论文标题
数据驱动梯度流
Data driven gradient flows
论文作者
论文摘要
我们提出了一个框架，该框架基于最小化运动（或Jordan-Kinderlehrer-otto）近似方案，使一般度量空间中梯度流的变异数据同化。在讨论了最一般情况下的稳定性之后，我们专门针对Wasserstein距离的概率措施空间。该设置涵盖了许多非线性偏微分方程（PDE），例如多孔培养基方程或一般漂移 - 扩散 - 聚集方程，可以通过我们的方法对它们的各自特性（例如传播的有限速度或爆炸）进行处理。然后，我们专注于使用原始偶算法的方法实现我们的方法。我们方法的优势在于一个事实，即通过简单地更改驱动功能，可以在无需采用数值方案的情况下处理广泛的PDE。我们通过介绍详细的数值示例来结束。
We present a framework enabling variational data assimilation for gradient flows in general metric spaces, based on the minimizing movement (or Jordan-Kinderlehrer-Otto) approximation scheme. After discussing stability properties in the most general case, we specialise to the space of probability measures endowed with the Wasserstein distance. This setting covers many non-linear partial differential equations (PDEs), such as the porous medium equation or general drift-diffusion-aggregation equations, which can be treated by our methods independent of their respective properties (such as finite speed of propagation or blow-up). We then focus on the numerical implementation of our approach using an primal-dual algorithm. The strength of our approach lies in the fact that by simply changing the driving functional, a wide range of PDEs can be treated without the need to adopt the numerical scheme. We conclude by presenting detailed numerical examples.
