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Despite their successes, machine learning techniques are often stochastic, errorprone and blackbox. How could they then be used in fields such as theoretical physics and pure mathematics for which errorfree results and deep understanding are a must? In this Perspective, we discuss techniques for obtaining zeroerror results with machine learning, with a focus on theoretical physics and pure mathematics. Nonrigorous methods can enable rigorous results via conjecture generation or verification by reinforcement learning. We survey applications of these techniquesforrigor ranging from string theory to the smooth 4D Poincaré conjecture in lowdimensional topology. We also discuss connections between machine learning theory and mathematics or theoretical physics such as a new approach to field theory motivated by neural network theory, and a theory of Riemannian metric flows induced by neural network gradient descent, which encompasses Perelman’s formulation of the Ricci flow that was used to solve the 3D Poincaré conjecture.more » « lessFree, publiclyaccessible full text available May 1, 2025

Abstract Both the path integral measure in field theory (FT) and ensembles of neural networks (NN) describe distributions over functions. When the central limit theorem can be applied in the infinitewidth (infinite
N ) limit, the ensemble of networks corresponds to a free FT. Although an expansion in corresponds to interactions in the FT, others, such as in a small breaking of the statistical independence of network parameters, can also lead to interacting theories. These other expansions can be advantageous over the $1/N$ expansion, for example by improved behavior with respect to the universal approximation theorem. Given the connected correlators of a FT, one can systematically reconstruct the action orderbyorder in the expansion parameter, using a new Feynman diagram prescription whose vertices are the connected correlators. This method is motivated by the Edgeworth expansion and allows one to derive actions for NN FT. Conversely, the correspondence allows one to engineer architectures realizing a given FT by representing action deformations as deformations of NN parameter densities. As an example, $1/N$φ ^{4}theory is realized as an infiniteN NN FT. 
Abstract We study infinite limits of neural network quantum states (
NNQS), which exhibit representation power through ensemble statistics, and also tractable gradient descent dynamics. Ensemble averages of entanglement entropies are expressed in terms of neural network correlators, and architectures that exhibit volumelaw entanglement are presented. The analytic calculations of entanglement entropy bound are tractable because the ensemble statistics are simplified in the Gaussian process limit. A general framework is developed for studying the gradient descent dynamics of neural network quantum states (NNQS), using a quantum state neural tangent kernel (QSNTK). For $\mathrm{\infty}$ NNQS the training dynamics is simplified, since the QSNTK becomes deterministic and constant. An analytic solution is derived for quantum state supervised learning, which allows an $\mathrm{\infty}$ NNQS to recover any target wavefunction. Numerical experiments on finite and infinite NNQS in the transverse field Ising model and Fermi Hubbard model demonstrate excellent agreement with theory. $\mathrm{\infty}$ NNQS opens up new opportunities for studying entanglement and training dynamics in other physics applications, such as in finding ground states. $\mathrm{\infty}$ 
A bstract We study electricmagnetic duality in compactifications of Mtheory on twisted connected sum (TCS) G 2 manifolds via duality with Ftheory. Specifically, we study the physics of the D3branes in Ftheory compactified on a CalabiYau fourfold Y , dual to a compactification of Mtheory on a TCS G 2 manifold X . $$ \mathcal{N} $$ N = 2 supersymmetry is restored in an appropriate geometric limit. In that limit, we demonstrate that the dual of D3branes probing sevenbranes corresponds to the shrinking of certain surfaces and curves, yielding light particles that may carry both electric and magnetic charges. We provide evidence that the MinahanNemeschansky theories with E n flavor symmetry may be realized in this way. The SL(2 , ℤ) monodromy of the 3/7brane system is dual to a FourierMukai transform of the dual IIA/Mtheory geometry in this limit, and we extrapolate this monodromy action to the global compactification. Away from the limit, the theory is broken to $$ \mathcal{N} $$ N = 1 supersymmetry by a Dterm.more » « less

A bstract In this paper we study the 6d localized charged matter spectrum of Ftheory directly on a singular elliptic CalabiYau 3fold, i.e. without smoothing via resolution or deformation of the entire fibration. Given only the base surface, discriminant locus, and the SL(2 , ℤ) local system, we propose a general prescription for determining the charged matter spectrum localized at intersections of sevenbranes, using the technology of string junctions. More precisely, at each codimension2 collision of sevenbranes, we determine the local sevenbrane content and compute the number of massless string junctions modulo the action of the SL(2 , ℤ) monodromy. We find agreement with the predicted results from 6d anomaly cancellation in all cases considered. Examples include a generic Weierstrass model with arbitrary Kodaira fiber intersecting an I 1 , as well as cases with jointly charged matter localized at intersections of nonabelian sevenbranes.more » « less

null (Ed.)A bstract Dark YangMills sectors, which are ubiquitous in the string landscape, may be reheated above their critical temperature and subsequently go through a confining firstorder phase transition that produces stochastic gravitational waves in the early universe. Taking into account constraints from lattice and from YangMills (center and Weyl) symmetries, we use a phenomenological model to construct an effective potential of the semi quarkgluon plasma phase, from which we compute the gravitational wave signal produced during confinement for numerous gauge groups. The signal is maximized when the dark sector dominates the energy density of the universe at the time of the phase transition. In that case, we find that it is within reach of the nexttonext generation of experiments (BBO, DECIGO) for a range of dark confinement scales near the weak scale.more » « less