Learning in complicated systems where we know that there is a conservation law in effect. Or, more advanced, learning a conservation law that we did not know was in effect. As seen in especially ML for physics. This is not AFAIK a particular challenge in traditional parametric statistics where we can impose conservation laws on a problem through the likelihood, but nonparametrics models, or models with overparameterisation such as neural nets this can get fiddly. Where does conservation of mass, momentum, energy etc reside in a convnet? Some types of conservation laws can be characterised by a looser constraint such as stability.

There is a particular type of conservation law which we frequently impose upon deep learning, specifically, energy-conservation in neural net signal propagation, which is not a conservation law in the regression model *per se*, but a conservation law that ensures the model itself is trainable.
This is the deep learning as dynamical system trick.
In fact, there are a whole bunch of conservation laws and symmetries implicit in what we do, for example in the potential theory, in the statistical mechanics of learning, in the use of conservation laws in Hamiltonian Monte Carlo but in deep learning these do not necessarily align with the symmetries and conservation laws of the subject matter.

I wonder if the Learning invariant representations idea could help.

Recent entrant in this area: Lagrangian Neural networks. C&C (M. Cranmer et al. 2020; Lutter, Ritter, and Peters 2019).

Also related learning with a PDE constraint.

## References

*Machine Learning and the Physical Sciences Workshop at the 33rd Conference on Neural Information Processing Systems (NeurIPS)*, 6.

*Machine Learning and the Physical Sciences Workshop at the 33rd Conference on Neural Information Processing Systems (NeurIPS)*, 6.

*Advances in Neural Information Processing Systems 32*, edited by H. Wallach, H. Larochelle, A. Beygelzimer, F. d\textquotesingle Alché-Buc, E. Fox, and R. Garnett, 15379–89. Curran Associates, Inc. http://papers.nips.cc/paper/9672-hamiltonian-neural-networks.pdf.

*Proceedings of the 20th Conference of the International Society for Music Information Retrieval*, 8. http://archives.ismir.net/ismir2019/paper/000085.pdf.

*Machine Learning and the Physical Sciences Workshop at the 33rd Conference on Neural Information Processing Systems (NeurIPS)*, 8.

*Journal of Computational Physics*378 (February): 686–707. https://doi.org/10.1016/j.jcp.2018.10.045.

*2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*, 2241–45. https://doi.org/10.1109/ICASSP.2018.8461686.

*Journal of Computational Physics*394 (October): 56–81. https://doi.org/10.1016/j.jcp.2019.05.024.

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