基于神经网络的粒子输运问题高效计算方法

马锐垚; 王鑫; 李树; 勇珩; 上官丹骅

doi:10.7498/aps.73.20231661

摘要

蒙特卡罗方法是求解粒子输运问题的有力工具之一, 其局限性在于为达到精度要求需模拟大量粒子, 计算耗时长, 这阻碍了该方法的进一步应用, 尤其在需快速响应的情形. 本文结合神经网络和若干蒙特卡罗方法基本原理发展了一种计算方法, 能够实现源分布可变, 几何、材料和目标计数不变的中子输运问题的快速准确求解. 首先, 为高效生成用于神经网络训练的数据, 利用重要性原理实现在同样模拟次数基础上有效扩充训练数据集容量, 在一定程度上克服了使用蒙特卡罗计算获取训练数据耗时长的缺点. 进而, 基于目标计数是源分布与重要性函数乘积积分的事实, 设计了利用神经网络实现快速输运计算的策略. 该网络的输入是中子源项, 输出是目标计数, 在几何、材料和目标计数固定的情况下, 该神经网络可重复使用, 根据新的源项快速准确得到目标计数. 本文所提出方法的原理和框架同样适用于其他种类粒子的同类型输运问题. 基于若干基准模型的验证表明, 训练得到的神经网络能在不到1 s的时间内得到目标计数, 且与蒙特卡罗大样本模拟得到基准结果的平均相对偏差均低于5%.

关键词:

Abstract

Monte Carlo (MC) method is a powerful tool for solving particle transport problems. However, it is extremely time-consuming to obtain results that meet the specified statistical error requirements, especially for large-scale refined models. This paper focuses on improving the computational efficiency of neutron transport simulations. Specifically, this study presents a novel method of efficiently calculating neutron fixed source problems, which has many applications. This type of particle transport problem aims at obtaining a fixed target tally corresponding to different source distributions for fixed geometry and material. First, an efficient simulation is achieved by treating the source distribution as the input to a neural network, with the estimated target tally as the output. This neural network is trained with data from MC simulations of diverse source distributions, ensuring its reusability. Second, since the data acquisition is time consuming, the importance principle of MC method is utilized to efficiently generate training data. This method has been tested on several benchmark models. The relative errors resulting from neural networks are less than 5% and the times needed to obtain these results are negligible compared with those for original Monte Carlo simulations. In conclusion, in this work we propose a method to train neural networks, with MC simulation results containing importance data and we also use this network to accelerate the computation of neutron fixed source problems.

Keywords:

作者及机构信息

1.
北京应用物理与计算数学研究所, 北京　100094

2.
中物院高性能数值模拟软件中心, 北京　100088

通信作者: 上官丹骅, sgdh@iapcm.ac.cn

基金项目: 国家自然科学基金(批准号:12305173, 12375164, 12331010)、国家自然科学基金委员会-中国工程物理研究院NSAF联合基金(批准号: U2230208)和核数据重点实验室基金 (批准号: JCKY2022201C155)资助的课题.

Authors and contacts

1.
Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

2.
CAEP Software Center for High Performance Numerical Simulation, Beijing 100088, China

Corresponding author: Shangguan Dan-Hua, sgdh@iapcm.ac.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12305173, 12375164, 12331010), the Joint Fund of the National Natural Science Foundation of China (Grant No. U2230208), and the Key Laboratory of Nuclear Data Foundation, China (Grant No. JCKY2022201C155).

