Solving Inverse PDE Problems using Minimization Methods and AI

Noura Al Helwani; Sophie Moufawad; Georges Sakr

math.NAMar 2, 2026

Solving Inverse PDE Problems using Minimization Methods and AI

Authors: Noura Al Helwani, Sophie Moufawad, Georges Sakr

Organizations: Department of Physics, American University of Beirut, Lebanon · Department of Mathematics, American University of Beirut, Lebanon · The authors would like to acknowledge

Abstract

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.

Figures & tables

Time Interval	Intervals	$K$	$p_{0}$	$r$	RK4 (RelErr)	ODE45 (RelErr)
2011 - 2022	100	10	20	0.079	$4.1640\times 10^{-3}$	$9.1448\times 10^{-8}$
450 - 500	200	90	10	0.05	$9.8764\times 10^{-3}$	$5.6998\times 10^{-8}$
1 - 100	100000	$10^{3}$	$10^{2}$	0.9	$5.4090\times 10^{-5}$	$3.5415\times 10^{-4}$

Table 1: Relative errors (RelErr) for RK4 and ODE45 with different parameters.

Parameter	Value / Description
Nonlinearity parameter $\beta$	$3.0$
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)
Boundary conditions	Dirichlet values from Barenblatt at $x=-1$ and $x=1$
Tolerance (Newton)	$10^{-6}$
Maximum Newton iterations	$20$ per time step

Table 2: Parameters used in the Newton-based PME solver with Barenblatt initial and boundary conditions.

Feature	Adam	L-BFGS
Type	First-order optimizer	Second-order (quasi-Newton) optimizer
Speed per iteration	Fast	Slower (due to line search and history)
Convergence behavior	Good at global search	Good at local refinement
Memory usage	Low	Higher (stores gradient history)
Common use in PINNs	Early training phase	Fine-tuning after Adam

Table 3: Comparison between Adam and L-BFGS optimizers

Figure 2: Direct Problem: PINN Prediction vs Analytical Solution for Case 1.

Figure 3: Direct Problem : PINN Prediction vs Analytical Solution for Case 2.

Figure 4: Direct problem: PINN Prediction vs Analytical Solution for Case 3.

Figure 5: PINN: direct problem for Case 3 after normalization.

Case	PINN Error	ODE45 Error	RK4 Error
Case 1	$2.07\times 10^{-4}$	$9.14\times 10^{-8}$	$4.16\times 10^{-3}$
Case 2	$9.29\times 10^{-4}$	$5.70\times 10^{-8}$	$9.88\times 10^{-3}$
Case 3	$9.82\times 10^{-4}$	$3.54\times 10^{-4}$	$5.41\times 10^{-5}$

Table 4: Comparison of errors between PINN, ODE45, and RK4 methods

Figure 6: Inverse problem over r for Case 1.

Figure 7: Inverse Problem over r for Case 2.

Figure 8: Inverse Problem over r for Case 3.

Figure 9: Inverse problem over r and k for Case 1.

Figure 10: Inverse problem over r and k for Case 2

Figure 11: Inverse problem over r and k for Case 3 after normalization.

Figure 12: Distribution of interior, boundary, and initial points sampled using a Sobol sequence.

Figure 13: Training loss (log scale) over epochs.

Figure 14: Comparison of exact solution (left) and learned PINN solution (right) in the $(t,x)$ domain.

Figure 15: Training loss (log scale) over epochs.

Figure 16: Comparison of exact solution (left) and learned PINN solution (right) in the $(t,x)$ domain.

Figure 17: Sobol-distributed training and measurement points across the domain.

Figure 18: Training loss (log scale) over epochs.

