In physics, typically, we are concerned with identifying True Parameters for Universal Laws, applicable without prejudice across all the cosmos. We are hunting something like the Platonic ideals that our experiments are poor shadows of. Especially, say, quantum physics or cosmology.

In machine learning, typically we want to make generic predictions for a given process, and quantify how good those predictions can be given how much data we have and the approximate kind of process we witness, and there is no notion of universal truth waiting around the corner to back up our wild fancies. On the other hand, we are less concerned about the noisy sublunary chaos of our experiments and don’t need to worry about how far our noise drives us from truth as long as we estimate it. But then, far from universality, we have weak and vague notions of how to generalise our models to new circumstances and new noise. That is, in the Platonic ideal of machine learning, there are no Platonic ideals to be found.

(This explanation does no justice to either physics or machine learning, but here is a mere framing rather than an essay in the history or philosophy of science.)

Can these areas have something to say to one another nevertheless? After an interesting conversation with Shane Keating about the difficulties of ocean dynamics, I am thinking about this in a new way; Generally, we might have notions from physics of what “truly” underlies a system, but where many unknown parameters, noisy measurements, computational intractability and complex or chaotic dynamics interfere with our ability to predict things using only known laws of physics; Here, we want to come up with a “best possible” stochastic model of a system given our uncertainties and constraints, which looks a lot like an ML problem.

At a basic level, it’s not controversial (I don’t think?) to use machine learning methods to analyse data in experiments, even with trendy deep neural networks. I understand that this is significant, e.g. in connectomics.

Perhaps a little more fringe is using machine learning to reduce computational burden, e.g. Carleo and Troyer (2017).

The thing that is especially interesting to me is learning the whole model from ML formalism, using physical laws as input to the learning process.

To be concrete, Shane specifically was discussing problems in predicting and interpolating “tracers”, such as chemical or heat, in oceanographic flows. Here we know lots of things about the fluids concerned, but less about the details of the ocean floor and have very imperfect measurements of the details. Nonetheless, we also know that there are certain invariants, conservation laws etc, so a truly “nonparametric” approach to dynamics is certainly throwing away information. We know that our

There is some cute work in this area, like the *SINDy* method, a
compressive-sensing
state filter
of Brunton, Proctor, and Kutz (2016); but it’s hard to imagine
scaling this up (at least not directly) to big things like large image sensor
arrays and other such weakly structured input.

Researchers like Chang et al. (2017) claim that learning “compositional object” models should be possible. The compositional models here are learnable objects with learnable pairwise interactions, and bear a passing resemblance to something like the physical laws that physics experiments hope to discover, although I’m not yet totally persuaded about the details of this particular framework. On the other hand, unmotivated appealing to autoencoders as descriptions of underlying dynamics of physical reality doesn’t seem sufficient.

There is an O’Reilly podcast and reflist about deep learning for science in particular.

Sample images of atmospheric rivers correctly classified (true positive) by our deep CNN model. Figure shows total column water vapor (color map) and land sea boundary (solid line). Liu et al. (2016):

## Emulation approaches

Reducing complciated physic-drvien simulations to simpler/or faster ones using ML techniques.

A recent hyped paper that exemplifies this approach is Kasim et al. (2020), which (somewhat implicitly) uses arguments from Gaussian process regression to produce quasi-Bayesian emulations of notoriously slow simulations. I am amazed that this works.

## The other direction: What does physics say about learning?

See why does deep learning work or the statistical mechanics of statistics.

## Addendum

Why not “statistics for physics”? No reason not to do that; but physics already uses lots of statistics, so we don’t need a new notebook for that. Moreover, there are certain interesting problems in statistics that seem to be especially obvious in physics-driven dynamical problems.

OTOH, I might be able to get more mileage out of Bayesian time series methods? I’ve never looked into estimation theory there.

Related, maybe: the recovery phase transition in compressed sensing.

TBC.

Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. 2016. “Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems.” *Proceedings of the National Academy of Sciences* 113 (15): 3932–7. https://doi.org/10.1073/pnas.1517384113.

Carleo, Giuseppe, and Matthias Troyer. 2017. “Solving the Quantum Many-Body Problem with Artificial Neural Networks.” *Science* 355 (6325): 602–6. https://doi.org/10.1126/science.aag2302.

Chang, Michael B., Tomer Ullman, Antonio Torralba, and Joshua B. Tenenbaum. 2017. “A Compositional Object-Based Approach to Learning Physical Dynamics.” In *Proceedings of ICLR*. http://arxiv.org/abs/1612.00341.

Kasim, M. F., D. Watson-Parris, L. Deaconu, S. Oliver, P. Hatfield, D. H. Froula, G. Gregori, et al. 2020. “Up to Two Billion Times Acceleration of Scientific Simulations with Deep Neural Architecture Search,” January. http://arxiv.org/abs/2001.08055.

Liu, Yunjie, Evan Racah, Prabhat, Joaquin Correa, Amir Khosrowshahi, David Lavers, Kenneth Kunkel, Michael Wehner, and William Collins. 2016. “Application of Deep Convolutional Neural Networks for Detecting Extreme Weather in Climate Datasets,” May. http://arxiv.org/abs/1605.01156.

Medasani, Bharat, Anthony Gamst, Hong Ding, Wei Chen, Kristin A. Persson, Mark Asta, Andrew Canning, and Maciej Haranczyk. 2016. “Predicting Defect Behavior in B2 Intermetallics by Merging Ab Initio Modeling and Machine Learning.” *Npj Computational Materials* 2 (1): 1. https://doi.org/10.1038/s41524-016-0001-z.

Pathak, Jaideep, Brian Hunt, Michelle Girvan, Zhixin Lu, and Edward Ott. 2018. “Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach.” *Physical Review Letters* 120 (2): 024102. https://doi.org/10.1103/PhysRevLett.120.024102.

Pathak, Jaideep, Zhixin Lu, Brian R. Hunt, Michelle Girvan, and Edward Ott. 2017. “Using Machine Learning to Replicate Chaotic Attractors and Calculate Lyapunov Exponents from Data.” *Chaos: An Interdisciplinary Journal of Nonlinear Science* 27 (12): 121102. https://doi.org/10.1063/1.5010300.

Saemundsson, Steindor, Alexander Terenin, Katja Hofmann, and Marc Peter Deisenroth. 2020. “Variational Integrator Networks for Physically Structured Embeddings,” March. http://arxiv.org/abs/1910.09349.

Sargsyan, Khachik, Bert Debusschere, Habib Najm, and Youssef Marzouk. 2009. “Bayesian Inference of Spectral Expansions for Predictability Assessment in Stochastic Reaction Networks.” *Journal of Computational and Theoretical Nanoscience* 6 (10): 2283–97. https://doi.org/10.1166/jctn.2009.1285.

Tompson, Jonathan, Kristofer Schlachter, Pablo Sprechmann, and Ken Perlin. 2017. “Accelerating Eulerian Fluid Simulation with Convolutional Networks.” In. https://openreview.net/pdf?id=ByH2gxrKl.