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

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.

• The paper introduces GeoDVF, a Fourier-based neural variational method that jointly optimizes the order parameter and domain dimensions, reducing artificial stress caused by mismatched periodic cells.
• A warmup spectral penalty destabilizes the disordered state and enables reliable HEX discovery from random initializations, achieving post-refinement errors near 10⁻⁹ and demonstrating strong performance across several 3D phases, including A15.
• Discovery remains barrier- and basin-dependent: DG and σ require guided initialization, while orthogonal-grid restrictions and limited spontaneous exploration constrain broader application to complex lattice geometries.

Motivation and problem setting

The Landau–Brazovskii (LB) model is a canonical continuum description of ordering at a finite characteristic wavenumber, with applications ranging from block-copolymer microphase separation to nuclear pasta. Its free-energy functional,

$$\mathcal{E}(\phi)=\int\left(\tfrac{1}{2}|(\Delta+1)\phi|^2+\tfrac{\tau}{2}\phi^2-\tfrac{\gamma}{6}\phi^3+\tfrac{1}{24}\phi^4\right)\mathrm{d}\bm{x},$$

subject to mass conservation $\int \phi\,\mathrm{d}\bm{x}=0$, admits a rich set of local minima: lamellar (LAM), hexagonal cylinders (HEX), BCC, FCC, double gyroid (DG), and Frank–Kasper phases such as A15 and $\sigma$. Numerically, the standard approach—the Crystalline Approximant Method (CAM)—restricts the system to a periodic box whose side lengths must match the natural periodicity of the target phase; mismatched domains impose artificial stress that traps the solver in high-energy metastable configurations.

The paper's central observation is that the domain geometry cannot be treated as a fixed background. The authors demonstrate this with a controlled sensitivity study on the 2D LB model using an alternating iterative scheme (minimize over $\phi$ at fixed $\Omega$, then update $\Omega$ via an analytically derived gradient). Initialized with the correct HEX topology but a square domain $[0,20]^2$, the solver converges to a distorted hexagonal phase (HEX*) rather than the stress-free ground state; from random noise on the same domain it collapses to the disordered phase entirely. This establishes that both the field initialization and the domain size must be prescribed accurately for traditional solvers to succeed—a strong dependence on prior knowledge that motivates the proposed method.

The GeoDVF framework

The Geometry-Adaptive Deep Variational Framework (GeoDVF) reformulates the problem as joint minimization of the rescaled free-energy density $f(\phi,\bm{\omega})$ over both the order parameter and the geometric parameters $\bm{\omega}=(L_1,\dots,L_d)$, which enter through the geometry-adapted Laplacian $\Delta_{\bm{\omega}}=\sum_i L_i^{-2}\partial_i^2$. Two design choices are notable:

• Reparameterization: the inverse lengths $\int \phi\,\mathrm{d}\bm{x}=0$0 serve as trainable variables, mapping domain sizes into $\int \phi\,\mathrm{d}\bm{x}=0$1 and converting rational geometric dependencies into polynomial terms, which stabilizes backpropagation.
• Architecture: the order parameter is represented by an FNO-inspired network—coordinate embedding, stacked Fourier layers with spectral truncation at $\int \phi\,\mathrm{d}\bm{x}=0$2, and a pointwise projection. The Fourier basis enforces periodicity by construction and yields mesh-invariant, resolution-independent representations.
• Hard constraint: mass conservation is imposed exactly via a zero-mean projection at the network output, avoiding penalty hyperparameters associated with soft constraints.

Derivatives are computed pseudospectrally in the frequency domain rather than by automatic differentiation, giving high accuracy for the fourth-order operator in the elastic term.

Warmup penalty and its analysis

A key obstacle for any Deep Ritz-style method on this problem is that small random network weights produce $\int \phi\,\mathrm{d}\bm{x}=0$3, i.e., the disordered phase—which is itself a local minimum for $\int \phi\,\mathrm{d}\bm{x}=0$4 and acts as a strong attractor. The authors introduce a warmup penalty that penalizes deviation of the order parameter's projection onto the critical spectral sphere from a target amplitude $\int \phi\,\mathrm{d}\bm{x}=0$5:

$\int \phi\,\mathrm{d}\bm{x}=0$6

active only during an initial window ($\int \phi\,\mathrm{d}\bm{x}=0$7 steps) before being annealed to zero.

