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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.
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Robust data-driven discovery of governing physical laws with error bars
Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier–Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.
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- PAR ID:
- 10188453
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 474
- Issue:
- 2217
- ISSN:
- 1364-5021
- Page Range / eLocation ID:
- 20180305
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rspa.2018.0305
