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

Noura Al Helwani; Sophie Moufawad; Nabil Nassif

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

Abstract

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.

Figures & tables

Parameter	Value
Spatial domain	$x\in[-1,1]$ with $N=100$ intervals ( $N+1$ grid points)
Time domain	$t\in[0,1]$ with $\Delta t=0.01$ ( $100$ time steps)
Tolerance (Newton)	$10^{-6}$
Maximum Newton iterations	$20$ per time step
Jacobian approximation	Finite differences with perturbation $h=10^{-6}$

Table 1: Parameters for Direct Classical Method

Component	Specification
Input layer	Size 2 (for $t$ and $x$ )
Hidden layers	4 layers, each with 20 neurons
Activation function	Hyperbolic tangent $\tanh$
Output layer	Single neuron returning $u_{\theta}(t,x)$

Table 2: Architecture of the neural network.

$m$	PINN vs Exact	Classical vs Exact	PINN vs Classial
2	1.774 $\times 10^{-2}$	2.475 $\times 10^{-2}$	2.591 $\times 10^{-2}$
3	5.378 $\times 10^{-4}$	1.557 $\times 10^{-2}$	1.315 $\times 10^{-2}$
4	3.394 $\times 10^{-4}$	7.232 $\times 10^{-3}$	6.404 $\times 10^{-3}$
5	1.575 $\times 10^{-4}$	3.652 $\times 10^{-3}$	3.322 $\times 10^{-3}$

Table 3: Relative

L^{2}

-errors for the Barenblatt Solution

Table 5

Panel of Figure 2 — Figure 2: visual comparison of Barenblatt numerical solutions for $m=3$

Figure 3: Zoomed-in view of the exact Barenblatt solution for $m=3$ for t $\in[0,0.3]$

Panel of Figure 4 — Figure 4: visual comparison of the Polynomial numerical solutions

$m$	PINN vs Exact	Classical vs Exact	PINN vs Classical
2	1.548 $\times 10^{-2}$	1.012 $\times 10^{-2}$	2.255 $\times 10^{-2}$
3	8.050 $\times 10^{-4}$	1.134 $\times 10^{-2}$	1.127 $\times 10^{-2}$
4	1.255 $\times 10^{-3}$	1.186 $\times 10^{-2}$	1.157 $\times 10^{-2}$
5	1.740 $\times 10^{-3}$	1.215 $\times 10^{-2}$	1.180 $\times 10^{-2}$

Table 6: Relative

L^{2}

-errors for the Polynomial solution

Table 10

Panel of Figure 5 — Figure 5: visual comparison of the Harmonic numerical solutions

$m$	PINN vs Exact	Classical vs Exact	PINN vs Classical
2	2.081 $\times 10^{-5}$	1.292 $\times 10^{-3}$	1.287 $\times 10^{-3}$
3	2.096 $\times 10^{-5}$	1.449 $\times 10^{-3}$	1.451 $\times 10^{-3}$
4	3.656 $\times 10^{-5}$	1.723 $\times 10^{-3}$	1.722 $\times 10^{-3}$
5	5.156 $\times 10^{-5}$	2.049 $\times 10^{-3}$	1.566 $\times 10^{-3}$

Table 9: Relative

L^{2}

-errors for the Damped Harmonic solution

Table 13

Panel of Figure 6 — Figure 6: visual comparison of the m-dependent numerical solutions

$m$	PINN vs Exact	Classical vs Exact	PINN vs Classical
2	1.586 $\times 10^{-3}$	2.715 $\times 10^{-3}$	1.636 $\times 10^{-3}$
3	1.651 $\times 10^{-2}$	8.514 $\times 10^{-3}$	1.311 $\times 10^{-2}$
4	7.679 $\times 10^{-2}$	1.437 $\times 10^{-2}$	7.074 $\times 10^{-2}$
5	1.505 $\times 10^{-1}$	1.802 $\times 10^{-2}$	1.420 $\times 10^{-1}$

Table 12: Relative

L^{2}

-errors for the m-dependent solution

Table 16

m	PINN relative error	PINN training time
2	1.716 $\times 10^{-2}$	60.74
3	8.214 $\times 10^{-4}$	207.00
4	1.008 $\times 10^{-3}$	193.70
5	1.622 $\times 10^{-3}$	191.69

Table 15: Relative

L^{2}

-errors and training time (second) for the direct PINN applied to the polynomial solution using exact data

m	PINN relative error	PINN training time
2	1.109 $\times 10^{-3}$	69.66
3	1.283 $\times 10^{-2}$	71.18
4	6.747 $\times 10^{-2}$	71.32
5	2.284 $\times 10^{-1}$	46.68

Table 16: Relative

L^{2}

-errors and training time (second) for the direct PINN applied to the m-dependent solution using exact data

Panel of Figure 7 — Figure 7: Relative error of the estimated parameter $m$ with respect to different initial guesses. Each color corresponds to a different true value of $m$ . The x-axis represents the initial guess $m_{0}$ , taken around the true value ( $\pm 50\%$ and $80\%$ 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.

