# Physics-Informed Neural Networks for Cantilever Dynamics and Fluid-Induced Excitation
## 4 Discussion  
This study systematically investigated strategies for constructing PINNs to resolve the structural dynamics of a cantilever, specifically in scenarios where the application of PINN solutions was previously limited. Initially, the study addressed the free vibration of a cantilevered beam, involving fourth-order differentials in space and second-order differentials in time. Subsequently, the fluid-induced flutter of a cantilevered cylinder was examined, utilizing the coupled dynamic equations from Euler-Bernoulli beam bending theory and [PERSON]'s elongated body theory. The primary contribution of this research is the development of reliable strategies that significantly enhance the capability of PINNs to model complex structural dynamics. Further considerations of inextensibility, large amplitude oscillation, and the integration of lift-driven and resistive forces could be the next challenge for the PINN in relevant areas. Additionally, replacing global scaling factors with sub-networks to address the inverse problem in our approach could be a potential extension of this strategy to further mitigate the vibration problem. The novel strategies proposed in this study have significantly aided in resolving structural dynamics problems. Firstly, the configuration of the PINN was optimized using scaling analysis, which considers the relative scales of the physical variables in terms of their characteristic scales. This approach helped balance the relative importance of constraints from PDEs, BCs, and ICs, resulting in significant improvements in both the accuracy of PINN predictions and the efficiency of training procedures. Secondly, an alternative activation function to tanh, \(x_{j}+\sin^{2}\!\left(x_{j}\right)\) activation, followed by a linear scaling layer \(wx_{j}\) \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(L1\) Absolute Error & \(L2\) Relative Error & **Ref** \\ \hline Regular PINN & \(5.42\times 10^{-1}\) & \(8.87\times 10^{-1}\) & [14] \\ Curriculum training & \(1.10\times 10^{-2}\) & \(2.02\times 10^{-2}\) & [14] \\ Model 1 (Scaling law) & \(2.57\times 10^{-3}\) & \(4.48\times 10^{-3}\) & Present \\ Model 2 (Scaling law with \(x+\sin^{2}\!x\) activation) & & & \\ Model 3 (Scaling law with \(x+\sin^{2}\!x\) in hidden layer, \(\sin x\) at the last layer) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Training errors of the tested PINN architectures for 1D advection problem. was employed. This function proved effective in capturing the second-order non-linearity of the input variables, enabling the network to model the periodic nature of the phenomena under study. Lastly, the mathematical interpretation of the problem informed the PINN architecture. For instance, by constructing the neural network architecture to reflect the decomposed bases of the desired solution, akin to a combination of parallels, the PINN effectively resolved structural dynamics problems, informed by physical interpretations from modal behaviors. Furthermore, the time domain considered in the PINN model can be extended to additional time steps by repetitively training neural networks using the results from the previous model, which is similar to extrapolating the time domain data. The strategies proposed in this study are expected to provide crucial guidelines for constructing PINNs in the realm of structural dynamics problems. Furthermore, these strategies, focusing on guided configuration and architecture designs, can be effectively integrated with recent advancements in PINN technology, including enhanced gradient flows [12], considerations of causality [10; 13], parallelization techniques [18], and optimized collocation points sampling [20]. By synergizing these strategies with recent developments, the potential of PINNs to resolve complex problems in various scientific domains can be significantly expanded.