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The applications and impact of high fidelity simulation of fluid flows are far-reaching. They include settling some long-standing and fundamental questions in turbulence. However, the computational resources required for such efforts are extensive. Here, we explore the possibility of employing the recent computing paradigm of quantum computing to simulate fluid flows. The lure of this new paradigm is the potentially exponential advantage in memory and speed, in comparison with classical computing. This field has recently witnessed a considerable uptick in excitement and contributions. In this work, we give a succinct discussion of the progress made so far, with focus on fluid flows, accompanied by an enumeration of challenges that require sustained efforts for progress. Quantum computing of fluid flows has a promising future, but the inherently nonlinear nature of flows requires serious efforts on resolving various bottlenecks, and on synthesising progress on theoretical, numerical and experimental fronts. We present certain critical details that have not yet attracted adequate attention., Comment: 17 pages, 12 figures

Construction workers exert intense physical effort and experience serious safety and health risks in hazardous working environments. Quiet stance and kneeling are among the most common postures performed by construction workers during their daily work. This paper analyzes lower-limb joint influence on neural balance control strategies using the frequency behavior of the intersection point of ground reaction forces. To evaluate the impact of elevation and wearable knee exoskeletons on postural balance and welding task performance, we design and integrate virtual- and mixed-reality (VR/MR) to simulate elevated environments and welding tasks. A linear quadratic regulator-controlled triple- and double-link inverted pendulum model is used for balance strategy quantification in quiet stance and kneeling, respectively. Extensive multi-subject experiments are conducted to evaluate the usability of occupational exoskeletons in destabilizing construction environments. The quantified balance strategies capture the significance of knee joint during balance control of quiet stance and kneeling gaits. Results show that center of pressure sway area reduced up to 62% in quiet stance and 39% in kneeling for subjects tested in high-elevation VR/MR worksites when provided knee exoskeleton assistance. The comprehensive balance and multitask evaluation methodology developed aims to reveal exoskeleton design considerations to mitigate the fall risk in construction., Comment: 14 pages, 15 figures, submitted to IEEE Transactions on Automation Science and Engineering

relentless is an open-source Python package that enables the optimization of objective functions computed using molecular dynamics simulations. It has a high-level, extensible interface for model parametrization; setting up, running, and analyzing simulations natively in established software packages; and gradient-based optimization. We describe the design and implementation of relentless in the context of relative entropy minimization, and we demonstrate its abilities to design pairwise interactions between particles that form targeted structures. relentless aims to streamline the development of computational materials design methodologies and promote the transparency and reproducibility of complex workflows integrating molecular dynamics simulations., Comment: 13 pages, 6 figures

Seismic mapping of the top of the inner core indicates two distinct areas of high P-wave velocity, the stronger one located beneath Asia, and the other located beneath the Atlantic. This two-fold pattern supports the idea that a lower-mantle heterogeneity can be transmitted to the inner core through outer core convection. In this study, a two-component convective dynamo model, where thermal convection is near critical and compositional convection is strongly supercritical, produces a substantial inner core heterogeneity in the rapidly rotating strongly driven regime of Earth's core. While the temperature profile that models secular cooling ensures that the mantle heterogeneity propagates as far as the inner core boundary (ICB), the distribution of heat flux at the ICB is determined by the strength of compositional buoyancy. A large heat flux variation $q^*$ of $O(10)$ at the core-mantle boundary (CMB), where $q^*$ is the ratio of the maximum heat flux difference to the mean heat flux at the CMB, produces a core flow regime of long-lived convection in the east and time-varying convection in the west. Here, the P-wave velocity estimated from the ICB heat flux in the dynamo is higher in the east than in the west, with the hemispherical difference of the same order as the observed lower bound, 0.5%. Additional observational constraints are satisfied in this regime -- the variability of high-latitude magnetic flux in the east is lower than that in the west; and the stratified F-layer at the base of the outer core, which is fed by the mass flux from regional melting of the inner core and magnetically damped, attains a steady-state height of $\sim$ 200 km., Comment: 29 pages, 4 figures, 4 tables