Parameter	Value
Optimization algorithm	Interior-point (MATLAB default)
Maximum iterations	50
Optimality tolerance	$10^{-6}$
Lower bound	$m=1$
Upper bound	$m=10$

Table 17: fmincon Configuration

Category	Parameter	Value/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
Domain	Time domain	[0, 1]
	Space domain	[-1, 1]

Table 18: Configurations and Parameters for Inverse PINN Method

Figure 9: Distribution of Training Points

true m	initial guess	fmincon relative error	PINN relative error	fmincon time	PINN training time
2	0.4	failed	1.74%	2.30	228.70
	1	failed	4.67%	0.70	238.43
	3	0.28%	1.07%	14.50	229.50
	3.6	0.28%	1.71%	19.88	124.91
3	0.6	failed	0.79%	0.76	230.82
	1.5	0.37%	0.52%	12.52	238.57

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

m

and initial guesses.

Panel of Figure 10 — Figure 10: Inverse Barenblatt solution results plots

Figure 11: Training loss and m evolution for case m=3 with initial guess =4.5

Panel of Figure 12 — 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 m	initial guess	fmincon relative error	PINN relative error	fmincon time	PINN training time
2	0.4	1.10%	0.51%	30.05	360.64
	1	1.10%	0.28%	29.46	356.92
	3	1.10%	0.15%	16.38	298.42
	3.6	1.10%	0.22%	34.53	354.22
3	0.6	0.07%	0.14%	33.93	360.07
	1.5	0.07%	0.20%	24.96	362.01

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

m

and initial guesses.

Panel of Figure 13 — Figure 13: Inverse Polynomial solution plots

true m	initial guess	fmincon relative error	PINN relative error	fmincon time	PINN training time
2	0.4	0.85%	0.94%	5.38	338.86
	1	0.85%	1.37%	7.32	338.78
	3	0.85%	0.15%	12.32	250.15
	3.6	0.85%	0.11%	14.43	341.01
3	0.6	0.70%	0.31%	9.31	352.13
	1.5	0.70%	0.33%	7.77	349.41

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

m

and initial guesses.

Panel of Figure 14 — Figure 14: Inverse Harmonic solution plots

true m	initial guess	fmincon relative error	PINN relative error	fmincon time	PINN training time
2	0.4	0.08%	6.79%	107.29	379.69
	1	0.08%	2.83%	110.53	382.94
	3	0.08%	49.85%	71.56	266.39
	3.6	0.08%	0.15%	55.11	366.76
3	0.6	failed	9.49%	26.84	370.63
	1.5	0.27%	29.66%	1248.56	376.23

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

m

and initial guesses.

Panel of Figure 15 — Figure 15: Inverse m-dependent solution plots

Panel of Figure 16 — 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

Panel of Figure 35 — Polynomial Solution

$m$	Barenblatt	Polynomial	Harmonic	m-dependent
2	0.078	0.087	0.182	0.134
3	0.040	0.053	0.033	0.054
4	0.388	0.529	0.496	0.938
5	0.378	0.524	0.491	1.133

Table 23: Computation time (seconds) for solving the direct problem using MATLAB for different manufactured solutions.

Parameter	Value
Noise model	Additive Gaussian, $\varepsilon_{i}\sim\mathcal{N}(0,\sigma^{2})$
Noise levels $\delta$	$0\%,\ 1\%,\ 5\%$
Standard deviation $\sigma$	$\delta\cdot\text{RMS}\left(u^{\text{PINN}}\right)$
Perturbed quantity	Measurement dataset only (from direct PINN)
Number of perturbed points	$N_{\text{meas}}=225$
Random seed	42

Table 24: Parameters of the data perturbation used in the inverse PINN experiments.

true m	initial guess	recovered m	PINN relative error	PINN training time s
2	0.4	2.00	0.22%	338.66
	1.0	2.00	0.01%	285.76
	3.0	2.00	0.30%	224.85
	3.6	2.00	0.35%	244.06
3	0.6	3.00	0.28%	341.31
	1.5	3.00	0.19%	337.40

Table 25: PINN inverse results for the polynomial solution using measurement data generated from the direct PINN with 0% additional noise

true m	initial guess	recovered m	PINN relative error pct	PINN training time s
2	0.4	2.00	0.04%	355.85
	1	1.99	0.22%	254.72
	3	1.99	0.02%	201.63
	3.6	2.00	0.40%	223.58
3	0.6	3.00	0.04%	359.60
	1.5	3.00	0.07%	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 19: Zoomed-in view of the exact Barenblatt solution for $m=2$ for t $\in[0,0.3]$

Figure 22: Zoomed-in view of the exact Barenblatt solution for $m=3$ for t $\in[0,0.3]$

Figure 25: Zoomed-in view of the exact Barenblatt solution for $m=4$ for t $\in[0,0.3]$

Figure 28: Zoomed-in view of the exact Barenblatt solution for $m=5$ for t $\in[0,0.3]$

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Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well-established numerical methods with new AI-based techniques, specifically Physics-Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed-form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.

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