Nesterov's Accelerated Gradient Flow Method for Block Copolymer Systems

ABSTRACT

Minimizers of phase‐field models are commonly computed by evolving gradient‐flow dynamics, such as , , or Wasserstein gradient flows, toward equilibrium. However, phase‐field energies often possess complicated energy landscapes, and the corresponding relaxation dynamics may involve long‐time, multi‐stage evolution. In many applications, the main objective is to identify the self‐assembled equilibrium morphology, rather than to resolve the entire dynamical pathway leading to it. In such cases, direct simulation of the underlying gradient flow can be unnecessarily expensive. Motivated by this observation, we introduce a Nesterov accelerated gradient flow framework for phase‐field models of block copolymer systems and develop corresponding Nesterov accelerated gradient descent schemes. We prove that a modified energy dissipation law, held at the continuous level, is inherited by the proposed discrete schemes. We further compare the accelerated scheme with a BDF2 discretization of the gradient flow. Numerical experiments show that the proposed Nesterov accelerated gradient descent scheme reduces the computational cost by a factor of approximately six, while retaining accuracy comparable to classical gradient‐flow solvers.