## 4 Conclusions  
This work aims to shed light on the common challenges encountered by PINNs when applied to high-frequency and multi-scale problems. We explored the potential of transfer learning as a viable solution to these problems. In the experiments, we observed that the PINN depicts ability in approximating the harmonic oscillator at a frequency of 20 Hz. However, as the frequency increases, a noticeable increase in computational cost follows, accompanied by increased convergence times.  
The application of the vanilla PINN, utilizing an identical neural network architecture as the 20 Hz case, proves unfeasible in achieving convergence at 40 Hz, 50 Hz, and 60 Hz with the same amount of collocation points. While the model performs well on low-frequency problems, it starts struggling when given higher frequencies. Through transfer learning, we were able to learn the 50 Hz and 60 Hz solutions, without adding more layers, or changing the number of collocation points. The results were promising as well, with a loss reaching an order of \(10^{-2}\).  
Similarly, in the context of the one-dimensional wave equation, with the use of transfer learning, we learnt the PINN solution for different wave velocities, starting from two up to four. The transfer learning method turned out to be effective. As for a higher wave velocity, the model achieved convergence significantly quicker.  
Transfer learning has proven to be an effective method for enhancing the efficiency and convergence characteristics of PINNs, preventing the necessity for modifications to the network architecture, which causes more parameters. A meaningful research direction is establishing a pre-trained model library for complex engineering problems so that transfer learning technology can be used to train models for specific problems rapidly and accurately.  
However, applying transfer learning techniques is not without its pitfalls. Negative transfer describes the phenomenon where inappropriately transferring knowledge from the source domain can inversely hurt the target performance [51]. Although we did not observe  
Figure 9: Results for \(c=1\) (L-BFGS): (**a**) approximated solution by PINN, (**b**) exact solution, (**c**) loss curve. Results for various values of \(c\) with transfer learning: (**d**) \(c=1.5\), (**e**) \(c=2\), (**f**) \(c=4\). Loss curves for: (**g**) \(c=1.5\), (**h**) \(c=2\), (**i**) \(c=4\).  
any negative effects resulting from the application of transfer learning in the experiments involved in this article, this does not mean that the technique can be applied to the training of PINNs without careful consideration. The important point here is to understand the limitations of transfer learning, that is, under what circumstances transfer learning might fail, and how such failures can be avoided. We believe this contributes not only to the theory of PINN training but also has practical significance in solving more complex real-world engineering problems. This also points to the direction of our future work.  
Conceptualization, H.L. and Z.R.; methodology, A.H.M., H.L. and F.W.; software, A.H.M. and H.L.; validation, A.H.M. and H.L.; formal analysis, A.H.M. and H.L.; investigation, A.H.M. and H.L.; data curation, A.H.M. and H.L.; writing--original draft preparation, A.H.M. and H.L.; writing--review and editing, A.H.M., H.L. and Z.R.; visualization, A.H.M. and H.L.; supervision, H.L. and Z.R.; project administration, H.L., F.W. and Z.R. All authors have read and agreed to the published version of the manuscript.  
Not applicable.  
The data points used for training PINN models can be sampled using the method mentioned in the article.  
Author [PERSON] was employed by the company GeoAnalysis Engineering GmbH. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.