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的科学机器学习方法日益普及。在此框架下，物理信息神经网络作为一种新型深度学习架构，可有效处理涉及非线性偏微分方程的初边值问题。近期研究表明，PINNs在多个应用领域已展现出良好性能。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，这是首批验证PINN框架对此类问题求解有效性的研究之一。针对部分已知解析解的测试问题，我们估算了相应的逼近误差。数值实验涵盖二维和三维空间域，充分证明了该方法在预测精确解方面的有效性。