文章全文 : translate this paragraph

参考文献

[1]	Wu Y C 2017 Fusion Neutronics (Singapore: Springer Singapore) p21
[2]	邓力, 李刚 2019 粒子输运问题的蒙特卡罗模拟方法与应用 (北京: 科学出版社) Deng L, Li G 2019 Monte Carlo Simulation Methods and Applications for Particle Transport Problems (Beijing: Science Press
[3]	Shangguan D H, Yan W H, Wei J X, Gao Z M, Chen Y B, Ji Z C 2023 Nucl. Sci. Tech. 34 58 Google Scholar
[4]	上官丹骅, 闫威华, 魏军侠, 高志明, 陈艺冰, 姬志成 2022 物理学报 71 090501 Google Scholar Shangguan D H, Yan W H, Wei J X, Gao Z M, Chen Y B, Ji Z C 2022 Acta. Phys. Sin. 71 090501 Google Scholar
[5]	Martin W R 2012 Nucl. Eng. Technol. 44 151 Google Scholar
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[7]	Hassan M 2020 Int. J. Sci. Res. 9 913 Google Scholar
[8]	Gul A 2016 Prog. Nucl. Energy 92 164 Google Scholar
[9]	Huang Z P, Cao L Z, He Q M, Wu H C 2022 J. Nucl. Sci. Technol. 59 1375 Google Scholar
[10]	García-Pareja S, Lallena A M, Salvat F 2021 Front. Phys. 9 633 Google Scholar
[11]	Pooneh S, Mahdi S, Claudio T (Victor C Ed.) 2013 Theory and Applications of Monte Carlo Simulations (Rijeka: IntechOpen) pp152–172
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[15]	Davis A, Turner A 2011 Fusion Eng. Des. 86 2698 Google Scholar
[16]	Wagner J C, Haghighat A 1998 Nucl. Sci. Eng. 128 186 Google Scholar
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[18]	Sun A K, Chen Z P, Li L M, Liu C W, Yu T 2022 Proceedings of the 23rd Pacific Basin Nuclear Conference Beijing, China, November 1–4, 2022 p821
[19]	Ramón J, Peña J 1995 Comput. Phys. Commun. 88 76 Google Scholar
[20]	LeCun Y, Bengio Y, Hinton G 2015 Nature 521 436 Google Scholar
[21]	Schmidhuber J 2015 Neural Networks 61 85 Google Scholar
[22]	Webb S 2018 Nature 554 555 Google Scholar
[23]	田十方, 李彪 2023 物理学报 72 100202 Google Scholar Tian S F, Li B 2023 Acta Phys. Sin. 72 100202 Google Scholar
[24]	Xiao M J, Yu T C, Zhang Y S, Yong H 2023 Comput. Fluids 266 106025 Google Scholar
[25]	Liu L, Liu S P, Xie H, Xiong F S, Yu T C, Xiao M J, Liu L F, Yong H 2024 J. Sci. Comput. 98 22 Google Scholar
[26]	Xie H, Zhai C L, Liu L, Yong H 2022 arXiv: 2205.06658v1 [math. NA]
[27]	胡泽华, 应阳君, 永珩, 续瑞瑞 2023 原子能科学技术 57 812 Google Scholar Hu Z H, Ying Y J, Yong H, Xu R R 2023 At. Energy Sci. Technol. 57 812 Google Scholar
[28]	Lu H Y, Li C H, Chen B B, Li W, Qi Y, Meng Z Y 2022 Chin. Phys. Lett. 39 050701 Google Scholar
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[32]	杨莹, 曹怀信 2023 物理学报 72 110301 Google Scholar Yang Y, Cao H X 2023 Acta Phys. Sin. 72 110301 Google Scholar
[33]	Berry J J, Gil-Delgado G G, Osborne A G S 2021 Ann. Nucl. Energy 160 108367 Google Scholar
[34]	Cao P, Gan Q, Song J, Long P C, Wang F, Hu L Q, Wu Y C 2020 Ann. Nucl. Energy 138 107 Google Scholar
[35]	Zhang G M, Song Y M, Zhang Z H, Yuan W W 2022 Ann. Nucl. Energy 175 109248 Google Scholar
[36]	张海明, 张昊春 2022 现代应用物理 13 020209 Google Scholar Zhang H M, Zhang H C 2022 Mod. Appl. Phys. 13 020209 Google Scholar
[37]	林海鹏, 李国栋, 陈法国, 韩毅, 梁润成 2020 辐射防护 40 516 Lin H P, Li G D, Chen F G, Han Y, Liang R C 2020 Radiat. Prot. 40 516
[38]	Osborne A, Dorville J, Romano P 2023 Energy AI 13 100247 Google Scholar
[39]	Kim S H, Shin S G, Han S, Kim M H, Pyeon C H 2020 Prog. Nucl. Energy 119 103183 Google Scholar
[40]	Zhou S K, Greenspan H, Davatzikos C, Duncan J S, Van Ginneken B, Madabhushi A, Prince J L, Rueckert D, Summers R M 2021 Proc. IEEE 109 820 Google Scholar
[41]	Peng Z, Shan H M, Liu T Y, Pei X, Wang G, Xu X G 2019 IEEE Access 7 76680 Google Scholar
[42]	Ma J H, Piao Z, Huang S, Duan X M, Qin G G, Zhou L H, Xu Y 2021 Photonics Res. 9 B45 Google Scholar
[43]	Romano P K, Horelik N E, Herman B R, Nelson A G, Forget B, Smith K 2015 Ann. Nucl. Energy 82 90 Google Scholar
[44]	Abadi M, Agarwal A, Barham P, et al. 2016 arXiv: 1603.04467 [cs. DC]
[45]	Kingma D P, Ba J L 2014 arXiv: 1412.6980 [cs. LG]
[46]	Kobayashi K, Sugimura N, Nagaya Y 2001 Prog. Nucl. Energy 39 119 Google Scholar
[47]	Igor Z, Richard S 2001 Prog. Nucl. Energy 39 207 Google Scholar
[48]	Remec I, Kam F B K 1998 H. B. Robinson-2 Pressure Vessel Benchmark Report (United States) p2
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施引文献

