Accurately evaluating configurational integrals for dense solids remains a central and difficult challenge in the statistical mechanics of condensed systems. Here, we present a tensor network approach that reformulates the high-dimensional configurational integral for identical-particle crystals into a sequence of computationally efficient summations. We represent the integrand as a high-dimensional tensor and apply tensor-train (TT) decomposition together with a custom TT-cross interpolation. This approach circumvents the need to explicitly construct the full tensor. We introduce tailored rank-1 and rank-2 schemes optimized for sharply peaked Boltzmann probability densities, typical for identical-particle crystals. When applied to the calculation of internal energy and pressure-temperature curves for crystalline Cu and Ar at high (GPa) pressures, as well as the alpha-to-beta phase transition diagram of Sn, our method accurately reproduces molecular dynamics simulation results using tight-binding, machine learning, hierarchical interacting particle–neural network, and modified embedded atom method potentials,all within seconds of computation time.

In this paper, we present a new space-time Galerkin-like method, where we treat the discretization of spatial and temporal domains simultaneously. This method utilizes a mixed formulation of the tensor-train (TT) and quantized tensor-train (QTT) (please see Section Tensor-Train Decomposition), designed for the finite element discretization (Q1-FEM) of the time-dependent convection–diffusion–reaction (CDR) equation. We reformulate the assembly process of the finite element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the finite element operators. Recognizing the banded structure inherent in the finite element framework’s discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example.

Tensor train (TT) methods have recently gained popularity for accelerating the solving of systems of PDEs. Here, we evaluate the performance of TT methods in the context of geophysical fluid dynamics (GFD) using the shallow water equations and a discretization scheme employed by the ocean component of the Energy Exascale Earth System Model (E3SM). Through a suite of four test cases of increasing complexity, we evaluate TT methods in terms of how much TT is able to compress the model state, the error incurred by the TT approximation, and the speedup obtained by TT versus an optimal standard non-TT implementation in a representative subproblem. We show that though TT is able to effectively compress and speed up simple flows, it struggles to efficiently represent more complex states that are common in realistic GFD applications.

A tensor provides a concise way to codify the interdependence of complex data. Treating a tensor as a d-way array, each entry records the interaction between the different indices. Clustering provides a way to parse the complexity of the data into more readily understandable information. Clustering methods are heavily dependent on the algorithm of choice, as well as the chosen hyperparameters of the algorithm. However, their sensitivity to data scales is largely unknown. In this work, we apply the discrete wavelet transform to analyze the effects of coarse-graining on clustering tensor data. We are particularly interested in understanding how scale affects clustering of the Earth’s climate system. The discrete wavelet transform allows classification of the Earth’s climate across a multitude of spatial-temporal scales. The discrete wavelet transform is used to produce an ensemble of classification estimates, as opposed to a single classification. Each element of the ensemble is a clustering at a different spatial-temporal scale. Information theoretic approaches are used to identify important scale lengths in clustering the L15 Climate Datset. We also discover a sub-collection of the ensemble that spans the majority of the variance observed, allowing for efficient consensus clustering techniques that can be used to identify climate biomes.

The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}_2$). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization $X = W \land H$ has a unique $W$, a unique $H$ (for a fixed $W$), and when both $W$ and $H$ are unique, given a rank constraint. We introduce a method for robust Boolean model selection, called BMF$k$, and show on numerical examples that BMF$k$ not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.