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As a mathematical statistics student, I was wondering about learning data science application for physics, in general probabilistic modelling for physics. I don't mean that much engineering perspective like robots learning classical mechanics. What are good resources to do so? What techniques are more relevant and specially taylored for physics research? Is there a lecture online, a book or some lecture notes etc.?

Answers :

  • People:
    Max Tegmark explores this field, with Feynman 2.0 project and his other papers. [3]
  • Fields:
    Neural Networks are on a quest for solving differential equations.$[2]$
    Symbolic regression is of interest for discovering new laws and an example would be Graph Neural Networks $[1]$.
    Statistical Mechanics (detailed answer below)
    Moddeling astronomicla constelations (detailed answer below)
    Bayesian statistics (detailed answer below)

Reference:

  1. M. Cranmer et al., "Discovering Symbolic Models from Deep Learning with Inductive Biases", Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020), [PDF].
  2. Z. Li et al., "Fourier Neural Operator for Parametric Partial Differential Equations" [PDF]
  3. [YouTube]
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For the data analysis in the fields I worked in (cosmology and gravitational waves), the mathematical foundation is Bayesian inference.

An excellent, dedicated book about Bayesian inference (but not focused on physics applications) is by Gelman et al: http://www.stat.columbia.edu/~gelman/book/

One resource I really liked is this review by Romano and Cornish: https://arxiv.org/abs/1608.06889. It focuses on stochastic gravitational-wave backgrounds, which is quite a specialized topic, but Sections 3 and to some extent 4 are about statistics and are more general. It covers the difference between Bayesian and frequentist statistics in detail.

You can find many resources googling Bayesian analysis and physics. I have found that mathematically, Bayesian analysis is delightfully simple. The devil is really in the details of how you choose a likelihood and prior for a given problem, the vast array of numerical methods to sample from the posterior, and how you go about checking that you can trust your result.

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In many domains physicists prefer deterministic models over probabilistic models, when possible. If the underlying system evolution is deterministic (as in Newton's laws, relativity, Shrodinger's equation, etc.), then the deterministic model will be more useful than an ad hoc, stochastic model. If you want to find probabilistic models in physics, you'll need to look for domains where the deterministic dynamics of the system are hard to figure out.

Statistical mechanics comes to mind as the obvious case where, because it is impractical to keep track of individual particle dynamics, probabilistic models are used. Statistical and Thermal Physics: With Computer Applications by Gould and Tobochnik is an undergraduate text that uses probabilistic simulations to study thermodynamic systems. This book is also available as a free online version.

Modeling astronomical populations is another place where data science appears in physics. Statistics, Data Mining, and Machine Learning in Astronomy by Ivezić, et al comes to mind. You might also check out a list of resources maintained by the Astrostatistics and Astroinformatics Portal (ASAIP).

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