Reparameterizing 4DVAR with neural fields

Four-dimensional variational data assimilation (4DVAR) is a cornerstone of numerical weather prediction, but its cost function is difficult to optimize and computationally intensive. We propose a neural field-based reformulation in which the full spatiotemporal state is represented as a continuous function parameterized by a neural network. This reparameterization removes the time-sequential dependency of classical 4DVAR, enabling parallel-in-time optimization in parameter space. Physical constraints are incorporated directly through a physics-informed loss, simplifying implementation and reducing computational cost. We evaluate the method on the two-dimensional incompressible Navier--Stokes equations with Kolmogorov forcing. Compared to a baseline 4DVAR implementation, the neural reparameterized variants produce more stable initial condition estimates without spurious oscillations. Notably, unlike most machine learning-based approaches, our framework does not require access to ground-truth states or reanalysis data, broadening its applicability to settings with limited reference information.