Random feature neural network approximations of the potential in Hamiltonian systems yield approximations of molecular dynamics correlation observables that have the expected error OK-1+J-1212, for networks with K nodes using J data points, provided the Hessians of the potential and the observables are bounded. The loss function is based on the least squares error of the potential and regularizations, with the data points sampled from the Gibbs density. The proof uses a new derivation of the generalization error for random feature networks that does not apply the Rademacher or related complexities.

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 Monte Carlo 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 built on Brownian bridges connecting permuted electron coordinates. The computational work for n electrons is O(n3), which reduces the computational complexity associated with the fermion sign problem. We analyze a bias resulting from this approximation and provide a rough computational error indicator. It remains to rigorously explain the surprisingly high accuracy for high temperatures. The method becomes infeasible at low temperatures due to a large sample variance.

Inexpensive numerical methods are key to enabling simulations of systems of a large number of particles of different shapes in Stokes flow and several approximate methods have been introduced for this purpose. We study the accuracy of the multiblob method for solving the Stokes mobility problem in free space, where the 3D geometry of a particle surface is discretised with spherical blobs and the pair-wise interaction between blobs is described by the RPY-tensor. The paper aims to investigate and improve on the magnitude of the error in the solution velocities of the Stokes mobility problem using a combination of two different techniques: an optimally chosen grid of blobs and a pair-correction inspired by Stokesian dynamics. Different optimisation strategies to determine a grid with a given number of blobs are presented with the aim of matching the hydrodynamic response of a single accurately described ideal particle, alone in the fluid. It is essential to obtain small errors in this self-interaction, as they determine the basic error level in a system of well-separated particles. With an optimised grid, reasonable accuracy can be obtained even with coarse blob-resolutions of the particle surfaces. The error in the self-interaction is however sensitive to the exact choice of grid parameters and simply hand-picking a suitable geometry of blobs can lead to errors several orders of magnitude larger in size. The pair-correction is local and cheap to apply, and reduces the error for moderately separated particles and particles in close proximity. Two different types of geometries are considered: spheres and axisymmetric rods with smooth caps. The error in solutions to mobility problems is quantified for particles of varying inter-particle distances for systems containing a few particles, comparing to an accurate solution based on a second kind BIE-formulation where the quadrature error is controlled by employing quadrature by expansion (QBX).

Admission of applicants to higher education in a fair, reliable, transparent, and efficient way is a real challenge, especially if there are more eligible applicants than available places and if there are applicants from many different educational systems. Previous research on best practices for admission to master’s programmes identified the key question about an applicant’s potential for success in studies, but was not able to provide an answer about how to rate the merits of the applicants. In this study, indicators for study success are analysed by comparing the study performance of 228 students in master’s programmes with their merits at the time of admission. The null hypothesis was that the applicant’s average grade at the time of admission is the only indictor for study success. After testing for potential bias using almost 20 possible other indicators, the null hypothesis had to be rejected for four indicators (in order of importance): (i) university ranking, (ii) length of bachelor’s studies within subject, (iii) English language test and (iv) subject matching between bachelor’s and master’s education. Evaluation of quality of prior education is tricky and results from this study clearly indicate that students from higher ranked universities possess better knowledge and stronger skills for our master’s programmes. Work is ongoing to improve the merit rating model by involving more master’s programmes at KTH and analysing performance data from a larger number of students.

Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be O(M-1), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and M is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain O(M-1) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace h := Tr(He-beta H)/Tr(e(-beta H)) with respect to the electron degrees of freedom and H is the Weyl symbol corresponding to a quantum many body Hamiltonian (sic). It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy O(M-1 + t epsilon(2)), for correlation time t where epsilon(2) is related to the variance of mean value approximation h. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.

Estimates of the generalization error are proved for a residual neural network with L random Fourier features layers z¯+1 = ¯z + ReK k=1 b¯k eiωkz¯ + ReK k=1 c¯k eiω k·x. An optimal distribution for the frequencies (ωk, ω k) of the random Fourier features eiωkz¯ and eiω k·x is derived. This derivation is based on the corresponding generalization error for the approximation of the function values f(x). The generalization error turns out to be smaller than the estimate ˆf 2 L1(Rd) /(KL) of the generalization error for random Fourier features, with one hidden layer and the same total number of nodes KL, in the case of the L∞-norm of f is much less than the L1-norm of its Fourier transform ˆf . This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network. Promising performance of the proposed new algorithm is demonstrated in computational experiments.

The supervised learning problem todetermine a neural network approximation $\mathbb{R}^d\ni x\mapsto\sum_{k=1}^K\hat\beta_k e^{{\mathrm{i}}\omega_k\cdot x}$with one hidden layer is studied asa random Fourier features algorithm.  The Fourier features, i.e., the frequencies $\omega_k\in\mathbb{R}^d$,are sampled using an adaptive Metropolis sampler.The Metropolis test accepts proposal frequencies $\omega_k'$, having corresponding amplitudes $\hat\beta_k'$, with the probability$\min\big\{1, (|\hat\beta_k'|/|\hat\beta_k|)^\gamma\big\}$,for a certain positive parameter $\gamma$, determined by minimizing the approximation error for given computational work.This adaptive, non-parametric stochastic method leads asymptotically, as $K\to\infty$, to equidistributed amplitudes $|\hat\beta_k|$, analogous  to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods.Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is testedboth on synthetic data and a real-world high-dimensional benchmark.

On page 2744 in [1], it is stated that the nonlinear eigenvalue problem (3.8).

In the classical work by Irving and Zwanzig [J.H. Irving and R.W. Zwanzig, J. Chem. Phys., 19, 1173-1180, 1951] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. In this work we derive the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the classical limit of the quantum observables in the conservation laws also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit are: (i) to approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a nonlinear eigenvalue problem, (ii) to define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) to determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) to construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.

It is known that ab initio molecular dynamics based on the electron ground state eigenvaluecan be used to approximate quantum observables in the canonical ensemble when the temperature is low compared tothe first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics,  corresponding to each electron eigenvalue, approximates quantum observables for any temperature.The proof uses the semi-classical Weyl law to show thatcanonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The resultincludes observables that depend on correlations in time. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's lawshows that the error estimate holds %for observables and Hamiltonian symbols  that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei positionand the observables are in diagonal form with respect to the electron eigenstates.