Solving Inverse PDE Problems using Minimization Methods and AI

The Porous Medium Equation (PME), given by $u_t = Δ(u^m)$ for $m > 1$, is a degenerate nonlinear parabolic partial differential equation that arises in various physical applications such as fluid flow in porous media, heat transfer in plasmas, and population dynamics. It is known for its nonlinear diffusion and finite propagation speed. In this paper, we study numerical solutions of the one-dimensional direct and inverse PME using Physics-Informed Neural Networks (PINNs), and compare them with classical numerical methods and available analytical and manufactured solutions. While PINNs provide a flexible framework for solving both forward and inverse problems, we show that the standard inverse formulation suffers from a strong sensitivity to the initial guess, leading to only local convergence. To address this issue, we propose a novel two-stage PINN training framework for the inverse problem, which significantly improves convergence stability and allows reliable recovery of the unknown parameter even for poor initial guesses. Overall, the proposed approach demonstrates that PINNs are a flexible and accurate alternative to classical methods for the 1D PME, and the introduced two-stage training strategy substantially improves their robustness in inverse problems, providing a solid basis for extensions to more complex geometries and higher-dimensional cases.

Solving inverse partial differential equation (PDE) problems is a fundamental topic in scientific research due to its broad significance across a wide range of real-world applications. Inverse PDE problems arise across medical imaging, geophysics, materials science, and aerodynamics, where the goal is to infer hidden causes, design structures, or control physical states. In this paper, we provide a comprehensive review of recent advances in solving inverse PDE problems using artificial intelligence (AI). We first introduce the basic formulation, key challenges, and traditional numerical foundations of inverse PDE problems, and then organize it into three major categories: inverse problems, inverse design, and control problems. For each category, we further present a methodological paradigms, and review representative state-of-the-art approaches from recent years. We then summarize representative applications across scientific and industrial domains, including mechanical systems, aerodynamic problems, thermal systems, full-waveform inversion, system identification, and medical imaging. Finally, we discuss open challenges and future prospects, such as physics-informed architectures, limited real-world data, uncertainty quantification, and inverse foundation models. This survey aims to provide the first unified and systematic perspective on AI for inverse PDE problems, demonstrating how modern learning-based methods are reshaping inverse problems, inverse design, and control problems in PDE-governed systems.

We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.