Related papers: Quantum Geometry insights in Deep Learning
The quantum geometry plays a crucial role in the nonlinear transport of quantum materials. Here, we use the Boltzmann transport formalism to study the magnetic control of nonlinear transport induced by the quantum metric in two-dimensional…
Recent advances in large-scale optimal transport have greatly extended its application scenarios in machine learning. However, existing methods either not explicitly learn the transport map or do not support general cost function. In this…
We discuss and demonstrate an unsupervised machine-learning procedure to detect topological order in quantum many-body systems. Using a restricted Boltzmann machine to define a variational ansatz for the low-energy spectrum, we sample wave…
The landscape of condensed matter physics is facing an unprecedented data surge driven by high-throughput ab initio workflows and rapidly expanding experimental datasets. Traditional first-principles methods such as Density Functional…
In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge-Amp\`{e}re equation \begin{eqnarray}…
The Matrix-Element Method (MEM) has long been a cornerstone of data analysis in high-energy physics. It leverages theoretical knowledge of parton-level processes and symmetries to evaluate the likelihood of observed events. In parallel, the…
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble…
Machine learning algorithms often take inspiration from established results and knowledge from statistical physics. A prototypical example is the Boltzmann machine algorithm for supervised learning, which utilizes knowledge of classical…
Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally…
The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of…
Quantum geometry quantifies how the single-particle Bloch wavefunction changes in phase and amplitude across the Brillouin Zone. In multi-orbital systems where bands have strongly mixed orbital composition, quantum geometry plays a vital…
This article investigates the quality of the estimator of the linear Monge mapping between distributions. We provide the first concentration result on the linear mapping operator and prove a sample complexity of $n^{-1/2}$ when using…
Deep generative models (DGM) are neural networks with many hidden layers trained to approximate complicated, high-dimensional probability distributions using a large number of samples. When trained successfully, we can use the DGMs to…
An important object appearing in the framework of the Tomita--Takesaki theory is an invariant cone under the modular automorphism group of von Neumann algebras. As a result of the connection between von Neumann algebras and quantum field…
A basic challenge in experimental physics is the extraction of information related to variables that are not directly measured. The challenge is particularly severe in quantum systems where one may be interested in correlations of operators…
Efficient sampling of the Boltzmann distribution of molecular systems is a long-standing challenge. Recently, instead of generating long molecular dynamics simulations, generative machine learning methods such as normalizing flows have been…
Existence and boundary regularity away from the corners are established for two-dimensional Monge-Amp\`{e}re equations on convex polytopes with Guillemin boundary conditions. An important step is to derive an expansion in terms of functions…
We employ machine learning techniques to provide accurate variational wavefunctions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. Variational quantum Monte Carlo is implemented with deep generative flows to…
We show that the discrete Sinkhorn algorithm - as applied in the setting of Optimal Transport on a compact manifold - converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter…
Starting from the Fermat's principle of least action, which governs classical and quantum mechanics and from the theory of exterior differential forms, which governs the geometry of curved manifolds, we show how to derive the equations…