control the network by parameters more easily.Taking the scattering parameter design of a two-port waveguide as an example,we present a new metal topology inverse design scheme and give a detailed explanation.In numerical experiments,we target a set of physically realizable scattering parameters and inversely design the metallic septum using a deep physics-informed neural network.The results show that the method can not only achieve the design target but also obtain solutions with different topologies.The establishment of multiple solutions is extremely valuable in the implementation of the inverse design.It can allow the designer to decide the size and location of the design area more freely while achieving the performance requirements.This scheme is expected to promote the application and development of the inverse design of electromagnetic devices.
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Dourado, Arinan, and Felipe A. C. Viana. "Physics-Informed Neural Networks for Corrosion-Fatigue Prognosis." Annual Conference of the PHM Society 11, no. 1 (2019). http://dx.doi.org/10.36001/phmconf.2019.v11i1.814.
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In this paper, we present a novel physics-informed neural network modeling approach for corrosion-fatigue. The hybrid approach is designed to merge physics- informed and data-driven layers within deep neural networks. The result is a cumulative damage model where the physics-informed layers are used to model the relatively well understood physics (crack growth through Paris law) and the data-driven layers account for the hard to model effects (bias in damage accumulation due to corrosion). A numerical experiment is used to present the main features of the proposed physics-informed recurrent neural network for damage accumulation. The test problem consists of predicting corrosion-fatigue of an Al 2024-T3 alloy used on panels of aircraft wing. Besides cyclic loading, the panels are also subjected to saline corrosion. The physics-informed neural network is trained using full observation of inputs (far-field loads, stress ratio and a corrosivity index – defined per airport) and very limited observation of outputs (crack length at inspection for only a small portion of the fleet). Results show that the physics-informed neural network is able to learn the correction in the original fatigue model due to corrosion and predictions are accurate enough for ranking damage in different airplanes in the fleet (which can be used to prioritizing inspection).
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Zapf, Bastian, Johannes Haubner, Miroslav Kuchta, Geir Ringstad, Per Kristian Eide, and Kent-Andre Mardal. "Investigating molecular transport in the human brain from MRI with physics-informed neural networks." Scientific Reports 12, no. 1 (2022). http://dx.doi.org/10.1038/s41598-022-19157-w.
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AbstractIn recent years, a plethora of methods combining neural networks and partial differential equations have been developed. A widely known example are physics-informed neural networks, which solve problems involving partial differential equations by training a neural network. We apply physics-informed neural networks and the finite element method to estimate the diffusion coefficient governing the long term spread of molecules in the human brain from magnetic resonance images. Synthetic testcases are created to demonstrate that the standard formulation of the physics-informed neural network faces challenges with noisy measurements in our application. Our numerical results demonstrate that the residual of the partial differential equation after training needs to be small for accurate parameter recovery. To achieve this, we tune the weights and the norms used in the loss function and use residual based adaptive refinement of training points. We find that the diffusion coefficient estimated from magnetic resonance images with physics-informed neural networks becomes consistent with results from a finite element based approach when the residuum after training becomes small. The observations presented here are an important first step towards solving inverse problems on cohorts of patients in a semi-automated fashion with physics-informed neural networks.
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Wang, Shihai, Jiangyong He, Congcong Liu, et al. "Physics-Informed Neural Network for Fiber Amplifiers." Journal of Lightwave Technology, 2025, 1–6. https://doi.org/10.1109/jlt.2025.3545238.
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