图 1 本文技术路线图

Fig. 1. Study framework of this paper.

下载: 全尺寸图片幻灯片

图 2 用于代理加速MC模拟的神经网络结构

Fig. 2. Data-driven neural network for Monte Carlo simulation.

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图 3 重要性结果和MC模拟结果对比(MC-MC计算结果, IMP-重要性原理计算结果)

Fig. 3. Comparison of results of MC simulation and importance data (MC- results of Monte Carlo simulation, IMP- results of importance data).

下载: 全尺寸图片幻灯片

图 4 网络训练损失曲线　(a) 使用 1000 个MC模拟结果样本进行训练; (b) 结合重要性数据和MC结果进行训练

Fig. 4. Network training loss curves: (a) Trained with 1000 results samples of MC simulation; (b) trained with a combination of importance data and MC results.

下载: 全尺寸图片幻灯片

图 5 Kobayashi-1模型几何示意图

Fig. 5. Diagram of Kobayashi-1 benchmark

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图 6 Kobayashi-1模型不同探测器的网络预测偏差分布

Fig. 6. Deviation distributions of predicted-results for different detectors by networks of Kobayashi-1.

下载: 全尺寸图片幻灯片

图 7 HBR-2模型几何结构示意图

Fig. 7. Detector geometry diagram of HBR-2 benchmark.

下载: 全尺寸图片幻灯片

图 8 HBR-2模型不同探测器的网络预测偏差分布

Fig. 8. Deviation distributions of predicted-results for different detectors by networks of HBR-2.

下载: 全尺寸图片幻灯片

表 1 简单模型相空间划分参数

Table 1. Phase space meshing parameters of the Fe model.

变量	范围	网格尺寸	网格数目
X/Y/Z	–3—3 cm	0.2 cm	30
E	9—10 MeV	0.2 MeV	10
U	1.04—2.05 rad	0.2 rad	5
W	1.04—2.05 rad	0.2 rad	5
注: X/Y/Z 为相空间网格空间维度在X/Y/Z 方向坐标; E 为相空间网格能量维度坐标; U/W 为相空间网格角度维度在极角和方位角的坐标

下载: 导出CSV

表 2 Kobayashi模型相空间划分参数

Table 2. Phase space division parameters of Kobayashi benchmark.

变量	范围	网格尺寸	网格数目
X/Y/Z	–10.1—10.1 cm	0.2 cm	100
E	—	—	—
U	0—1.57 rad	0.79 rad	11
W	0—1.57 rad	0.79 rad	11

下载: 导出CSV

表 3 Kobayashi模型不同探测器的网络预测结果

Table 3. Prediction results for different detectors by networks of Kobayashi-1.

基准题	探测器位置/cm	训练偏差	验证偏差	测试偏差
Kobayashi-1-i	(15, 15, 15)	0.0424	0.0458	0.0408
	(25, 25, 25)	0.0426	0.0431	0.0442
	(35, 35, 35)	0.0473	0.0478	0.0458
	(45, 45, 45)	0.0473	0.0483	0.0452
Kobayashi-1-ii	(15, 15, 15)	0.0392	0.0401	0.0387
	(25, 25, 25)	0.0394	0.0404	0.0364
	(35, 35, 35)	0.0402	0.0456	0.0443
	(45, 45, 45)	0.0413	0.0455	0.0425

下载: 导出CSV

表 4 Kobayashi-1模型不同探测器的网络预测偏差分布

Table 4. Deviation distributions of predicted-results for different detectors by networks of Kobayashi-1.

