The design of structures of materials is one of the most important issues in various fields of physical science, as their structures are related to their physical properties. The structures are often characterized by periodic or quasiperiodic order. These ordered structures, which we call a pattern, are ubiquitous in nature ranging from fluid convection to the microphase separation of block copolymers and atomic and molecular crystals. Surprisingly, the same pattern appears in different systems with completely different length scales. Recently, we have been working on estimating a model and non-equilibrium process to reproduce a desired structure[1,2].

Continuum models described by partial differential equations (PDE) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying pattern formation. Although many PDE models have been proposed, they rely on the pre-request knowledge of physical laws and symmetries, and developing a model to reproduce a given desired pattern remains difficult. We propose a method to estimate the best dynamical PDE for one snapshot of a target pattern under the stationary state without ground truth[3]. We apply our method to nontrivial patterns, such as quasi-crystals (QCs), a double gyroid and Frank Kasper structures recently found in soft materials. Our method works for noisy patterns and the pattern synthesised without the ground truth parameters, which are required for the application toward experimental data.

We also discuss our recent efforts to estimate a coarse-grained model from data of microscopic dynamics in a data-driven manner[4]. Specifically, we consider a hydrodynamic description of active matter systems from their particle dynamics.

References:
[1] U. Tu Lieu and N. Yoshinaga, J. Chem. Phys. 156 (2022) 054901.
[2] U. Tu Lieu and N. Yoshinaga, Soft Matter 21 (2025) 514.
[3] N. Yoshinaga and S. Tokuda, Phys. Rev. E 106 (2022) 065301.
[4] B. Roy and N. Yoshinaga, arXiv:2411.03783 (2024).

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