Figure 19: Comparison of exact and learned solutions $u(t,x)$ in the inverse problem.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

fminunc
$nr_{0}$	iter	$r$	$\lvert r-r_{\text{exact}}\rvert/r_{\text{exact}}$	feval	Interp. error	Extrap. error
$0.25$	3	$1.200\,\mathrm{E}{-1}$	$1.976\,\mathrm{E}{-7}$	$1.111\,\mathrm{E}{-14}$	$1.397\,\mathrm{E}{-7}$	$1.101\,\mathrm{E}{-10}$
0.5	7	$1.300\mathrm{E}{-1}$	$5.465\mathrm{E}{-7}$	$8.500\mathrm{E}{-14}$	$3.864\mathrm{E}{-7}$	3 $.047\mathrm{E}{-10}$
0.75	6	$1.300\mathrm{E}{-1}$	$6.360\mathrm{E}{-8}$	$1.151\mathrm{E}{-15}$	$4.496\mathrm{E}{-8}$	$3.545\mathrm{E}{-11}$
0.9	5	$1.300\mathrm{E}{-1}$	$5.630\mathrm{E}{-8}$	$9.020\mathrm{E}{-16}$	$3.981\mathrm{E}{-8}$	$3.135\mathrm{E}{-11}$
1.10	4	$1.300\mathrm{E}{-1}$	$5.918\mathrm{E}{-8}$	$9.965\mathrm{E}{-16}$	$4.184\mathrm{E}{-8}$	$3.299\mathrm{E}{-11}$

Appendix

Table 5: Optimization results without noise

Newton with symbolic diff (NewtonS)
$nr_{0}$	iter	$r$	$\lvert r-r_{\text{exact}}\rvert/r_{\text{exact}}$	feval	Interp. error	Extrap. error
0.25	3	$-3.000\mathrm{E}{-03}$	$1.024\mathrm{E}{+00}$	$5.590\mathrm{E}{-01}$	$9.910\mathrm{E}{-01}$	$9.940\mathrm{E}{-01}$
0.5	7	$1.300\mathrm{E}{-01}$	$5.503\mathrm{E}{-12}$	$8.616\mathrm{E}{-24}$	$3.891\mathrm{E}{-12}$	$3.085\mathrm{E}{-15}$
0.75	5	$1.300\mathrm{E}{-01}$	$7.150\mathrm{E}{-07}$	$1.455\mathrm{E}{-13}$	$5.055\mathrm{E}{-07}$	$3.986\mathrm{E}{-10}$
0.9	5	$1.300\mathrm{E}{-01}$	$3.516\mathrm{E}{-12}$	$3.518\mathrm{E}{-24}$	$2.486\mathrm{E}{-12}$	$1.956\mathrm{E}{-15}$
1.1	5	$1.300\mathrm{E}{-01}$	$9.852\mathrm{E}{-11}$	$2.762\mathrm{E}{-21}$	$6.965\mathrm{E}{-11}$	$5.491\mathrm{E}{-14}$

Appendix

Table 6: Optimization results without noise (additional solvers)

$nr_{0}$	fminunc	fmincon	Secant	NewtonS	NewtonE	SD
0.25	$1.238\mathrm{E{-}03}$	$8.849\mathrm{E{-}03}$	$1.588\mathrm{E{-}03}$	$1.301\mathrm{E{-}02}$	$1.833\mathrm{E{-}04}$	$2.944$
0.5	$1.389\mathrm{E{-}03}$	$1.010\mathrm{E{-}02}$	$2.403\mathrm{E{-}04}$	$4.552\mathrm{E{-}04}$	$5.035\mathrm{E{-}04}$	$1.801$
0.75	$7.652\mathrm{E{-}04}$	$7.296\mathrm{E{-}03}$	$1.315\mathrm{E{-}04}$	$4.075\mathrm{E{-}04}$	$4.215\mathrm{E{-}04}$	$1.994$
0.9	$6.263\mathrm{E{-}04}$	$7.064\mathrm{E{-}03}$	$8.965\mathrm{E{-}05}$	$3.989\mathrm{E{-}04}$	$3.789\mathrm{E{-}04}$	$1.480$
1.1	$6.322\mathrm{E{-}04}$	$7.920\mathrm{E{-}03}$	$3.872\mathrm{E{-}04}$	$3.198\mathrm{E{-}04}$	$2.813\mathrm{E{-}04}$	$1.721$
1.5	$6.338\mathrm{E{-}04}$	$5.625\mathrm{E{-}03}$	$2.172\mathrm{E{-}04}$	$1.666\mathrm{E{-}04}$	$1.872\mathrm{E{-}04}$	$2.153$

