# NSGA-PINN: A Multi-Objective Optimization Method for Physics-Informed Neural Network Training
## 4 Numerical Experiments
### Burgers Equation  
This experiment uses the Burgers equation to study the effectiveness of the proposed NSGA-PINN algorithm on a PDE problem. The Burgers equation is defined as follows:  
\[\frac{du}{dt}+u\frac{du}{dx}=v\frac{d^{2}u}{dx^{2}},\quad x\in[-1,1],t\in[0,1] \tag{10}\] \[u(0,x)=-\sin(\pi x)\] \[u(t,-1)=u(t,1)=0\]  
Here, \(u\) is the PDE solution, \(\Omega=[-1,1]\) is the spatial domain, and \(v=0.01/\pi\) is the diffusion coefficient.  
The nonlinearity in the convection term causes the solution to become steep, due to the small value of the diffusion coefficient \(v\). To address this problem, we utilized a neural  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & **Residual Loss** & **Initial Loss** & **Data Loss** & **Total Loss** \\ \hline ADAM & 0.0028 & 0.0028 & 0.0114 & 0.017 \\ NSGA-PINN & 0.0010 & 0.0011 & 0.0112 & 0.0133 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Inverse pendulum problem with data noise: each loss value from NN trained by different training methods with noisy data.  
Figure 6: Inverse pendulum problem with data noise: The figure shows the comparing of the result from PINN trained by NSGA-PINN method and ADAM method with noisy input data.  
network for PINN, which consisted of 8 hidden layers with 20 neurons each. The hyperbolic tangent activation function was used to activate the neurons in each layer. We sampled 100 data points on the boundaries and 10,000 collocation data points for PINN training.  
For the proposed NSGA-PINN method, the original population size was set to 20 neural networks, and the algorithm ran for 20 generations. The loss function in the Burgers' equation can be defined as follows:  
\[\mathcal{L}=\mathcal{L}_{u}+\mathcal{L}_{b}+\mathcal{L}_{ics}. \tag{11}\]  
Here, the total loss value is the combination of the residual loss, the initial condition loss, and the boundary loss.  
We can observe the effectiveness of the proposed NSGA-PINN algorithm by examining the loss values depicted in Figure 7 and Table 3. In particular, Table 3 compares the loss values of PINNs trained by the NSGA-PINN algorithm and the traditional ADAM method. Noticeably, the loss value trained by the NSGA-PINN framework is \(3.746\times 10^{-5}\), which is much lower than the traditional ADAM method, which has the loss value as 0.0003.  
Finally, Figure 8 displays contour plots of the solution to Burgers' equation. The top figure shows the result predicted using the proposed NSGA-PINN algorithm. The bottom row compares the exact value with the values from the proposed algorithm and the ADAM method at t = 0.25, 0.50, and 0.75. Based on this comparison, both the NSGA-PINN algorithm and the ADAM method predict values that are close to the true values.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & **Residual Loss** & **Boundary Loss** & **Total Loss** \\ \hline ADAM & 0.0002 & \(9.4213\times 10^{-5}\) & 0.0003 \\ NSGA-PINN & \(2.89\times 10^{-5}\) & \(8.56\times 10^{-6}\) & \(3.746\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Burgers equation: Comparison of the loss value from NN trained by ADAM method and NSGA-PINN method for Burgers equation.  
Figure 7: Burgers equation: group of figures from left to right columns show each loss value by using different training method.  
We show the error contour plot for the Burgers equation in Figure 9 to visualize the accuracy of the predicted solution.