Balance equations for physics-informed machine learning.

Area of Science:

• Physics-informed machine learning (PIML)
• Computational science
• Applied mathematics

Background:

• Traditional machine learning (ML) models may violate physical laws.
• Physics-based models constrain ML using physical laws via loss terms.
• Deriving complex differential equations for physical laws in ML is challenging.

Purpose of the Study:

• To propose a systematic framework for constructing residual loss terms in PIML.
• To ensure ML solutions are consistent with the laws of physics.
• To advance PIML development across multiple domains.

Main Methods:

• Developed a new framework based on balance equations for PIML.
• Proposed a unified approach to derive differential equations from a generic balance equation.
• Demonstrated practical application with worked examples.

Main Results:

• The balance equation method provides a systematic approach to formulating residual loss terms.
• A unified framework can incorporate physical laws into ML models.
• This method ensures physical integrity in partial differential equation (PDE) solutions.

Conclusions:

• The proposed balance equation method offers a generalized approach to PIML.
• This framework unifies the treatment of fundamental physical equations.
• It may lead to more efficient development of physics-based ML for complex systems.