Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics
Abstract
:1. Introduction
2. Materials and Methods
2.1. Single-Relaxation-Time Lattice Boltzmann Method (SRT-LBM)
2.2. Network Structure
2.3. Loss Function and Optimization Method
2.4. Dataset Construction
3. Results
3.1. Sod Shock Tube
3.2. Lid-Driven Cavity Flow
3.3. Flow around Circular Cylinder
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Dataset | Xexact | ρexact | Uexact | T |
---|---|---|---|---|
Sod shock tube | Ns × D1 | Ns × Ts | Ns × D1 × Ts | Ts |
Lid-driven cavity flow | NL × D2 | NL × TL | NL × D2 × TL | TL |
Flow around circular cylinder | Ncy × D2 | Ncy × Tcy | Ncy × D2 × Tcy | Tcy |
Model | erru (%) | errp (%) | errρ (%) |
---|---|---|---|
PINN-SRT-LBM-I | 6.47% | 1.64% | 1.85% |
PINN-SRT-LBM-II | 15.87% | 10.56% | 14.87% |
DNNs | 18.87% | 5.66% | 5.75% |
Model | erru (%) | errv (%) | errρ (%) |
---|---|---|---|
DNNs | 2.01% | 1.76% | 0.23% |
PINN-SRT-LBM-I | 0.30% | 0.26% | 0.05% |
PINN-SRT-LBM-II | 0.08% | 0.06% | 0.03% |
Re | A | B | C | D |
---|---|---|---|---|
400 | (0.5563, 0.6000) | (0.5547, 0.6055) | (0.5608, 0.6078) | (0.5556, 0.6000) |
1000 | (0.5438, 0.5625) | (0.5313, 0.5625) | (0.5333, 0.5647) | (0.5327, 0.5652) |
2000 | (0.5226, 0.5482) | (0.5255, 0.5490) | (0.5250, 0.5500) | (0.5254, 0.5499) |
5000 | (0.5125, 0.5313) | (0.5117, 0.5352) | (0.5176, 0.5373) | (0.5137, 0.5424) |
Re | erru (%) | errv (%) | errρ (%) |
---|---|---|---|
400 | 0.06% | 0.09% | 0.05% |
1000 | 0.08% | 0.06% | 0.04% |
2000 | 0.09% | 0.06% | 0.04% |
5000 | 0.13% | 0.05% | 0.04% |
Model | erru (%) | errv (%) |
---|---|---|
DNNs | 13.48% | 20.51% |
PINN-SRT-LBM-I | 2.71% | 2.47% |
PINN-SRT-LBM-II | 0.49% | 0.51% |
PINN-NS | 0.35% | 0.72% |
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Liu, Z.; Chen, Y.; Song, G.; Song, W.; Xu, J. Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics. Mathematics 2023, 11, 4147. https://doi.org/10.3390/math11194147
Liu Z, Chen Y, Song G, Song W, Xu J. Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics. Mathematics. 2023; 11(19):4147. https://doi.org/10.3390/math11194147
Chicago/Turabian StyleLiu, Zhixiang, Yuanji Chen, Ge Song, Wei Song, and Jingxiang Xu. 2023. "Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics" Mathematics 11, no. 19: 4147. https://doi.org/10.3390/math11194147
APA StyleLiu, Z., Chen, Y., Song, G., Song, W., & Xu, J. (2023). Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics. Mathematics, 11(19), 4147. https://doi.org/10.3390/math11194147