Function Approximation Research Articles

Linear Regression-Based Traffic Flow Simulation: Vehicle Density and Speed Analysis on Buah Batu Road

Traffic congestion has become an increasingly severe problem in many major cities around the world, including in the city of Bandung. Population growth and increased vehicle use exacerbate congestion. Jalan Buah Batu, one of the main roads in the city of Bandung, often experiences congestion due to high density. This study explains the traffic flow simulation using the Lighthill-Whitham-Richards (LWR) model with a speed-density function obtained from observation data on Jalan Buah Batu, Bandung. The data included the relationship between vehicle density and speed which was then analyzed using the linear regression method. The approximation of the velocity-density function obtained from linear regression is v(ρ)= -6.904+4.302. Traffic flow simulations were carried out with a road length of 60 meters, a total time of 5 minutes, and high resolution with 300 grid points. At the beginning of the simulation, a peak density of 0.70 occurred in a 15-25 meter road segment. Over time, the peak density shifted and decreased: 0.65 at 20-30 meters at 1.25 minutes, 0.60 at 25-35 meters at 2.5 minutes, and 0.50 at 30-50 meters at the end of the simulation (5 minutes). These results show the movement of vehicles that reduce congestion and improve the smooth flow of traffic. In conclusion, linear regression is effective in determining the velocity-density function.

Double polarized deuteron–deuteron scattering and test of T-invariance

A component of the total cross section of the deuteron-deuteron interaction connected with the vector polarized ([Formula: see text]) one initial deuteron and tensor polarized ([Formula: see text]) another one constitutes a null-test signal of the T-invariance violation but P-parity conservation. The energy dependence of this signal is calculated here at beam kinetic energies per nucleon of 0.1–1[Formula: see text]GeV within the Glauber spin-dependent theory of multiple scattering in the S-wave approximation for the deuteron wave function.

Shannon Wavelet‐Based Approximation Scheme for Information Entropy Integrals in Confined Domain

ABSTRACTIn this work, the author attempted to develop a Shannon wavelet‐based numerical scheme to approximate the information entropies in both configuration and momentum space corresponding to the ground and adjacent excited energy states of one‐dimensional Schrödinger equation appearing in non‐relativistic quantum mechanics. The development of this scheme is based on the judicious use of sinc scale functions as an approximation basis and a suitable numerical quadrature to approximate entropies in position and momentum spaces. Priori and posteriori errors appearing in the approximations of wave functions and entropy integrals have been discussed. The scheme (coded in Python) has been subsequently exercised for various exactly solvable and quasi‐exactly solvable non‐relativistic quantum mechanical models in confined domain.

Effect of Fluorine Substitution on Magnetizabilities: Insights from Density Functional Theory and Benchmark Coupled-Cluster Calculations.

The quantitative calculation of the magnetizability tensor of fluorine-containing molecules has been a longstanding challenge for quantum chemistry. We report a benchmark study on the effect of fluorine substitution on the magnetizability (both isotropic and anisotropic) of 24 small closed-shell molecules using coupled-cluster (CCSD(T)) theory, large basis sets, and London atomic orbitals. By extrapolation to a complete basis set (CBS) result, we establish the CCSD(T)/CBS limit and take the opportunity to assess the performance of various density functional theory approximations. Correcting for zero-point vibration further allows us to directly compare theory with (gas-phase) experimental data. We revisit Flygare's hypothesis on molecular magnetizabilities, which relates the successive replacement of hydrogen by fluorine atoms to changes in the magnetizability anisotropy. Some exceptions to this hypothesis are documented, and we rationalize these observations on the basis of hybridization on carbon.

Random matrix approach to time-dependent forcing in many-body quantum systems

Changing some of its parameters over time is a paradigmatic way of driving an otherwise isolated many-body quantum system out of equilibrium, and a vital ingredient for building quantum computers and simulators. Here, we further develop a recently proposed nonlinear response theory which is based on typicality and random-matrix methods, and which is applicable to a wide variety of such parametrically perturbed systems in and out of equilibrium: We derive analytical approximations of the characteristic response function for the two limiting cases of fast driving and of strong and short-ranged-in-energy driving. Furthermore, we work out implications and predictions for common applications, including finite-time quenches and time-dependent forcing that breaks conservation laws of the underlying undriven system. Finally, we verify all predictions by numerical examples and discuss the theory's scope and limitations. Published by the American Physical Society 2024

Structural and energetic properties of cluster models of anatase-supported single late transition metal atoms: a density functional theory benchmark study.

