math.OCAug 17, 2026

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

Authors: Noura Al Helwani, Sophie Moufawad, Nabil Nassif

Organizations: Department of Physics, American University of Beirut, Lebanon · Department of Mathematics, American University of Beirut, Lebanon

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Abstract

The Porous Medium Equation (PME), given by ut=Δ(um)u_t = Δ(u^m) for m>1m > 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.

Figures & tables

ParameterValue
Spatial domainx∈[−1,1]x\in[-1,1] with N=100N=100 intervals ( N+1N+1 grid points)
Time domaint∈[0,1]t\in[0,1] with Δt=0.01\Delta t=0.01 ( 100100 time steps)
Tolerance (Newton)10−610^{-6}
Maximum Newton iterations2020 per time step
Jacobian approximationFinite differences with perturbation h=10−6h=10^{-6}
Table 1: Parameters for Direct Classical Method
Figure 1: Schematic of the direct PINN
ComponentSpecification
Input layerSize 2 (for tt and xx )
Hidden layers4 layers, each with 20 neurons
Activation functionHyperbolic tangent tanh⁡\tanh
Output layerSingle neuron returning uθ(t,x)u_{\theta}(t,x)
Table 2: Architecture of the neural network.
mmPINN vs ExactClassical vs ExactPINN vs Classial
21.774 ×10−2\times 10^{-2}2.475 ×10−2\times 10^{-2}2.591 ×10−2\times 10^{-2}
35.378 ×10−4\times 10^{-4}1.557 ×10−2\times 10^{-2}1.315 ×10−2\times 10^{-2}
43.394 ×10−4\times 10^{-4}7.232 ×10−3\times 10^{-3}6.404 ×10−3\times 10^{-3}
51.575 ×10−4\times 10^{-4}3.652 ×10−3\times 10^{-3}3.322 ×10−3\times 10^{-3}
Table 3: Relative L2L^{2} -errors for the Barenblatt Solution
Table 5
Figure 2: visual comparison of Barenblatt numerical solutions for m=3m=3
Figure 3: Zoomed-in view of the exact Barenblatt solution for m=3m=3 for t ∈[0,0.3]\in[0,0.3]
Figure 4: visual comparison of the Polynomial numerical solutions
mmPINN vs ExactClassical vs ExactPINN vs Classical
21.548 ×10−2\times 10^{-2}1.012 ×10−2\times 10^{-2}2.255 ×10−2\times 10^{-2}
38.050 ×10−4\times 10^{-4}1.134 ×10−2\times 10^{-2}1.127 ×10−2\times 10^{-2}
41.255 ×10−3\times 10^{-3}1.186 ×10−2\times 10^{-2}1.157 ×10−2\times 10^{-2}
51.740 ×10−3\times 10^{-3}1.215 ×10−2\times 10^{-2}1.180 ×10−2\times 10^{-2}
Table 6: Relative L2L^{2} -errors for the Polynomial solution
Table 10
Figure 5: visual comparison of the Harmonic numerical solutions
mmPINN vs ExactClassical vs ExactPINN vs Classical
22.081 ×10−5\times 10^{-5}1.292 ×10−3\times 10^{-3}1.287 ×10−3\times 10^{-3}
32.096 ×10−5\times 10^{-5}1.449 ×10−3\times 10^{-3}1.451 ×10−3\times 10^{-3}
43.656 ×10−5\times 10^{-5}1.723 ×10−3\times 10^{-3}1.722 ×10−3\times 10^{-3}
55.156 ×10−5\times 10^{-5}2.049 ×10−3\times 10^{-3}1.566 ×10−3\times 10^{-3}
Table 9: Relative L2L^{2} -errors for the Damped Harmonic solution
Table 13
Figure 6: visual comparison of the m-dependent numerical solutions
mmPINN vs ExactClassical vs ExactPINN vs Classical
21.586 ×10−3\times 10^{-3}2.715 ×10−3\times 10^{-3}1.636 ×10−3\times 10^{-3}
31.651 ×10−2\times 10^{-2}8.514 ×10−3\times 10^{-3}1.311 ×10−2\times 10^{-2}
47.679 ×10−2\times 10^{-2}1.437 ×10−2\times 10^{-2}7.074 ×10−2\times 10^{-2}
51.505 ×10−1\times 10^{-1}1.802 ×10−2\times 10^{-2}1.420 ×10−1\times 10^{-1}
Table 12: Relative L2L^{2} -errors for the m-dependent solution
Table 16
mPINN relative errorPINN training time
21.716 ×10−2\times 10^{-2}60.74
38.214 ×10−4\times 10^{-4}207.00
41.008 ×10−3\times 10^{-3}193.70
51.622 ×10−3\times 10^{-3}191.69
Table 15: Relative L2L^{2} -errors and training time (second) for the direct PINN applied to the polynomial solution using exact data
mPINN relative errorPINN training time
21.109 ×10−3\times 10^{-3}69.66
31.283 ×10−2\times 10^{-2}71.18
46.747 ×10−2\times 10^{-2}71.32
52.284 ×10−1\times 10^{-1}46.68
Table 16: Relative L2L^{2} -errors and training time (second) for the direct PINN applied to the m-dependent solution using exact data
Figure 7: Relative error of the estimated parameter mm with respect to different initial guesses. Each color corresponds to a different true value of mm . The x-axis represents the initial guess m0m_{0} , taken around the true value ( ±50%\pm 50\% and 80%80\% perturbations). The y-axis shows the relative error (%). The left plot shows the standard inverse PINN, where the solution is accurate only when the initial guess is close to the true value, while the right plot shows the results after applying the 2-stage training PINN, where the method achieves significantly lower and more stable errors across different initial guesses.
Figure 8: Training Diagram
ParameterValue
Optimization algorithmInterior-point (MATLAB default)
Maximum iterations50
Optimality tolerance10−610^{-6}
Lower boundm=1m=1
Upper boundm=10m=10
Table 17: fmincon Configuration
CategoryParameterValue/Description
