Machine learning methods provide new perspective for more convenient and efficient prediction of groundwater flow. In this study, a deep learning method “GW-PINN” without labeled data for solving groundwater flow equations with wells was proposed. GW-PINN takes the physics inform neural network (PINN) as the backbone and uses either the hard or soft constraint in the loss function for training. A locally refined sampling strategy (LRS) is adopted to generate the consistent spatial sampling points for problems with strong hydraulic head change, and then combined with an appropriate temporal sampling scheme to obtain the final spatial-temporal sampling points. A snowball-style two-stage training strategy by dividing the temporal domain into two subdomains is designed to decrease the sampling points. Five cases were designed to test the training performance of GW-PINN under different sampling strategies and two constraints. The predicted results of GW-PINN were compared with MODFLOW and the analytical solution. The results demonstrate that GW-PINN possesses strong ability in capturing the hydraulic head change for both confined and un-confined aquifers. The hard constraint owns more robust learning ability than the soft constraint. The LRS strategy can generate more accurate results with much fewer sampling points than traditional sampling strategies, and the snowball-style two-stage training strategy is significantly efficient for problems with the drastic change of hydraulic head. Furthermore, the application of GW-PINN as a surrogate model for parameterized groundwater flow equations is illustrated. This study provides an option tool for efficient groundwater flow simulation, especially for those with local refinements are needed.

机器学习方法为更方便、更有效地预测地下水流量提供了新的视角。在这项研究中，提出了一种无标记数据的深度学习方法“GW-PINN”，用于求解带井的地下水流动方程。GW-PINN 以物理信息神经网络（PINN）为骨干，并在损失函数中使用硬约束或软约束进行训练。针对水头变化强的问题，采用局部细化采样策略（LRS）生成一致的空间采样点，然后结合适当的时间采样方案得到最终的时空采样点。通过将时间域划分为两个子域的雪球式两阶段训练策略旨在减少采样点。设计了五个案例来测试 GW-PINN 在不同采样策略和两个约束条件下的训练性能。GW-PINN的预测结果与MODFLOW和解析解进行了比较。结果表明，GW-PINN对承压含水层和非承压含水层的水头变化都有很强的捕捉能力。硬约束比软约束具有更强的学习能力。LRS策略比传统的采样策略可以用更少的采样点产生更准确的结果，而滚雪球式的两阶段训练策略对于水头急剧变化的问题效率显着。此外，还说明了 GW-PINN 作为参数化地下水流量方程的替代模型的应用。