Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well-established numerical methods with new AI-based techniques, specifically Physics-Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed-form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.

翻译：许多物理与工程系统既需要求解正问题以预测系统行为，又需要通过反问题从测量数据中确定未知参数。本文针对微分方程控制的系统，对比研究成熟的数值方法与新兴的基于人工智能的技术——特别是物理信息神经网络（PINNs）。我们首先分析逻辑斯蒂微分方程，利用其解析解验证数值格式并评估PINN的性能。随后针对无一般解析解的非线性偏微分方程——多孔介质方程（PME），构建了强健的正问题求解器，并测试了反问题中参数估计的技术。结果表明，PINNs能以具有竞争力的计算成本精确逼近解，从而为复杂系统的正反问题求解提供了有效工具。