非梯度系统的熵正则化
Entropic regularisation of non-gradient systems
摘要：
概率测度空间中的Wasserstein梯度流理论在过去的20年里取得了巨大的进展。它构成了一个统一的和强大的框架，在耗散偏微分方程（PDE）的研究提供了手段，以证明适定性，正则性，稳定性和定量收敛到平衡。最近发展的熵正则化技术为求解这些梯度流的快速有效的数值方法铺平了道路。然而，许多感兴趣的偏微分方程不具有梯度流结构，并且先验地，该理论不适用。在本文中，我们开发了一个时间离散熵正则变分方案的一般类这样的非梯度偏微分方程。我们证明了该方案的收敛性，并说明了所提出的框架的广度与具体的例子，包括非线性动力学福克-普朗克（Kramers）方程和非线性退化扩散的Kolmogorov型。还提供了数值模拟。
英文摘要：
The theory of Wasserstein gradient flows in the space of probability measures has made an enormous progress over the last twenty years. It constitutes a unified and powerful framework in the study of dissipative partial differential equations (PDEs) providing the means to prove well-posedness, regularity, stability and quantitative convergence to the equilibrium. The recently developed entropic regularisation technique paves the way for fast and efficient numerical methods for solving these gradient flows. However, many PDEs of interest do not have a gradient flow structure and, a priori, the theory is not applicable. In this paper, we develop a time-discrete entropy regularised variational scheme for a general class of such non-gradient PDEs. We prove the convergence of the scheme and illustrate the breadth of the proposed framework with concrete examples including the non-linear kinetic Fokker-Planck (Kramers) equation and a non-linear degenerate diffusion of Kolmogorov type. Numerical simulations are also provided.
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