基于深度学习的扩散方程扩散系数反问题求解

doi:10.19596/j.cnki.1001-246x.8892

摘要/Abstract

摘要：

物理信息神经网络(PINN)为偏微分方程正反问题数值求解开创了一条具有广阔应用前景的新途径。本文聚焦于扩散方程的扩散系数反演问题。针对固定系数、各向异性系数、空间依赖系数、时空依赖系数以及非线性扩散系数等问题展开了系统研究, 提出了求解各类问题所需的网络结构及求解方法。数值实验表明, PINN方法在求解扩散系数反问题时只需较少的数据即可反演出较为精确的未知系数, 并在一定噪声水平下表现出较强的稳健性。

关键词: 物理信息神经网络, 微分方程反问题, 扩散方程, 数值解

Abstract:

Physics-Informed Neural Networks (PINN) have provided a new way of numerically solving forward and inverse problems of partial differential equations with promising applications. This paper focuses on the diffusion coefficient inverse problem of the diffusion equation. A systematic study is carried out for the problems of fixed coefficients, anisotropic coefficients, spatial dependence coefficients, spatio-temporal dependence coefficients, and nonlinear diffusion coefficients, and the neural network structure and solution method required for solving each type of problem are proposed. Numerical experiments show that the PINN method can reconstruct the unknown coefficients accurately with less data and is robust under a certain noise level.

Key words: physics-informed neural networks, inverse problem for partial differential equations, diffusion equation, numerical solution

张延庆, 谷同祥. 基于深度学习的扩散方程扩散系数反问题求解[J]. 计算物理, 2025, 42(2): 146-159.

Yanqing ZHANG, Tongxiang GU. Deep Learning Method for Solving Inverse Problem of Diffusion Coefficients for Diffusion Equation[J]. Chinese Journal of Computational Physics, 2025, 42(2): 146-159.

图/表 27

图1 求解常扩散系数反演问题的神经网络结构

Fig.1 Neural network structure for solving inverse problems with constant diffusion coefficients

图2 求解时空依赖扩散系数反演问题的网络结构

Fig.2 Neural network structure for solving inverse problems with spatio-temporal dependent diffusion coefficient

图3 仅空间依赖扩散系数反演的网络结构

Fig.3 Neural network structure for solving inverse problems with space dependent diffusion coefficient

表1 PINN方法反演的扩散系数及相对误差

Table 1 Diffusion coefficients and relative errors by PINN method

结果	噪声水平/%
结果	0	1	5	10
$ \hat{\lambda}$	6.246	6.236	6.374	5.866
$\|\lambda-\hat{\lambda} / \lambda\| $	0.048	0.071	1.994	6.133

图4 t=0.5时刻方程(13)的(a)精确解；(b)PINN方法预测解；(c)逐点误差

Fig.4 (a)Exact solution of Eq.(13) at t=5; (b) Predicted solution by PINN method; (c) Point-wise error

表2 PINN方法预测解的误差

Table 2 Errors in predict solutions by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.115 6	0.676 5	2.170	10.20
‖e‖_∞	0.375 0	0.902 6	1.579	6.889

表3 PINN方法反演扩散系数在不同数据点数及噪声水平下的相对误差

Table 3 Relative errors of diffusion coefficients by PINN method in different number of data points and noise levels

数据点数	噪声水平/%
数据点数	0	1	5	10
N=5	0.062	0.192	0.053	1.972
N=10	0.162	0.366	2.954	1.567
N=20	0.058	0.298	0.164	2.026
N=50	0.012	0.242	0.605	0.249
N=100	0.012	0.398	0.281	3.912

表4 PINN方法反演扩散系数Λ1的误差

Table 4 Errors in diffusion coefficients Λ1 by PINN method

误差	噪声水平/%
误差	0	1	5	10
$ \left\|\left(a_{11}-\hat{a}_{11}\right) / a_{11}\right\|$	0.059	0.063	2.833	1.077
$ \left\|\left(a_{22}-\hat{a}_{22}\right) / a_{22}\right\|$	1.094	1.522	22.603	14.105
$ \left\|a_{12}-\hat{a}_{12}\right\|$	0.136 3	7.593	6.216	19.400

图5 (a) 方程(14)的精确解；(b) PINN方法预测解；(c) 逐点误差

Fig.5 (a) Exact solution of Eq.(14); (b) Predicted solution by PINN method; (c) Point-wise error

表5 PINN方法在Λ1参数下方程解的误差

Table 5 Error in predict solutions by PINN method in Λ1

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.036 89	0.214 2	1.842	1.648
‖e‖_∞	0.058 85	0.277 1	1.254	1.221