基准题	探测器位置/cm	测试偏差
基准题	探测器位置/cm	均值	最大值	标准差	< 0.05比例^*/%
Kobayashi-1-i	(15, 15, 15)	0.0408	0.152	0.0078	95.5
	(25, 25, 25)	0.0442	0.178	0.0116	94.2
	(35, 35, 35)	0.0458	0.214	0.0256	92.6
	(45, 45, 45)	0.0452	0.220	0.0250	92.2
Kobayashi-1-ii	(15, 15, 15)	0.0387	0.126	0.0106	96.2
	(25, 25, 25)	0.0364	0.211	0.0230	95.7
	(35, 35, 35)	0.0443	0.245	0.0288	93.4
	(45, 45, 45)	0.0425	0.253	0.0298	93.3
注: *预测结果与真值相对偏差小于5%的样本占总测试样本量的比例

下载: 导出CSV

表 5 HBR-2模型不同探测器的网络预测结果

Table 5. Prediction results for different location detector networks of HBR-2 benchmark.

探测器	训练偏差	验证偏差	测试偏差
①	0.0413	0.0402	0.0398
②	0.0425	0.0412	0.0442

下载: 导出CSV

表 6 HBR-2模型不同探测器的网络预测偏差分布

Table 6. Deviation distributions of predicted-results for different detectors by networks of HBR-2.

探测器	测试偏差
探测器	均值	最大值	标准差	< 0.05比例^*
①	0.0398	0.0826	0.0096	92.3%
②	0.0442	0.0965	0.0132	90.2%
*预测结果与真值相对偏差小于5%的样本占总测试样本量的比例

