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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 PME problems require reliable parameter recovery, but standard PINNs converge only locally from favorable initial guesses. This paper develops a two-stage PINN framework for one-dimensional direct and inverse PME problems, improving convergence robustness and reducing dependence on the initial parameter guess.

  • Problem

    Reliable parameter recovery for inverse one-dimensional PME problems remains limited because standard PINNs depend strongly on the initial guess and converge only locally.

  • Method

    The paper develops a PINN framework for direct and inverse one-dimensional PME problems, including a two-stage training strategy for inverse parameter estimation.

  • Results

    The two-stage strategy improves inverse-problem convergence robustness and reduces dependence on the initial parameter guess, while PINN errors for direct and inverse problems are mostly below 5%.

  • Takeaways & Limitations

    PINNs provide a viable and flexible alternative for one-dimensional PME problems and a foundation for extensions to higher-dimensional geometries.

  • Takeaways & Limitations

    Standard inverse PINNs remain sensitive to the initial parameter guess, achieving accurate recovery only when initialization is close to the true value.

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

This paper develops an improved PINN framework for direct and inverse 1D PME problems, addressing initialization-sensitive local convergence through a novel two-stage training strategy. The framework is accurate and flexible, while establishing a foundation for higher-dimensional and more complex geometries.

  • Motivation: PINNs provide a mesh-free and flexible framework for solving inverse PME problems, with easier extension to higher dimensions than classical discretization-based methods.Classical inverse problems are generally more expensive and sometimes ill-conditioned than direct problems, whereas PINNs can address both within one framework.
  • Contributions: The paper improves prior 1D PME PINN work by covering arbitrary m, addressing direct and inverse problems, and testing three manufactured solutions.The manufactured solutions are used to verify the method and test its robustness.
  • Main results: Most direct and inverse PINN errors are below 5%, demonstrating accuracy and flexibility relative to classical methods despite higher computational cost.The resulting 1D framework is intended as a reliable foundation for higher-dimensional problems and more complex geometries.
  • Inverse-problem limitation: Standard inverse PINNs exhibit local convergence because accurate parameter recovery strongly depends on the initial guess.The paper explicitly investigates initialization, training strategy, stability techniques, and loss scheduling as factors affecting inverse PINN performance.
  • Two-stage learning: The proposed two-stage training framework improves convergence robustness and stabilizes parameter estimation across different initial guesses, including guesses far from the true value.The strategy is presented as adaptable to other PDE-based inverse problems through application-specific stages and structures.

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

The paper formulates the one-dimensional PME direct problem on a bounded interval with prescribed initial and homogeneous boundary conditions, then solves it using both a classical finite-difference method and a PINN. The PINN enforces the PDE and condition residuals through automatic differentiation and composite-loss optimization, providing the foundation for the later inverse methodology.

  • 2 The Direct Problem of the 1D PME: The direct PME is posed as u_t = Δ(u^m), m > 1, with nonnegative density, initial data, and zero boundary conditions.For the computational domain, x ∈ [−1, 1] and t ∈ [0, 1].
  • 2 The Direct Problem of the 1D PME: The PME is degenerate because its diffusion coefficient D(u) = mum−1 vanishes at u = 0.The equation models nonlinear diffusion phenomena in physics, biology, and engineering.
  • 2 The Direct Problem of the 1D PME: The study solves the direct problem with a classical numerical method and a PINN, then compares results with four reference solutions.The direct-problem results are presented in Section 2.3, while the reference solutions are introduced in Section 2.2.
  • 2.1.1 Classical Numerical Scheme: The classical reference scheme uses backward Euler in time, central differences in space, and Newton’s method for the nonlinear system at each time step.The Jacobian is approximated using finite differences.
  • 2.1.2 PINN Approach: PINNs approximate the solution with a feedforward neural network trained using automatic differentiation and optimizers such as ADAM and L-BFGS.The inverse framework builds on these direct PME solvers, which were introduced in earlier work.
  • 2.1.2 PINN Approach: For the direct problem, the PINN loss combines residuals for the PDE, initial value, and boundary conditions, with optional physical constraints.Residual terms can be weighted by hyperparameters, including λ in the total loss.
  • 2.1.2 PINN Approach: The PINN uses a fully connected network uθ(t, x), Xavier initialization with tanh activation, and a logarithmic composite loss based on mean-squared residuals.Gradients are computed through automatic differentiation.
  • 2.1.2 PINN Approach: Training applies ADAM for up to 2000 epochs followed by L-BFGS for up to 500 epochs, with early stopping when validation loss improvement remains below 10^-6 for 200 epochs.Sobol points provide quasi-random space-time coverage, using 256 interior, 64 per spatial boundary, and 64 initial-condition points.

