# Physics-Informed Neural Networks for Cantilever Dynamics and Fluid-Induced Excitation
## 2 Proposed Approach
### Scaling Analysis
One choice to meet this requirement could be to set \(\alpha_{IC,1}=1\) and \(\alpha_{IC,2}=O\big{(}\tau^{2}\big{)}\). The above argument can be generalized using the notations of \(\mathcal{L}_{p,i}\) and \(\alpha_{p,i}\) for the loss components and regularization parameters, respectively. Here, \(v\) e {PDE, IC, BC} corresponds to the category of the loss, and \(i\) is the loss index in each category. The total loss is then rewritten as a weighted summation of loss components with regularization coefficients: \(\mathcal{L}_{tot}=\sum_{v}\sum_{i}\alpha_{v,i}\mathcal{L}_{v,i}\). The basic idea is to adjust the coefficients \(\alpha_{v,i}\) by enforcing additional conditions that the scales of the weighted loss components be of the same order of magnitude; for example, \(\alpha_{v,i}\mathcal{L}_{v,i}\sim O(1)\). Here, the order-of magnitude of the target physical variable \(O(u_{v,i})\) can be determined from the scaling analysis for the given physical problems, which is exemplified in the later sections. Suppose that the loss component \(\mathcal{L}_{v,i}\) considers the MSE of a physical variable, \(u_{v,i}\), with a desired relative tolerance, \(\varepsilon_{v,i}\), for the PINN solution. The prediction error can be scaled as \(E_{v,i}\sim\varepsilon_{v,i}O(u_{v,i})\), leading to the loss component expressed as \(\mathcal{L}_{v,i}\sim(E_{v,i})^{2}\sim\varepsilon_{v,i}^{2}O(u_{v,i}^{2})\). Therefore, the regularization coefficients \(\alpha_{v,i}\) have to be scaled as \(1/\varepsilon_{v,i}^{2}O(u_{v,i}^{2})\) to adjust the scales of the weighted loss components to be of \(O(1)\). Finally, the alternative description of the total loss is obtained as  
\[\mathcal{L}_{tot}=\sum_{v}\sum_{i}\frac{\mathcal{L}_{v,i}}{\varepsilon_{v,i}^ {2}O\big{(}u_{v,i}^{2}\big{)}} \tag{3}\]  
Here, the order of magnitude of the target physical variable, \(O(u_{v,i})\), can be determined from the scaling analysis for the given physical problems, which is exemplified in the later sections. As practical guidance, this study suggests using an initial guess of desired tolerances \(\varepsilon_{v,i}\) as \(10^{-2}\) for PDEs and \(10^{-3}\) for BCs and ICs. The PDE loss tends to encourage convergence to a trivial solution of the PDE, such as a zero-valued function over the entire spacetime domain. On the other hand, the loss from ICs and BCs typically causes neural networks to reproduce ICs, regardless of the time change. Therefore, finding an optimal solution between two misguided points is desired. One effective way found in this study is to guide neural networks to learn temporal variations obeying the PDE from the ICs/BCs. Then, the suggestion of the PDE tolerance being 10 times smaller than that of ICs and BCs usually works well to find a solution for a given problem. One of the easiest ways to satisfy the requirements of scaling analysis is to start with governing equations in dimensionless form, which configures most of the dimensional variables and their derivatives to be scaled as \(O(1)\). However, when the problem involves multiple characteristic scales in the same physical dimension (for example, when the characteristic length scales in the x- and y-dimensions are distinct), the proposed scaling analysis could be adopted to resolve the scaling imbalances in a more rigorous way. Furthermore, the scale of the dimensionless number in the governing equation could involve an additional imbalance compared to ICs and BCs, which have to be properly resolved based on scaling analysis.