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

Yuchen Xie, Jianyuan Yin, Lei Zhang

TL;DR

A Geometry-Adaptive Deep Variational Framework 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, and provides a robust and geometry-consistent variational solver capable of identifying both stable and metastable states without prior knowledge.

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.

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

TL;DR

A Geometry-Adaptive Deep Variational Framework 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, and provides a robust and geometry-consistent variational solver capable of identifying both stable and metastable states without prior knowledge.

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.
Paper Structure (16 sections, 1 theorem, 36 equations, 6 figures, 5 tables)

This paper contains 16 sections, 1 theorem, 36 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Landau--Brazovskii model
  3. Geometry-Adaptive Deep Variational Framework
  4. Neural network architecture
  5. Mass-conservation constraint
  6. Loss function and training strategies
  7. Theoretical analysis of the warmup penalty
  8. Numerical experiments
  9. Implementation details
  10. Phase identification and statistics
  11. Evaluation metrics
  12. Escaping the disordered phase
  13. Ablation study
  14. Global minimum selection
  15. Discovery of complex 3D ordered structures
  16. ...and 1 more sections

Key Result

Theorem 3.1

Consider the Landau--Brazovskii free-energy functional $f(\phi)$ with $\tau > 0$ subject to the mass-conservation constraint, and then the disordered phase $\phi = 0$ is a local minimum of the free energy $f(\phi)$. For $w_{\text{p}} > 0$ and $C > 0$, the disordered phase is strictly unstable for th

Figures (6)

  • Figure 1: Representative stationary states of the 2D LB model. (a) The disordered phase. (b) LAM phase. (c) HEX phase obtained within an optimal computational domain. (d) A distorted hexagonal phase (denoted as HEX*). This is a high-energy metastable state where the lattice structure is deformed due to the artificial stress caused by domain mismatch.
  • Figure 2: An overview of GeoDVF. (a) The workflow of GeoDVF. The system takes spatial coordinates $\bm{x}$ as input and simultaneously optimizes the network parameters $\theta$ (for the order parameter field $\phi_{\theta}$) and the inverse geometric parameters $\bm{\beta}$ (for the domain geometry). The zero-mean projection enforces the mass conservation constraint, converting the raw network output $N_{\theta}(\bm{x})$ into the physical order parameter $\phi_{\theta}$. The total loss $\mathcal{L}(\Theta)$ integrates the physical free-energy density $f$ with auxiliary terms from the warmup penalty or guided initialization. The dashed line indicates that the unified parameter set $\Theta = (\theta, \bm{\beta})$ is updated jointly via backpropagation. (b) Detailed architecture of the neural network $N_{\theta}$. The coordinate embedding layer $P$ maps the input coordinates $\bm{x}$ directly into a high-dimensional feature function. This representation evolves through a stack of Fourier layers $\mathcal{H}_l$ and is mapped back to the unconstrained scalar field $N_{\theta}$ by the projection layer $Q$. The inset illustrates the operations within a single Fourier layer.
  • Figure 3: Comparison of phase discovery results on the 2D LB model. (a) The standard DRM gets trapped in the disordered phase. (b) Our method successfully nucleates the hexagonal phase $\phi_{\text{pred}}$. (c) The high-precision reference solution $\phi_{\text{ref}}$. Panels (a)--(c) share the common colorbar located to the right of panel (c). (d) The pointwise difference map $\phi_{\text{pred}} - \phi_{\text{ref}}$, showing the spatial distribution of the signed error with a magnitude of $\mathcal{O}(10^{-4})$, corresponding to the colorbar on the far right.
  • Figure 4: Illustration of two scheduler functions. (a) Hard cutoff. (b) Linear decay. The blue regions indicate when the spectral forcing is active.
  • Figure 5: Transition pathways between the LAM and HEX phases for (a) $(\tau, \gamma) = (-0.4, 0.3)$, (b) $(-0.4, 0.35)$, (c) $(-0.4, 0.4)$, and (d) $(-0.4, 0.45)$. In each plot, the vertical axis denotes the free-energy density $f$, and the pathway connects the local minimum (left), the transition state (TS, middle), and the global minimum (right). Note that in (a) and (b), the LAM phase is the global minimum, whereas in (c) and (d), the HEX phase becomes the global minimum. The white circles highlight the critical nucleus region within the transition states.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 3.1: Local instability of the disordered phase
  • proof
  • Remark 3.1
  • Remark 3.2