Surrogate-Based Physics-Informed Neural Networks for Elliptic Partial Differential Equations

Zhi, Peng; Wu, Yuching; Qi, Cheng; Zhu, Tao; Wu, Xiao; Wu, Hongyu

doi:10.3390/math11122723

Cited by 4 publications

(2 citation statements)

References 36 publications

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“…The deep learning technique has been broadly investigated, and the surrogate approach has also been widely explored and applied in different areas [ 30 ]. But, corresponding mathematical theories seem to be limited.…”

Section: Theory Of the Surrogate Finite Element Methodsmentioning

confidence: 99%

Structural optimization of single-layer domes using surrogate-based physics-informed neural networks

Wu,

Zhi

et al. 2023

Heliyon

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Section: Theory Of the Surrogate Finite Element Methodsmentioning

confidence: 99%

Structural optimization of single-layer domes using surrogate-based physics-informed neural networks

Wu,

Zhi

et al. 2023

Heliyon

Self Cite

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“…In recent years, PINN has led to significant changes in numerical simulation technology, and the method for solving PDEs based on PINN not only enables fast forward modeling and inversion modeling [45][46], but also effectively solves nonlinear problems [47][48][49], and can solve more complex and high-dimensional PDEs [50][51][52]. Falini et al (2023) and Zhi et al (2023) have even argued that as PINN has better stability, it can be used as an alternative to the traditional FEM in solving PDEs [53][54].…”

Section: Solving Pdes Based On Pinnmentioning

confidence: 99%

Compressible Non-Newtonian Fluid Based Road Traffic Flow Equation Solved by Physical-Informed Rational Neural Network

Yang,

Li,

Nai

et al. 2024

IEEE Access

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The study of road traffic flow theory utilizes physics and applied mathematics to analyze relevant parameters and their relationships quanlitatively and quantitatively, in order to explore their dynamic changes. The fluid dynamics model used for traffic flow analysis is highly favored by scholars due to its solid mathematical foundation and good simulation results. However, existing models have two main shortcomings: firstly, existing research is mostly limited to non-viscoelastic fluid equation or incompressible non-Newtonian fluid equation, making it difficult to accurately describe the viscosity state and micro cluster properties of the actual traffic flow; secondly, the existing non-Newtonian fluid partial differential equations (PDEs) rely heavily on the finite element method (FEM) for solving, requiring higher computational cost, larger storage space, and more constraint conditions. Thus, in this paper, a traffic flow equation based on compressible non-Newtonian fluid has been constructed, and it has been solved by using physical-informed rational neural network (PIRNN) and noise heavy-ball acceleration gradient descent (NHAGD) to ensure learning and training speed and accuracy. Numerical results indicate that the proposed method can truly reflect the gradual change process in the viscosity of traffic flow, and has better solving performance than traditional FEM and physical-informed neural network (PINN) with activation functions; under the same conditions, the prediction error of the proposed method is also smaller than that of traditional traffic flow models.

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Sequence-to-Sequence Stacked Gate Recurrent Unit Networks for Approximating the Forward Problem of Partial Differential Equations

Zhang,

Wang

2024

IEEE Access

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We proposed an optimisation algorithm based on the sequence-to-sequence (Seq2Seq) stacking of the gate recurrent unit (GRU) model to characterise and approximate the forward problem of partial differential equations (PDEs). Unlike traditional methods based on mesh differential approximation and no parameters, this is a meshless approach based on parametric semi-supervised learning. Specifically, the algorithm employs the ability of deep feedback neural networks to approximate continuous dynamical systems, which is enhanced by stacked GRU modules to capture the evolution of the PDEs over time and thus enrich the representations of the sequences in the hidden space. The loss function of the model incorporates partial physical knowledge as an a priori condition to guide the optimisation direction, i.e., transforming the numerical iterative problem into a non-convex optimisation problem. In addition, each round of training of the model incorporates data resampling to prevent it from overfitting. We evaluated the ability of the proposed algorithm to solve mathematical physics equations for a variety of differential operators and constraints, including the heat, wave, Burgers, Schrodinger, diffusion, and Kovasnay flow equations. The experimental results confirmed the outstanding prediction precision and generalisation capability of the proposed algorithm.INDEX TERMS Forward problem of partial differential equations, Deep Learning, Gate recurrent unit, Sequence-to-sequence.

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Surrogate-Based Physics-Informed Neural Networks for Elliptic Partial Differential Equations

Cited by 4 publications

References 36 publications

Structural optimization of single-layer domes using surrogate-based physics-informed neural networks

Structural optimization of single-layer domes using surrogate-based physics-informed neural networks

Compressible Non-Newtonian Fluid Based Road Traffic Flow Equation Solved by Physical-Informed Rational Neural Network

Sequence-to-Sequence Stacked Gate Recurrent Unit Networks for Approximating the Forward Problem of Partial Differential Equations

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