  Neural networks are a promising technique for parameterizing sub-grid-scale
physics (e.g. moist atmospheric convection) in coarse-resolution climate
models, but their lack of interpretability and reliability prevents widespread
adoption. For instance, it is not fully understood why neural network
parameterizations often cause dramatic instability when coupled to atmospheric
fluid dynamics. This paper introduces tools for interpreting their behavior
that are customized to the parameterization task. First, we assess the
nonlinear sensitivity of a neural network to lower-tropospheric stability and
the mid-tropospheric moisture, two widely-studied controls of moist convection.
Second, we couple the linearized response functions of these neural networks to
simplified gravity-wave dynamics, and analytically diagnose the corresponding
phase speeds, growth rates, wavelengths, and spatial structures. To demonstrate
their versatility, these techniques are tested on two sets of neural networks,
one trained with a super-parametrized version of the Community Atmosphere Model
(SPCAM) and the second with a near-global cloud-resolving model (GCRM). Even
though the SPCAM simulation has a warmer climate than the cloud-resolving
model, both neural networks predict stronger heating/drying in moist and
unstable environments, which is consistent with observations. Moreover, the
spectral analysis can predict that instability occurs when GCMs are coupled to
networks that support gravity waves that are unstable and have phase speeds
larger than 5 m/s. In contrast, standing unstable modes do not cause
catastrophic instability. Using these tools, differences between the SPCAM- vs.
GCRM- trained neural networks are analyzed, and strategies to incrementally
improve both of their coupled online performance unveiled.
