Application of physics-informed neural networks to inverse problems in unsaturated groundwater flow

Figures & data

Figure 1. PINN illustration adapted from Lu et al. (2019).

Figure 2. Solution of the Richards equation in terms of ψ.

Figure 3. Solution of the Richards equation in terms of θ.

Figure 4. Inverse problem solution for the pressure head PINN formulation.

Table 1. Loss values with the estimates of input parameters for a range of ${\mathcal{T}}_i$ values.

${\mathcal{T}}_i$	${\mathcal{L}}_i$	${\mathcal{L}}_f$	${\hat{k}}_s$ [m/s]	$\hat{n}$	$\hat{\alpha }$ [1/m]	${\sigma }_{k_s}$ [m/s]	${\sigma }_n$	${\sigma }_{\alpha }$ [1/m]
500	$3.05\cdot {10}^{-4}$	$4.02\cdot {10}^{-13}$	$6.498\cdot {10}^{-8}$	5.896	0.569	$1.137\cdot {10}^{-8}$	1.017	0.126
1000	$3.86\cdot {10}^{-4}$	$2.67\cdot {10}^{-12}$	$7.044\cdot {10}^{-8}$	$4.213$	$0.547$	$1.055\cdot {10}^{-9}$	0.697	0.102
2000	$2.842\cdot {10}^{-4}$	$8.455\cdot {10}^{-13}$	$7.206\cdot {10}^{-8}$	$5.512$	$0.5638$	$7.815\cdot {10}^{-9}$	0.796	0.096

Figure 5. Inverse problem solution for the pressure head PINN formulation with noisy data and ${\sigma }_e=0.05$.

Table 2. Loss values with the estimates of input parameters for a range of ${\sigma }_e$ values.

${\sigma }_e$	${\mathcal{L}}_i$	${\mathcal{L}}_f$	${\hat{k}}_s$ [m/s]	$\hat{n}$	$\hat{\alpha }$ [1/m]	${\sigma }_{k_s}$ [m/s]	${\sigma }_n$	${\sigma }_{\alpha }$ [1/m]
0.025	$3.548\cdot {10}^{-4}$	$1.408\cdot {10}^{-12}$	$7.107\cdot {10}^{-8}$	$5.238$	$0.547$	$1.083\cdot {10}^{-8}$	0.363	0.058
0.05	$3.638\cdot {10}^{-4}$	$1.975\cdot {10}^{-12}$	$5.202\cdot {10}^{-8}$	4.802	0.296	$1.095\cdot {10}^{-8}$	0.545	0.267
0.075	$7.155\cdot {10}^{-3}$	$3.664\cdot {10}^{-10}$	$5.243\cdot {10}^{-8}$	$3.923$	$0.601$	$2.727\cdot {10}^{-8}$	0.525	0.107

Figure 6. Inverse problem solution for the VWC-PINN formulation.

Table 3. Loss values with the estimates of input parameters for a range of ${\mathcal{T}}_i$ values.

${\mathcal{T}}_i$	${\mathcal{L}}_i$	${\mathcal{L}}_f$	${\hat{k}}_s$ [m/s]	$\hat{n}$	$\hat{\alpha }$ [1/m]	${\sigma }_{k_s}$ [m/s]	${\sigma }_n$	${\sigma }_{\alpha }$ [1/m]
500	$1.769\cdot {10}^{-6}$	$1.372\cdot {10}^{-9}$	$7.820\cdot {10}^{-8}$	5.806	0.623	$9.734\cdot {10}^{-9}$	2.279	0.068
1000	$1.560\cdot {10}^{-6}$	$1.405\cdot {10}^{-9}$	$5.776\cdot {10}^{-8}$	$7.005$	$0.586$	$1.125\cdot {10}^{-9}$	1.127	0.063
2000	$3.644\cdot {10}^{-6}$	$7.444\cdot {10}^{-9}$	$8.511\cdot {10}^{-8}$	$4.617$	$0.578$	$4.527\cdot {10}^{-9}$	3.700	0.067

Figure 7. Inverse problem solution for the VWC-PINN formulation with a noisy data, specified with ${\sigma }_e=0.04$.

Table 4. Loss values with the estimates of input parameters for a range of ${\sigma }_e$ values.

${\sigma }_e$	${\mathcal{L}}_i$	${\mathcal{L}}_f$	${\hat{k}}_s$ [m/s]	$\hat{n}$	$\hat{\alpha }$ [1/m]	${\sigma }_{k_s}$ [m/s]	${\sigma }_n$	${\sigma }_{\alpha }$ [1/m]
0.01	$1.151\cdot {10}^{-6}$	$1.019\cdot {10}^{-9}$	$4.207\cdot {10}^{-8}$	4.269	0.449	$2.591\cdot {10}^{-9}$	0.479	0.148
0.02	$2.185\cdot {10}^{-6}$	$2.208\cdot {10}^{-9}$	$6.057\cdot {10}^{-8}$	$5.895$	$0.342$	$1.417\cdot {10}^{-8}$	0.236	0.122
0.04	$1.456\cdot {10}^{-6}$	$9.987\cdot {10}^{-10}$	$5.534\cdot {10}^{-8}$	$4.669$	$0.477$	$2.036\cdot {10}^{-8}$	0.943	0.161

Figure 8. Water infiltration column setup with the PINN prediction of the solution and comparisons with the measurements.

Figure 9. Water infiltration column setup with the PINN prediction of the solution and comparisons with the measurements and unobserved data.

Lu, Lu, Xuhui Meng, Zhiping Mao, and George E. Karniadakis. 2019. “DeepXDE: A Deep Learning Library for Solving Differential Equations.” ArXiv preprint arXiv:1907.04502.

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