Training Points ( see Fig 9 )n_int (interior points)2048
 n_sb (spatial boundary points per side)256
 n_tb (temporal boundary points)256
 n_meas (measurement points)15x15 = 225
DomainTime domain[0, 1]
 Space domain[-1, 1]
Table 18: Configurations and Parameters for Inverse PINN Method
Figure 9: Distribution of Training Points
true minitial guessfmincon relative errorPINN relative errorfmincon timePINN training time
20.4failed1.74%2.30228.70
1failed4.67%0.70238.43
30.28%1.07%14.50229.50
3.60.28%1.71%19.88124.91
30.6failed0.79%0.76230.82
1.50.37%0.52%12.52238.57
Table 19: Relative error and computational time (seconds) comparison between the classical method and the PINN approach for the Barenblatt solution with different values of mm and initial guesses.
Figure 10: Inverse Barenblatt solution results plots
Figure 11: Training loss and m evolution for case m=3 with initial guess =4.5
Figure 12: Training loss and m evolution for case m=4 with initial guess =7.5, before and after removing the early stopping function
true minitial guessfmincon relative errorPINN relative errorfmincon timePINN training time
20.41.10%0.51%30.05360.64
11.10%0.28%29.46356.92
31.10%0.15%16.38298.42
3.61.10%0.22%34.53354.22
30.60.07%0.14%33.93360.07
1.50.07%0.20%24.96362.01
Table 20: Relative error and computational time (seconds) comparison between the classical method and the PINN approach for the polynomial solution with different values of mm and initial guesses.
Figure 13: Inverse Polynomial solution plots
true minitial guessfmincon relative errorPINN relative errorfmincon timePINN training time
20.40.85%0.94%5.38338.86
10.85%1.37%7.32338.78
30.85%0.15%12.32250.15
3.60.85%0.11%14.43341.01
30.60.70%0.31%9.31352.13
1.50.70%0.33%7.77349.41
Table 21: Relative error and computational time (seconds) comparison between the classical method and the inverse PINN approach for the harmonic solution with different values of mm and initial guesses.
Figure 14: Inverse Harmonic solution plots
true minitial guessfmincon relative errorPINN relative errorfmincon timePINN training time
20.40.08%6.79%107.29379.69
10.08%2.83%110.53382.94
30.08%49.85%71.56266.39
3.60.08%0.15%55.11366.76
30.6failed9.49%26.84370.63
1.50.27%29.66%1248.56376.23
Table 22: Relative error and computational time (seconds) comparison between the classical method and the PINN approach for the m_dependent solution with different values of mm and initial guesses.
Figure 15: Inverse m-dependent solution plots
Figure 16: Training loss and m evolution for case m=2 with initial guess =3, after increasing the epochs and removing the early stopping, for the m-dependent solution
Polynomial Solution
mmBarenblattPolynomialHarmonicm-dependent
20.0780.0870.1820.134
30.0400.0530.0330.054
40.3880.5290.4960.938
50.3780.5240.4911.133
Table 23: Computation time (seconds) for solving the direct problem using MATLAB for different manufactured solutions.
ParameterValue
Noise modelAdditive Gaussian, εi∼N(0,σ2)\varepsilon_{i}\sim\mathcal{N}(0,\sigma^{2})
Noise levels δ\delta0%, 1%, 5%0\%,\ 1\%,\ 5\%
Standard deviation σ\sigmaδ⋅RMS(uPINN)\delta\cdot\text{RMS}\left(u^{\text{PINN}}\right)
Perturbed quantityMeasurement dataset only (from direct PINN)
Number of perturbed pointsNmeas=225N_{\text{meas}}=225
Random seed42
Table 24: Parameters of the data perturbation used in the inverse PINN experiments.
true minitial guessrecovered mPINN relative errorPINN training time s
20.42.000.22%338.66
1.02.000.01%285.76
3.02.000.30%224.85
3.62.000.35%244.06
30.63.000.28%341.31
1.53.000.19%337.40
Table 25: PINN inverse results for the polynomial solution using measurement data generated from the direct PINN with 0% additional noise
true minitial guessrecovered mPINN relative error pctPINN training time s
20.42.000.04%355.85
11.990.22%254.72
31.990.02%201.63
3.62.000.40%223.58
30.63.000.04%359.60
1.53.000.07%360.32
Table 26: PINN inverse results for the polynomial solution using measurement data generated from the direct PINN with 1% additional noise
Appendix figures & tables40 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 40
Figure 19: Zoomed-in view of the exact Barenblatt solution for m=2m=2 for t ∈[0,0.3]\in[0,0.3]
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Figure 22: Zoomed-in view of the exact Barenblatt solution for m=3m=3 for t ∈[0,0.3]\in[0,0.3]
Figure 45
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Figure 25: Zoomed-in view of the exact Barenblatt solution for m=4m=4 for t ∈[0,0.3]\in[0,0.3]
Figure 48
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Figure 28: Zoomed-in view of the exact Barenblatt solution for m=5m=5 for t ∈[0,0.3]\in[0,0.3]
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