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

Noura Al Helwani, Sophie Moufawad, Nabil Nassif

arXiv:2608.16475v1math.OCcs.AI

TL;DR

Inverse PINN solutions for the one-dimensional Porous Medium Equation can depend strongly on the initial parameter guess. This paper introduces a two-stage training framework that improves convergence robustness and parameter recovery across different initial guesses.

  • Problem

    The paper seeks a robust PINN methodology for inverse one-dimensional Porous Medium Equation problems and a foundation for higher-dimensional extensions.

    Source: §1 · p. 3, §4 · p. 35

  • Method

    The authors develop PINNs for direct and inverse one-dimensional PME problems and introduce a two-stage training framework for robust inverse parameter estimation.

    Source: §1 · p. 4, §2.1.2 · p. 7, §4 · p. 35

  • Results

    The framework solves direct and inverse PME problems accurately across manufactured solutions and m values, with most errors below 5%.

    Source: §1 · p. 3, §4 · p. 35

  • Takeaways & Limitations

    The approach provides a foundation for extending PINNs to higher-dimensional PME geometries, though it has higher computational cost than classical methods.

    Source: §1 · p. 3

    Standard inverse PINNs converge only locally, with accurate parameter recovery requiring an initial guess close to the true value.

    Source: §1 · p. 4, Standard Inverse PINN Model · p. 19

Abstract

from arXiv · show

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.

1 Introduction

The paper develops a PINN framework for direct and inverse 1D Porous Medium Equation problems, emphasizing robust inverse parameter recovery and future extension to higher-dimensional geometries. It identifies initialization-sensitive local convergence in standard inverse PINNs and introduces two-stage training to improve robustness.

Motivation

  • PINNs are motivated by their mesh-free, flexible, sparse-data capability and unified treatment of direct and inverse PDE problems, despite higher computational cost than classical methods.Classical approaches become more expensive, complex, and potentially ill-conditioned for inverse or higher-dimensional problems, while PINNs retain the same main structure with changes mainly to inputs and collocation sampling.Source: §1 · pp. 3–4

Overall findings

  • Across direct and inverse problems, the PINN framework achieves mostly below-5% errors and is a viable, flexible alternative to classical methods for the 1D PME.The reported results emphasize accuracy and flexibility while acknowledging higher computational cost and room for improvement.Source: §1 · p. 3

Scope and objectives

  • The study develops an improved PINN framework for direct and inverse 1D PME problems, using classical numerical solvers as validation benchmarks across manufactured solutions and values of m.The framework is intended to establish a reliable foundation before extending to more complex geometries and higher dimensions.Source: §1 · pp. 3–4

Inverse-problem limitation

  • Standard inverse PINNs exhibit local convergence because accurate parameter recovery strongly depends on the initial guess.The paper therefore investigates initialization, training strategy, stability techniques, and loss scheduling for inverse PINN performance.Source: §1 · p. 4

Two-stage inverse training

  • The proposed two-stage training framework improves convergence robustness and stabilizes inverse parameter estimation across different initial guesses, including guesses far from the true solution.The strategy is presented as adaptable to other PDE-based inverse problems through problem-specific stage structures.Source: §1 · pp. 3–4

Future extension

  • The work establishes a robust 1D inverse PINN foundation for higher-dimensional and more complex geometries, where classical methods typically require new discretizations, meshes, or solvers.This foundation is the stated prerequisite for extending the methodology beyond the one-dimensional setting.Source: §1 · pp. 3–4

2 The Direct Problem of the 1D PME · 2.1 Methodologies

The paper formulates the one-dimensional direct PME with degenerate nonlinear diffusion, finite-speed propagation, and prescribed initial and boundary data. It summarizes classical finite differences and PINNs, including their physics-informed losses, optimization, and evaluation against analytical solutions.

