Neural ensemble Kalman filter: Data assimilation for compressible flows with shocks
This work provides a more robust data assimilation method for researchers and engineers working with compressible flows and shock phenomena, particularly in fields like aerospace or fluid dynamics, by mitigating common numerical instabilities.
This paper addresses the challenge of data assimilation for compressible flows with shocks, where standard ensemble Kalman filters (EnKF) often produce spurious oscillations due to bimodal forecast distributions. The authors introduce the neural EnKF, which maps the forecast ensemble of shocked flows to the parameter space of a deep neural network, allowing for data assimilation in that space. This method, combined with physics-informed transfer learning, successfully avoids the spurious oscillations and nonphysical features seen in standard EnKF.
Data assimilation (DA) for compressible flows with shocks is challenging because many classical DA methods generate spurious oscillations and nonphysical features near uncertain shocks. We focus here on the ensemble Kalman filter (EnKF). We show that the poor performance of the standard EnKF may be attributed to the bimodal forecast distribution that can arise in the vicinity of an uncertain shock location; this violates the assumptions underpinning the EnKF, which assume a forecast which is close to Gaussian. To address this issue we introduce the new neural EnKF. The basic idea is to systematically embed neural function approximations within ensemble DA by mapping the forecast ensemble of shocked flows to the parameter space (weights and biases) of a deep neural network (NN) and to subsequently perform DA in that space. The nonlinear mapping encodes sharp and smooth flow features in an ensemble of NN parameters. Neural EnKF updates are therefore well-behaved only if the NN parameters vary smoothly within the neural representation of the forecast ensemble. We show that such a smooth variation of network parameters can be enforced via physics-informed transfer learning, and demonstrate that in so-doing the neural EnKF avoids the spurious oscillations and nonphysical features that plague the standard EnKF. The applicability of the neural EnKF is demonstrated through a series of systematic numerical experiments with an inviscid Burgers' equation, Sod's shock tube, and a two-dimensional blast wave.