Appendix

Table 7: Convergence time (in seconds)

fminunc
$nr_{0}$ iter	$r$	$\lvert r-r_{\text{exact}}\rvert/r_{\text{exact}}$	feval	Interp. error	Extrap. error
0.25	3	$1.300\mathrm{E{-}01}$	$1.995\mathrm{E{-}07}$	$1.151\mathrm{E{-}14}$	$1.422\mathrm{E{-}07}$	$2.276\mathrm{E{-}08}$
0.5	7	$1.300\mathrm{E{-}01}$	$5.474\mathrm{E{-}07}$	$8.545\mathrm{E{-}14}$	$3.875\mathrm{E{-}07}$	$2.346\mathrm{E{-}08}$
0.75	6	$1.300\mathrm{E{-}01}$	$6.352\mathrm{E{-}08}$	$1.548\mathrm{E{-}15}$	$5.215\mathrm{E{-}08}$	$2.108\mathrm{E{-}08}$
0.9	5	$1.300\mathrm{E{-}01}$	$5.352\mathrm{E{-}08}$	$1.367\mathrm{E{-}15}$	$4.901\mathrm{E{-}08}$	$2.135\mathrm{E{-}08}$
1.1	4	$1.300\mathrm{E{-}01}$	$5.862\mathrm{E{-}08}$	$1.443\mathrm{E{-}15}$	$5.035\mathrm{E{-}08}$	$1.992\mathrm{E{-}08}$

Appendix

Table 8: Optimization results with

3\%

noise

$nro$	fminunc	fmincon	Secant	NewtonS	NewtonE	SD
0.25	$8.254\mathrm{E{-}04}$	$6.130\mathrm{E{-}03}$	$1.118\mathrm{E{-}03}$	$1.001\mathrm{E{-}02}$	$1.352\mathrm{E{-}04}$	$2.638\mathrm{E{+}00}$
0.5	$8.158\mathrm{E{-}04}$	$1.087\mathrm{E{-}02}$	$1.769\mathrm{E{-}04}$	$3.429\mathrm{E{-}04}$	$3.720\mathrm{E{-}04}$	$1.816\mathrm{E{+}00}$
0.75	$6.854\mathrm{E{-}04}$	$7.862\mathrm{E{-}03}$	$1.354\mathrm{E{-}04}$	$2.483\mathrm{E{-}04}$	$2.596\mathrm{E{-}04}$	$2.004\mathrm{E{+}00}$
0.9	$6.594\mathrm{E{-}04}$	$7.635\mathrm{E{-}03}$	$8.859\mathrm{E{-}05}$	$2.477\mathrm{E{-}04}$	$2.665\mathrm{E{-}04}$	$1.572\mathrm{E{+}00}$
1.1	$5.858\mathrm{E{-}04}$	$6.007\mathrm{E{-}03}$	$2.270\mathrm{E{-}04}$	$2.568\mathrm{E{-}04}$	$2.506\mathrm{E{-}04}$	$1.882\mathrm{E{+}00}$
1.5	$6.069\mathrm{E{-}04}$	$5.653\mathrm{E{-}03}$	$2.161\mathrm{E{-}04}$	$1.574\mathrm{E{-}02}$	$1.057\mathrm{E{-}04}$	$2.504\mathrm{E{+}00}$

