Physics-informed neural networks (PINNs) are receiving increased attention in modeling flow in porous media because they can surpass purely data-driven approaches. However, in heterogeneous domains, PINNs often face convergence challenges due to discontinuities in rock properties. A promising alternative is the mixed formulation of PINNs, which utilizes pressure head and velocity fields as primary variables. This formulation introduces a multi-term loss function whose terms must be carefully balanced to ensure effective convergence during training. The main goal of this work is to identify the most suitable weighting technique to overcome convergence issues and enhance the applicability of the mixed formulation of PINNs for modeling flow in heterogeneous porous media. Thus, we implement and adapt different global and local weighting techniques and evaluate their performance through multiple test scenarios, involving stochastic and block heterogeneity. The results reveal that the most appropriate weighting strategy is the max-average technique. In the case of stochastic heterogeneity, this technique allows for improving the convergence of the training algorithm. In the case of discontinuous heterogeneity, the max-average method is the only strategy that achieved convergence, highlighting its robustness. The results also show that under high heterogeneity, using an appropriate weighting technique becomes imperative because baseline PINN failed to converge. Implementing an optimal weighting strategy can improve convergence and yield accurate solutions with fewer learnable parameters, thereby enhancing overall model performance.

中文翻译：

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使用物理信息神经网络模拟非均质多孔介质中的流体流动：混合压力水头-速度公式的加权策略


物理信息神经网络 （PINN） 在多孔介质中的流动建模中受到越来越多的关注，因为它们可以超越纯粹的数据驱动方法。然而，在非均质域中，由于岩石特性的不连续性，PINN 经常面临收敛挑战。一个很有前途的替代方案是 PINN 的混合公式，它利用压力水头和速度场作为主要变量。该公式引入了一个多项损失函数，其项必须仔细平衡，以确保在训练期间有效收敛。这项工作的主要目标是确定最合适的加权技术来克服收敛问题并提高 PINN 混合公式在非均质多孔介质中流动建模的适用性。因此，我们实施和调整了不同的全局和局部加权技术，并通过涉及随机和块异构性的多个测试场景评估它们的性能。结果表明，最合适的加权策略是 max-average 技术。在随机异质性的情况下，该技术可以提高训练算法的收敛性。在不连续异质性的情况下，max-average 方法是唯一实现收敛的策略，突出了其稳健性。结果还表明，在高异质性下，使用适当的加权技术变得势在必行，因为基线 PINN 未能收敛。实施最佳加权策略可以提高收敛性，并以更少的可学习参数生成准确的解决方案，从而提高模型的整体性能。