PE-EK-PINN: Physics Embedding with Evolving Kernel for Scalable Physics-Informed Neural Networks

Problem

This work addresses the limitations of existing Physics-Informed Neural Networks (PINNs) in effectively modeling oscillatory wave behavior and practical radiation problems. The authors highlight that traditional PINNs struggle with these complex phenomena, necessitating a more robust approach. The paper is a preprint and has not undergone peer review.

Method

The proposed architecture, PE-EK-PINN (Physics Embedded with Evolving Kernels), innovatively treats physics kernels as reusable learned representations. This method significantly reduces training costs from $O(N)$ to $O( ext{log } N)$, enhancing computational efficiency. The architecture involves a converged subsystem field that is frozen and subsequently promoted to an evolved kernel, allowing for more effective representation of the underlying physics in the neural network framework.

Results

PE-EK-PINN demonstrates a substantial improvement in training efficiency, solving a $256$-dipole array more than 30 times faster than the direct PE-PINN approach. Additionally, it achieves reduced or comparable relative $L_2$ error when benchmarked against previous methods, indicating its effectiveness in maintaining accuracy while enhancing speed.

Limitations

The authors note that the kernel dictionary must be manually constructed, which could introduce challenges in scalability and usability. Furthermore, there are scaling issues associated with the number of elementary units in hierarchically structured systems, which may limit the applicability of the method in more complex scenarios.

Why it matters

The implications of this work are significant for the field of physics-informed machine learning, particularly in applications requiring efficient modeling of complex physical phenomena. By improving the scalability and efficiency of PINNs, this research paves the way for more practical applications in engineering and scientific computing, potentially enabling real-time simulations and analyses in various domains.