Physics-informed neural networks with hard constraints for hydraulic conductivity field inversion

doi:10.13745/j.esf.sf.2025.10.38

Abstract

Abstract:

In recent years, Physics-Informed Neural Networks (PINNs) have been widely applied to numerically solve partial differential equations and in computational fluid dynamics, having demonstrated promising potential for groundwater modeling. Existing studies typically handle boundary conditions in groundwater models using soft-constraint algorithms, which satisfy physical constraints approximately by minimizing the boundary condition error. However, the application of hard-constraint algorithms, which can enhance solving accuracy and stability by design, remains limited in groundwater modeling. To address this gap, this paper introduces a hard-constraint PINNs algorithm that simultaneously enforces constant-head and no-flow boundary conditions. Using the inverse problem of estimating the hydraulic conductivity field in a two-dimensional confined aquifer as a case study, we demonstrate the superior performance of the hard-constraint approach over its soft-constraint counterpart. The results show that the proposed method reduces the average relative inversion error by 75% compared to soft-constraint PINNs and by 60% compared to hard-constraint PINNs considering constant-head boundaries alone. Furthermore, the proposed method reduces the number of required training samples and alleviates the need for extensive manual hyperparameter tuning, thereby improving training efficiency and robustness. Therefore, the hard-constraint PINNs method proves to be a highly accurate and efficient approach for reconstructing hydraulic conductivity fields, demonstrating significant potential for broader application.

Key words: physics-informed neural networks, hard-constraint PINNs, hydraulic conductivity field inversion, groundwater modeling, confined aquifer

CLC Number:

P641.2

SHU Wei, JIANG Jianguo, WU Jichun. Physics-informed neural networks with hard constraints for hydraulic conductivity field inversion[J]. Earth Science Frontiers, 2026, 33(1): 500-510.

Figures/Tables 8

Fig.1 Schematic diagram of the PINNs architecture and the solution process for partial differential equations

Fig.2 Schematic diagram of the soft-constraint and hard-constraint PINNs algorithms

Fig.3 Schematic diagram of the study area

Fig.4 Schematic diagram of the PINNs hard-constraint algorithm for permeability field inversion

Table 1 Computational hyperparameters used in the PINNs examples for the two-dimensional confined groundwater flow model

PINNs算法	N_f	N_B	N_D	NN₁	NN₂/NN₃	NN_K
PINNs-S	400	200	25	100×6		60×6
PINNs-H-I	400	200	25	100×6		60×6
PINNs-H-II	400	200	25	100×6	100×3	60×6
PINNs-H-III	400		25	100×6	100×3	60×6

Fig.5 Comparison of the inversion results of different PINNs algorithms for the K field in a two-dimensional confined aquifer

Fig.6 Model performance evaluation metrics of the K field inverted by four different PINNs algorithms

Fig.7 Variation of the average absolute error (AAE) between the inverted and reference K fields with the number of training iterations for the four PINNs algorithms

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