A Geometry-Adaptive Deep Variational Framework for Phase Discovery in the Landau-Brazovskii Model
Authors: Yuchen Xie, Jianyuan Yin, Lei Zhang
Organizations: School of Mathematical Sciences, Peking University, Beijing, 100871, China · School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, 100875, China · School of Mathematical Sciences, Beijing International Center for Mathematical Research, Center for Quantitative Biology, Center for Machine Learning Research, Peking University, Beijing, 100871, China
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.
Figures & tables
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.
(τ,γ)
Initial guess (ϕ0,ω0)
(ϕHEX0,ωopt)
(ϕHEX0,ωs)
(ϕLAM0,ωopt)
(ϕLAM0,ωs)
(ϕrandn0,ωopt)
(ϕrandn0,ωs)
(0.001, 0.6)
HEX
HEX*
disorder
disorder
HEX
disorder
(-0.4, 0.3)
HEX
LAM
LAM
LAM
LAM
LAM
Table 1: Convergence outcomes for the alternating iterative solver under different parameter sets (τ,γ) and initial guesses.
Figure 2: An overview of GeoDVF. (a) The workflow of GeoDVF. The system takes spatial coordinates x as input and simultaneously optimizes the network parameters θ (for the order parameter field ϕθ ) and the inverse geometric parameters β (for the domain geometry). The zero-mean projection enforces the mass conservation constraint, converting the raw network output Nθ(x) into the physical order parameter ϕθ . The total loss L(Θ) 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 Θ=(θ,β) is updated jointly via backpropagation. (b) Detailed architecture of the neural network Nθ . The coordinate embedding layer P maps the input coordinates x directly into a high-dimensional feature function. This representation evolves through a stack of Fourier layers Hl and is mapped back to the unconstrained scalar field Nθ 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 ϕpred . (c) The high-precision reference solution ϕref . Panels (a)–(c) share the common colorbar located to the right of panel (c). (d) The pointwise difference map ϕpred−ϕref , showing the spatial distribution of the signed error with a magnitude of 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.
Schedule
wmax
Convergence rate
Error metrics
HEX
HEX *
disorder
Ediff
Eres
Eϕ
Edom
Baseline
0
0%
0%
100%
-
-
-
-
Hard cutoff
0.0001
0%
3%
97%
-
-
-
-
0.001
56%
44%
0%
3.7×10−6
5.1×10−4
4.1×10−8
2.4×10−11
0.01
70%
30%
0%
1.8×10−6
1.3×10−3
6.6×10−10
3.7×10−11
0.1
18%
0%
82%
9.5×10−6
7.4×10−2
6.1×10−8
2.8×10−10
Table 2: Comparison of phase discovery success rates and error metrics under different penalty schedules and peak magnitudes.
Figure 5: Transition pathways between the LAM and HEX phases for (a) (τ,γ)=(−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.
Property
(τ,γ)
(−0.4,0.3)
(−0.4,0.35)
(−0.4,0.4)
(−0.4,0.45)
Energy landscape
Global minimum
LAM
LAM
HEX
HEX
Energy barrier Δf (LAM → HEX)
1.6×10−2
7.7×10−3
2.4×10−3
4.9×10−4
HEX critical nucleus
large
\makebox[85.35826pt]
small
Energy barrier Δf (HEX → LAM)
1.0×10−3
2.8×10−3
7.9×10−3
1.7×10−2
Table 3: Quantitative analysis of phase selection. A lower energy barrier for nucleation associates with a higher success rate of locating the global minimum.
Figure 6: 3D ordered structures identified by our GeoDVF. The gallery displays a diverse set of local minima in the 3D LB model: (top row) LAM, HEX, BCC, FCC; (bottom row) A15, DG, σ , and HCP. The red volumetric regions correspond to where the order parameter is high. The LAM, HEX, BCC, FCC, A15 and HCP phases can be successfully nucleated from random initializations using strategy I.
Phase
(τ,γ)
Ediff
Eres
Eϕ
Edom
Strategy I: warmup penalty
LAM
(−0.4,0.22)
1.8×10−7
2.1×10−1
6.9×10−11
5.7×10−12
HEX
(−0.3,0.8)
3.8×10−7
3.2×10−1
2.4×10−10
1.0×10−11
BCC
(−0.008,0.8)
6.9×10−6
2.6×10−2
6.7×10−10
8.7×10−10
FCC
(0.1,1.6)
1.3×10−6
1.5×10−2
7.2×10−10
4.6×10−11
A15
(−0.008,1.2)
3.2×10−6
6.7×10−3
7.4×10−10
2.1×10−11
Table 4: Quantitative evaluation of the discovered 3D phases.
(τ,γ)
Energy density f
Global minimum
BCC
FCC
HCP
(−0.008,0.8)
−0.02081
−0.01948
−0.02001
BCC
(−0.008,1.0)
−0.05159
−0.05149
−0.05169
σ
(−0.008,1.2)
−0.1104
−0.1126
−0.1125
A15
Table 5: Comparison of free-energy densities f for the competing BCC, FCC, and HCP phases.
Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.
School of Mathematical Sciences, Peking University, Beijing 100871, China · Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA · Center for Applied Mathematics, Renmin University of China, Beijing 100872, China +1