  In typical machine learning tasks and applications, it is necessary to obtain
or create large labeled datasets in order to to achieve high performance.
Unfortunately, large labeled datasets are not always available and can be
expensive to source, creating a bottleneck towards more widely applicable
machine learning. The paradigm of weak supervision offers an alternative that
allows for integration of domain-specific knowledge by enforcing constraints
that a correct solution to the learning problem will obey over the output
space. In this work, we explore the application of this paradigm to 2-D
physical systems governed by non-linear differential equations. We demonstrate
that knowledge of the partial differential equations governing a system can be
encoded into the loss function of a neural network via an appropriately chosen
convolutional kernel. We demonstrate this by showing that the steady-state
solution to the 2-D heat equation can be learned directly from initial
conditions by a convolutional neural network, in the absence of labeled
training data. We also extend recent work in the progressive growing of fully
convolutional networks to achieve high accuracy (< 1.5% error) at multiple
scales of the heat-flow problem, including at the very large scale (1024x1024).
Finally, we demonstrate that this method can be used to speed up exact
calculation of the solution to the differential equations via finite
difference.
