# A Two-Stage Learning PINN Approach for Solving the Inverse Problem of the 1D Porous Medium Equation

arXiv: https://arxiv.org/abs/2608.16475 · HTML: https://mlsanity.com/paper/2608.16475/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 50 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Transfer Learning in Physics-Informed Neural Networks: Full Fine-Tuning, Lightweight Fine-Tuning, and Low-Rank Adaptation](https://mlsanity.com/paper/2502.00782/) (2025)
- [Stiff Transfer Learning for Physics-Informed Neural Networks](https://mlsanity.com/paper/2501.17281/) (2025)
- [Physics-informed neural networks with curriculum training for poroelastic flow and deformation processes](https://mlsanity.com/paper/2404.13909/) (2024)
- [Rethinking the shape convention of an MLP](https://mlsanity.com/paper/2510.01796/) (2025)
- [Solving Inverse PDE Problems using Minimization Methods and AI](https://mlsanity.com/paper/2603.01731/) (2026)

## Abstract (arXiv)

The Porous Medium Equation (PME), given by ut = Delta(um) 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.