Understanding and predicting complex physical systems remain significant challenges in scientific research and engineering. Machine learning models, while powerful, often fail to follow the fundamental rules of physics, leading to inaccurate or unphysical results. To address this, physics-informed machine learning has emerged as a solution by embedding these rules into machine learning models. However, creating precise conditions that enforce these rules is a difficult task, especially when dealing with complex mathematical equations. Researchers Dr. Sandor Molnar from Academia Sinica and Professor Joseph Godfrey and Dr. Binyang Song from Virginia Tech have introduced a new approach that unifies various physical laws under a single framework. Their work, published in the journal Heliyon, proposes a balance equation method to systematically integrate physics into machine learning models.
Traditional physics-informed machine learning methods rely on additional correction terms derived from governing equations to ensure compliance with physical laws. However, defining these correction terms is often inconsistent and lacks a universal guideline. The proposed balance equation framework addresses this by deriving all fundamental equations of classical physics—such as those describing how fluids move, how electric fields behave, how materials stretch, and how heat transfers—from a single balance equation. This equation accounts for the conservation and movement of physical quantities like mass, force, and energy. By applying specific material relationships, researchers can adapt the balance equation 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 known as the generic balance equation, in conjunction with specific constitutive relations that bind the balance equation to a particular domain.” This approach provides a more structured and universal method for incorporating physics into machine learning.
One major benefit of this approach is its ability to systematically enforce physical rules without needing extra adjustments for different types of equations. The researchers showed that their method accurately captures how complex systems behave by solving both prediction problems and reverse engineering problems in physics-informed machine learning. Prediction problems involve forecasting how a system will change over time based on known physical laws, while reverse engineering problems involve discovering the unknown rules governing a system by analyzing real-world data. Their method allows both types of problems to be tackled 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 wide range of applications across different scientific fields. The balance equation method can be used to model how liquids and gases flow, how chemical reactions occur, and how electrical forces interact, among other applications. By bringing together different physical principles under one equation, this approach not only simplifies the process of integrating physics into machine learning models but also provides a more reliable and adaptable method. The researchers provided practical examples showing how their framework can be applied, demonstrating its flexibility and usefulness in real-world situations.
Highlighting the significance of their findings, Professor Godfrey stated, “Our approach suggests that a single framework can be followed to incorporate physics into machine learning models. This level of generalization may provide the basis for more efficient methods of developing physics-based machine learning for complex systems.”
As machine learning continues to play a major role in scientific research, ensuring that its predictions align with physical reality is essential. The balance equation framework presents an important step toward more reliable and understandable machine learning models for complex systems. Professor Godfrey emphasized the broader implications of their work, saying, “The balance equation framework enables the communication of physical constraints to a physics-informed neural network (PINN) by specifying the balance equations and the associated constitutive equations. These equations can be combined into a single partial differential equation or a system of such equations.” “The future is physics-informed machine learning” added Dr. Molnar.
By offering a structured and universal method for incorporating physics into machine learning, this work lays the foundation for future improvements in computational modeling. It opens the door for more precise simulations, better predictions, and deeper insights into the behavior of natural and engineered systems.
Journal Reference
Molnar S.M., Godfrey J., Song B. “Balance equations for physics-informed machine learning.” Heliyon, 2024; 10: e38799. DOI: https://doi.org/10.1016/j.heliyon.2024.e38799
About the Authors
Joseph R Godfrey was born on 15 April 1958 in San Jose, Costa Rica. He received his BS degree in Mathematics from The University of Chicago in 1979 and his PhD in High Energy Physics from the University of Notre Dame in 1987. Professor Godfrey is currently the Director of the Masters of Engineering Administration (MEA) program, under the Grado Department of Industrial and Systems Engineering at Virginia Tech. His responsibilities include managing and developing the program, recruiting students, and developing partnerships with public and private institutions.
Sandor M. Molnar was born on 27 August 1955 in Budapest, Hungary. He received his Diploma in Astronomy from Eotvos University, Budapest, Hungary, in 1979, and two MSc degrees (in Physics, Astronomy) from the University of Massachusetts at Amherst in 1993 and 1995. He received his PhD from the University of Bristol, UK, in 1998. After his PhD, he spent three years as a research associate at NASA, Goddard Space Flight Center (two years as a National Academy of Sciences/National Research Council Research Associate). He then held postdoctoral positions at several universities (Rutgers, Washington State University, University of Zurich) before joining the Institute of Astronomy and Astrophysics at Academia Sinica in Taipei, Taiwan, as a Visiting Scientist in 2007. Dr Molnar officially retired from the Institute of Astronomy and Astrophysics in 2020 but has continued his research as a Visiting Scholar. He has more than 70 publications in astrophysics and cosmology on galaxy clusters and related topics. Dr Molnar published a book in 2015 entitled Cosmology with Clusters of Galaxies (by Nova).