The paper provides a formal justification: Theorem 3.1 shows that although $\int \phi\,\mathrm{d}\bm{x}=0$8 is a local minimum of the physical energy, it is strictly unstable under the penalized loss. The mechanism rests on a scaling asymmetry—the physical energy grows quadratically near zero while the penalty contributes a negative linear directional derivative $\int \phi\,\mathrm{d}\bm{x}=0$9 along any mass-conserving perturbation with nonzero projection intensity $\sigma$0. A remark further notes that the absolute value in $\sigma$1 creates a gradient singularity at the origin, yielding a nonvanishing constant driving force even for infinitesimal perturbations, which accelerates escape relative to classical linear instability where the force vanishes as $\sigma$2. This is a clean theoretical result, though it addresses local instability only; it does not characterize which ordered basin the trajectory enters.

A second strategy, guided initialization, adds a time-limited data-fidelity term $\sigma$3 to steer optimization toward phases with narrow basins of attraction, after which purely physical relaxation refines the solution.

Numerical results

Escaping the disordered phase. On the 2D benchmark $\sigma$4, the standard DRM fails in all runs (converging to disorder), whereas GeoDVF with the warmup penalty consistently nucleates the global HEX minimum from random initializations. After refinement by the traditional solver, the state approximation error reaches $\sigma$5 and the domain error $\sigma$6, with pointwise errors of order $\sigma$7. The raw network output attains an E-L residual of $\sigma$8, positioning GeoDVF as a structural predictor to be polished by high-precision solvers rather than a replacement for them—an honest framing of precision limitations.

Ablation. Across 100 random runs, the scheduler design matters substantially: at peak weight $\sigma$9, a hard cutoff achieves only 18% success versus 71% for linear decay, because abrupt removal of strong forcing ejects trajectories from the HEX basin. This identifies the annealing schedule as a critical hyperparameter rather than an incidental detail.

Global-minimum selection. Sweeping across the LAM–HEX phase boundary, the authors correlate discovery rates with nucleation barriers computed via string-method transition pathways. As $\phi$0 increases from 0.3 to 0.45, the LAM→HEX barrier drops from $\phi$1 to $\phi$2, and the HEX discovery rate rises from 1% to roughly 72%, while LAM discovery falls from 97% to 20%. The interpretation—that spectral forcing supplies fluctuations sufficient to form small critical nuclei—is physically coherent but implies a genuine limitation: when barriers are large, the method preferentially finds whichever phase nucleates easily, not necessarily the global minimum.

3D phases. From random initializations alone, GeoDVF discovers LAM, HEX, BCC, FCC, HCP, and—remarkably—the A15 phase (eight spheres per unit cell), with relative energy differences against reference solutions between $\phi$3 and $\phi$4. The DG and $\phi$5 phases could not be nucleated spontaneously and required guided initialization; the authors correctly attribute this to the intrinsic narrowness of their basins, noting that traditional solvers fail identically from noise. Once guided, the network refines low-quality initial guesses to $\phi$6 and $\phi$7, demonstrating that the failure is one of optimization dynamics rather than architectural expressivity. Energetic hierarchies among BCC/FCC/HCP across varying $\phi$8 match known phase-field-crystal phase diagrams, confirming correct resolution of subtle stability competition.

Limitations and open questions

The paper is explicit about several constraints. Spontaneous discovery of topologically intricate phases (DG, $\phi$9) remains out of reach without guidance; whether a modified exploration strategy can access these narrow basins autonomously is left open. The reliance on discrete Fourier transforms restricts the framework to orthogonal box domains on uniform grids, so non-orthogonal lattice symmetries cannot be fully relaxed—and the orthogonal constraint itself introduces spurious local minima such as HEX*. Finally, the demonstrated success rate for the global minimum degrades precisely when nucleation barriers are large, so the method's selectivity is barrier-dependent rather than universal.

Conclusion

GeoDVF contributes a principled coupling of neural variational optimization with trainable domain geometry, eliminating the artificial-stress pathology of fixed-domain CAM solvers. The warmup penalty is supported by a rigorous local-instability result and empirically enables spontaneous nucleation of complex 3D phases—including A15—from pure noise, with post-refinement accuracies approaching machine precision. The framework's applicability to self-consistent field theory, phase-field crystal models, and Landau–de Gennes systems is identified as the natural extension, contingent on resolving the orthogonality restriction and the narrow-basin discovery problem.