[joshlk]

For some systems even with the Lagrangian/Hamiltonian setup your solving differential equations with numerical techniques that has error. It might be that the neural networks has less error than the standard techniques. This is a guess.

[seesawtron]

Hamiltonian NNs are not a new thing. There was a NIPS 2019 paper [0] that attempted to do that same for some toy problems.

In general the idea of including model or context-based information into neural networks goes along the line of Kahneman's System I and System II of the human mind. System I is the "emotional" brain that is fast and makes decisions quickly while System II is the "rational" brain that is slow and expensive and takes time to compute a response. Researchers have been trying to develop ML models that utilize this dichotomy by building corresponding dual modules but the major challenge remains in efficiently embedding the assumptions of the world dynamics into the models.

[0] https://arxiv.org/abs/1906.01563 [1] https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow

[noobermin]

To be frank, this should be the reference, compare to numerical integration and see which is better.