2.2 Reference Solutions

The section defines four reference PME solutions for testing the proposed methods: the analytical Barenblatt benchmark and manufactured solutions designed to assess recovery under prescribed source terms and boundary conditions.

  • Reference-solution set: Four PME solutions are introduced as reference cases for testing the numerical methods.The set includes the Barenblatt solution and manufactured solutions.
  • Barenblatt solution: The Barenblatt solution provides an explicit self-similar benchmark for validating numerical methods.It models evolution from point-concentrated initial mass and has compact support determined by the constant C.
  • Manufactured solutions: Manufactured solutions are inserted into the PME to compute source terms, enabling exact-solution recovery tests without requiring a closed-form original solution.This construction supports direct accuracy evaluation of the solver.
  • Manufactured solutions: The reference cases include a smooth polynomial solution that vanishes at the boundaries and therefore satisfies the homogeneous Dirichlet test conditions.Its forcing term is obtained by substituting the function into the governing equation.
  • Manufactured solutions: A time-decaying harmonic solution and an additional m-dependent manufactured solution broaden the validation set beyond solutions independent of m.Both are paired with corresponding source terms; the harmonic case does not use homogeneous Dirichlet boundary conditions.

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

Direct PINNs accurately solve the tested 1D PME cases, generally outperforming classical methods on some solutions but struggling with the m-dependent case. Adding exact measurement data usually preserves error magnitude while increasing training cost, so it provides little benefit when standard PINNs are already accurate.

  • 2.3 Direct PINN Testing without Data: The standard direct PINN uses the PDE residual together with initial and boundary conditions, without measurement data in its loss.
  • Barenblatt Solution Results 2.2.1: Relative error below 5% in all Barenblatt cases shows that both methods are scientifically accurate, while the PINN consistently achieves higher accuracy.Accuracy improves as m increases for both methods; m = 2 produces the largest errors.
  • Polynomial Solution Results 2.2.2: As m increases for the polynomial solution, the number of epochs needed to reach minimum training loss increases, although convergence remains consistent.The solution itself is independent of m, so the tests assess the effect of increasing nonlinearity on training.
  • Damped Harmonic solution Results 2.2.2: The PINN outperforms the classical method in all damped harmonic cases by two orders of magnitude.
  • m-dependent solution Results 2.2.2: The m-dependent solution is harder for both methods: error increases with m, and for m = 2 their errors are of the same order, with the PINN slightly more accurate.The PINN triggers early stopping in all cases, while the classical method takes longer as m increases.
  • 2.4 Direct PINN Testing with Data: In the data-augmented direct PINN, exact measurements are added through a measurement loss weighted λmeas = 10 using Nmeas = 225 values on a uniform 15×15 grid.The grid covers (x, t) ∈[-1, 1]×[0, 1].
  • 2.4 Direct PINN Testing with Data: For the polynomial solution, measurement data slightly reduce errors for m = 4 and m = 5 but increase them for m = 2 and m = 3.The changes are 1.255 × 10−3 to 1.008 × 10−3, 1.740 × 10−3 to 1.622 × 10−3, 1.548 × 10−2 to 1.716 × 10−2, and 8.050 × 10−4 to 8.214 × 10−4, respectively.
  • 2.4 Direct PINN Testing with Data: For the m-dependent solution, measurement data do not resolve convergence difficulty: m = 4 and m = 5 remain above 5%, m = 5 rises from 1.505 × 10−1 to 2.284 × 10−1, and training time increases in every case.Overall, the classical method is always faster, while PINN and classical accuracy varies by solution except for the larger gap in the m-dependent case.

