We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the symmetry transformations and the corresponding generators. The constructed loss functions ensure that the applied transformations are symmetries and the corresponding set of generators forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different types of conserved quantities preserved by symmetry. In the process of deriving the full set of symmetries, we analyze the complete subgroup structure of the rotation groups SO(2), SO(3), and SO(4), and of the Lorentz group SO(1,3) . Other examples include squeeze mapping, piecewise discontinuous labels, and SO(10), demonstrating that our method is completely general, with many possible applications in physics and data science. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.

https://iopscience.iop.org/article/10.1088/2632-2153/acd989/pdf

Deep learning symmetries and their Lie groups, algebras, and subalgebras from first principles

This paper investigates the failure cases and out-of-distribution behavior of transformers trained on matrix inversion and eigenvalue decomposition. I show that incorrect model predictions still retain deep mathematical properties of the solution (e.g. correct eigenvalues, unit norm of eigenvectors), and that almost all model failures can be attributed to, and predicted from, properties of the problem or solution. This demonstrates that, when in doubt, math transformers do not hallucinate absurd solutions (as was sometimes proposed) but remain “roughly right”. I also show that the careful choice of a training dataset can accelerate training, while allowing the model to generalize out of its training distribution, invalidating the idea that transformers “merely interpolate” from memorized examples.

What is my math transformer doing? - Three results on interpretability and generalization

We demonstrate the use of symbolic regression in deriving analytical formulas, which are needed at various stages of a typical experimental analysis in collider phenomenology. As a first application, we consider kinematic variables like the stransverse mass, $M_{T2}$, which are defined algorithmically through an optimization procedure and not in terms of an analytical formula. We then train a symbolic regression and obtain the correct analytical expressions for all known special cases of $M_{T2}$ in the literature. As a second application, we reproduce the correct analytical expression for a next-to-leading order (NLO) kinematic distribution from data, which is simulated with a NLO event generator. Finally, we derive analytical approximations for the NLO kinematic distributions after detector simulation, for which no known analytical formulas currently exist.

Is the Machine Smarter than the Theorist: Deriving Formulas for Particle Kinematics with Symbolic Regression

https://link.springer.com/content/pdf/10.1007/JHEP09(2022)006.pdf

Analytic Computation of three-point energy correlator in QCD

The Dickman function $\rho(u)$ gives the asymptotic probability that a large integer $N$ has no prime divisor exceeding $N^{1/u}$. We expand it in terms of rapidly computable multiple polylogarithms, as defined by Goncharov and intensively used for evaluations of Feynman integrals in quantum field theory. In parallel, we solve Buchstab's differential-delay equation, which concerns large integers $N$ divisible by no prime less than $N^{1/u}$. Discussion of the latter problem occurred in letters to the journal Nature during the second world war, from the physicists Frederick Lindemann and Wendell Furry. We recount how Furry evaluated a dilogarithm in reply to a puzzle resulting from Mertens' third theorem, raised by Lindemann. We refine Furry's analysis to include multiple polylogarithms of weights up to 200.

Dickman multiple polylogarithms and the Lindemann-Furry letters

Polylogrithmic functions, such as the logarithm or dilogarithm, satisfy a number of algebraic identities. For the logarithm, all the identities follow from the product rule. For the dilogarithm and higher-weight classical polylogarithms, the identities can involve five functions or more. In many calculations relevant to particle physics, complicated combinations of polylogarithms often arise from Feynman integrals. Although the initial expressions resulting from the integration usually simplify, it is often difficult to know which identities to apply and in what order. To address this bottleneck, we explore to what extent machine learning methods can help. We consider both a reinforcement learning approach, where the identities are analogous to moves in a game, and a transformer network approach, where the problem is viewed analogously to a language-translation task. While both methods are effective, the transformer network appears more powerful and holds promise for practical use in symbolic manipulation tasks in mathematical physics.

Simplifying Polylogarithms with Machine Learning

We consider the field theory that defines a perfect incompressible 2D fluid. One distinctive property of this system is that the quadratic action for fluctuations around the ground state features neither mass nor gradient term. Quantum mechanically this poses a technical puzzle, as it implies the Hilbert space of fluctuations is not a Fock space and perturbation theory is useless. As we show, the proper treatment must instead use that the configuration space is the area preserving Lie group SDiff. Quantum mechanics on Lie groups is basically a group theory problem, but a harder one in our case, since SDiff is infinite dimensional. Focusing on a fluid on the 2-torus T 2, we could however exploit the well known result SDiff(T 2) ∼ SU(N) for N → ∞, reducing for finite N to a tractable case. SU(N) offers a UV-regulation, but physical quantities can be robustly defined in the continuum limit N →∞. The main result of our study is the existence of ungapped localized excitations, the vortons, satisfying a dispersion ω ∝ k2 and carrying a vorticity dipole. The vortons are also characterized by very distinctive derivative interactions whose structure is fixed by symmetry. Departing from the original incompressible fluid, we constructed a class of field theories where the vortons appear, right from the start, as the quanta of either bosonic or fermionic local fields. ar X iv :2 21 1. 09 82 0v 1 [ he pth ] 1 7 N ov 2 02 2

Aur'elien Dersy

Papers

Simplifying Polylogarithms with Machine Learning

The Quantum Perfect Fluid in 2D