High-Order Variable-Scaled Energetic Variational Neural Networks for Phase-Field Gradient Flows

This paper presents a high-order variable-scaled energetic variational neural network (VS-EVNN) method specifically designed for phase-field gradient flows. The approach centers and normalizes spatial inputs by the interface scale, while maintaining the original energy and movement penalty formulations. This rescaling strategy is derived from a coordinate transformation in the energy functional, enabling improved discretization accuracy across different interface scales.