3 The 2-stage Learning Inverse PINN

The inverse PINN recovers the PME parameter m but standard joint training is highly sensitive to initialization and converges only locally. A proposed two-stage schedule first learns the solution structure and then updates m, substantially improving robustness across initial guesses and perturbed data.

  • Inverse PINN limitations: Standard inverse PINN training converged only locally, accurately approximating m mainly when the initial guess was close to the true value.Large approximation errors occurred for distant initial guesses.
  • Training improvements: Xavier normal initialization produced better convergence and more accurate parameter recovery than Xavier uniform initialization in the inverse setting.Both schemes have the same variance, but Xavier normal’s unbounded support and variance characteristics were more effective for training.
  • Two-stage training framework: The two-stage PINN separates learning the solution structure from learning m, first guiding the network toward a physically meaningful solution profile before updating the parameter.This avoids optimizing uθ and m simultaneously from the beginning.
  • Training improvements: 79% to 1.38%: removing early stopping reduced the error for m = 4 with an initial guess of 7.5 after training stopped on a loss plateau.The result suggests that premature stopping can prevent the later loss decrease needed for accurate recovery.
  • Perturbed-data robustness: 1.5%: across three perturbed-data scenarios, the recovered m had relative error no greater than 1.5%, except when true m = 5 and the initial guess was 1.That exception retained errors between 76% and 77% across perturbation levels and already failed in the noise-free case with 76.74% error.

4 Conclusion

The paper establishes a robust PINN framework for direct and inverse one-dimensional Porous Medium Equation problems, centered on a two-stage strategy that improves inverse convergence and parameter recovery. The framework also provides a foundation for higher-dimensional extensions requiring only minor code modifications.

  • 4 Conclusion: The study develops a PINN framework for the inverse one-dimensional Porous Medium Equation and proposes two-stage learning to improve convergence robustness and parameter recovery.The methodology investigates factors affecting inverse PINN performance and aims to support future higher-dimensional PME extensions.
  • 4 Conclusion: The proposed framework accurately solves direct and inverse PME problems across different manufactured solutions and values of m.The two-stage strategy improves inverse robustness and reduces dependence on the initial parameter guess.
  • 4 Conclusion: Future work extends the framework to two-dimensional and higher-dimensional PME problems using Barenblatt and numerically generated solutions for validation, comparison, and data collection.These extensions require mainly adding extra spatial coordinates to the network inputs, with only minor modifications to the existing code.

A Visual Comparisons of Direct Solutions

This section visualizes direct-solution comparisons across each solution and m, showing PINN, exact, and classical solutions alongside PINN training-loss evolution.

  • Heatmaps compare the PINN, exact, and classical solutions for each solution and m, together with the evolution of the PINN training loss.

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].

  • A.1 Barenblatt solution: The exact Barenblatt solution for m = 2 is shown over t ∈[0, 0.3].The solution is presented in a zoomed-in view.
  • A.1 Barenblatt solution: The exact Barenblatt solution for m = 3 is shown over t ∈[0, 0.3].The solution is presented in a zoomed-in view.
  • A.1 Barenblatt solution: The exact Barenblatt solution for m = 4 is shown over t ∈[0, 0.3].The solution is presented in a zoomed-in view.
  • A.1 Barenblatt solution: The exact Barenblatt solution for m = 5 is shown over t ∈[0, 0.3].The solution is presented in a zoomed-in view.

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

The inverse PINN’s m-dependent solution is evaluated by tracking the learned 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 (IG), the inverse PINN tracks the evolution of the m parameter and training loss over the epochs.The passage describes the monitored quantities and experimental conditions but reports no numerical results.

C Inverse PINN results before the 2-stage PINN model

This section presents inverse PINN results obtained after standard modifications but before introducing the two-stage PINN model. Results are shown for different reference solutions from Section 2.2 using the same plot layout as Figure 7.

  • The section evaluates the inverse PINN after standard modifications and before the two-stage PINN model.
  • Results are reported for different reference solutions described in Section 2.2.
  • The results follow the same plot layout used in Figure 7.
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