From PINNs to PIKANs: recent advances in physics-informed machine learning.

Area of Science:

• Scientific Machine Learning
• Computational Mathematics
• Applied Physics

Background:

• Physics-Informed Neural Networks (PINNs) have become crucial for solving differential equations since 2017.
• PINNs efficiently utilize sparse measurements for solving ordinary and partial differential equations.
• Recent progress has focused on training, optimization, network architectures, and adaptive techniques.

Purpose of the Study:

• To provide a comprehensive review of recent advancements in PINNs.
• To highlight new developments like Physics-Informed Kolmogorov-Arnold Networks (PIKANS).
• To survey applications and available computational tools for PINNs.

Main Methods:

• Review of literature on PINN advancements.
• Focus on network design, feature expansion, and optimization strategies.
• Exploration of uncertainty quantification and theoretical underpinnings.

Main Results:

• Significant improvements in PINN training and optimization techniques.
• Introduction of PIKANS as a promising alternative to traditional PINNs.
• Broad applicability demonstrated across diverse scientific and engineering fields.

Conclusions:

• PINNs and PIKANS represent a rapidly evolving field with substantial potential.
• Continued research is enhancing the capabilities and applicability of physics-informed machine learning.
• Development of robust computational frameworks supports the adoption of these methods.