Turbulent convection in the interiors of the Sun and the Earth occurs at high Rayleigh numbers $Ra$, low Prandtl numbers $Pr$, and different levels of rotation rates. To understand the combined effects better, we study rotating turbulent convection for $Pr = 0.021$ (for which some laboratory data corresponding to liquid metals are available), and varying Rossby numbers $Ro$, using direct numerical simulations (DNS) in a slender cylinder of aspect ratio 0.1; this confinement allows us to attain high enough Rayleigh numbers. We are motivated by the earlier finding in the absence of rotation that heat transport at high enough $Ra$ is similar between confined and extended domains. We make comparisons with higher aspect ratio data where possible. We study the effects of rotation on the global transport of heat and momentum as well as flow structures (a) for increasing rotation at a few fixed values of $Ra$ and (b) for increasing $Ra$ (up to $10^{10}$) at the fixed, low Ekman number of $1.45 \times 10^{-6}$. We compare the results with those from unity $Pr$ simulations for the same range of $Ra$ and $Ro$, and with the non-rotating case over the same range of $Ra$ and low $Pr$. We find that the effects of rotation diminish with increasing $Ra$. These results and comparison studies suggest that, for high enough $Ra$, rotation alters convective flows in a similar manner for small and large aspect ratios, and so useful insights on the effects of high thermal forcing on convection can be obtained by considering slender domains., Comment: In Press, Journal of Fluid Mechanics

Turbulent wall-flows are the most important means for understanding the effects of boundary conditions and fluid viscosity on turbulent fluctuations. There has been considerable recent research on mean-square fluctuations. Here, we present expressions for high-order moments of streamwise velocity fluctuation $u$, in the form $ \langle u^{+2q} \rangle^{1/q}=\alpha_q-\beta_q y^{\ast1/4}$; $q$ is an integer, $\alpha_q$ and $\beta_q$ are constants independent of the friction Reynolds number $Re_\tau$, and $y^{\ast} = y/\delta$ is the distance away from the wall, normalized by the flow thickness $\delta$; in particular, $\alpha_q =\mu+\sigma q$ according to the `linear q-norm Gaussian' process, where $\mu$ and $\sigma$ are flow-independent constants. Excellent agreement is found between these formulae and available data in pipes, channels and boundary layers for $1 \leq q \leq 5$. For fixed $y^+ = y^*Re_\tau$, the present {formulation} leads to the bounded state $\langle u^{+2q} \rangle^{1/q}=\alpha_q$ as $Re_\tau\rightarrow\infty$ while the attached eddy model predicts that {the} moments continually grow as log Reynolds number., Comment: 5 pages, 4 figures

A direct dynamical test of the sunspot-cycle is carried out which indicates that a stochastically forced non-linear oscillator characterizes its dynamics. The sunspot series is then decomposed into its eigen time-delay coordinates. The analysis of these coordinates reveals that the sunspot series exhibits bistability, and suggests the possibility of modeling the solar cycle as a stochastically and periodically forced bistable oscillator, accounting for the Poloidal and Toroidal modes of the solar magnetic field. Such a representation of the sunspot series in terms of stochastic bistable dynamical system enables us to conjecture stochastic resonance as the key mechanism in amplifying the planetary influence of Jupiter on the sun, and that extreme events, due to turbulent convection noise inside the sun, dictate crucial phases of the sunspot cycle, such as the Maunder minimum., Comment: 6 pages, 6 figures

In this work, we investigate whether small language models can determine high-quality subsets of large-scale text datasets that improve the performance of larger language models. While existing work has shown that pruning based on the perplexity of a larger model can yield high-quality data, we investigate whether smaller models can be used for perplexity-based pruning and how pruning is affected by the domain composition of the data being pruned. We demonstrate that for multiple dataset compositions, perplexity-based pruning of pretraining data can \emph{significantly} improve downstream task performance: pruning based on perplexities computed with a 125 million parameter model improves the average performance on downstream tasks of a 3 billion parameter model by up to 2.04 and achieves up to a $1.45\times$ reduction in pretraining steps to reach commensurate baseline performance. Furthermore, we demonstrate that such perplexity-based data pruning also yields downstream performance gains in the over-trained and data-constrained regimes.

