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

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.

Learning task-specific predictive models for scientific computing

We consider learning a predictive model to be subsequently used for a given downstream task (described by an algorithm) that requires access to the model evaluation. This task need not be prediction, and this situation is frequently encountered in machine-learning-augmented scientific computing. We show that this setting differs from classical supervised learning, and in general it cannot be solved by minimizing the mean square error of the model predictions as is frequently performed in the literature. Instead, we find that the maximum prediction error on the support of the downstream task algorithm can serve as an effective estimate for the subsequent task performance. With this insight, we formulate a task-specific supervised learning problem based on the given sampling measure, whose solution serves as a reliable surrogate model for the downstream task. Then, we discretize the empirical risk based on training data, and develop an iterative algorithm to solve the task-specific supervised learning problem. Three illustrative numerical examples on trajectory prediction, optimal control and minimum energy path computation demonstrate the effectiveness of the approach.