# Physics-Informed Neural Networks for Cantilever Dynamics and Fluid-Induced Excitation
## 2 Proposed Approach
### Scaling Analysis  
The convergence and accuracy of PINNs are significantly impacted due to various problem configurations, such as loss-regularization coefficients, physical variables in PDEs, and reference values in PINNs. However, configuring PINNs is often challenging due to imbalances in physical dimensions and scales that can arise. The loss function is constructed from different physical dimensions in each loss category of PDEs, ICs, and BCs, and there is no guarantee that the scales of loss components are well balanced even within the same loss category. For instance, differential equations with second-order time derivatives require two initial conditions that have different physical dimensions, which can result in scale imbalances in \(\mathcal{L}_{IC}\). To address these issues, scaling analysis can provide an advantageous initial estimate for the PINN configuration and help resolve any scale imbalances. Let us assume that we want to use a PINN, \(\Phi(x,t\mid\theta)\), to solve the differential operator \(\mathcal{D}\), which can be decomposed into spatial differential \(\mathcal{D}_{x}\) and temporal differential \(\mathcal{D}_{t}\), in the domain of in \(-\frac{t}{2}<x<\frac{t}{2}\) and \(0<t<\tau_{h}\). We can scale the PDE residuals, \(\mathcal{D}_{x}(\Phi(x,t))\) and \(\mathcal{D}_{t}(\Phi(x,t))\), with the physical variables of the problem, such as the characteristic length scale, \(L\), and the desired time horizon, \(\tau_{h}\). For example, if the PDE is constructed from first-order spatial and temporal differentials, with \(\mathcal{D}_{x}=\partial/\partial x\) and \(\mathcal{D}_{t}=\partial/\partial t\), the scales of the differential operators correspond to \(\partial/\partial x\sim~{}1/L\) and \(\partial/\partial t\sim~{}1/\tau_{h}\), respectively. If the characteristic length scale is much larger than the time scale \(L\gg\tau_{h}\), the differential operator can be approximated as \(\mathcal{D}\approx\mathcal{D}_{t}\). Then, a trivial solution that remains constant in time would be found. One way to prevent this problem is to adjust the time scale of the PINN to ensure that the scales of the differential operators are of the same order of magnitude, \(O(\mathcal{D}_{x})\sim O(\mathcal{D}_{t})\). This can be achieved by adopting a time-marching strategy [14; 15], which iteratively trains a series of PINNs with a time step of \(\tau\) until the desired time horizon, \(\tau_{h}\), is reached. The time step \(\tau\) is then selected to balance the scales between differential operators. For the exemplified first-order differential operators, the scaling as \(O(\tau)\sim O(L)\) is obtained from the requirement \(O(\mathcal{D}_{x})\sim O(\mathcal{D}_{t})\). In addition, the scale imbalance in the loss function must be further addressed. The conventional regularization coefficients \(\alpha_{PDE}\), \(\alpha_{IC}\), and \(\alpha_{BC}\) in Equation (2) are insufficient for resolving the scale imbalance in second- or higher-order differential equations. For example, when solving a second-order ODE (ordinary differential equation) for \(u(t)\) in time \(t\), the ICs can be given as \(u(0)\) and \(\left(\left.du/dt\right)\right|_{t=0}\). If the MSE (mean-squared error) is used as the loss metric for \(\mathcal{L}\), the resulting scales in each loss component, \(\mathcal{L}_{IC,1}(\Phi(t)|_{t=0},u(0))\) and \(\mathcal{L}_{IC,2}(\mathrm{d}\Phi(t)/\mathrm{d}t|_{t=0},\left(du/dt\right) \right|_{t=0})\), correspond to \(O(E(u)^{2})\) and \(O(E(u)^{2})/\tau^{2}\), respectively, with the operator \(E\) denoting the prediction error. Thus, a significant scale imbalance could arise due to the time step \(\tau\). To address this scale imbalance in \(\mathcal{L}_{IC}\), additional regularization coefficients, such as \(\alpha_{IC,1}\) and \(\alpha_{IC,2}\), can be employed by enforcing the scales in each loss component to be of the same order of magnitude with an additional requirement, \(\alpha_{IC,1}\mathcal{L}_{IC,1}\sim\alpha_{IC,2}\mathcal{L}_{IC,2}\).