Single-atom catalytic systems constitute an intriguing research topic due to their inherently different chemical behavior as compared to classic heterogeneous catalysts. In this study, cluster systems representing single late transition metal atoms adsorbed on anatase were constructed starting from previously generated periodic models and subjected to a density functional theory (DFT) benchmark study. The ability of different density functional approximations representing all rungs of the Jacob's Ladder classification to accurately describe bond lengths and adsorption energies was assessed for these clusters with the aim of revealing the functional that allows to retain the structural characteristics of the initial periodic system, while also delivering reliable energetics. In this regard, our results indicate that optimisation of the clusters with the meta-GGA functionals TPSS or RevTPSS provides the lowest mean unsigned error and root-mean-square deviations with respect to the periodic models. Moreover, these functionals and, to a slightly lesser degree, PW91 were also found to provide adsorption energies that are statistically the least deviating from the CCSD(T) reference data. More complex hybrid functionals appear to be performing less well. Cluster geometries were determined at the Kohn-Sham DFT level using the LANL2DZ basis set for the transition metals and the Pople 6-31G(d) basis set for O and H. The density functional approximations considered were SVWN, PBE, BP86, BLYP, PW91, TPSS, RevTPSS, M06L, M11L, B3LYP, PBE0, M06, M06-2X, MN15, ωB97X-D, CAM-B3LYP, M11, and MN12-SX. Reference adsorption energies of the metals on the support cluster were obtained at the CCSD(T)/LANL2TZ (transition metals)/6-311 + + G(d,p)//RevTPSS/LANLD2DZ (transition metals)/6-31G*. Besides the above-mentioned functionals, energy calculations using the double-hybrid functionals, DSDPBEP86, PBE0-DH, and B2PLYP, were also performed. All adsorption energy calculations were carried out on the RevTPSS geometries.

Ground and Excited Electronic Structure Analysis of FeH with Correlated Wave Function Theory and Density Functional Approximations.

FeH is one of the most challenging diatomic molecules to study under electronic structure theory. Here, we have successfully studied 22 electronic states of FeH using ab initio multireference configuration interaction (MRCI), Davidson-corrected MRCI (MRCI+Q), and coupled cluster singles, doubles, and perturbative triples [CCSD(T)] levels of theory. We report their potential energy curves (PECs), excitation energies, dissociation energies, equilibrium electronic configurations, and a series of spectroscopic constants with the use of augmented triple-ζ, quadruple-ζ, and quintuple-ζ quality correlation consistent basis sets. The scalar relativistic effects and active space and core electron correlation contribution on the properties of FeH are also explored. The use of a large CASSCF active space that includes 4s, 4p, 3d, and 4d orbitals of Fe and the 1s of H is critical for producing accurate full PECs with proper dissociations and predicting the exact order of the electronic states. Our findings are in harmony with the experimental results available in the literature and will serve as reference values for future studies of FeH. Furthermore, with the use of PECs, the total internal partition function sum (TIPS) of FeH was calculated across a range of temperatures. Finally, we exploited the single-reference nature of the a6Δ of FeH and its ionized product FeH+ (X5Δ) to evaluate the associated density functional theory (DFT) errors on their dissociation energies and spectroscopic parameters.

Thermal cross section of poly(2-hydroxyethyl methacrylate) hydrogels for neutron dose calculation

Phantom materials are used to design radiation safety and protection strategies by means of numerical dose calculation. In the case of thermal neutrons, the radiation transport in the phantom relies on mass attenuation coefficients and total cross sections which are dependent on the physical and chemical properties of the material. Specifically for medical applications, such as neutron capture therapy, the neutron-induced dose is mainly related to the neutron absorption by hydrogen and nitrogen in the human tissue and body. Here, we investigate the use of poly(2-hydroxyethyl methacrylate) as a phantom material by experimental measurement and modeling of its total neutron scattering cross section and mass attenuation coefficient. We show that by varying the hydration level from 10 to 40 w%, one can obtain a neutron attenuation coefficient similar to polymethyl methacrylate or more representative of the human body, respectively. By benchmarking our phenomenological model on the experimental data, we provide a new example of how to use the average functional group approximation to accurately model total neutron cross sections in the framework of personalized medicine approaches.

PUMA: Deep Metric Imitation Learning for Stable Motion Primitives

Imitation learning (IL) facilitates intuitive robotic programming. However, ensuring the reliability of learned behaviors remains a challenge. In the context of reaching motions, a robot should consistently reach its goal, regardless of its initial conditions. To meet this requirement, IL methods often employ specialized function approximators that guarantee this property by construction. Although effective, these approaches come with some limitations: 1) they are typically restricted in the range of motions they can model, resulting in suboptimal IL capabilities, and 2) they require explicit extensions to account for the geometry of motions that consider orientations. To address these challenges, we introduce a novel stability loss function that does not constrain the function approximator's architecture and enables learning policies that yield accurate results. Furthermore, it is not restricted to a specific state space geometry; therefore, it can easily incorporate the geometry of the robot's state space. Proof of the stability properties induced by this loss is provided and the method is empirically validated in various settings. These settings include Euclidean and non‐Euclidean state spaces, as well as first‐order and second‐order motions, both in simulation and with real robots. More details about the experimental results can be found at https://youtu.be/ZWKLGntCI6w.

