Machine learning, particularly neural networks, has rapidly permeated most
activities and work where data has a story to tell. Recently, deep learning has
started to be used for solving differential equations with input from physics,
also known as Physics Informed Neural Networks (PINNs). We present a study
showing the efficacy of PINNs for solving the Zerilli and the Regge-Wheeler
equations in the time domain to calculate the quasi-normal oscillation modes of
a Schwarzschild black hole. We compare the extracted modes with those obtained
with finite difference methods. Although the PINN results are competitive, with
a few percent differences in the quasi-normal modes estimates relative to those
computed with finite difference methods, the real power of PINNs will emerge
when applied to large dimensionality problems.
Machine learning, especially deep learning and neural networks, has become pervasive in data-driven activities and work. A recent development in this field is the use of Physics Informed Neural Networks (PINNs) to solve differential equations with input from physics. In this study, we demonstrate the effectiveness of PINNs in solving the Zerilli and Regge-Wheeler equations in the time domain to calculate the quasi-normal oscillation modes of a Schwarzschild black hole. We compare the results obtained using PINNs with those obtained using finite difference methods.
The results show that PINNs can provide competitive estimates of the quasi-normal modes, with only a few percent difference compared to the finite difference methods. However, the true potential of PINNs will be realized when they are applied to problems with large dimensionality.
Future Roadmap
In the future, there are several key challenges and opportunities that lie ahead in the field of PINNs:
- Scaling to Large Dimensionality: One of the main advantages of PINNs is their ability to handle high-dimensional problems. As we apply PINNs to larger and more complex systems, it will be important to ensure their scalability and efficiency. This may require further research and development of novel architectures and algorithms.
- Improving Accuracy: Although PINNs provide competitive results for the specific problem studied in this article, there is always room for improvement. Researchers should explore ways to enhance the accuracy of PINNs, perhaps through better optimization techniques, regularization methods, or model architectures.
- Robustness to Noisy and Incomplete Data: Real-world data is often noisy and incomplete. PINNs should be able to handle such data effectively and provide reliable results. Developing techniques to make PINNs more robust to noise and missing data will be crucial for their widespread application.
- Interpretability and Explainability: Neural networks, including PINNs, are often considered black boxes due to their complex and opaque nature. It is important to develop methods to interpret and explain the results obtained from PINNs. This will enable researchers and practitioners to gain insights into the underlying physics and improve trust in the models.
- Integration with Other Fields: PINNs have the potential to be integrated with other fields such as computational physics, numerical methods, and optimization. Collaborations and interdisciplinary research can lead to new breakthroughs and applications.
In conclusion, PINNs have shown promising results in solving differential equations with input from physics. As the field progresses, addressing the challenges and capitalizing on the opportunities will pave the way for the widespread adoption of PINNs in various scientific and engineering domains.