用AI方法求解微分方程正问题和逆问题，效果接近传统方法且更高效。
Solving Inverse PDE Problems using Minimization Methods and AI
- 结合物理信息神经网络与数值方法，统一求解微分方程的正/逆问题
- 在无解析解的非线性庞加莱介质方程上，PINN逼近精度高且计算成本低
- 适合需要快速参数反演的复杂系统建模与仿真研究者使用
许多物理与工程系统需解决两类问题：一是预测系统行为的正问题，二是根据观测数据反推未知参数的逆问题。本文针对由微分方程描述的系统，对比了经典的数值方法与新兴的AI技术（特别是物理信息神经网络，PINNs）。首先以逻辑斯蒂微分方程为例，利用其闭式解验证数值方案并评估PINN性能；随后处理无一般闭式解的非线性庞加莱介质方程（PME），构建高效的直接问题求解器，并测试逆问题中的参数估计方法。结果表明，PINNs能在可接受的计算开销下精准逼近解，为复杂系统同时求解正问题与逆问题提供了有效工具。
原文摘要 · Abstract (English)
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.
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