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Mean-field molecular dynamics based on path integrals is used to approximate canonical quantum observables for particle systems consisting of nuclei and electrons. A computational bottleneck is the sampling from the Gibbs density of the electron operator, which due to the fermion sign problem has a computational complexity that scales exponentially with the number of electrons. In this work we construct an algorithm that approximates the mean-field Hamiltonian by path integrals for fermions. The algorithm is based on the determinant of a matrix with components based on Brownian bridges connecting permuted electron coordinates. The computational work for $n$ electrons is $\mathcal O(n^3)$, which reduces the computational complexity associated with the fermion sign problem. We analyze a bias resulting from this approximation and provide a computational error indicator. It remains to rigorously explain the surprisingly high accuracy.

We construct a family of finite element sub-complexes of the conformal complex on tetrahedral meshes. This complex includes vector fields and symmetric and traceless tensor fields, interlinked through the conformal Killing operator, the linearized Cotton-York operator, and the divergence operator, respectively. This leads to discrete versions of transverse traceless (TT) tensors and York splits in general relativity. We provide bubble complexes and investigate supersmoothness to facilitate the construction. We show the exactness of the finite element complex on contractible domains.

Let $\mathscr{F}_{n,d}$ be the class of all functions $f:\{-1,1\}^n\to[-1,1]$ on the $n$-dimensional discrete hypercube of degree at most $d$. In the first part of this paper, we prove that any (deterministic or randomized) algorithm which learns $\mathscr{F}_{n,d}$ with $L_2$-accuracy $\varepsilon$ requires at least $\Omega((1-\sqrt{\varepsilon})2^d\log n)$ queries for large enough $n$, thus establishing the sharpness as $n\to\infty$ of a recent upper bound of Eskenazis and Ivanisvili (2021). To do this, we show that the $L_2$-packing numbers $\mathsf{M}(\mathscr{F}_{n,d},\|\cdot\|_{L_2},\varepsilon)$ of the concept class $\mathscr{F}_{n,d}$ satisfy the two-sided estimate $$c(1-\varepsilon)2^d\log n \leq \log \mathsf{M}(\mathscr{F}_{n,d},\|\cdot\|_{L_2},\varepsilon) \leq \frac{2^{Cd}\log n}{\varepsilon^4}$$ for large enough $n$, where $c, C>0$ are universal constants. In the second part of the paper, we present a logarithmic upper bound for the randomized query complexity of classes of bounded approximate polynomials whose Fourier spectra are concentrated on few subsets. As an application, we prove new estimates for the number of random queries required to learn approximate juntas of a given degree, functions with rapidly decaying Fourier tails and constant depth circuits of given size. Finally, we obtain bounds for the number of queries required to learn the polynomial class $\mathscr{F}_{n,d}$ without error in the query and random example models.

This paper develops a two-stage stochastic model to investigate evolution of random fields on the unit sphere $\bS^2$ in $\R^3$. The model is defined by a time-fractional stochastic diffusion equation on $\bS^2$ governed by a diffusion operator with the time-fractional derivative defined in the Riemann-Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on $\bS^2$ as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on $\bS^2$. The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree $L\geq1$. The rate of convergence of the truncation errors as a function of $L$ and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic H\"{o}lder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.

A system of coupled oscillators on an arbitrary graph is locally driven by the tendency to mutual synchronization between nearby oscillators, but can and often exhibit nonlinear behavior on the whole graph. Understanding such nonlinear behavior has been a key challenge in predicting whether all oscillators in such a system will eventually synchronize. In this paper, we demonstrate that, surprisingly, such nonlinear behavior of coupled oscillators can be effectively linearized in certain latent dynamic spaces. The key insight is that there is a small number of `latent dynamics filters', each with a specific association with synchronizing and non-synchronizing dynamics on subgraphs so that any observed dynamics on subgraphs can be approximated by a suitable linear combination of such elementary dynamic patterns. Taking an ensemble of subgraph-level predictions provides an interpretable predictor for whether the system on the whole graph reaches global synchronization. We propose algorithms based on supervised matrix factorization to learn such latent dynamics filters. We demonstrate that our method performs competitively in synchronization prediction tasks against baselines and black-box classification algorithms, despite its simple and interpretable architecture.

We propose a threshold-type algorithm to the $L^2$-gradient flow of the Canham-Helfrich functional generalized to $\mathbb{R}^N$. The algorithm to the Willmore flow is derived as a special case in $\mathbb{R}^2$ or $\mathbb{R}^3$. This algorithm is constructed based on an asymptotic expansion of the solution to the initial value problem for a fourth order linear parabolic partial differential equation whose initial data is the indicator function on the compact set $\Omega_0$. The crucial points are to prove that the boundary $\partial\Omega_1$ of the new set $\Omega_1$ generated by our algorithm is included in $O(t)$-neighborhood from $\partial\Omega_0$ for small time $t>0$ and to show that the derivative of the threshold function in the normal direction for $\partial\Omega_0$ is far from zero in the small time interval. Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our threshold-type algorithm.

For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in $H^s(\mathbb T^2)$, where $s>0$, convergence of order $\mathcal O(\tau^{s/2}+N^{-s})$ is proved in $L^2$. Here $\tau$ denotes the time step size and $N$ the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in $L^2$. The stated convergence behavior is illustrated by several numerical examples.

Common regularization algorithms for linear regression, such as LASSO and Ridge regression, rely on a regularization hyperparameter that balances the tradeoff between minimizing the fitting error and the norm of the learned model coefficients. As this hyperparameter is scalar, it can be easily selected via random or grid search optimizing a cross-validation criterion. However, using a scalar hyperparameter limits the algorithm's flexibility and potential for better generalization. In this paper, we address the problem of linear regression with l2-regularization, where a different regularization hyperparameter is associated with each input variable. We optimize these hyperparameters using a gradient-based approach, wherein the gradient of a cross-validation criterion with respect to the regularization hyperparameters is computed analytically through matrix differential calculus. Additionally, we introduce two strategies tailored for sparse model learning problems aiming at reducing the risk of overfitting to the validation data. Numerical examples demonstrate that our multi-hyperparameter regularization approach outperforms LASSO, Ridge, and Elastic Net regression. Moreover, the analytical computation of the gradient proves to be more efficient in terms of computational time compared to automatic differentiation, especially when handling a large number of input variables. Application to the identification of over-parameterized Linear Parameter-Varying models is also presented.

We provide numerical bounds on the Crouzeix ratiofor KLS matrices $A$ which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials $p$ of the spectral norm of $p(A)$ dividedby the maximum absolute value of $p$ on the numerical range of $A$.Our bounds confirm the conjecture that this ratiois less than or equal to $2$. We also give a precise description of these numerical ranges.

We explore the maximum likelihood degree of a homogeneous polynomial $F$ on a projective variety $X$, $\mathrm{MLD}_F(X)$, which generalizes the concept of Gaussian maximum likelihood degree. We show that $\mathrm{MLD}_F(X)$ is equal to the count of critical points of a rational function on $X$, and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.