  Accurate representations of unknown and sub-grid physical processes through
parameterizations (or closure) in numerical simulations with quantified
uncertainty are critical for resolving the coarse-grained partial differential
equations that govern many problems ranging from weather and climate prediction
to turbulence simulations. Recent advances have seen machine learning (ML)
increasingly applied to model these subgrid processes, resulting in the
development of hybrid physics-ML models through the integration with numerical
solvers. In this work, we introduce a novel framework for the joint estimation
of physical parameters and machine learning parameterizations with uncertainty
quantification. Our framework incorporates online training and efficient
Bayesian inference within a high-dimensional parameter space, facilitated by
differentiable programming. This proof of concept underscores the substantial
potential of differentiable programming in synergistically combining machine
learning with differential equations, thereby enhancing the capabilities of
hybrid physics-ML modeling.
