Bayesian modeling of pattern formation from one snapshot of pattern.

Area of Science:

• Computational Physics
• Materials Science
• Mathematical Modeling

Background:

• Partial differential equations (PDEs) are crucial for modeling natural patterns but require prior knowledge of physical laws and symmetries.
• Developing accurate PDE models for specific patterns remains challenging due to the complexity of pattern formation mechanisms.
• Existing methods often necessitate ground truth data or detailed physical insights, limiting their applicability.

Purpose of the Study:

• To propose a novel method, Bayesian modeling of PDEs (BM-PDEs), for estimating dynamical PDEs from a single pattern snapshot.
• To enable the discovery of underlying PDE models without requiring ground truth parameters or knowledge of physical laws.
• To demonstrate the method's efficacy on complex, non-trivial patterns relevant to materials science and physics.

Main Methods:

• Developed the Bayesian modeling of PDEs (BM-PDEs) framework to infer PDE models from observed patterns.
• Applied BM-PDEs to reconstruct patterns including quasicrystals (QCs), double gyroid, and Frank-Kasper structures.
• Utilized estimated parameters from Frank-Kasper A15 structures to generate novel 3D dodecagonal QCs.

Main Results:

• Successfully estimated optimal dynamical PDEs for various complex patterns from single, stationary snapshots.
• Generated three-dimensional dodecagonal quasicrystals using a PDE model derived from Frank-Kasper structures.
• Demonstrated robustness of the BM-PDEs method with noisy data and in the absence of ground-truth parameters.

Conclusions:

• BM-PDEs offers a powerful data-driven approach to discovering pattern-forming PDE models without prior physical constraints.
• The method facilitates the synthesis of complex materials structures, such as quasicrystals, through computational modeling.
• This technique significantly advances the ability to model and generate intricate patterns observed in nature and materials science.