Abstract
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 emerging research field. However, current
methods are restricted in various ways: they require prior knowledge about the
governing equations, and are limited to linear or first-order equations. In
this work we propose NeuralPDE, a model which combines convolutional neural
networks (CNNs) with differentiable ODE solvers to model dynamical systems. We
show that the Method of Lines used in standard PDE solvers can be represented
using convolutions which makes CNNs the natural choice to parametrize arbitrary
PDE dynamics. Our model can be applied to any data without requiring any prior
knowledge about the governing PDE. We evaluate NeuralPDE on datasets generated
by solving a wide variety of PDEs, covering higher orders, non-linear equations
and multiple spatial dimensions.
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