## 3 Results and Discussion
### Simple Harmonic Oscillator (SHM)  
The damped harmonic oscillator is a classic problem in mechanics that describes the motion of a mechanical oscillator (e.g., a spring pendulum) under the influence of a restoring force and friction. The governing equation for the damped harmonic oscillator is given by Equation (10):  
\[F(x)=m\frac{d^{2}u}{dt^{2}}+\mu\frac{du}{dt}+ku \tag{10}\]  
where  
\(m\): mass of the oscillator;  
\(\mu\): coefficient of friction;  
\(k\): spring constant. In this paper, we focus on the under-damped state, i.e., where the oscillation is slowly damped by friction occurs when \(\delta<\omega_{0}\), where \(\delta=\frac{\mu}{2m}\) and \(\omega_{0}=\sqrt{\frac{k}{m}}\). The following initial conditions are applied:  
\[u(t=0)=1,\quad\frac{du}{dt}(t=0)=0\]  
The exact solution of the above setup is given by Equation (11):  
\[u(t)=e^{-\delta t}(2A\cos(\phi+\omega t)), \tag{11}\]  
where \(\omega=\sqrt{\omega_{0}^{2}-\delta^{2}}\). The interior residual is given by Equation (12):  
\[r_{\text{int},\theta}(t):=m\frac{d^{2}\theta_{\theta}}{dt^{2}}+\mu\frac{d \theta_{\theta}}{dt}+kl_{\theta} \tag{12}\]  
This is the exact solution of the oscillator with \(w_{0}\) as 20 Hz. With an increasing frequency (\(\omega_{0}\)), the damped harmonic oscillator function becomes more complicated for PINNs to approach. Figure 2 illustrates the exact solution of the oscillator for four frequencies \(\omega_{0}=20,40,50,60\). In the experiment, our PINN model is used to approximate the solutions of the oscillator for the above four frequencies. The selected source terms yield uncomplicated solutions that demonstrate how the F-principle affects the convergence of PINNs to the numerical solution. According to the F-principle, the low-frequency or large-scale characteristics of the solution are initially manifested in the PINNs, while it may take multiple training epochs to retrieve high-frequency or small-scale features [34]. We expect that the vanilla PINN will converge faster and achieve better accuracy in learning the damped harmonic oscillator for lower-frequency components, e.g., \(\omega_{0}=20\) than for higher-frequency components (\(\omega_{0}=40,50,60\)). The experiment results that are presented in the following part are aligned with the expectations. The PINN model we used in experiments for this case comprises a fully connected network (FCN) with five fully connected layers, each consisting of 64 neurons, totalling 4321 parameters. We trained the PINN model using two optimizers, Adam and L-BFGS, which are mentioned in most PINN papers. To populate the computational domain, we utilized a total of 100 equidistant points. It is worth noting that the selection of the number of points within the domain is a decision that is dependent on the user. Figure 2: Exact solution for different values of \(\omega\). The results of training PINN models for \(\omega_{0}=20\) and \(\omega_{0}=30\) are given in Figures 3 and 4. For \(\omega_{0}=20\), the PINN was able to fit well where the loss reaches the order of \(10^{-3}\) with both Adam and L-BFGS optimizers (Figure 3a,c). With the LBFGS optimizer, it converged at around 2200 epochs(Figure 3b), while with the Adam optimizer, it converged at around 7000 epochs (Figure 3d). For \(\omega_{0}=30\), the PINN was also able to reach the order of \(10^{-3}\) loss with both Adam and L-BFGS optimizers (Figures 3c and 4a). With the L-BFGS optimizer, it converged at around 7000 epochs (Figure 4b). The Adam optimizer needs around 22,000 epochs to converge (Figure 4d). In comparing the convergence behaviour of the Adam optimizer and the L-BFGS optimizer at a frequency of 40 Hz, it becomes apparent that both algorithms exhibit different characteristics and performances, particularly in their speed of convergence and stability during optimization. The Adam optimizer eventually converges; however, this is achieved after a significant number of iterations. This observation raises concerns as we transition  
Figure 3: Comparisons of Adam optimizer vs. L-BFGS optimizer at 20 Hz.