Understanding and predicting complex physical systems remains a major challenge in scientific research and engineering. Machine learning models, while powerful, often fail to follow the basic rules of physics, resulting in inaccurate or non-physical results. To solve this problem, by embedding these rules into machine learning models, machine learning of physical knowledge has become a solution. However, creating precise conditions that execute these rules is a difficult task, especially when dealing with complex mathematical equations. Researchers, Dr. Sandor Molnar from Sinica academia, Professor Joseph Godfrey of Virginia Tech and Binyang Song, introduce a new approach that unifies various laws of physics within a single framework. Their work, published in the journal Heliyon, proposes an equilibrium equation approach that integrates physical systems into machine learning models.
The machine learning approach of traditional physics knowledge relies on other corrective terms derived from management equations to ensure compliance with the laws of physics. However, defining these correction terms is often inconsistent and lacks general guidelines. The proposed framework of equilibrium equations solves this problem by derive all the fundamental equations of classical physics, such as describing how fluids move, how electric fields behave, how materials are stretched and how heat is transferred. This equation is the protection and motion of physical quantities such as mass, force and energy. By applying specific material relationships, researchers can adjust the equilibrium equations to different scientific fields, making it easier to integrate physics into machine learning models.
Professor Godfrey explained: “We show that all of these equations can be derived from a single equation called a universal equilibrium equation and combined with a specific constitutive relationship that combines the equilibrium equation with a specific domain.” This approach provides a more structured and general approach to incorporating physics into machine learning.
One of the main benefits of this approach is its ability to execute physical rules systematically without additional adjustments to different types of equations. The researchers show that their approach accurately captures the behavior of complex systems by solving predictive problems and physical knowledge in machine learning. The prediction problem involves how a prediction system changes over time according to known laws of physics, while the reverse engineering problem involves discovering unknown rules of the control system by analyzing real-world data. Their approach allows the solution of both types of problems using the same approach, significantly improving the efficiency and accuracy of machine learning models designed to work with physical systems.
One of the most important aspects of this research is its widespread use in different scientific fields. The equilibrium equation method can be used to simulate the flow of liquids and gases, the way chemical reactions occur, and the interaction of electricity, and other applications. By summarizing different physics principles in one equation, this approach not only simplifies the process of integrating physics into machine learning models, but also provides a more reliable and adaptable approach. The researchers provide practical examples showing how their framework can be applied to demonstrate its flexibility and practicality in real-world situations.
Professor Godfrey noted: “Our approach emphasizes the importance of their findings,” he said. “Our approach shows that a framework can be followed to incorporate physics into machine learning models. This level of generalization can provide the basis for more efficient development of methods for complex systems based on physics-based machine learning. ”
As machine learning continues to play an important role in scientific research, it is crucial to ensure that its predictions are consistent with physical reality. The equilibrium equation framework takes an important step for more reliable and understandable machine learning models for complex systems. Prof Godfrey highlights the broader implications of their work, saying: “The equilibrium equation framework can communicate physical constraints with neural networks (PINNs) of physical knowledge by specifying equilibrium equations and related constitutive equations. These equations can be combined into a single-part differential equation or a system of such equations.” Dr. Mornal added: “The future is machine learning of physical knowledge.”
By providing a structured, universal approach that physics can be incorporated into machine learning, this work lays the foundation for future improvements in computing modeling. It opens the door to more precise simulations, better predictions, and deeper insights into the behavior of natural and engineering systems.
Journal Reference
Molnar SM, Godfrey J., Song B. “The equilibrium equation of machine learning for physical knowledge.” Heliyon, 2024; 10:E38799. doi:
About the Author
Joseph R Godfrey Born on April 15, 1958 in San Jose, Costa Rica. He received a bachelor’s degree in mathematics from the University of Chicago in 1979 and a doctorate in high energy physics from Notre Dame in 1987. Professor Goldfrey is currently the director of the Grado Master of Engineering Management (MEA) program in the Department of Industrial and Systems Engineering at Virginia Tech. His responsibilities include managing and developing the program, recruiting students, and establishing partnerships with public and private institutions.
Sandor M. Molnar Born on August 27, 1955 in Budapest, Hungary. He received a diploma in astronomy from the Eotvos University in Budapest, Hungary in 1979 and received two MSC degrees (Physics, Astronomy) from the University of Massachusetts Amherst in 1993 and 1995. He received his PhD from the University of Bristol in 1998. He won the University of Bristol, England in 1998, and he received three years of research school in 1998. Research Assistant to the Science/National Research Council). He then held postdoctoral positions at several universities in Academia at Sinica, Taipei, Taiwan (Rugs, Washington State University, Washington State University, Washington State University, Washington State University). He owns more than 70 publications in astral physics and cosmology, covering the Milky Way and related topics. Dr. Mornard published a book in 2015 titled Cosmology with a group of galaxies (Nova).