Appendix

Table 9: Convergence time (in seconds) when adding 3% white noise

NewtonS
$nr_{0}$	iter	$r$	$\lvert r-r_{\text{exact}}\rvert/r_{\text{exact}}$	feval	Interp. error	Extrap. error
0.25	3	$-3.070\mathrm{E{-}03}$	$1.024\mathrm{E{+}00}$	$5.587\mathrm{E{-}01}$	$9.908\mathrm{E{-}01}$	$9.937\mathrm{E{-}01}$
0.5	7	$1.300\mathrm{E{-}01}$	$8.064\mathrm{E{-}10}$	$4.513\mathrm{E{-}16}$	$2.816\mathrm{E{-}08}$	$2.351\mathrm{E{-}08}$
0.75	5	$1.300\mathrm{E{-}01}$	$7.149\mathrm{E{-}07}$	$1.459\mathrm{E{-}13}$	$5.062\mathrm{E{-}07}$	$2.109\mathrm{E{-}08}$
0.9	5	$1.300\mathrm{E{-}01}$	$2.784\mathrm{E{-}09}$	$4.654\mathrm{E{-}16}$	$2.859\mathrm{E{-}08}$	$2.135\mathrm{E{-}08}$
1.1	5	$1.300\mathrm{E{-}01}$	$4.672\mathrm{E{-}10}$	$4.466\mathrm{E{-}16}$	$2.801\mathrm{E{-}08}$	$1.991\mathrm{E{-}08}$

Appendix

Table 10: Optimization results with

3\%

noise

Panel of Figure 20 — Figure 20: Loss surface seen from different angles. The data has been generated using $r$ =0.13 and $k$ = $10^{6}$ .

fminunc
$nr_{0}$	iter	$r$	$k$	$\lvert r-r_{\text{exact}}\rvert/r_{\text{exact}}$	$\lvert k-k_{\text{exact}}\rvert/k_{\text{exact}}$	Interp. error	Extrap. error
0.25	3	$2.038\mathrm{E{-}01}$	$2.500\mathrm{E{+}05}$	$5.679\mathrm{E{-}01}$	$7.500\mathrm{E{-}01}$	$7.269\mathrm{E{+}05}$	$7.498\mathrm{E{+}05}$
0.5	3	$1.693\mathrm{E{-}01}$	$5.000\mathrm{E{+}05}$	$3.020\mathrm{E{-}01}$	$5.000\mathrm{E{-}01}$	$4.691\mathrm{E{+}05}$	$4.999\mathrm{E{+}05}$
0.75	5	$1.480\mathrm{E{-}01}$	$7.500\mathrm{E{+}05}$	$1.382\mathrm{E{-}01}$	$2.500\mathrm{E{-}01}$	$2.258\mathrm{E{+}05}$	$2.499\mathrm{E{+}05}$
0.9	6	$1.370\mathrm{E{-}01}$	$9.000\mathrm{E{+}05}$	$5.421\mathrm{E{-}02}$	$1.000\mathrm{E{-}01}$	$8.790\mathrm{E{+}04}$	$9.997\mathrm{E{+}04}$
1.1	5	$1.229\mathrm{E{-}01}$	$1.100\mathrm{E{+}06}$	$5.450\mathrm{E{-}02}$	$1.000\mathrm{E{-}01}$	$8.402\mathrm{E{+}04}$	$9.995\mathrm{E{+}04}$

Appendix

Table 11: Optimization results for

r

and

k

without noise

$nr_{0}$	fminunc	fmincon	SD
0.25	$2.053\mathrm{E{-}03}$	$8.941\mathrm{E{-}03}$	$4.496\mathrm{E{-}03}$
0.5	$1.601\mathrm{E{-}03}$	$1.050\mathrm{E{-}02}$	$4.957\mathrm{E{-}03}$
0.75	$1.289\mathrm{E{-}03}$	$7.894\mathrm{E{-}03}$	$4.600\mathrm{E{-}03}$
0.9	$1.221\mathrm{E{-}03}$	$8.352\mathrm{E{-}03}$	$4.983\mathrm{E{-}03}$
1.1	$1.353\mathrm{E{-}03}$	$1.550\mathrm{E{-}02}$	$4.384\mathrm{E{-}03}$
1.5	$2.055\mathrm{E{-}03}$	$1.421\mathrm{E{-}02}$	$3.903\mathrm{E{-}03}$

