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In typical machine learning tasks and applications, it is necessary to obtainor create large labeled datasets in order to to achieve high performance.Unfortunately, large labeled datasets are not always available and can beexpensive to source, creating a bottleneck towards more widely applicablemachine learning. The paradigm of weak supervision offers an alternative thatallows for integration of domain-specific knowledge by enforcing constraintsthat a correct solution to the learning problem will obey over the outputspace. In this work, we explore the application of this paradigm to 2-Dphysical systems governed by non-linear differential equations. We demonstratethat knowledge of the partial differential equations governing a system can beencoded into the loss function of a neural network via an appropriately chosenconvolutional kernel. We demonstrate this by showing that the steady-statesolution to the 2-D heat equation can be learned directly from initialconditions by a convolutional neural network, in the absence of labeledtraining data. We also extend recent work in the progressive growing of fullyconvolutional networks to achieve high accuracy (< 1.5% error) at multiplescales of the heat-flow problem, including at the very large scale (1024x1024).Finally, we demonstrate that this method can be used to speed up exactcalculation of the solution to the differential equations via finitedifference.

Weakly-Supervised Deep Learning of Heat Transport via Physics Informed Loss