Countercurrent spontaneous imbibition (COUCSI) is a process in porous materials in which a wetting phase displaces non-wetting phase. In this work, we investigate for the first time the application of Physics-Informed Neural Networks (PINNs) in solving the 1D COUCSI problem in both early (ET) and late (LT) times. Also novel, we examine the Change-of-Variables technique for improving the performance of PINNs. We formulated the COUCSI problem in three equivalent forms by changing the independent variables: XT-, XY-, and Z-formulations. 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 ET). 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. No synthetical or experimental data were involved in the training. All three formulations could closely approximate the correct solutions (obtained by fine-grid numerical simulations), with water saturation mean absolute errors (MAE) around 0.019 and 0.009 for XT and XY formulations and 0.012 for the Z formulation at ET. The Z formulation perfectly captured the self-similarity of the system at ET. This was less captured by XT and XY formulations. The total variation (TV) of saturation was preserved in the Z formulation, and it was better preserved with XY- than XT formulation. It was demonstrated that redefining the problem based on 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：逆流自发渗吸（COUCSI）是多孔介质中润湿相驱替非润湿相的过程。本研究首次探讨了物理信息神经网络（PINNs）在求解一维COUCSI问题早期（ET）和晚期（LT）的应用。另一创新之处在于，我们检验了变量变换技术对提升PINNs性能的作用。通过改变自变量，我们以三种等价形式构建了COUCSI问题：XT形式、XY形式和Z形式。第一种形式将饱和度描述为归一化位置X和时间T的函数；第二种形式将其描述为X和Y=T^0.5的函数；第三种形式则将其视为仅依赖于Z=X/T^0.5的单一函数（仅适用于早期）。PINN模型采用前馈神经网络生成，并通过最小化加权损失函数进行训练，该损失函数包含基于物理信息的损失项以及对应于初始条件和边界条件的项。训练过程中未使用合成数据或实验数据。三种形式均能紧密逼近精确解（通过精细网格数值模拟获得），在早期，XT形式和XY形式的水饱和度平均绝对误差（MAE）分别约为0.019和0.009，Z形式约为0.012。Z形式完美捕捉了系统在早期的自相似性，而XT形式和XY形式对此捕捉较弱。Z形式保留了饱和度的总变差（TV），且XY形式比XT形式能更好地保留该性质。研究表明，基于物理启发变量重新定义问题可降低问题的非线性程度，从而获得更高的求解精度、更凸的损失景观、更少的配点需求、更小的网络规模以及更高的计算效率。