1. State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, China 2. Department of Mechanical Engineering and Science, Kyoto University, Kyoto 6158540, Japan
Two typical cases including the stiff system of ordinary differential equations ROBER problem and the steady-state mixture fraction equation in jet flame were selected in order to efficiently embed the complex physicochemical information of turbulent combustion into physics-informed neural networks (PINNs). The potential of PINNs in solving combustion chemical differential equations was explored. Results show that the PINNs model can correctly capture the evolution of the zero-dimensional stiff reaction system. PINNs solution accorded well with the conventional numerical solution for steady jet flame. The selection of residual points was particularly important for solving complex differential equations in the field of combustion and chemistry, which should be considered based on the specific configuration in detail.
Tab.1Training loss for different hidden layers for ROBER problem (Te = 1 s)
层类型
激活函数
输出类型
ROBER问题
混合分数PDE
Input
—
(None, 1): t
(None, 2): x, y
Dense
tanh
(None, 100)
(None, 200)
Dense
tanh
(None, 100)
(None, 200)
Dense
tanh
(None, 100)
(None, 200)
Dense
tanh
(None, 100)
(None, 200)
Output
#
(None, 3): y1, y2, y3
(None, 1): Z
Tab.2Structure of sequential PINNs model
Fig.2Evolutionary process of species in dynamic system (Te = 1 s)
Fig.3Evolutionary process of species in dynamic system (Te = 103 s)
Fig.4Two-dimensional configuration of jet diffusion flame
Fig.5Comparison of mixture fraction distributions
Fig.6Comparison of mixture fraction profiles at different locations
Fig.7Comparison of temperature distributions
Fig.8Distribution of residual points
案例
采样方式
ttr/s
L
ROBER问题
基于构型
621.64
4.40×10?7
ROBER问题
均匀
619.89
3.36×10?4
ROBER问题
伪随机
607.62
2.46×10?4
混合分数PDE
基于构型
691.61
5.48×10?3
混合分数PDE
均匀
700.52
4.67×10?3
混合分数PDE
伪随机
700.82
2.55×10?3
Tab.3Training time and loss value for different sampling approaches
Fig.9Loss value vs. iterations for different sampling approaches
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