Many claims of computational advantages have been made for quantum computing over classical, but they have not been demonstrated for practical problems. Here, we present algorithms for solving time-dependent PDEs, with particular reference to fluid equations. We build on an idea based on linear combination of unitaries to simulate non-unitary, non-Hermitian quantum systems, and generate hybrid quantum-classical algorithms that efficiently perform iterative matrix-vector multiplication and matrix inversion operations. Though these algorithms are end-to-end, they require relatively low-depth quantum circuits and protect quantum advantage, with the best-case asymptotic complexities, which we show} are near-optimal. We demonstrate the performance of the algorithms by conducting: (a) fully gate level, state-vector simulations using an in-house, high performance, quantum simulator called QFlowS; (b) experiments on a real quantum device; and (c) noisy simulations using Qiskit Aer. We also provide device specifications such as error-rates (noise) and state sampling (measurement) to accurately perform convergent flow simulations on noisy devices. The results offer evidence that the proposed algorithm is amenable for use on near-term quantum devices., Comment: 31 pages, 10 figures, 1 table

We study the dynamics of thermal and momentum boundary regions in three-dimensional direct numerical simulations of Rayleigh-B\'enard convection for the Rayleigh number range $10^5 \le Ra \le 10^{11}$ and $Pr=0.7$. Using a Cartesian slab with horizontal periodic boundary conditions and an aspect ratio of 4, we obtain statistical homogeneity in the horizontal $x$- and $y$-directions, thus approximating best an extended convection layer relevant for most geo- and astrophysical flow applications. We observe upon canonical use of combined long-time and area averages, with averaging periods of at least 100 free-fall times, that a global coherent mean flow is practically absent and that the magnitude of the velocity fluctuations is larger than the mean by up to 2 orders of magnitude. The velocity field close to the wall is a collection of differently oriented local shear-dominated flow patches interspersed by extensive shear-free incoherent regions which can be as large as the whole cross section, unlike for a closed cylindrical convection cell of aspect ratio of the order 1. The incoherent regions occupy a $60\%$ area fraction for all Rayleigh numbers investigated here. Rather than resulting in a pronounced mean with small fluctuations about such a mean, as found in small-aspect-ratio convection, the velocity field is dominated by strong fluctuations of all three components around a non-existent or weak mean. We discuss the consequences of these observations for convection layers with larger aspect ratios, including boundary layer instabilities and the resulting turbulent heat transport., Comment: 18 pages, 10 figures

We show theoretically that the mean turbulent dynamics can be described by a kinetic theory representation with a single free relaxation time that depends on space and time. A proper kinetic equation is constructed from averaging the Klimontovich-type equation for fluid elements satisfying the Navier-Stokes hydrodynamics exactly. The turbulent kinetic energy plays the role of temperature in standard molecular thermodynamics. We show that the dynamics of turbulent fluctuations resembles a collision process that asymptotically drives the mean distribution towards a Gaussian (Maxwell-Boltzmann) equilibrium form. Non-Gaussianity arises directly from non-equilibrium shear effects. The present framework overcomes the bane of most conventional turbulence models and theoretical frameworks arising from the lack of scale separation between the mean and fluctuating scales of the Navier-Stokes equation with an eddy viscous term. An averaged turbulent flow in the present framework behaves more like a flow of finite Knudsen number with finite relaxation time, and is thus more suitably described in a kinetic theory representation., Comment: 10 pages

Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection-diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithms (QLSA) based on the Harrow--Hassidim--Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to $N=64$ grid points on the unit interval. Both algorithms are of hybrid nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the sizes of an additional quantum register for the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement and decoherence circuit noise. We reflect the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer., Comment: 45 pages, 20 figures

In Nagib, Chauhan and Monkewitz~\cite{NCM07} we concluded that nearly all available $C_f$ relations for zero-pressure-gradient boundary layers are in remarkable agreement over the entire range $Re_\theta$ $<$ O($10^8$), provided one coefficient is adjusted in each relation by anchoring it to accurate measurements. Regarding the peak of the streamwise turbulence intensity $^+_P$, we conclude here that accurate measurements in flows with $Re_\tau$ $>$ O($10^6$) are required, especially when looking only at the peak $^+_P$ to discriminate between recently proposed trends. We also find remarkable agreement between the three analyses of Monkewitz \cite{M22}, Chen and Sreenivasan \cite{CS22} and Monkewitz and Nagib \cite{MN15}, with some coefficients slightly modified, by underpinning them with the same accurate measurements of $^+_P$ from reliable channel and boundary layer data. All the three analyses conclude that the inner peak of $^+$ remains finite in the limit of infinite Reynolds number, which is at variance with the unlimited growth of $^+_P$ as $\ln{Re}$ $_\tau$ predicted by the attached eddy model \cite{MM19}. Accurate measurements of high-order moments and the guidance of consistent asymptotic expansions may help clarify the issue at lower $Re_\tau$ values., Comment: 6 pages, 6 figures

