Deep Learning Method for Solving Inverse Problem of Diffusion Coefficients for Diffusion Equation

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

Abstract

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

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.

Figures/Tables 27

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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