  We introduce a new strategy designed to help physicists discover hidden laws
governing dynamical systems. We propose to use machine learning automatic
differentiation libraries to develop hybrid numerical models that combine
components based on prior physical knowledge with components based on neural
networks. In these architectures, named Deep Neural Numerical Models (DNNMs),
the neural network components are used as building-blocks then deployed for
learning hidden variables of underlying physical laws governing dynamical
systems. In this paper, we illustrate an application of DNNMs to upper ocean
dynamics, more precisely the dynamics of a sea surface tracer, the Sea Surface
Height (SSH). We develop an advection-based fully differentiable numerical
scheme, where parts of the computations can be replaced with learnable
ConvNets, and make connections with the single-layer Quasi-Geostrophic (QG)
model, a baseline theory in physical oceanography developed decades ago.