Appendix

Table 12: Convergence time for fminunc, fmincon, and steepest descent

fminunc
$rn_{0}$	iter	$r$	$k$	$\lvert r-r_{\text{exact}}\rvert/r_{\text{exact}}$	$\lvert k-k_{\text{exact}}\rvert/k_{\text{exact}}$	feval	Interp. error	Extrap. error
0.25	3	$1.300\mathrm{E{-}01}$	$1.000\mathrm{E{+}06}$	$1.976\mathrm{E{-}07}$	$1.044\mathrm{E{-}11}$	$1.111\mathrm{E{-}14}$	$1.397\mathrm{E{-}01}$	$1.148\mathrm{E{-}04}$
0.5	7	$1.300\mathrm{E{-}01}$	$1.000\mathrm{E{+}06}$	$5.466\mathrm{E{-}07}$	$5.588\mathrm{E{-}15}$	$8.500\mathrm{E{-}14}$	$3.864\mathrm{E{-}01}$	$3.046\mathrm{E{-}04}$
0.75	6	$1.300\mathrm{E{-}01}$	$1.000\mathrm{E{+}06}$	$6.360\mathrm{E{-}08}$	$8.149\mathrm{E{-}16}$	$1.151\mathrm{E{-}15}$	$4.495\mathrm{E{-}02}$	$3.544\mathrm{E{-}05}$
0.9	5	$1.300\mathrm{E{-}01}$	$1.000\mathrm{E{+}06}$	$5.630\mathrm{E{-}08}$	$6.985\mathrm{E{-}16}$	$9.020\mathrm{E{-}16}$	$3.980\mathrm{E{-}02}$	$3.138\mathrm{E{-}05}$
1.1	4	$1.300\mathrm{E{-}01}$	$1.000\mathrm{E{+}06}$	$5.918\mathrm{E{-}08}$	$1.164\mathrm{E{-}15}$	$9.965\mathrm{E{-}16}$	$4.183\mathrm{E{-}02}$	$3.298\mathrm{E{-}05}$

Appendix

Table 13: Optimization results for

r

and

k

without noise, with

k_{0}=10^{6}=k_{exact}

$nr_{0}$	fminunc	fmincon	SD
0.25	$2.099\mathrm{E{-}03}$	$1.921\mathrm{E{-}02}$	$4.347\mathrm{E{-}03}$
0.5	$1.214\mathrm{E{-}03}$	$1.590\mathrm{E{-}02}$	$4.314\mathrm{E{-}03}$
0.75	$1.501\mathrm{E{-}03}$	$1.743\mathrm{E{-}02}$	$4.554\mathrm{E{-}03}$
0.9	$1.589\mathrm{E{-}03}$	$1.479\mathrm{E{-}02}$	$4.728\mathrm{E{-}03}$
1.1	$1.089\mathrm{E{-}03}$	$1.792\mathrm{E{-}02}$	$4.403\mathrm{E{-}03}$
1.5	$1.304\mathrm{E{-}03}$	$1.626\mathrm{E{-}02}$	$5.578\mathrm{E{-}03}$

Appendix

Table 14: Convergence time (in seconds) for fminunc, fmincon, and steepest descent (methods which did not converge omited)

Figure 21: Loss function with $r=0.02$ fixed and $k$ as the variable.

$nr_{0}$	fminunc	fmincon	Secant	NewtonS	NewtonE	Steepest Descent
0.25	$2.274\mathrm{E}{-3}$	$2.329\mathrm{E}{-2}$	$5.347\mathrm{E}{-3}$	$7.958\mathrm{E}{-3}$	$6.390\mathrm{E}{-2}$	$6.062\mathrm{E}{-3}$
0.5	$3.760\mathrm{E}{-3}$	$2.451\mathrm{E}{-2}$	$8.170\mathrm{E}{-3}$	$1.085\mathrm{E}{-2}$	$1.192\mathrm{E}{-1}$	$1.037\mathrm{E}{-2}$
0.75	$4.996\mathrm{E}{-3}$	$3.950\mathrm{E}{-2}$	$1.249\mathrm{E}{-2}$	$2.348\mathrm{E}{-2}$	$8.248\mathrm{E}{-2}$	$9.238\mathrm{E}{-3}$
0.9	$2.941\mathrm{E}{-3}$	$3.708\mathrm{E}{-2}$	$1.215\mathrm{E}{-2}$	$1.834\mathrm{E}{-2}$	$8.456\mathrm{E}{-2}$	$8.972\mathrm{E}{-3}$
1.1	$8.772\mathrm{E}{-4}$	$2.645\mathrm{E}{-2}$	$1.029\mathrm{E}{-2}$	$1.295\mathrm{E}{-3}$	$1.307\mathrm{E}{-2}$	$8.784\mathrm{E}{-3}$
1.5	$9.005\mathrm{E}{-4}$	$2.894\mathrm{E}{-2}$	$9.021\mathrm{E}{-3}$	$1.579\mathrm{E}{-2}$	$8.211\mathrm{E}{-2}$	$1.017\mathrm{E}{-2}$

