In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning frameworks for solving initial-boundary value problems involving nonlinear PDEs. Recently, PINNs have shown promising results in several application fields. Motivated by applications to gas filtration problems, here we present and evaluate a PINN-based approach to predict solutions to $strongly\,\,degenerate\,\,parabolic\,\,problems\,\,with\,\,asymptotic\,\,structure\,\,of\,\,Laplacian\,\,type$. To the best of our knowledge, this is one of the first papers demonstrating the efficacy of the PINN framework for solving such kind of problems. In particular, we estimate an appropriate approximation error for some test problems whose analytical solutions are fortunately known. The numerical experiments discussed include two and three-dimensional spatial domains, emphasizing the effectiveness of this approach in predicting accurate solutions.

翻译：近年来，用于求解偏微分方程（PDEs）的科学机器学习（SciML）方法日益流行。在此范式下，物理信息神经网络（PINNs）是解决涉及非线性PDEs的初边值问题的新型深度学习框架。近期，PINNs已在多个应用领域展现出显著成效。受气体过滤问题的应用驱动，本文提出并评估了一种基于PINN的方法，用于预测具有拉普拉斯型渐近结构的强退化抛物问题的解。据我们所知，这是首批证明PINN框架对此类问题求解有效性的论文之一。特别地，我们针对部分已知解析解的测试问题估算了适当的近似误差。所讨论的数值实验涵盖二维和三维空间域，重点展示了该方法在预测精确解方面的有效性。