2 The Direct Problem of the 1D PME

  • The 1D PME models nonnegative density u(x,t) with exponent m > 1, zero boundary conditions, and prescribed initial data on a bounded interval.Its degeneracy causes finite-speed propagation and moving interfaces, while the Barenblatt solution provides an exact reference for numerical verification.Source: §2 · pp. 5–6
  • The computational domain is x ∈ [-1, 1], t ∈ [0, 1], with homogeneous spatial boundary conditions and initial condition u(0,x) = u0(x).These conditions define the direct PME problem used throughout the numerical formulations.Source: §2 · p. 6

2.1 Methodologies

  • The direct problem is solved using both a classical numerical method and a PINN, with reference solutions introduced for subsequent comparison.The paper positions these direct solvers as the foundation for the later inverse methodology.Source: §2 · p. 6, §2.1 · p. 6

2.1.1 Classical Numerical Scheme

  • The classical scheme uses backward Euler time discretization, central spatial differences, and Newton’s method to solve the nonlinear system at each time step.Its implementation uses finite-difference Jacobian approximations and Barenblatt-based initial and boundary conditions.Source: §2.1.1 · pp. 6–7

2.1.2 PINN Approach

  • PINNs approximate u(t,x) with a neural network and penalize violations of the PDE, initial conditions, and boundary conditions through automatic differentiation.For inverse problems, the loss additionally includes data misfit, whereas the direct formulation uses PDE, initial-value, and boundary residuals.Source: §2.1.2 · pp. 7–8, weights · p. 9
  • The fully connected network uses Xavier initialization with tanh activation, trains with ADAM for up to 2000 epochs, then L-BFGS for up to 500 epochs, and uses early stopping.Training halts when validation loss fails to improve by 10−6 over a 200-epoch patience window.Source: Neural Network Architecture · p. 8, weights · p. 9
  • The direct PINN minimizes a logarithmic composite loss combining boundary, initial-condition, and PDE residuals computed as mean squared errors.The formulation samples initial, boundary, and interior points and computes gradients using automatic differentiation.Source: weights · p. 9
  • PINN accuracy is assessed against the analytical solution using relative L2-error on the same grid points used by the classical Newton method.This shared grid supports direct comparison between the PINN and classical numerical solution.Source: weights · p. 9

2.2 Reference Solutions

The methods are tested against four PME reference solutions, combining the analytical Barenblatt benchmark with manufactured solutions designed for exact solver validation and boundary-condition testing.

2.2 Reference Solutions

  • Four PME solutions provide the reference set for testing the proposed methods.These include the Barenblatt solution and three manufactured solutions.Source: §2.2 · p. 10

2.2.1 Barenblatt Solution

  • The Barenblatt solution is an explicit self-similar benchmark describing evolution from point-concentrated initial mass.Its compact support depends on the total-mass constant C, and it does not use homogeneous Dirichlet boundary conditions.Source: §2.2.1 · p. 10

2.2.2 Manufactured Solution

Polynomial Solution

  • The polynomial manufactured solution is smooth in time and space, and vanishes at the boundaries to satisfy the test setup’s boundary conditions.Its forcing term is obtained by substituting the solution into the PME.Source: Polynomial Solution · p. 10

Damped Harmonic Solution

  • The damped harmonic manufactured solution provides a time-decaying test case but does not use homogeneous Dirichlet boundary conditions.A corresponding source term is derived for the equation.Source: Damped Harmonic Solution · p. 10

m-Dependent manufactured solution

  • An additional manufactured solution depends explicitly on m, complementing the earlier manufactured solutions that are independent of the nonlinearity parameter.Its corresponding source term is used in validation comparisons.Source: m-Dependent manufactured solution · p. 11

2.3 Direct PINN Testing without Data · 2.4 Direct PINN Testing with Data

Direct PINN tests across four reference solutions show generally accurate results, with performance depending on the solution and nonlinearity parameter m. Adding exact measurement data to the loss does not meaningfully improve accuracy and increases training time, including for cases where the standard PINN struggles.

2.3 Direct PINN Testing without Data

  • The no-data direct-PINN evaluation compares relative L2-errors, approximate plateau epochs, and runtimes for m ∈ {2, 3, 4, 5} against four reference solutions.Visual comparisons are shown for m = 3, while other cases appear in Appendix A; epochs denote the first loss plateau rather than final training epochs.Source: §2.3 · p. 11

Barenblatt Solution Results 2.2.1

  • For the Barenblatt solution, both methods have relative errors below 5%, with PINNs consistently more accurate and both methods improving as m increases.The largest errors occur at m = 2, while larger m produces more localized profiles that are easier to approximate.Source: Barenblatt Solution Results 2.2.1 · p. 11, Table 3 · p. 12

Polynomial Solution Results 2.2.2

Damped Harmonic solution Results 2.2.2

m-dependent solution Results 2.2.2

  • The m-dependent solution is harder for both methods: errors increase with m, while classical errors remain about 0.8%–2% and PINN errors range from about 2%–15% for m > 2.At m = 2, the PINN is slightly more accurate and both methods have errors of the same order; for larger m, the classical method is more accurate.Source: m-dependent solution Results 2.2.2 · p. 15, Table 12 · p. 15

