Deep learning-based methods for solving partial differential equations have become a research hotspot. The approach builds on the previous work of applying deep learning methods to partial differential equations, which avoid the need for meshing and linearization. However, deep learning-based methods face difficulties in effectively solving complex turbulent systems without using labeled data. Moreover, issues such as failure to converge and unstable solution are frequently encountered. In light of this objective, this paper presents an approximation-correction model designed for solving the seepage equation featuring unsteady boundaries. The model consists of two neural networks. The first network acts as an asymptotic block, estimating the progression of the solution based on its asymptotic form. The second network serves to fine-tune any errors identified in the asymptotic block. The solution to the unsteady boundary problem is achieved by superimposing these progressive blocks. In numerical experiments, both a constant flow scenario and a three-stage flow scenario in reservoir exploitation are considered. The obtained results show the method's effectiveness when compared to numerical solutions. Furthermore, the error analysis reveals that this method exhibits superior solution accuracy compared to other baseline methods.

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基于深度学习的非定常边界下渗流方程求解方法

基于深度学习的偏微分方程求解方法已成为研究热点。该方法建立在将深度学习方法应用于偏微分方程的先前工作的基础上，从而避免了网格划分和线性化的需要。然而，基于深度学习的方法在不使用标记数据的情况下有效解决复杂的湍流系统面临困难。此外，经常遇到收敛失败、解不稳定等问题。鉴于此，本文提出了一种用于求解非稳态渗流方程的近似修正模型。该模型由两个神经网络组成。第一个网络充当渐近块，根据其渐近形式估计解的进展。第二个网络用于微调渐近块中识别的任何错误。非稳态边界问题的解决是通过叠加这些渐进块来实现的。在数值实验中，考虑了油藏开发中的恒流场景和三阶段流场景。与数值解相比，获得的结果显示了该方法的有效性。此外，误差分析表明，与其他基线方法相比，该方法具有更高的求解精度。