Appendix

Table 15: Convergence time (in seconds) for all optimization algorithms

fminunc
$rn_{0}$	iter	$r$	$k$	$\lvert r-r_{\text{exact}}\rvert/r_{\text{exact}}$	$\lvert k-k_{\text{exact}}\rvert/k_{\text{exact}}$	feval	Interp. error	Extrap. error
0.25	1	$-3.152\mathrm{E}{-5}$	$3.172\mathrm{E}{1}$	$1.000\mathrm{E}{0}$	$1.000\mathrm{E}{0}$	$4.901\mathrm{E}{-1}$	$9.277\mathrm{E}{5}$	$1.203\mathrm{E}{6}$
0.5	1	$-1.099\mathrm{E}{-3}$	$1.048\mathrm{E}{3}$	$1.008\mathrm{E}{0}$	$9.990\mathrm{E}{-1}$	$4.924\mathrm{E}{-1}$	$9.299\mathrm{E}{5}$	$1.156\mathrm{E}{6}$
0.75	9	$1.300\mathrm{E}{-1}$	$1.000\mathrm{E}{6}$	$1.569\mathrm{E}{-9}$	$1.109\mathrm{E}{-7}$	$6.269\mathrm{E}{-15}$	$1.049\mathrm{E}{-1}$	$1.109\mathrm{E}{-1}$
0.9	11	$1.300\mathrm{E}{-1}$	$1.000\mathrm{E}{6}$	$4.566\mathrm{E}{-9}$	$1.064\mathrm{E}{-7}$	$5.872\mathrm{E}{-15}$	$1.015\mathrm{E}{-1}$	$1.063\mathrm{E}{-1}$
1.1	1	$-8.570\mathrm{E}{-1}$	$2.940\mathrm{E}{6}$	$7.592\mathrm{E}{0}$	$1.940\mathrm{E}{0}$	$5.692\mathrm{E}{-1}$	$9.998\mathrm{E}{5}$	$9.998\mathrm{E}{5}$

Appendix

Table 16: Optimization results for

r

and

log(k)

without noise

fminunc
$nr_{0}$	iter	$r$	$k$	rel. error $r$	rel. error $k$	feval	Interp. error	Extrap. error
0.25	17	$1.300\mathrm{E}{-1}$	$1.000\mathrm{E}{6}$	$5.090\mathrm{E}{-7}$	$1.483\mathrm{E}{-7}$	$6.159\mathrm{E}{-14}$	$3.289\mathrm{E}{-1}$	$1.482\mathrm{E}{-1}$
0.5	11	$1.300\mathrm{E}{-1}$	$1.000\mathrm{E}{6}$	$8.372\mathrm{E}{-9}$	$9.694\mathrm{E}{-8}$	$5.018\mathrm{E}{-15}$	$9.387\mathrm{E}{-2}$	$9.692\mathrm{E}{-2}$
0.75	12	$1.300\mathrm{E}{-1}$	$1.000\mathrm{E}{6}$	$6.098\mathrm{E}{-8}$	$3.602\mathrm{E}{-7}$	$5.986\mathrm{E}{-14}$	$3.242\mathrm{E}{-1}$	$3.601\mathrm{E}{-1}$
0.9	11	$1.300\mathrm{E}{-1}$	$9.999\mathrm{E}{5}$	$1.880\mathrm{E}{-7}$	$8.244\mathrm{E}{-7}$	$3.057\mathrm{E}{-13}$	$7.327\mathrm{E}{-1}$	$8.242\mathrm{E}{-1}$
1.1	10	$1.300\mathrm{E}{-1}$	$1.000\mathrm{E}{6}$	$1.905\mathrm{E}{-8}$	$2.012\mathrm{E}{-7}$	$2.173\mathrm{E}{-14}$	$1.954\mathrm{E}{-1}$	$2.011\mathrm{E}{-1}$

