Gradient-weighted physics-informed neural networks for one-dimensional Euler equation

Xiong, Fuqin; Liu, Li; Liu, Shengping; Wang, Han; Huang, Yong

doi:10.36227/techrxiv.20099957

Cited by 3 publications

(3 citation statements)

References 25 publications

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“…where κ IC , κ BC and κ f are constants and denote the weights assigned to the initial, boundary training points and collocation points, respectively. In addition, many differential equations have complex forms of solutions, which makes it challenging to train deep neural networks in the key region [18]. Researchers have tried many methods to overcome this difficulty, such as clustering method [27], RAR method [28], Weighted equation method [17,18], etc., and they have all achieved good results.…”

Section: Lossmentioning

confidence: 99%

“…In addition, many differential equations have complex forms of solutions, which makes it challenging to train deep neural networks in the key region [18]. Researchers have tried many methods to overcome this difficulty, such as clustering method [27], RAR method [28], Weighted equation method [17,18], etc., and they have all achieved good results. Therefore, in the second weighting strategy, we develop the weighted equation method in order to avoid the situation that some key regions are difficult to train.…”

Section: Lossmentioning

confidence: 99%

“…Liu et al [17] proposed a strategy to weaken the representation of neural networks in the vicinity of discontinuities by adding a physics-dependent parameter into the physical information locally at each residual point. Xiong et al [18] introduced the gradient-related weight function and sequence-to-sequence learning into PINN to solve four cases of Riemann problems. Recently, neural network-based methods have been used to solve fractional differential equations [19,20].…”

Section: Introductionmentioning

confidence: 99%

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Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Liu

Fan

et al. 2023

Phys. Scr.

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In this article, we introduce the modiﬁed physics-informed neural network (PINN) method for ﬁnding data-driven solutions of three classes of time-fractional Burgers-type equations under the conformable sense. Since conformable derivative satisﬁes the chain rule, automatic diﬀerentiation can be applied to compute it directly to avoid truncation and other numerical discretization. In addition, the locally adaptive activation function and two effective weighting strategies are introduced to improve solution accuracy. As a result, three numerical examples indicate that the modiﬁed PINN method gives an eﬃcient and reliable solution.

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Section: Lossmentioning

confidence: 99%

Section: Lossmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Liu

Fan

et al. 2023

Phys. Scr.

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Continuous and discontinuous compressible flows in a converging–diverging channel solved by physics-informed neural networks without exogenous data

Liang,

Song,

Zhao

et al. 2024

Sci Rep

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Physics-informed neural networks (PINNs) are employed to solve the classical compressible flow problem in a converging–diverging nozzle. This problem represents a typical example described by the Euler equations, a thorough understanding of which serves as a guide for solving more general compressible flows. Given a geometry of the channel, analytical solutions for the steady states do indeed exist, and they depend on the ratio between the back pressure of the outlet and the stagnation pressure of the inlet. Moreover, in the diverging region, the solution may branch into subsonic flow, supersonic flow, or a mixture of both with a discontinuous transition where a normal shock occurs. Classical numerical schemes with shock fitting and capturing methods have been developed to solve this type of problem effectively, whereas the original PINNs are unable to predict the flows correctly. We make a first attempt to exploit the power of PINNs to solve this problem directly by adjusting the weights of different components of the loss function to acquire physical solutions and in the meantime, avoid trivial solutions. With a universal setting yet no exogenous data, we are able to solve this problem accurately; that is, for different given pressure ratios, PINNs provide different branches of solutions at both steady and unsteady states, some of which are discontinuous in nature. For an inverse problem such as unknown specific-heat ratio, it works effectively as well.

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Applications of Symmetry-Enhanced Physics-Informed Neural Networks in High-Pressure Gas Flow Simulations in Pipelines

Alpar,

Faizulin,

Tokmukhamedova

et al. 2024

Symmetry

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This article presents a detailed examination of the methodology and modeling tools utilized to analyze gas flows in pipelines, rooted in the fundamental principles of gas dynamics. The methodology integrates numerical simulations with modern neural network techniques, particularly focusing on the PINN utilizing the continuous symmetry data inherent in PDEs, which is called the symmetry-enhanced Physics-Informed Neural Network. This innovative approach combines artificial neural networks (ANNs) integrating physical equations, which provide enhanced efficiency and accuracy when modeling various complex processes related to physics with a symmetric and asymmetric nature. The presented mathematical model, based on the system of Euler equations, has been carefully implemented using Python language. Verification with analytical solutions ensures the accuracy and reliability of the computations. In this research, a comparative and comprehensive analysis was carried out comparing the outcomes obtained using the symmetry-enhanced PINN method and those from conventional computational fluid dynamics (CFD) approaches. The analysis highlighted the advantages of the symmetry-enhanced PINN method, which produced smoother pressure and velocity fluctuation profiles while reducing the computation time, demonstrating its capacity as a revolutionary modeling tool. The estimated results derived from this study are of paramount importance for ensuring ongoing energy supply reliability and can also be used to create predictive models related to gas behavior in pipelines. The application of modeling techniques for gas flow simulations has the potential to improve the integrity of our energy infrastructure and utilization of gas resources, contributing to advancing our understanding of symmetry principles in nature. However, it is crucial to emphasize that the effectiveness of such models relies on continuous monitoring and frequent updates to ensure alignment with real-world conditions. This research not only contributes to a deeper understanding of compressible gas flows but also underscores the crucial role of advanced modeling methodologies in the sustainable management of gas resources for both current and future generations. The numerical data covered the physics of the process related to the modeling of high-pressure gas flows in pipelines with regard to density, velocity and pressure, where the PINN model was able to outperform the classical CFD method for velocity by 170% and for pressure by 360%, based on L∞ values.

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Gradient-weighted physics-informed neural networks for one-dimensional Euler equation

Cited by 3 publications

References 25 publications

Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Continuous and discontinuous compressible flows in a converging–diverging channel solved by physics-informed neural networks without exogenous data

Applications of Symmetry-Enhanced Physics-Informed Neural Networks in High-Pressure Gas Flow Simulations in Pipelines

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