Learning the governing partial differential equation of solid diffusion from atomistic simulations

W Nadew and HR Wang, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 481, 20250383 (2025).

DOI: 10.1098/rspa.2025.0383

Governing partial differential equations (PDEs) play a critical role in materials research and applications, as they describe essential physics underlying materials behaviour. Traditionally, these equations are developed through phenomenological modelling of experimental results or first principle analysis based on conservation laws. In addition, molecular dynamics (MD) simulations capture atomistic-scale behaviour with detailed physics. However, translating atomistic insights into continuum-scale governing equations remains a significant challenge. Empowered by recent advances in data-driven modelling, we develop a computational framework to learn governing PDEs directly from atomistic simulation data. The framework integrates numerical differentiation of MD data with the identification of constitutive relationships. It proves effective and efficient in learning governing PDEs from noisy and limited MD datasets, without requiring prior knowledge of the final PDEs. Using this framework, we identify a nonlinear PDE governing solid- state diffusion in nickel-hydrogen alloys. This PDE reveals a highly concentration-dependent diffusivity that varies over an order of magnitude. Our data-driven computational framework paves the way for cross-scale constitutive modelling.

Return to Publications page