Self-Scalable Tanh (Stan): Multi-Scale Solutions for Physics-Informed Neural Networks.

Area of Science:

• Computational Mathematics
• Deep Learning Applications
• Scientific Computing

Background:

• Differential equations are crucial for modeling physical systems across various domains.
• Traditional numerical methods approximate solutions, facing limitations with complex systems.
• Physics-informed Neural Networks (PINNs) emerge as a powerful mesh-free alternative for solving differential equations and inverse problems.

Purpose of the Study:

• To address the training limitations of Physics-informed Neural Networks (PINNs) caused by conventional activation functions.
• To introduce a novel, scalable activation function designed to enhance the learning process in PINNs.
• To improve the accuracy and efficiency of solving differential equations using PINNs.

Main Methods:

• Development of a novel Self-scalable tanh (Stan) activation function for PINNs.
• The Stan function is smooth, non-saturating, and incorporates a trainable parameter for adaptive scaling.
• Evaluation through forward problems (solving differential equations) and inverse problems (parameter identification).

Main Results:

• The proposed Stan activation function facilitates smoother gradient flow and systematic input-output mapping adjustments during training.
• Demonstrated superior training performance and prediction accuracy compared to existing activation functions in PINNs.
• Successfully applied to solve complex differential equations and identify system parameters.

Conclusions:

• The Stan activation function represents a significant advancement for Physics-informed Neural Networks.
• It enhances the capability of PINNs to accurately and efficiently model physical systems governed by differential equations.
• The Stan function offers a promising direction for future research in physics-informed machine learning.