Solving Inverse PDE Problems using Minimization Methods and AI
This work addresses parameter estimation in physical and engineering systems, but it is incremental as it applies existing PINN methods to new equations.
The paper tackled solving both direct and inverse problems for systems governed by differential equations, comparing traditional numerical methods with Physics-Informed Neural Networks (PINNs). Results showed that PINNs can closely estimate solutions at competitive computational cost for equations like the logistic differential equation and the Porous Medium Equation.
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.