Normalizing Flow Filter: Observation-Only Learning for Non-Gaussian Data Assimilation

State filtering is central to data assimilation and model-based probabilistic forecasting in high-dimensional dynamical systems. Many scientific applications require filtering methods that can simultaneously: (i) represent non-Gaussian posteriors, (ii) scale to high dimensions, (iii) train from observations alone, and (iv) retain explicit transition and observation models for calibrating physical parameters and sensor biases. To directly address this setting, we introduce the Normalizing Flow Filter (NFF), an amortized filter with $\tau$-step memory that represents the adjacent-state posterior using two conditional normalizing flows. The training objective combines a recursive filtering evidence lower bound with the continuous ranked probability score on short-horizon future observations. For a fixed, correctly specified state-space model including the initial-state prior, exact $\tau$-step memory, and sufficient flow expressivity, every global minimizer of the NFF loss averaged over data-generating observation sequences recovers the exact filtering posterior. NFF closely matches a $10^6$-particle-filter reference for a four-dimensional coupled oscillator, remains reliable under highly ambiguous nonlinear observations in 40-dimensional Lorenz–96, and scales to joint state, physical-parameter, and sensor-bias inference in 16,384-dimensional Kolmogorov flow.