The application of Physics-Informed Neural Networks (PINNs) is investigated for the first time in solving the one-dimensional Countercurrent spontaneous imbibition (COUCSI) problem at both early and late time (i.e., before and after the imbibition front meets the no-flow boundary). We introduce utilization of Change-of-Variables as a technique for improving performance of PINNs. We formulated the COUCSI problem in three equivalent forms by changing the independent variables. The first describes saturation as function of normalized position X and time T; the second as function of X and Y=T^0.5; and the third as a sole function of Z=X/T^0.5 (valid only at early time). The PINN model was generated using a feed-forward neural network and trained based on minimizing a weighted loss function, including the physics-informed loss term and terms corresponding to the initial and boundary conditions. All three formulations could closely approximate the correct solutions, with water saturation mean absolute errors around 0.019 and 0.009 for XT and XY formulations and 0.012 for the Z formulation at early time. The Z formulation perfectly captured the self-similarity of the system at early time. This was less captured by XT and XY formulations. The total variation of saturation was preserved in the Z formulation, and it was better preserved with XY- than XT formulation. Redefining the problem based on the physics-inspired variables reduced the non-linearity of the problem and allowed higher solution accuracies, a higher degree of loss-landscape convexity, a lower number of required collocation points, smaller network sizes, and more computationally efficient solutions.

翻译：本文首次研究了物理信息神经网络（PINNs）在求解一维逆向自发渗吸（COUCSI）问题早期和晚期（即渗吸前锋接触无流动边界前后）的应用。我们引入了变量替换技术以提升PINNs性能。通过改变自变量，将COUCSI问题转化为三种等价形式：第一种描述饱和度关于归一化位置X和时间T的函数；第二种描述为关于X和Y=T^0.5的函数；第三种则表示为关于Z=X/T^0.5的单一函数（仅适用于早期阶段）。PINN模型采用前馈神经网络构建，并基于加权损失函数（包含物理信息损失项及初始条件与边界条件对应项）进行最小化训练。三种公式均能精确逼近正确解：XT和XY公式的含水饱和度平均绝对误差约为0.019和0.009，早期Z公式误差为0.012。Z公式完美捕捉了系统在早期的自相似性，而XT和XY公式对此表现较弱。饱和度总变差在Z公式中得以保持，且XY公式的保持效果优于XT公式。基于物理启发变量重新定义问题降低了问题的非线性程度，实现了更高的求解精度、更凸的损失函数曲面、更少的配置点需求、更小的网络规模以及更高的计算效率。