# Physics-Informed Neural Networks for Cantilever Dynamics and Fluid-Induced Excitation
## 3 Experiments
### Vibration of a Cantilever
#### 3.1.2 Configuration  
The characteristic time scale, \(\tau\), and characteristic length scale of deflection, \(\omega_{\text{Ref}}\), are considered at the beginning of the configuration. The time scale, \(\tau\), can be described  
Figure 1: (**a**) Schematic illustration of the free vibration of a cantilevered beam. (**b**) Neural network architecture reflecting the STD approach employed to resolve the vibration of a cantilever.  
by balancing the inertia and restoring forces in the governing Equation (5), resulting in \(\tau\sim\left(\frac{\mu}{L}\epsilon\right)^{1/2}\). \(\omega_{\text{Ref}}\) can be scaled by balancing the initial load and the restoring force, giving \(\omega_{\text{Ref}}\sim PL^{3}/B\). These characteristic scales allow for the expression of the following conditions for each loss component.  
The PDE error \(E_{PDE,1}\) represents the difference between \(\mu\frac{\partial^{2}\omega(x,t)}{\partial t^{2}}\) and \(B\frac{\partial^{4}\omega(x,t)}{\partial x^{4}}\) in Equation (5) in the absence of the external load \(q(x,t)\). The PDE loss can be scaled using a desired relative tolerance \(\epsilon_{PDE,1}\) such that \(E_{PDE,1}\sim\epsilon_{PDE,1}\mu\frac{\partial^{2}\omega(x,t)}{\partial t^{2 }}\sim\frac{\epsilon_{PDE,1}\mu_{\text{Ref}}}{\tau^{2}}\), leading to the scaling of the PDE loss as \(\mathcal{L}_{PDE,1}\sim E_{PDE,1}{}^{2}\sim\frac{(\epsilon_{PDE,1}\mu_{ \text{Ref}})^{2}}{\tau^{2}}\). Similar arguments can be made for the BCs and ICs. Specifically, \(\mathcal{L}_{BC,1}\sim\left(\epsilon_{BC,1}\omega_{\text{Ref}}\right)^{2}\), \(\mathcal{L}_{BC,2}\sim\left(\epsilon_{BC,2}\frac{\omega_{\text{Ref}}}{2} \right)^{2}\), \(\mathcal{L}_{BC,3}\sim(\epsilon_{BC,3}\frac{\omega_{\text{Ref}}}{2})^{2}\), and \(\mathcal{L}_{BC,4}\sim\left(\epsilon_{BC,4}\frac{\omega_{\text{Ref}}}{L^{3}} \right)^{2}\) represent the loss due to boundary conditions, while \(\mathcal{L}_{IC,1}\sim\left(\epsilon_{IC,1}\omega_{\text{Ref}}\right)^{2}\) and \(\mathcal{L}_{IC,2}\sim\left(\epsilon_{IC,2}\frac{\omega_{\text{Ref}}}{\tau} \right)^{2}\) represent the loss due to initial conditions.  
By integrating the above scaling relations into Equation (3), the total loss \(\mathcal{L}_{tot}\) can be rewritten as follows:  
\[\mathcal{L}_{tot}=\widetilde{\mathcal{L}}_{PDE}\ \ +\widetilde{\mathcal{L}}_{IC} +\widetilde{\mathcal{L}}_{BC} \tag{6}\]  
where the scaled losses are defined as follows:  
\[\widetilde{\mathcal{L}}_{PDE}=\frac{\tau^{4}}{\left(\epsilon_{ PDE,1}\omega_{\text{Ref}}\right)^{2}}\mathcal{L}_{PDE}\] \[\widetilde{\mathcal{L}}_{IC}=\frac{1}{\left(\epsilon_{IC,1}\omega _{\text{Ref}}\right)^{2}}\mathcal{L}_{IC,1}+\frac{\tau^{2}}{\left(\epsilon_{ IC,2}\omega_{\text{Ref}}\right)^{2}}\mathcal{L}_{IC,2}\] \[\widetilde{\mathcal{L}}_{BC}=\frac{1}{\left(\epsilon_{BC,1}\omega _{\text{Ref}}\right)^{2}}\mathcal{L}_{BC,1}+\frac{L^{2}}{\left(\epsilon_{BC,2 }\omega_{\text{Ref}}\right)^{2}}\mathcal{L}_{BC,2}+\frac{L^{4}}{\left(\epsilon _{BC,3}\omega_{\text{Ref}}\right)^{2}}\mathcal{L}_{BC,3}+\frac{L^{6}}{\left( \epsilon_{BC,4}\omega_{\text{Ref}}\right)^{2}}\mathcal{L}_{BC,4}\]  
The relative tolerances are adopted as \(\epsilon_{IC}=\epsilon_{BC}=10^{-3}\) and \(\epsilon_{PDE}=10^{-2}\), corresponding to the guidance for the initial guess in the former section except for the elaboration made for \(\epsilon_{BC,1}=10^{-4}\). The loss construction based on the suggested scaling law dramatically improves the convergence of the PINN. It is worth noting that the hyperparameter search based on the regular configuration from Equation (2) could not obtain a reliable solution. This is supported by the absolute lack of the PINN solution for the dynamic beam bending problem [23], despite its abundance in engineering applications.