Solving Inverse PDE Problems using Minimization Methods and AI

TL;DR

This work first analyzes the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance, and addresses the Porous Medium Equation, a nonlinear partial differential equation with no general closed-form solution.

Abstract

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.

Figures (16)

• Figure 1: Fully Connected NN example
• Figure 2: Direct Problem: PINN Prediction vs Analytical Solution for Case 1.
• Figure 3: Direct Problem : PINN Prediction vs Analytical Solution for Case 2.
• Figure 4: Direct problem: PINN Prediction vs Analytical Solution for Case 3.
• Figure 5: PINN: direct problem for Case 3 after normalization.