基于守恒约束物理信息神经网络的刚性化学动力学长时模拟
doi: 10.11728/cjss2025.02.2024-0149 cstr: 32142.14.cjss.2024-0149
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方涵敏 女, 1998年12月出生于安徽省合肥市, 现为安徽工业大学计算机科学与技术学院硕士研究生, 主要研究方向为AI for Science和智能科学计算. E-mail: fanghanmin@ahut.edu.cn -
黄文龙 男, 1988年8月出生于安徽省铜陵市, 现为安徽工业大学计算机科学与技术学院副教授, 硕士生导师, 主要研究方向为AI for Science、空间等离子体物理、聚变等离子体物理等. E-mail: whuang@ahut.edu.cn
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Long-time Simulation of Stiff Chemical Kinetics Using Conservation-constrained Physics-informed Neural Network
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摘要: 刚性化学动力学方程在空间科学、大气科学等领域具有重要意义. 近年来, 物理信息神经网络(Physics-Informed Neural Network, PINN)作为一种融合物理定律与深度学习的框架, 被广泛应用于求解各种偏微分方程. 然而, 在求解刚性化学动力学方程时, PINN将出现优化失败, 难以有效求解. 为解决该问题, 本文提出一种新的守恒约束PINN方法. 该方法利用共享–分支网络有效处理耦合问题, 通过引入物质守恒约束显著提升刚性化学动力学方程的求解性能. 同时, 分段采样策略进一步增强长时模拟的精度和稳定性. 实验结果表明, 该方法能够在多时间尺度的复杂系统中实现长时稳定模拟, 为解决空间科学中的问题(例如无碰撞等离子体波动和星际物质化学反应)提供了一种新的方法.Abstract: Long-term simulation of Partial Differential Equations (PDEs) holds significant applications across various fields, including space physics and atmospheric science. Conventional numerical techniques, such as the finite difference, finite element, and finite volume methods have been extensively employed to solve PDEs across various disciplines. However, these methods often struggle with dimensional curse and complex geometry. In recent years, Physics-Informed Neural Network (PINN), which integrates physical laws within deep learning frameworks, has emerged as a powerful alternative for solving PDEs. Since PINN and its variants are mesh-free, they can avoid dimensional curse to a certain degree. Nonetheless, deep learning related approaches frequently encounter optimization challenges, particularly when applied to multi-time scale issues such as stiff chemical kinetics equations, which involve multiple reactions with different rates, leading to both fast and slow dynamics coexisting. To address these issues, this study introduces a novel Conservation-Constrained Physics-Informed Neural Network (CC-PINN) approach. This method combines shared-branch networks with a segmented sampling strategy. First, the shared-branch networks can effectively deal with coupling equations and reduce the difficulties during the optimization of neural networks. On the other hand, the conservation constraint is embedded into the loss function, ensuring the conservation of physical laws and the accuracy of the simulation results, which significantly improves the performance of PINN. At the same time, according to the dynamics of chemical kinetics in different time intervals, the segmented sampling strategy is adopted, which further improves the accuracy and stability of long-term simulation. In addition, the influence of different expressions of conservation constraints has also been discussed. Experimental results clearly show that, by combining the shared-branch networks and segmented sampling strategy, the new proposed CC-PINN can accurately integrate the stiff chemical kinetics equations in a long-time scale. In summary, this research contributes a new tool for solving problems, such as collisionless plasma fluctuations and interstellar matter chemical reaction, in space science.
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图 3 无守恒约束条件下的预测结果(a)与损失函数(b)
Figure 3. Predicted result (a) and loss function (b) without conservation constraint
图 4 加入守恒约束后的预测结果(a)和损失函数(b)
Figure 4. Predicted result (a) and loss function (b) with conservation constraint
图 5 无物质守恒约束 (a) 和有物质守恒约束 (b) 守恒量E的预测值和理论值
Figure 5. Predicted solution and theoretical one of conservation quantity E without (a) and with (b) conservation constraint
图 7 守恒约束PINN在随机采样策略下的预测结果(a)(c)和损失函数(b)(d). (a) (b)采样点数为1000, (c) (d)采样点数为20000
Figure 7. Predicted results (a)(c) and loss functions (b)(d) with conservation constraint of PINN under random sampling strategy. The number of sampling points in panels (a) and (b) is 1000, while in panels (c) and (d) is 20000
图 8 随机采样策略 (a) 和分段采样策略 (b)
Figure 8. Schematic of random sampling strategy (a) and segmented sampling strategy (b)
图 9 守恒约束PINN在分段采样策略下的预测结果(a)和损失函数(b)
Figure 9. Predicted results (a) and loss function (b) with conservation constraint of PINN under segmented sampling strategy
表 1 随机采样策略和分段采样策略的相对误差
Table 1. Relative error of the random sampling strategy and the segmented sampling strategy
采样策略 采样点数量 Adam L-BFGS $ {s}_{1} $相对误差/(%) $ {s}_{2} $相对误差/(%) $ {s}_{3} $相对误差/(%) 随机采样 20000, t ∈[0, 500] 35000 5000 1.22 99.99 1.42 分段采样 12000, t ∈ [0, 50];
8000, t ∈ [50, 500]35000 5000 0.40 10.72 0.47 
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表 2 三种不同形式物质守恒约束对应的预测结果
Table 2. Predicted results for three different forms of material conservation constraint
守恒形式 Adam L-BFGS 时间/min $ {s}_{1} $相对误差/(%) $ {s}_{2} $相对误差/(%) $ {s}_{3} $相对误差/(%) $ E = 1 $ 35000 5000 13.58 0.40 10.72 0.46 $ {E^2} = 1 $ 35000 5000 12.26 85.38 99.99 99.71 $ \sqrt E = 1 $ 35000 5000 13.59 0.52 99.99 0.64
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