下载: 导出CSV

[1]	Wu Y C 2017 Fusion Neutronics (Singapore: Springer Singapore) p21
[2]	邓力, 李刚 2019 粒子输运问题的蒙特卡罗模拟方法与应用 (北京: 科学出版社) Deng L, Li G 2019 Monte Carlo Simulation Methods and Applications for Particle Transport Problems (Beijing: Science Press
[3]	Shangguan D H, Yan W H, Wei J X, Gao Z M, Chen Y B, Ji Z C 2023 Nucl. Sci. Tech. 34 58 Google Scholar
[4]	上官丹骅, 闫威华, 魏军侠, 高志明, 陈艺冰, 姬志成 2022 物理学报 71 090501 Google Scholar Shangguan D H, Yan W H, Wei J X, Gao Z M, Chen Y B, Ji Z C 2022 Acta. Phys. Sin. 71 090501 Google Scholar
[5]	Martin W R 2012 Nucl. Eng. Technol. 44 151 Google Scholar
[6]	Noh K, Lee D 2013 Proceedings of the KNS Fall Meeting, Gyeongju, Korea, 2013, October 23–25
[7]	Hassan M 2020 Int. J. Sci. Res. 9 913 Google Scholar
[8]	Gul A 2016 Prog. Nucl. Energy 92 164 Google Scholar
[9]	Huang Z P, Cao L Z, He Q M, Wu H C 2022 J. Nucl. Sci. Technol. 59 1375 Google Scholar
[10]	García-Pareja S, Lallena A M, Salvat F 2021 Front. Phys. 9 633 Google Scholar
[11]	Pooneh S, Mahdi S, Claudio T (Victor C Ed.) 2013 Theory and Applications of Monte Carlo Simulations (Rijeka: IntechOpen) pp152–172
[12]	Dubi A 1985 Transport Theor. Stat. 14 167 Google Scholar
[13]	Dubi A 1985 Transport Theor. Stat 14 195 Google Scholar
[14]	Booth T E, Hendricks J S 1984 Fusion Sci. Technol. 5 90 Google Scholar
[15]	Davis A, Turner A 2011 Fusion Eng. Des. 86 2698 Google Scholar
[16]	Wagner J C, Haghighat A 1998 Nucl. Sci. Eng. 128 186 Google Scholar
[17]	Wagner J C, Peplow D E, Mosher S W 2014 Nucl. Sci. Eng. 176 37 Google Scholar
[18]	Sun A K, Chen Z P, Li L M, Liu C W, Yu T 2022 Proceedings of the 23rd Pacific Basin Nuclear Conference Beijing, China, November 1–4, 2022 p821
[19]	Ramón J, Peña J 1995 Comput. Phys. Commun. 88 76 Google Scholar
[20]	LeCun Y, Bengio Y, Hinton G 2015 Nature 521 436 Google Scholar
[21]	Schmidhuber J 2015 Neural Networks 61 85 Google Scholar
[22]	Webb S 2018 Nature 554 555 Google Scholar
[23]	田十方, 李彪 2023 物理学报 72 100202 Google Scholar Tian S F, Li B 2023 Acta Phys. Sin. 72 100202 Google Scholar
[24]	Xiao M J, Yu T C, Zhang Y S, Yong H 2023 Comput. Fluids 266 106025 Google Scholar
[25]	Liu L, Liu S P, Xie H, Xiong F S, Yu T C, Xiao M J, Liu L F, Yong H 2024 J. Sci. Comput. 98 22 Google Scholar
[26]	Xie H, Zhai C L, Liu L, Yong H 2022 arXiv: 2205.06658v1 [math. NA]
[27]	胡泽华, 应阳君, 永珩, 续瑞瑞 2023 原子能科学技术 57 812 Google Scholar Hu Z H, Ying Y J, Yong H, Xu R R 2023 At. Energy Sci. Technol. 57 812 Google Scholar
[28]	Lu H Y, Li C H, Chen B B, Li W, Qi Y, Meng Z Y 2022 Chin. Phys. Lett. 39 050701 Google Scholar
[29]	Ma Y G, Pang L G, Wang R, Zhou K 2023 Chin. Phys. Lett. 40 122101 Google Scholar
[30]	Sun K W, Wang F 2023 Chin. Phys. B 32 070705 Google Scholar
[31]	武长春, 周莆钧, 王俊杰, 李国, 胡绍刚, 于奇, 刘洋 2022 物理学报 71 148401 Google Scholar Wu C C, Zhou P J, Wang J J, Li G, Hu S G, Yu Q, Liu Y 2022 Acta Phys. Sin. 71 148401 Google Scholar
[32]	杨莹, 曹怀信 2023 物理学报 72 110301 Google Scholar Yang Y, Cao H X 2023 Acta Phys. Sin. 72 110301 Google Scholar
[33]	Berry J J, Gil-Delgado G G, Osborne A G S 2021 Ann. Nucl. Energy 160 108367 Google Scholar
[34]	Cao P, Gan Q, Song J, Long P C, Wang F, Hu L Q, Wu Y C 2020 Ann. Nucl. Energy 138 107 Google Scholar
[35]	Zhang G M, Song Y M, Zhang Z H, Yuan W W 2022 Ann. Nucl. Energy 175 109248 Google Scholar
[36]	张海明, 张昊春 2022 现代应用物理 13 020209 Google Scholar Zhang H M, Zhang H C 2022 Mod. Appl. Phys. 13 020209 Google Scholar
[37]	林海鹏, 李国栋, 陈法国, 韩毅, 梁润成 2020 辐射防护 40 516 Lin H P, Li G D, Chen F G, Han Y, Liang R C 2020 Radiat. Prot. 40 516
[38]	Osborne A, Dorville J, Romano P 2023 Energy AI 13 100247 Google Scholar
[39]	Kim S H, Shin S G, Han S, Kim M H, Pyeon C H 2020 Prog. Nucl. Energy 119 103183 Google Scholar
[40]	Zhou S K, Greenspan H, Davatzikos C, Duncan J S, Van Ginneken B, Madabhushi A, Prince J L, Rueckert D, Summers R M 2021 Proc. IEEE 109 820 Google Scholar
[41]	Peng Z, Shan H M, Liu T Y, Pei X, Wang G, Xu X G 2019 IEEE Access 7 76680 Google Scholar
[42]	Ma J H, Piao Z, Huang S, Duan X M, Qin G G, Zhou L H, Xu Y 2021 Photonics Res. 9 B45 Google Scholar
[43]	Romano P K, Horelik N E, Herman B R, Nelson A G, Forget B, Smith K 2015 Ann. Nucl. Energy 82 90 Google Scholar
[44]	Abadi M, Agarwal A, Barham P, et al. 2016 arXiv: 1603.04467 [cs. DC]
[45]	Kingma D P, Ba J L 2014 arXiv: 1412.6980 [cs. LG]
[46]	Kobayashi K, Sugimura N, Nagaya Y 2001 Prog. Nucl. Energy 39 119 Google Scholar
[47]	Igor Z, Richard S 2001 Prog. Nucl. Energy 39 207 Google Scholar
[48]	Remec I, Kam F B K 1998 H. B. Robinson-2 Pressure Vessel Benchmark Report (United States) p2
[49]	Roberto O 2020 Nucl. Eng. Technol. 52 2 Google Scholar

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