CARE-SAV：用于梯度流能量稳定模拟的条件感知随机特征框架
CARE-SAV: A Conditioning-Aware Random-Feature Framework for Energy-Stable Simulation of Gradient Flows
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中文总结 AI 辅助
该研究提出CARE-SAV框架，用于梯度流的能量稳定模拟，可保留离散能量耗散结构，兼具准确性、鲁棒性与计算效率，为梯度流保结构离散化提供新范式。
中文摘要 AI 辅助
梯度流模型具有内在的能量耗散结构，在离散层面忠实地保留该结构对稳定可靠的长时间模拟至关重要。为此，我们开发了条件感知标量辅助变量表示增强（CARE-SAV）框架，该框架从灵活的候选特征构建紧凑的空间近似空间，并在此空间内直接演化梯度流动力学。所得的全离散格式保留了离散能量耗散定律，同时为传统的规定空间离散化提供了灵活的替代方案。严格分析确立了所提方法的近似能力、可解性、稳定性和收敛性。对代表性梯度流问题的数值实验证明了其准确性、鲁棒性和计算效率。我们认为，CARE-SAV可为梯度流问题的保结构离散化提供一种简单、灵活且计算高效的范式。
英文摘要
Gradient-flow models are characterized by an intrinsic energy-dissipation structure, and faithfully preserving this structure at the discrete level is important for stable and reliable long-time simulation. To this end, we develop a Conditioning-Aware Representation Enhancement with Scalar Auxiliary Variable (CARE-SAV) framework, which constructs a compact spatial approximation space from flexible candidate features and evolves the gradient-flow dynamics directly within this space. The resulting fully discrete scheme preserves the discrete energy-dissipation law while providing a flexible alternative to conventional prescribed spatial discretizations. Rigorous analysis establishes the approximation capability, solvability, stability and convergence of the proposed method. Numerical experiments on representative gradient-flow problems demonstrate its accuracy, robustness and computational efficiency. We believe that CARE-SAV could provide a simple, flexible, and computationally efficient paradigm for structure-preserving discretization of gradient-flow problems.
