Solving differential equations with physics-informed machine learning

Title	Solving differential equations with physics-informed machine learning
Authors	A. M. Abrashkin¹ ¹Ulyanovsk State University
Annotation	The article discusses physics-informed machine learning (PIML) -- a new approach to solving partial differential equations at the intersection of classical numerical methods and machine learning. Including information about physical laws in the loss function of neural networks allows us to successfully solve problems even with a limited volume and noisy data. The use of universal function approximators on clouds of randomly selected points allows us to bypass the complexities of computational grids and flexibly integrate experimental data. The limitations of PIML associated with advection, rigid boundary conditions, and optimization local minima are described, as well as prospects for further development of PIML in engineering, physics, and quantum computing.
Keywords	machine learning, differential equations, partial derivatives.
Citation	Abrashkin A. M. ''Solving differential equations with physics-informed machine learning'' [Electronic resource]. Proceedings of the International Scientific Youth School-Seminar "Mathematical Modeling, Numerical Methods and Software complexes" named after E.V. Voskresensky (Saransk, July 29-31, 2025). Saransk: SVMO Publ, 2025. - pp. 15-21. Available at: https://conf.svmo.ru/files/2025/papers/paper01.pdf. - Date of access: 30.08.2025.