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This is a huge topic in applying ML in physics and chemistry where we already have a lot of prior detailed knwoledge about the systems we want to describe and it would be silly not to build it into the ML models.

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What's the current state of art in this direction? Is there a way to encode equations explicitly prior to training?

People now try to use ML anywhere and everywhere so it's wild west a little. Three examples: [1] uses a standard neural net to represent a many-body wave function, with all the machinery of quantum mechanics on top of that, and reinforcement learning to find the true ground state. [2] uses a handcrafted neural net, which by construction already takes advantage of a lot of prior knowledge, to directly predict molecular energies. [3] uses a simple kernel ridge regression coupled with a sophisticated handcrafted scheme to automatically construct a good basis (set of features) for a given input, to predict molecular energies.

In all these cases, the ML itself is not the target problem, but only a tool, and most effort goes into figuring out where exactly to use ML as a part of a larger problem, and how to encode prior knowledge, either via feature construction or neural net handcrafting.

[1] http://sci-hub.io/10.1126/science.aag2302

[2] http://sci-hub.io/10.1038/ncomms13890

[3] https://arxiv.org/pdf/1707.04146.pdf

Variational inference

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