A competitive baseline for deep learning enhanced data assimilation using conditional Gaussian ensemble Kalman filtering

Ensemble Kalman Filtering (EnKF) is a popular technique for data assimilation, with far ranging applications. However, the vanilla EnKF framework is not well-defined when perturbations are nonlinear. We study two non-linear extensions of the vanilla EnKF - dubbed the conditional-Gaussian EnKF (CG-EnKF) and the normal score EnKF (NS-EnKF) - which sidestep assumptions of linearity by constructing the Kalman gain matrix with the `conditional Gaussian' update formula in place of the traditional one. We then compare these models against a state-of-the-art deep learning based particle filter called the score filter (SF). This model uses an expensive score diffusion model for estimating densities and also requires a strong assumption on the perturbation operator for validity. In our comparison, we find that CG-EnKF and NS-EnKF dramatically outperform SF for a canonical problem in high-dimensional multiscale data assimilation given by the Lorenz-96 system. Our analysis also demonstrates that the CG-EnKF and NS-EnKF can handle highly non-Gaussian additive noise perturbations, with the latter typically outperforming the former.

Generative Modeling by Minimizing the Wasserstein-2 Loss

This paper approaches the unsupervised learning problem by minimizing the second-order Wasserstein loss (the $W_2$ loss). The minimization is characterized by a distribution-dependent ordinary differential equation (ODE), whose dynamics involves the Kantorovich potential between a current estimated distribution and the true data distribution. A main result shows that the time-marginal law of the ODE converges exponentially to the true data distribution. To prove that the ODE has a unique solution, we first construct explicitly a solution to the associated nonlinear Fokker-Planck equation and show that it coincides with the unique gradient flow for the $W_2$ loss. Based on this, a unique solution to the ODE is built from Trevisan's superposition principle and the exponential convergence results. An Euler scheme is proposed for the distribution-dependent ODE and it is shown to correctly recover the gradient flow for the $W_2$ loss in the limit. An algorithm is designed by following the scheme and applying persistent training, which is natural in our gradient-flow framework. In both low- and high-dimensional experiments, our algorithm converges much faster than and outperforms Wasserstein generative adversarial networks, by increasing the level of persistent training appropriately.

A Differential Equation Approach for Wasserstein GANs and Beyond

We propose a new theoretical lens to view Wasserstein generative adversarial networks (WGANs). In our framework, we define a discretization inspired by a distribution-dependent ordinary differential equation (ODE). We show that such a discretization is convergent and propose a viable class of adversarial training methods to implement this discretization, which we call W1 Forward Euler (W1-FE). In particular, the ODE framework allows us to implement persistent training, a novel training technique that cannot be applied to typical WGAN algorithms without the ODE interpretation. Remarkably, when we do not implement persistent training, we prove that our algorithms simplify to existing WGAN algorithms; when we increase the level of persistent training appropriately, our algorithms outperform existing WGAN algorithms in both low- and high-dimensional examples.