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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. 

Included in this dataset are: 1. Monte Carlo (MC) simulation data used for PDE identification 2. Molecular dynamics (MD) simulation data used for PDE identification 3. LAMMPS input file for PDE validation, Case 1: linear initial C profile 4. LAMMPS input file for PDE validation, Case 2: exponential initial C profile They are used in our manuscript, Learning the Governing PDE of Solid Diffusion from Atomistic Simulations, by Wongelemengist Nadew and Haoran Wang 

The mechanical response of Al-substituted LLZO to compressive forces was measured using instrumented indentation testing. Large correlated variations in compressive strength are observed across microscale regions of the solid electrolyte.