The secular variation of the geomagnetic field suggests that there are anticyclonic polar vortices in the Earth's core. Under the influence of a magnetic field, the polar azimuthal flow is thought to be produced by one or more coherent upwellings within the tangent cylinder, offset from the rotation axis. In this study, convection within the tangent cylinder in rapidly rotating dynamos is investigated through the analysis of forced magnetic waves. The first part of the study investigates the evolution of an isolated buoyancy disturbance in an unstably stratified rotating fluid subjected to an axial magnetic field. It is shown that the axial flow intensity of the slow Magnetic-Archimedean-Coriolis (MAC) waves becomes comparable to that of the fast MAC waves when $|\omega_M/\omega_C| \sim 0.1$, where $\omega_M$ and $\omega_C$ are the Alfv\'en wave and inertial wave frequencies respectively. In spherical shell dynamo simulations, the isolated upwellings within the tangent cylinder are shown to originate from the localized excitation of slow MAC waves in the dipole-dominated regime. Axial flow measurements in turn reveal the approximate parity between the slow and fast wave intensities in this regime, which corresponds to the existence of strong polar vortices in the Earth's core. To obtain the observed peak azimuthal motions of $0.6$--$0.9^\circ$yr$^{-1}$, the Rayleigh number in the low-inertia geodynamo must be $\sim 10^3$ times the Rayleigh number for the onset of nonmagnetic convection. However, if the forcing is so strong as to cause polarity reversals, the field within the tangent cylinder decays away, and the convection takes the form of an ensemble of plumes supported entirely by the fast waves of frequency $\omega \sim \omega_C$. The resulting weak polar circulation is comparable to that obtained in nonmagnetic convection., Comment: 30 pages, 13 figures, 3 tables. arXiv admin note: text overlap with arXiv:2305.17640

Inertial wave modes in the Sun are of interest owing to their potential to reveal new insight into the solar interior. These predominantly retrograde-propagating modes in the solar subsurface appear to deviate from the thin-shell Rossby-Haurwitz model at high azimuthal orders. We present new measurements of sectoral equatorial inertial modes at $m>15$ where the modes appear to become progressively less retrograde compared to the canonical Rossby-Haurwitz dispersion relation in a co-rotating frame. We use a spectral eigenvalue solver to compute the spectrum of solar inertial modes in the presence of differential rotation. Focussing specifically on equatorial Rossby modes, we find that the numerically obtained mode frequencies lie along distinct ridges, one of which lies strikingly close to the observed mode frequencies in the Sun. We also find that the $n=0$ ridge is deflected strongly in the retrograde direction. This suggests that the solar measurements may not correspond to the fundamental $n=0$ Rossby-Haurwitz solutions as was initially suspected, but to a those for a higher $n$. The numerically obtained eigenfunctions also appear to sit deep within the convection zone -- unlike those for the $n=0$ modes -- which differs substantially from solar measurements and complicates inference., Comment: 16 pages, 12 figures

Contrastive learning has gained significant attention as a method for self-supervised learning. The contrastive loss function ensures that embeddings of positive sample pairs (e.g., different samples from the same class or different views of the same object) are similar, while embeddings of negative pairs are dissimilar. Practical constraints such as large memory requirements make it challenging to consider all possible positive and negative pairs, leading to the use of mini-batch optimization. In this paper, we investigate the theoretical aspects of mini-batch optimization in contrastive learning. We show that mini-batch optimization is equivalent to full-batch optimization if and only if all $\binom{N}{B}$ mini-batches are selected, while sub-optimality may arise when examining only a subset. We then demonstrate that utilizing high-loss mini-batches can speed up SGD convergence and propose a spectral clustering-based approach for identifying these high-loss mini-batches. Our experimental results validate our theoretical findings and demonstrate that our proposed algorithm outperforms vanilla SGD in practically relevant settings, providing a better understanding of mini-batch optimization in contrastive learning.

