用可学习的计算域大小,自动发现复杂有序相。
A Geometry-Adaptive Deep Variational Framework for Phase Discovery in the Landau-Brazovskii Model
- 让计算域尺寸可训练,消除人为应力干扰。
- 通过预热惩罚,从随机初始化中自发形成三维有序结构。
- 适合研究复杂相变与能量局域态的科研人员。
在模式形成系统(如Landau-Brazovskii模型)中,有序结构的发现常受数值求解器对计算域尺寸敏感的影响。不匹配的域会导致人工应力,使系统陷入高能亚稳态。为此,我们提出几何自适应深度变分框架(GeoDVF),联合优化由神经网络参数化的无限维序参量和有限维计算域几何参数。通过在变分形式中显式将域尺寸设为可训练变量,GeoDVF在训练过程中自然消除人工应力。为克服小初始值下对无序相的吸引,引入预热惩罚机制,有效破坏无序相稳定性,实现从随机初始化中自发成核复杂三维有序相。此外,设计引导初始化协议以解决拓扑复杂、吸引盆狭窄的相。大量数值实验表明,GeoDVF是一种鲁棒且几何一致的变分求解器,可在无需先验知识的情况下识别稳定态与亚稳态。
原文摘要 · Abstract (English)
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.
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