  Many physical processes such as weather phenomena or fluid mechanics are
governed by partial differential equations (PDEs). Modelling such dynamical
systems using Neural Networks is an active research field. However, current
methods are still very limited, as they do not exploit the knowledge about the
dynamical nature of the system, require extensive prior knowledge about the
governing equations or are limited to linear or first-order equations. In this
work we make the observation that the Method of Lines used to solve PDEs can be
represented using convolutions which makes convolutional neural networks (CNNs)
the natural choice to parametrize arbitrary PDE dynamics. We combine this
parametrization with differentiable ODE solvers to form the NeuralPDE Model,
which explicitly takes into account the fact that the data is governed by
differential equations. We show in several experiments on toy and real-world
data that our model consistently outperforms state-of-the-art models used to
learn dynamical systems.