表6 PINN方法反演扩散系数Λ2的相对误差

Table 6 Relative errors in diffusion coefficients Λ2 by PINN method

误差	噪声水平/%
误差	0	1	5	10
$ \left\|\left(a_{11}-\hat{a}_{11}\right) / a_{11}\right\|$	0.115	0.251	1.022	0.785
$ \left\|\left(a_{22}-\hat{a}_{22}\right) / a_{22}\right\| $	1.564	7.680	25.419	21.586
$ \left\|\left(a_{12}-\hat{a}_{12}\right) / a_{12}\right\|$	0.025	0.204	1.312	13.617

图6 (a) 方程(14)的精确解；(b) PINN方法预测解；(c) 逐点误差

Fig.6 (a) Exact solution of Eq.(14); (b) Predicted solution by PINN method; (c) Point-wise error

表7 PINN方法在Λ2参数下方程解的误差

Table 7 Error in predict solutions by PINN method in Λ2

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.078 35	0.326 6	1.899	2.187
‖e‖_∞	0.112 60	0.304 1	1.102	1.552

图7 (a) t=0.8时刻扩散系数精确值；(b) PINN方法预测的扩散系数; (c) 逐点误差

Fig.7 (a) Exact values of diffusion coefficients at t=0.8; (b) Diffusion coefficients predicted by PINN method; (c) Point-wise error

表8 PINN方法反演扩散系数a(t, x, y)的误差

Table 8 Errors in the inversion of diffusion coefficients a(t, x, y) by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	1.249	1.245	2.662	41.04
‖e‖_∞	7.189	16.18	22.09	282.7

表9 PINN方法得到预测解的误差

Table 9 Errors in predict solutions by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.204 4	0.140 5	0.612 6	4.451
‖e‖_∞	0.658 7	0.289 1	0.915 0	11.81

图8 (a) 扩散系数精确值; (b) PINN方法预测的扩散系数; (c) 逐点误差

Fig.8 (a)Exact values of diffusion coefficients; (b) Diffusion coefficients predicted by PINN method; (c) Point-wise error

表10 PINN方法反演扩散系数a(x)的误差

Table 10 Errors in the inversion of diffusion coefficients a(x) by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.798	4.049	9.125	32.17
‖e‖_∞	2.188	10.88	22.68	71.21

表11 PINN方法得到方程解的误差

Table 11 Errors in predict solutions by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.071 1	0.360 5	1.201	2.733
‖e‖_∞	0.133 5	0.438 8	1.324	3.991

图9 (a) 扩散系数精确值; (b) PINN方法预测的扩散系数; (c) 逐点误差

Fig.9 (a) Exact values of diffusion coefficients; (b) Diffusion coefficients predicted by PINN method; (c) Point-wise error

表12 PINN方法反演扩散系数a(x)的误差

Table 12 Errors in the inversion of diffusion coefficients a(x) by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.347 0	0.529 0	1.861	7.185
‖e‖_∞	0.503 6	0.617 5	0.259 5	8.057

表13 PINN方法得到方程解的误差

Table 13 Errors in predict solutions by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.110 7	0.199 8	0.497 8	1.638
‖e‖_∞	0.143 0	0.153 7	0.447 4	1.316

图10 (a) 扩散系数精确值; (b) PINN方法预测的扩散系数; (c) 逐点误差

Fig.10 (a) Exact values of diffusion coefficients; (b) Diffusion coefficients predicted by PINN method; (c) Point-wise error

表14 PINN方法反演扩散系数a(u)的误差

Table 14 Errors in inversion of diffusion coefficients a(u) by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	1.585	2.995	11.65	26.14
‖e‖_∞	3.176	5.971	16.07	38.16

表15 PINN方法得到方程解的误差

Table 15 Errors in predict solutions by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.127 5	0.525 2	2.650	3.614
‖e‖_∞	0.189 7	0.814 7	4.539	7.781

表16 PINN方法反演的扩散系数及相对误差

Table 16 Diffusion coefficients and relative errors by PINN method

误差	噪声水平/%
误差	0	1	5	10
$ \|(\mu-\hat{\mu}) / \mu\| $	2.848	7.241	7.902	6.307
$ \|(k-\hat{k}) / k\|$	2.858	1.937	1.018	5.982

表17 PINN方法得到方程解的误差

Table 17 Errors in predict solutions by PINN method

误差	噪声水平/%
误差	0	1	5	10
‖e‖₂	0.078 80	0.225 1	0.466 9	5.636
‖e‖_∞	0.270 4	0.674 8	2.190	2.966

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