基于物理信息神经网络的内部声场正反问题数值计算

doi:10.19596/j.cnki.1001-246x.8520

摘要/Abstract

摘要：

针对频域内部声场正反问题的数值模拟, 建立基于物理信息的神经网络架构。与基于数据驱动的神经网络不同, 将声学问题的Helmholtz方程及其对应的边界条件引入神经网络, 所建立的神经网络算法不仅能够反映训练数据样本的分布规律, 而且也遵循由偏微分方程描述的物理定律。考虑到频域声学问题中含有复数部分, 建立两种网络架构, 并进行验证和比较分析。该方法无需网格划分和数值积分等繁琐的数值计算过程, 可自由地处理不规则区域和非均匀分布情形。数值实验考察二维和三维复杂几何结构的声学正问题及反问题, 结果表明所建立的物理信息神经网络算法具有较高的精确度、收敛性和鲁棒性。

关键词: 物理信息神经网络, 声学问题, Helmholtz方程, 正问题, 反问题

Abstract:

A physics-informed neural network (PINN) is proposed for numerical simulation of forward and inverse problems associated with internal sound field in frequency domain. Unlike data-driven neural network, Helmholtz equation of a acoustic problem and corresponding boundary conditions are embedded in the neural network. The developed neural network reflects the distribution law of training data samples, and follows the physical law described by partial differential equations as well. For frequency acoustic problem with complex numbers, two types of networks are established. Verification and comparison are performed. Tedious numerical calculation processes such as meshing and numerical integration are not needed, and irregular domain and non-uniformly distributed nodes are freely addressed. Numerical examples, including the forward and inverse problems in two-dimensional and three-dimensional complex geometric structures, are provided to investigate effectiveness of the method. It shows that the PINN has good accuracy, convergence and robustness.

Key words: physics-informed neural network, acoustic problems, Helmholtz equation, forward problems, inverse problems

吴国正, 王发杰, 程隋福, 张成鑫. 基于物理信息神经网络的内部声场正反问题数值计算[J]. 计算物理, 2022, 39(6): 687-698.

Guozheng WU, Fajie WANG, Suifu CHENG, Chengxin ZHANG. Numerical Simulation of Forward and Inverse Problems of Internal Sound Field Based on Physics-informed Neural Network[J]. Chinese Journal of Computational Physics, 2022, 39(6): 687-698.

图/表 18

图1 内部声场正反问题示意图(a) 正问题; (b) 反问题

Fig.1 A schematic of forward and inverse problems associated with internal sound field (a) forward problem; (b) inverse problem

图2 第一种物理神经网络(PINN1)的结构示意图

Fig.2 A sketch of the first physics-informed neural network (PINN1)

图3 第二种物理神经网络(PINN2)的结构示意图

Fig.3 A sketch of the second physics-informed neural network (PINN2)

图4 管道声腔模型

Fig.4 Pipe acoustic model

图5 矩形声腔内声压分布的准确值与预测值

Fig.5 Accurate and predicted sound pressures in a rectangular acoustic cavity

图6 两种神经网络的绝对误差分布

Fig.6 Absolute error distributions of two neural networks

图7 管道底部轴线上的声压分布

Fig.7 Distributions of sound pressure at the lower boundary

表1 相同节点分布下PINN1、PINN2、LMFS和GFDM的计算误差

Table 1 Numerical errors of PINN1, PINN2, LMFS and GFDM under same nodal distribution

方法	最大绝对误差	最大相对误差	均方根误差
PINN1	3.000 9 × 10^-5	1.692 3 × 10^-5	6.874 2 × 10^-7
PINN2	8.163 5 × 10^-4	7.302 9 × 10^-3	9.252 3 × 10^-5
LMFS	7.753 1 × 10^-6	4.183 7 × 10^-5	5.004 9 × 10^-6
GFDM	5.859 3 × 10^-2	1.593 6 × 10^-1	3.631 3 × 10^-2

图8 多联通区域反问题几何模型

Fig.8 Geometry model of a inverse problem with multiple connected domain

图9 两种损失函数的计算结果(a) 计算结果；(b)绝对误差

Fig.9 Calculation results of the two loss functions (a) calculation results; (b) absolute error

图10 两种神经网络预测结果的绝对误差云图(a) PINN1；(b) PINN2

Fig.10 Contours of absolute error of two neural networks (a) PINN1; (b) PINN2

图11 三个不可测边界上声压实际值与PINN1的预测值

Fig.11 Exact results and predicted values with PINN1 on three unmeasurable boundaries

图12 三个不可测边界上声压实际值与PINN2的预测值

Fig.12 Exact results and predicted values with PINN2 on three unmeasurable boundaries

图13 不同噪声水平下两种网络的绝对误差分布

Fig.13 Distributions of absolute error of two networks under different noise levels

图14 两种网络随噪声水平增加的均方根误差曲线

Fig.14 RMSEs of PINN1 and PINN2 with increasing noisy level

图15 三维封闭声腔结构

Fig.15 Three-dimensional acoustic cavity structures

图16 三种模型内部节点处预测值的绝对误差

Fig.16 Absolute errors of predicted values at internal nodes in three models

表2 三种声腔结构声学分析中PINN1的最大相对误差和均方根误差

Table 2 Maximum relative errors and root mean square errors of PINN1 in acoustic analysis for three cavity structures

	最大相对误差	均方根误差
区域(a)	2.609 7×10^-5	4.009 5×10^-8
区域(b)	1.114 1×10^-4	2.989 6×10^-7
区域(c)	4.537 6×10^-4	1.149 2×10^-6

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