2.4 Direct PINN Testing with Data

  • With exact data, the polynomial test uses 225 values on a uniform 15×15 grid and measurement-loss weight λmeas = 10, matching the initial and boundary-condition weight.The added loss term augments the standard PDE, initial-condition, and boundary-condition losses.Source: §2.4 · p. 16
  • Adding measurement data does not meaningfully improve direct-PINN accuracy and increases training time, including when the standard PINN struggles on the m-dependent solution.For m = 5 in the m-dependent case, the error increases from 1.505 × 10^-1 to 2.284 × 10^-1, while training time increases in all four cases.Source: §2.4 · pp. 16–17, Table 16 · p. 17
  • Across direct tests, PINNs achieve accurate solutions close to classical methods, but classical methods are always faster and remain more accurate for the m-dependent solution.The comparison includes early-stopping time in PINN runtimes, including a 200-epoch patience period.Source: §2.4 · p. 17

3 The 2-stage Learning Inverse PINN

This section examines the inverse PME problem of estimating the nonlinear parameter from reference measurements using classical optimization and PINNs, focusing on the proposed two-stage training framework and stabilization choices. It evaluates accuracy, robustness to measurement perturbations and initialization, behavior for larger nonlinearities, and computational cost.

3.1 Classical Approach: fmincon

  • The classical fmincon method estimates m by minimizing numerical–reference error under 1 ≤ m ≤ 10, solving the PDE with Newton’s method for each candidate.Synthetic data are generated from known-m reference solutions before constrained iterative optimization.Source: §3.1 · p. 18

3.2 PINN Approach

  • The inverse PINN retains the direct model’s architecture, treats m as trainable, and adds measurement loss to the PDE and boundary/initial-condition losses.Measurement points are generated from the corresponding exact solution, while training otherwise follows the direct PINN procedure.Source: §3.2 · p. 18, Standard Inverse PINN Model · p. 18

Standard Inverse PINN Model

  • The two-stage PINN substantially improves robustness over the standard inverse PINN, enabling accurate recovery of m across a wider range of initial guesses.The standard formulation converges well only when the initial guess is near the true value, whereas the two-stage schedule stabilizes optimization by separating solution learning from parameter learning.Source: Standard Inverse PINN Model · p. 19, Figure 7 · p. 19

3.3 Inverse PINN Design Experiments

  • Design experiments retained the original direct-problem architecture, selected Xavier normal initialization, and used positivity parameterization plus L2 gradient clipping for stability.Alternative network widths and depths showed no clear improvement; Xavier normal gave better convergence and parameter recovery, while clipping with max norm = 1 prevented NaN values when m was large.Source: Neural Network Architecture · p. 20, Weights · p. 20, Stability · pp. 20–21

3.3.1 The Proposed Two-Stage Training Inverse PINN Model

  • The proposed schedule freezes m at m0 for 20% of epochs, trains uθ with PDE and boundary conditions, then jointly optimizes θ and m.This warm-up reduces incorrect parameter updates caused by an initially poorly learned solution; training uses ADAM for 10,000 total epochs.Source: §3.3.1 · p. 21
  • The two-stage method also anneals the measurement-loss weight λm from 10 during warm-up to a final value of 100, after fixed-weight training proved sensitive and learnable weights proved unstable.The model used 2048 interior, 256 spatial-boundary-per-side, 256 temporal-boundary, and 225 measurement points.Source: §3.3.1 · pp. 21–22

