Scaled asymptotic solution nets for unlabeled seepage equation solutions with variable well flow

The seepage equation is essential for understanding fluid flow in porous media, crucial for analyzing fluid behavior in various pore structures and supporting reservoir engineering. However, solving this equation under complex conditions, such as variable well flow rates, poses significant challenges. Although physics-informed neural networks have been effective in addressing partial differential equations, they often struggle with the complexities of such physical phenomena. This paper presents an improved method using physical asymptotic solution nets combined with scaling before activation (SBA) and gradient constraints to solve the seepage equation in porous media under varying well flow rates without labeled data. The model consists of two neural networks: one that approximates the asymptotic solution of the seepage equation and another that corrects approximation errors to ensure both mathematical and physical accuracy. When the well flow rate changes, the network may fail to fully satisfy the asymptotic solution due to pressure distribution variations, resulting in sub-optimal outcomes. To address this, we incorporate gradient information into the loss function to reinforce physical constraints and utilize the SBA method to enhance the approximation. This gradient information is derived from the pressure distribution at the previous flow rate, and the SBA method regulates weight adjustments through an adjustment coefficient constrained by the loss function, preventing sub-optimal local minima during optimization. Experimental results show that our method achieves an accuracy range of 10−4 to 10−2.