A Geometry-Adaptive Deep Variational Framework for Phase Discovery in the Landau-Brazovskii Model

TL;DR

This paper introduces GeoDVF, a neural network-based framework that adaptively optimizes domain geometry to reliably discover stable and metastable phases in pattern-forming systems like the Landau-Brazovskii model, overcoming domain sensitivity issues.

Contribution

The paper presents a novel geometry-adaptive variational framework that jointly optimizes order parameters and domain geometry, enabling robust phase discovery without prior domain size assumptions.

Findings

Successfully identifies stable and metastable phases.

Eliminates artificial stress caused by domain mismatch.

Enables spontaneous nucleation of complex phases from random initializations.

Abstract

The discovery of ordered structures in pattern-forming systems, such as the Landau-Brazovskii (LB) model, is often limited by the sensitivity of numerical solvers to the prescribed computational domain size. Incompatible domains induce artificial stress, frequently trapping the system in high-energy metastable configurations. To resolve this issue, we propose a Geometry-Adaptive Deep Variational Framework (GeoDVF) that jointly optimizes the infinite-dimensional order parameter, which is parameterized by a neural network, and the finite-dimensional geometric parameters of the computational domain. By explicitly treating the domain size as trainable variables within the variational formulation, GeoDVF naturally eliminates artificial stress during training. To escape the attraction basin of the disordered phase under small initializations, we introduce a warmup penalty mechanism, which effectively destabilizes the disordered phase, enabling the spontaneous nucleation of complex three-dimensional ordered phases from random initializations. Furthermore, we design a guided initialization protocol to resolve topologically intricate phases associated with narrow basins of attraction. Extensive numerical experiments show that GeoDVF provides a robust and geometry-consistent variational solver capable of identifying both stable and metastable states without prior knowledge.