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

2026-03-05

AI Summary

This work proposes a geometrically adaptive deep variational framework (GeoDVF) to overcome the limitations of traditional numerical solvers, which are sensitive to computational domain size and prone to introducing spurious stresses due to domain mismatch, thereby trapping the system in high-energy metastable states and obscuring true ordered phases. GeoDVF uniquely treats the geometric parameters of the computational domain as trainable variables, jointly optimizing them with neural network–parameterized order parameters to eliminate artificial stresses. Combined with thermal annealing penalties and a guided initialization strategy, the method enables spontaneous, prior-free emergence of complex three-dimensional ordered phases from random initial conditions. Applied to the Landau–Brazovskii model, GeoDVF successfully identifies multiple stable and metastable phases, demonstrating its robustness and geometric consistency.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: 3D Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• phase discovery
• Landau-Brazovskii model
• computational domain
• artificial stress
• metastable states

Innovation

• Geometry-Adaptive Optimization
• Deep Variational Method
• Phase Discovery
• Neural Representation of Order Parameter
• Domain Size as Trainable Variable