We present a pedagogical introduction to the current state of quantum computing algorithms for the simulation of classical fluids. Different strategies, along with their potential merits and liabilities, are discussed and commented on., Comment: 7 pages

Inertial range scaling exponents for both Lagrangian and Eulerian structure functions are obtained from direct numerical simulations of isotropic turbulence in triply periodic domains at Taylor-scale Reynolds number up to 1300. We reaffirm that transverse Eulerian scaling exponents saturate at $\approx 2.1$ for moment orders $p \ge 10$, significantly differing from the longitudinal exponents (which are predicted to saturate at $\approx 7.3$ for $p \ge 30$ from a recent theory). The Lagrangian scaling exponents likewise saturate at $\approx 2$ for $p \ge 8$. The saturation of Lagrangian exponents and transverse Eulerian exponents is related by the same multifractal spectrum by utilizing the well known frozen hypothesis to relate spatial and temporal scales. Furthermore, this spectrum is different from the known spectra for Eulerian longitudinal exponents, suggesting that that Lagrangian intermittency is characterized solely by transverse Eulerian intermittency. We discuss possible implication of this outlook when extending multifractal predictions to the dissipation range, especially for Lagrangian acceleration., Comment: 6 pages, 6 figures

Large language models like GPT-4 exhibit emergent capabilities across general-purpose tasks, such as basic arithmetic, when trained on extensive text data, even though these tasks are not explicitly encoded by the unsupervised, next-token prediction objective. This study investigates how small transformers, trained from random initialization, can efficiently learn arithmetic operations such as addition, multiplication, and elementary functions like square root, using the next-token prediction objective. We first demonstrate that conventional training data is not the most effective for arithmetic learning, and simple formatting changes can significantly improve accuracy. This leads to sharp phase transitions as a function of training data scale, which, in some cases, can be explained through connections to low-rank matrix completion. Building on prior work, we then train on chain-of-thought style data that includes intermediate step results. Even in the complete absence of pretraining, this approach significantly and simultaneously improves accuracy, sample complexity, and convergence speed. We also study the interplay between arithmetic and text data during training and examine the effects of few-shot prompting, pretraining, and model scale. Additionally, we discuss length generalization challenges. Our work highlights the importance of high-quality, instructive data that considers the particular characteristics of the next-word prediction objective for rapidly eliciting arithmetic capabilities.

For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with an end-to-end practical applications -- in this instance, fluid dynamics. We debut here a high performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates $T_{0}$ (a parameter crucial to Hamiltonian simulations) to the condition number $\kappa$ of the simulation matrix, and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for (a) the functional form or sparse quantum state preparation, and (b) \textit{in-situ} quantum post-processing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be $\mathcal{O}(\textrm{polylog} (N/\epsilon)\kappa/\epsilon_{QPP})$, where $N$ is the size of the linear system of equations, $\epsilon$ is the the solution error and $\epsilon_{QPP}$ is the error in post processing. This work suggests a path towards quantum simulation of fluid flows, and highlights the special considerations needed at the gate level implementation of QC., Comment: 21 pages, 10 figures, 1 table

In furtherance of our earlier work (Chen \& Sreenivasan, {\it J. Fluid Mech.} {\bf 908}, 2021, p. R3; {\bf 933}, 2022, p. A20 -- together referred to as CS hereafter), we present a self-consistent Reynolds number asymptotics for wall-normal profiles of variances of streamwise and spanwise velocity fluctuations as well as root-mean-square pressure, across the entire flow region of channel and pipe flows and flat-plate boundary layers. It is first shown that, when normalized by peak values, the Reynolds number dependence and wall-normal variation of all three profiles can be decoupled, in excellent agreement with available data, sharing the common inner expansion of the type $\phi^+(y^+)=f_0(y^+)+f_1(y^+)/Re^{1/4}_\tau$, where $\phi^+$ is one of the quantities just mentioned, and the functions $f_0$ and $f_1$ depend only on $y^+$. Here, the superscript $+$ indicates normalization by wall variables. Secondly, by matching the above inner expansion and the inviscid outer flow similarity form, a bounded variation $\phi^+(y^\ast)=\alpha_\phi-\beta_{\phi}y^{\ast{{1}/{4}}}$ is derived beyond the peak where, for each $\phi^+$, the constants $\alpha_\phi$ and $\beta_{\phi}$ are independent of $Re_\tau$ and $y^\ast$ ($=y^+/Re_\tau$, $Re_\tau$ being the Reynolds number based on the friction velocity) -- also in excellent agreement with simulations and experimental data. One of the predictions of the analysis is that, for asymptotically high Reynolds numbers, a finite plateau $\phi^+\approx\alpha_\phi$ appears in the outer region. This result sheds light on the intriguing issue of the outer shoulder of the variance of the streamwise velocity fluctuation, which should be bounded by the asymptotic plateau of about 10.