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Overparameterized models have proven to be powerful tools for solving various machine learning tasks. However, overparameterization often leads to a substantial increase in computational and memory costs, which in turn requires extensive…

The dynamics of gradient-based training in neural networks often exhibit nontrivial structures; hence, understanding them remains a central challenge in theoretical machine learning. In particular, a concept of feature unlearning, in which…

Density functional theory (DFT) plays a pivotal role for the chemical and materials science due to its relatively high predictive power, applicability, versatility and computational efficiency. We review recent progress in machine learning…

The methodology developed in this article is motivated by a wide range of prediction and uncertainty quantification problems that arise in Statistics, Machine Learning and Applied Mathematics, such as non-parametric regression, multi-class…

This review article provides an overview of random matrix theory (RMT) with a focus on its growing impact on the formulation and inference of statistical models and methodologies. Emphasizing applications within high-dimensional statistics,…

Dynamic Mode Decomposition (DMD) is an unsupervised machine learning method that has attracted considerable attention in recent years owing to its equation-free structure, ability to easily identify coherent spatio-temporal structures in…

Deep Boltzmann machines (DBMs), one of the first ``deep'' learning methods ever studied, are multi-layered probabilistic models governed by a pairwise energy function that describes the likelihood of all variables/nodes in the network. In…

The stochastic density functional theory (DFT) [Phys. Rev. Lett. 111, 106402 (2013)] is a valuable linear scaling approach to Kohn-Sham DFT that does not rely on the sparsity of the density matrix. Linear (and often sub-linear) scaling is…

Decentralized stochastic optimization has emerged as a fundamental paradigm for large-scale machine learning. However, practical implementations often rely on biased gradient estimators arising from communication compression or inexact…

The dynamical mean-field theory (DMFT) is a widely applicable approximation scheme for the investigation of correlated quantum many-particle systems on a lattice, e.g., electrons in solids and cold atoms in optical lattices. In particular,…

We analyze in a closed form the learning dynamics of stochastic gradient descent (SGD) for a single-layer neural network classifying a high-dimensional Gaussian mixture where each cluster is assigned one of two labels. This problem provides…

This work introduces a new framework integrating port-Hamiltonian systems (PHS) and neural network architectures. This framework bridges the gap between deterministic and stochastic modeling of complex dynamical systems. We introduce new…

A machine learning technique is proposed for quantifying uncertainty in power system dynamics with spatiotemporally correlated stochastic forcing. We learn one-dimensional linear partial differential equations for the probability density…

Using dynamical density functional theory (DDFT) methods we investigate the laning instability of a sheared colloidal suspension. The nonequilibrium ordering at the laning transition is driven by non-affine particle motion arising from…

In the pursuit of autonomous learning systems, the foundational assumption of stationarity, the premise that data distributions and model behaviors remain constant, is fundamentally untenable. Historically, the research community has…

The evolution of a large class of biological, physical and engineering systems can be studied through both dynamical systems theory and Hamiltonian mechanics. The former theory, in particular its specialization to study systems with…

In applications of dynamical systems, situations can arise where it is desired to predict the onset of synchronization as it can lead to characteristic and significant changes in the system performance and behaviors, for better or worse. In…

Density Functional Theory (DFT) has become a cornerstone in the modeling of metals. However, accurately simulating metals, particularly under extreme conditions, presents two significant challenges. First, simulating complex metallic…

The generic linear evolution of the density matrix of a system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear…

The Deep Material Network (DMN) has emerged as a powerful framework for multiscale materials modeling, enabling efficient and accurate prediction of material behavior across different length scales. Unlike conventional data-driven…