3.4.1 Inverse PINN testing without Noisy Data

  • In noiseless tests, PINNs produced results for all cases, while many classical runs failed; successful classical runs were usually more accurate, and most errors were below 5%.The comparison covers m ∈ {2, 3, 4, 5} and initial guesses perturbed by ±50% and ±80%.Source: §3.4 · p. 23, Barenblatt Solution · p. 25, Table 19 · p. 25
  • Removing early stopping reduced the m = 4, initial guess 7.5 error from 79% to 1.38%.The run had stopped during a loss plateau at 4000 epochs, whereas comparable cases later showed another loss decrease.Source: Barenblatt Solution · p. 26, Figure 12 · p. 26
  • For the m-dependent solution, inverse PINN errors increase with nonlinearity: learning is best at m = 2 and worst at m = 5, with some runs stuck at their initial guesses.At m = 2, most initial guesses stay near the 5% error threshold, whereas higher m values produce errors of 20–80%; some m = 3–5 cases equal the initial offset.Source: m-dependent solution · p. 28, Table 22 · p. 29
  • Increasing training epochs from 10,000 to 20,000 reduced the m = 2, 50%-high initial-guess error to 2.9%, while flat losses may require rescaling, L-BFGS, or improved training.Early stopping or exhausted epochs often occurred as the loss began decreasing, suggesting insufficient training in several difficult cases.Source: m-dependent solution · p. 29, Figure 16 · p. 30
  • The inverse PINN performs comparably to the classical method overall, producing results even for challenging m-dependent cases where numerical instabilities and singularities prevent classical solutions.The classical method succeeds accurately only for m = 2 across all initial guesses and m = 3 with an initial guess of 1.5 (-50%).Source: m-dependent solution · p. 30

Time comparison

  • The classical method is consistently faster, averaging about 7–36 seconds, or 107 seconds for the m-dependent solution, versus roughly 300–340 seconds for the PINN.A Python–MATLAB comparison attributes much of the timing gap to nested-loop execution speed: the harmonic direct solve took 26.34 seconds in Python versus 1.21 seconds in MATLAB.Source: Time comparison · pp. 30, 32, Figure 17 · p. 31, Table 23 · p. 32

3.4.2 Inverse PINN testing with Noisy Data

  • Across perturbed-data scenarios, the recovered m has relative error at most 1.5%, except true m = 5 with initial guess 1, where error remains 76–77%.The exception already fails without noise, while other estimates remain stable as noise increases, indicating robustness to measurement perturbations.Source: Table 25 · p. 33, §3.4.2 · pp. 34–35, Table 26 · p. 34

4 Conclusion

The paper develops a robust PINN framework for direct and inverse one-dimensional PME problems, centered on two-stage learning to improve inverse convergence and parameter recovery. The framework establishes a foundation for higher-dimensional extensions.

Conclusion

  • The study develops a robust inverse-PINN methodology by investigating performance factors and introducing two-stage learning to improve convergence robustness and parameter recovery.The framework is intended as a foundation for future higher-dimensional PME extensions.Source: §4 · p. 35
  • The proposed two-stage PINN framework successfully solves direct and inverse 1D PME problems accurately across manufactured solutions and different m values, while reducing dependence on the initial parameter guess.This strategy improves inverse-problem robustness and parameter recovery.Source: §4 · p. 35

Future work

  • Future work will extend the framework to two-dimensional and higher-dimensional PME problems using Barenblatt, finite-difference, and other numerical solutions for validation, comparison, and data collection.These extensions require mainly adding spatial coordinates, updating the PDE residual, and sampling collocation points in new domains.Source: §4 · p. 35

A Visual Comparisons of Direct Solutions

This section visualizes PINN, exact, and classical solutions for each solution case and exponent m, alongside the evolution of PINN training loss.

A Visual Comparisons of Direct Solutions

A.1 Barenblatt solution

The section presents zoomed-in views of the exact Barenblatt solution for m = 2, 3, 4, and 5 over t ∈[0, 0.3].

  • The exact Barenblatt solution for m = 2 is shown over t ∈[0, 0.3].Source: Figure 19 · p. 40
  • The exact Barenblatt solution for m = 3 is shown over t ∈[0, 0.3].Source: Figure 22 · p. 41
  • The exact Barenblatt solution for m = 4 is shown over t ∈[0, 0.3].Source: Figure 25 · p. 42
  • The exact Barenblatt solution for m = 5 is shown over t ∈[0, 0.3].Source: Figure 28 · p. 43

A.2 Polynomial solution … B Inverse PINN’s training loss of the m-dependent solution

The inverse PINN study tracks the evolution of the unknown m parameter and training loss across epochs for each m and initial guess.

B Inverse PINN’s training loss of the m-dependent solution

  • For each m and initial guess, the inverse PINN’s m parameter and training loss are tracked over the training epochs.The supplied passage does not provide numerical trajectories or comparisons among initial guesses.Source: B Inverse PINN’s training loss of the… · p. 51

C Inverse PINN results before the 2-stage PINN model

This section presents inverse PINN results after standard modifications but before the two-stage model, across reference solutions using the Figure 7 plot layout.

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