Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, Fokker-Planck equations, which model the diffusion of probability measures, can be understood as gradient descent over entropy functionals in Wasserstein space. This equivalence, introduced by Jordan, Kinderlehrer and Otto, inspired the so-called JKO scheme to approximate these diffusion processes via an implicit discretization of the gradient flow in Wasserstein space. Solving the optimization problem associated to each JKO step, however, presents serious computational challenges. We introduce a scalable method to approximate Wasserstein gradient flows, targeted to machine learning applications. Our approach relies on input-convex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent. Unlike previous work, our method does not require domain discretization or particle simulation. As a result, we can sample from the measure at each time step of the diffusion and compute its probability density. We demonstrate our algorithm's performance by computing diffusions following the Fokker-Planck equation and apply it to unnormalized density sampling as well as nonlinear filtering.

翻译：Vasserstein 梯度流提供了一种强大的理解和解决许多扩散方程式的手段。 具体地说, Fokker- Planck 方程式是概率测量的传播量的模型,可以被理解为瓦塞尔斯坦空间的星系功能的梯度下降。 由约旦、 Kinderhead 和 Otto 引入的这一等值激励了所谓的JKO 方案,通过瓦塞尔斯坦空间的梯度流的隐性离散来接近这些扩散过程。 然而,解决与JKO 步骤相关的优化问题,则带来了严重的计算挑战。 我们引入了一种可缩放方法,以机器学习应用为对象,接近瓦塞尔斯坦梯度流。 我们的方法依靠输入- convex 神经网络( ICNNS) 来将JKO 步骤分解, 可以通过随机梯度下降优化。 与以往的工作不同, 我们的方法不需要区域离散或粒子模拟。 结果, 我们可以在扩散的每个阶段从测量中取样, 并计算其概率密度密度。 我们通过按照 Fokker- Planckcock 等方程式计算, 将我们的算算算法的性表现为非正常密度取样。