# Physics-Informed Neural Networks for Cantilever Dynamics and Fluid-Induced Excitation
###### Abstract  
Physics-informed neural networks (PINNs) represent a continuous and differentiable mapping function, approximating solution curves for given differential equations. Recent studies have demonstrated the significant potential of PINNs as an alternative or complementary approach to conventional numerical methods. However, their application in structural dynamics, such as cantilever dynamics and fluid-induced excitations, poses challenges. In particular, limited accuracy and robustness in resolving high-order differential equations, including fourth-order differential equations encountered in structural dynamics, are major problems with PINNs. To address these challenges, this study explores optimal strategies for constructing PINNs in the context of cantilever dynamics: (1) performing scaling analysis for the configuration, (2) incorporating the second-order non-linear term of the input variables, and (3) utilizing a neural network architecture that reflects a series solution of decomposed bases. These proposed methods have significantly enhanced the predictive capabilities of PINNs, showing an order-of-magnitude improvement in accuracy compared to standard PINNs in resolving the dynamic oscillation of cantilevers and fluid-induced excitation driven by added mass forces. Furthermore, this study extends to the domain of fluid-induced excitation in cantilever dynamics, representing an extreme case of coupled dynamics in fluid-structure interaction. This research is expected to establish crucial baselines for the further development of PINNs in structural dynamics, with potential applicability to high-order coupled differential equations. physics-informed neural networks; dynamic oscillation of the cantilever; fluid-structure interaction; deep neural networks; structural dynamics +
Footnote †: journal: Applied Sciences  
0000-0002-323  
fourth-order in space and second-order in time, could limit the acquisition of a reliable solution based on PINNs. This limitation will be confirmed later in this study. Resolving high-order differential equations remains one of the challenges with PINNs, prompting an active search for advanced strategies, including elaboration in loss construction, the implementation of adaptive activation, and domain decomposition. Previous studies have highlighted that the solution of PINNs can be dramatically influenced by the weighted combination of loss components from ICs, BCs, and PDEs [11; 27; 28]. Consequently, the strategy to balance each loss component has been systematically investigated, as they can be integrated into the training process using a self-adaptive loss-balance approach. Adaptive weight balancing based on the eigenvalues of the neural tangent kernel (NTK) is one representative example of self-adaptive loss balancing [11]. However, it is noteworthy that the adjustment of the weights of loss components relies on the estimated scale of loss components during the training process, not on the physical interpretation of each loss component. In the implementation of physics-informed neural networks (PINNs), the activation function plays a crucial role in ensuring a differentiable solution curve that satisfies the governing equations presented in differential form. Popular choices for activation functions in PINNs, such as tanh or sine, are favored for their multiple differentiability at the expense of poor expression capability [29; 30; 31]. [PERSON] et al. (2020) proposed that PINNs can be guided to avoid suboptimal points by incorporating a scaling factor with a positive trainable parameter before a non-linear activation [32]. This adaptive activation strategy can be generalized based on the Kronecker product, incorporating trainable scaling parameters for weight matrices and bias vectors of varying dimensions, a concept introduced in Kronecker neural networks [33].