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Abstract Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by “baking” the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers’, and Navier-Stokes equations.
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This content will become publicly available on September 17, 2026
Learning the governing partial differential equation of solid diffusion from atomistic simulations
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.
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- Award ID(s):
- 2138431
- PAR ID:
- 10636345
- Publisher / Repository:
- The Royal Soceity
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 481
- Issue:
- 2322
- ISSN:
- 1364-5021
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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