Appendix

Table 17: Optimization results for

r

and

log(k)

without noise, with

k_{0}=log(10^{6})=log(k_{exact})

$nr_{0}$	fminunc	fmincon	NewtonS	NewtonE	SD
0.25	$3.577\mathrm{E}{-3}$	$2.026\mathrm{E}{-2}$	$6.091\mathrm{E}{-4}$	$8.350\mathrm{E}{-3}$	$6.559\mathrm{E}{-3}$
0.5	$1.180\mathrm{E}{-3}$	$1.681\mathrm{E}{-2}$	$4.829\mathrm{E}{-3}$	$5.453\mathrm{E}{-2}$	$5.658\mathrm{E}{-3}$
0.75	$1.951\mathrm{E}{-3}$	$1.752\mathrm{E}{-2}$	$3.064\mathrm{E}{-4}$	$3.864\mathrm{E}{-3}$	$5.285\mathrm{E}{-3}$
0.9	$1.814\mathrm{E}{-3}$	$2.282\mathrm{E}{-2}$	$3.232\mathrm{E}{-4}$	$6.507\mathrm{E}{-2}$	$5.754\mathrm{E}{-3}$
1.1	$1.556\mathrm{E}{-3}$	$1.499\mathrm{E}{-2}$	$3.094\mathrm{E}{-4}$	$6.708\mathrm{E}{-2}$	$5.042\mathrm{E}{-3}$
1.5	$1.117\mathrm{E}{-3}$	$1.774\mathrm{E}{-2}$	$1.873\mathrm{E}{-3}$	$1.690\mathrm{E}{-2}$	$6.408\mathrm{E}{-3}$

Appendix

Table 18: Convergence time (in seconds) excluding Secant method(did not converge)

$nr_{0}$	fminunc	fmincon	NewtonS	NewtonE	SD
0.25	$2.966\mathrm{E}{-3}$	$1.828\mathrm{E}{-2}$	$4.765\mathrm{E}{-4}$	$7.216\mathrm{E}{-3}$	$5.583\mathrm{E}{-3}$
0.5	$1.145\mathrm{E}{-3}$	$1.514\mathrm{E}{-2}$	$5.036\mathrm{E}{-3}$	$5.859\mathrm{E}{-2}$	$5.025\mathrm{E}{-3}$
0.75	$1.500\mathrm{E}{-3}$	$1.537\mathrm{E}{-2}$	$3.769\mathrm{E}{-4}$	$4.394\mathrm{E}{-3}$	$5.836\mathrm{E}{-3}$
0.9	$1.299\mathrm{E}{-3}$	$2.550\mathrm{E}{-2}$	$3.912\mathrm{E}{-4}$	$7.757\mathrm{E}{-2}$	$5.374\mathrm{E}{-3}$
1.1	$1.138\mathrm{E}{-3}$	$1.298\mathrm{E}{-2}$	$3.259\mathrm{E}{-4}$	$8.939\mathrm{E}{-2}$	$6.953\mathrm{E}{-3}$
1.5	$1.216\mathrm{E}{-3}$	$1.668\mathrm{E}{-2}$	$4.714\mathrm{E}{-3}$	$1.659\mathrm{E}{-2}$	$5.421\mathrm{E}{-3}$

Appendix

Table 19: Convergence time (in seconds) excluding Secant method (did not converge) for the case with noise

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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

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

PDE Solving PDE Inverse Problems Parameter Estimation Physics-Informed ML

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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.

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