Bayesian inference for geophysical fluid dynamics using generative models

Abstract

Data assimilation plays a crucial role in numerical modeling, enabling the integration of real-world observations into mathematical models to enhance the accuracy and predictive capabilities of simulations. This approach is widely applied in fields such as meteorology, oceanography, and environmental science, where the dynamic nature of systems demands continuous updates to model states. However, the calibration of models in these high-dimensional, nonlinear systems poses significant challenges. In this paper, we explore a novel calibration methodology using diffusion generative models. We generate synthetic data that statistically aligns with a given set of observations (in this case the increments of the numerical approximation of a solution of a partial differential equation). This allows us to efficiently implement a model reduction and assimilate data from a reference system state modeled by a highly resolved numerical solution of the rotating shallow water equation of order 104 degrees of freedom into a stochastic system having two orders of magnitude less degrees of freedom. To do so, the new samples are incorporated into a particle filtering methodology augmented with tempering and jittering for dynamic state estimation, a method particularly suited for handling complex and multimodal distributions. This work demonstrates how generative models can be used to improve the predictive accuracy for particle filters, providing a more computationally efficient solution for data assimilation and model calibration.

Figures (14)

• Figure 1: Filtering and Calibration. Data assimilation and model calibration. Top: The predictive distribution is advanced using a forecast model and then updated via a nonlinear operation based on observed data. Bottom: Before time $0$ the forecast model is calibrated. Filtering begins strictly after the calibration process is finshed.
• Figure 2: Classical Particle Filter. The resampling procedure ensures that particles with low weights are replaced with particles with higher weights. Following the resampling, an ensemble of equal-weighted particles is obtained. In high-dimensional spaces: one particle gains a weight close to one while all the others have weights close to zero and are discarded.
• Figure 3: Initial Condition. Initial condition for the shallow water simulations.
• Figure 4: Shallow Water Snapshots. Plots of the full and filtered (rows) non-dimensional shallow water variables (columns). The variables (a+d) height $\eta$, (b+e) zonal velocity $U$, and (c+f) meridional velocity $V$ are plotted on the fine grid of resolution $128\times 128$ grid points. The top row (a-c) shows the fully resolved fields and the bottom row (d-f) shows the coarsened fields after applying the low-pass filter with a maximum frequency of $16$ gridpoints. The solution is obtained after running $4000$ timesteps up to time $t=0.4$ started from the prescribed initial condition.
• Figure 5: Distribution of the Noise Data obtained from (a) the deterministic shallow water model and produced by (b) the trained generative model.