Learning PDE to Model Self-Organization of Matter

Abstract

A self-organization hydrodynamic process has recently been proposed to partially explain the formation of femtosecond laser-induced nanopatterns on Nickel, which have important applications in optics, microbiology, medicine, etc. Exploring laser pattern space is difficult, however, which simultaneously (i) motivates using machine learning (ML) to search for novel patterns and (ii) hinders it, because of the few data available from costly and time-consuming experiments. In this paper, we use ML to predict novel patterns by integrating partial physical knowledge in the form of the Swift-Hohenberg (SH) partial differential equation (PDE). To do so, we propose a framework to learn with few data, in the absence of initial conditions, by benefiting from background knowledge in the form of a PDE solver. We show that in the case of a self-organization process, a feature mapping exists in which initial conditions can safely be ignored and patterns can be described in terms of PDE parameters alone, which drastically simplifies the problem. In order to apply this framework, we develop a second-order pseudospectral solver of the SH equation which offers a good compromise between accuracy and speed. Our method allows us to predict new nanopatterns in good agreement with experimental data. Moreover, we show that pattern features are related, which imposes constraints on novel pattern design, and suggest an efficient procedure of acquiring experimental data iteratively to improve the generalization of the learned model. It also allows us to identify the limitations of the SH equation as a partial model and suggests an improvement to the physical model itself.

Beyond the Microscale: Advances in Surface Nanopatterning by Laser‐Driven Self‐Organization

“…To deepen the comprehension of complex dynamics in the presence of fluctuations during fluid instability onset, a recent advancement involves a nonlinear mathematical approach capable of predicting Turing-like patterns that lead to self-organization. [93] The self-organization process was first proposed in the 1990s to theoretically explain a specific class of periodic laser-induced structures that differed qualitatively from LIPSS in that their orientation was unrelated to the laser polarization and the structure period was not directly correlated to the exciting radiation wavelength. Later, Reif et al employed self-organization to explain the creation of LIPSS on wide bandgap materials using a nonlinear-dynamic erosion/smoothing model, based on Kuramoto-Sivashinsky equation commonly used in ion sputtering, to simulate surface pattering during multi-pulse femtosecond laser ablation [63,116,117] :…”

“…It reduces experimental irradiation parameters to simple model coefficients, which can then be optimized and extrapolated for surface pattern engineering. The generalized Swift-Hohenberg equation was derived in an adimensional form as [93,120] :…”

“…This ML approach reduces experimental irradiation parameters to simple model coefficients, which can then be optimized and extrapolated for surface pattern engineering. [93,120]…”

“…In the context of 2D‐LIPSS, understanding the nonlinear mechanisms governing symmetry breaking and pattern selection in convective instabilities remains a complex challenge, making the transition conditions from one self‐organization regime to another difficult to predict. [ 93 ] In a general sense, these nanostructures emerge during the interplay of pressure gradient forces, mainly driven by surface tension and cavitation processes (through rarefaction), along with the rapid resolidification of the molten surface region. Thermo‐convective instabilities can induce fluid flow due to comparable transverse and longitudinal thermal gradients.…”

“…To deepen the comprehension of complex dynamics in the presence of fluctuations during fluid instability onset, a recent advancement involves a nonlinear mathematical approach capable of predicting Turing‐like patterns that lead to self‐organization. [ 93 ]…”