一类结合神经算子网络与贝叶斯神经网络的主动学习算法: 从微观数据学习集群行为的宏观模型

doi:10.19596/j.cnki.1001-246x.8979

计算物理 ›› 2024, Vol. 41 ›› Issue (6): 783-796.DOI: 10.19596/j.cnki.1001-246x.8979

一类结合神经算子网络与贝叶斯神经网络的主动学习算法: 从微观数据学习集群行为的宏观模型

高正雅¹(), 毛志平²^,*()

1. 厦门大学数学科学学院, 福建厦门 361005
2. 宁波东方理工大学(暂名)数学科学学院, 浙江宁波 315200

收稿日期:2024-07-04 出版日期:2024-11-25 发布日期:2024-12-26
通讯作者: 毛志平
作者简介:
高正雅, 硕士研究生, 研究方向为深度学习与微分方程数值解, E-mail: 2900549129@qq.com
基金资助:
国家科技部重点研发计划(2022YFA1004500); 国家自然科学基金面上项目(12171404)

Active Learning Algorithm Using Neural Operator Networks and Bayesian Neural Networks: Learning Macroscale Models for Collective Behavior from Microscale Data

Zhengya GAO¹(), Zhiping MAO²^,*()

1. School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China
2. School of Mathematical Sciences, Eastern Institute of Technology, Ningbo, Zhejiang 315200, China

Received:2024-07-04 Online:2024-11-25 Published:2024-12-26
Contact: Zhiping MAO

摘要/Abstract

摘要：

随着人工智能和科学计算的发展, 深度学习在数学建模中发挥着越来越重要的作用。本文发展了一类结合微观数据的主动学习算法对集群行为建立的宏观模型。具体来说, 针对Cucker-Smale模型, 结合微观粒子数据与部分机理, 发展了一类结合神经算子网络与贝叶斯神经网络的主动学习算法。该算法可通过群体行为的微观数据高效地建立对应的宏观Euler模型。最后通过一维和二维数值模拟验证了主动学习算法的有效性。

关键词: 数学建模, 非局部欧拉方程, 贝叶斯神经网络, 主动学习算法, Cucker-Smale模型

Abstract:

With the development of artificial intelligence and scientific computing, deep learning plays a significant role in mathematical modeling. In this work we develop an active learning algorithm that uses microscopic data to establish a macroscopic model for collective behavior. Specifically, we take the Cucker-Smale model in this work and develop the corresponding active learning algorithm that integrates neural operator networks and Bayesian neural networks by utilizing microscopic particle data and partial physics. This algorithm is used to efficiently establishes the corresponding macroscopic Euler model through microscopic data. Finally, the effectiveness of the active learning algorithm is validated through one-dimensional and two-dimensional numerical simulations.

Key words: mathematical modeling, nonlocal Euler equations, Bayesian neural networks, active learning algorithm, Cucker-Smale model

高正雅, 毛志平. 一类结合神经算子网络与贝叶斯神经网络的主动学习算法: 从微观数据学习集群行为的宏观模型[J]. 计算物理, 2024, 41(6): 783-796.

Zhengya GAO, Zhiping MAO. Active Learning Algorithm Using Neural Operator Networks and Bayesian Neural Networks: Learning Macroscale Models for Collective Behavior from Microscale Data[J]. Chinese Journal of Computational Physics, 2024, 41(6): 783-796.

图/表 8

图1 DeepONet网络示意图

Fig.1 Schematic of DeepONet

图2 一维欧拉方程算子网络的预测结果(a) 误差；(b)α=1时u的预测值；(c) α=1时，t=0.5, 1.5, 2.5的结果

Fig.2 Prediction of DeepONet for one-dimensional Euler equation (a)loss of train and test; (b) p rediction of u with α=1; (c)exact and predict solution of u with α=1 at t=0.5, 1.5, 2.5

图3 二维欧拉方程算子网络的预测结果(a)u、v二维误差图；(b) α=1时u、v分别在t=0.5, 1, 2的实验误差图

Fig.3 Prediction of the DeepONet for two-dimensional Euler equation (a)train and test loss of u and v; (b) test loss of u and v with α=1 and t=0.5, 1.2

图4 贝叶斯神经网络示意图

Fig.4 Schematic of Bayesian neural networks

表1 一维情形下参数α推断值

Table 1 Inferred value of α for the one-dimensional case

$\hat{\alpha}$	α	F(α)
0.5	0.481 5	3.04×10^-2
1.2	1.210 4	5.05×10^-2

图5 $\hat{\alpha}$=0.5, 1.2学习过程

Fig.5 Learning process of $\hat{\alpha}$=0.5, 1.2

表2 二维情形下，参数α推断值

Table 2 Inferred value of α for the two-dimensional case

$\hat{\alpha}$	α	F(α)
0.5	0.550 2	1.34×10^-1
1.2	1.253 8	8.79×10^-2

图6 $\hat{\alpha}$=0.5, 1.2学习过程

Fig.6 Learning process of $\hat{\alpha}$ =0.5, 1.2

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一类结合神经算子网络与贝叶斯神经网络的主动学习算法: 从微观数据学习集群行为的宏观模型

Active Learning Algorithm Using Neural Operator Networks and Bayesian Neural Networks: Learning Macroscale Models for Collective Behavior from Microscale Data

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