A singularly perturbed ruin problem for a two-dimensional Brownian motion in the positive quadrant

Abstract We consider the following problem: the drift of the wealth process of two companies, modelled by a two-dimensional Brownian motion, is controllable such that the total drift adds up to a constant. The aim is to maximize the probability that both companies survive. We assume that the volatility of one company is small with respect to the other, and use methods from singular perturbation theory to construct a formal approximation of the value function. Moreover, we validate this formal result by explicitly constructing a strategy that provides a target functional, approximating the value function uniformly on the whole state space.

Algorithm XXXX: The Delaunay Density Diagnostic

Accurate approximation of a real-valued function depends on two aspects of the available data: the density of inputs within the domain of interest and the variation of the outputs over that domain. There are few methods for assessing whether the density of inputs is sufficient to identify the relevant variations in outputs—i.e., the “geometric scale” of the function—despite the fact that sampling density is closely tied to the success or failure of an approximation method. In this paper, we introduce a general purpose, computational approach to detecting the geometric scale of real-valued functions over a fixed domain using a deterministic interpolation technique from computational geometry. The algorithm is intended to work on scalar data in moderate dimensions (2-10). Our algorithm is based on the observation that a sequence of piecewise linear interpolants will converge to a continuous function at a quadratic rate (in \(L^{2}\) norm) if and only if the data are sampled densely enough to distinguish the feature from noise (assuming sufficiently regular sampling). We present numerical experiments demonstrating how our method can identify feature scale, estimate uncertainty in feature scale, and assess the sampling density for fixed (i.e. static) datasets of input–output pairs. We include analytical results in support of our numerical findings and have released lightweight code that can be adapted for use in a variety of data science settings.

Development of a software solution to an applied mechanics problem based on numerical methods

The present work delves into the significance of material selection in the manufacturing of tools for milling operations and determines their suitability through a detailed analysis of strength and behavior during material processing. To enhance longevity and optimize production, the utilization of mathematical models and numerical methods, notably the least squares method and the solution of systems of linear algebraic equations (SLAE) employing the Gaussian elimination method with pivot selection, is proposed. These methods are applied for the approximation of experimental data and the analysis of material characteristics, ensuring precision in the evaluation of its properties. Situations where only partial data is known are investigated, along with simplifications in the computation of known functions. The study encompasses software for numerical computation and visualization of various problem types, effectively addressed through the aforementioned methods. The software algorithm for data approximation involves storing information in a text file, user input of variables, and selection of the quantity and type of basis functions. After the user inputs of parameters, the program formulates a system of equations based on selected functions, determines approximation coefficients, and generates a graph for an objective assessment of results. With a user-friendly interface, users can easily interact with the program, input values, and analyze results through graphical representation, streamlining the workflow and facilitating the comprehension of obtained data. The approximation of functions using numerical methods proves to be efficiently applicable in diverse fields for addressing applied mechanics problems.

A new perspective on optimizers: leveraging moreau-yosida approximation in gradient-based learning

Machine Learning (ML) heavily relies on optimization techniques built upon gradient descent. Numerous gradient-based update methods have been proposed in the scientific literature, particularly in the context of neural networks, and have gained widespread adoption as optimizers in ML software libraries. This paper introduces a novel perspective by framing gradient-based update strategies using the Moreau-Yosida (MY) approximation of the loss function. Leveraging a first-order Taylor expansion, we demonstrate the concrete exploitability of the MY approximation to generalize the model update process. This enables the evaluation and comparison of regularization properties underlying popular optimizers like gradient descent with momentum, ADAGRAD, RMSprop, and ADAM. The MY-based unifying view opens up possibilities for designing new update schemes with customizable regularization properties. To illustrate this potential, we propose a case study that redefines the concept of closeness in the parameter space using network outputs. We present a proof-of-concept experimental procedure, demonstrating the effectiveness of this approach in continual learning scenarios. Specifically, we employ the well-known permuted MNIST dataset, a progressively-permuted MNIST and CIFAR-10 benchmarks, and a non i.i.d. stream. Additionally, we validate the update scheme’s efficacy in an offline-learning scenario. By embracing the MY-based unifying view, we pave the way for advancements in optimization techniques for machine learning.

Data-driven Approximation of Cumulative Distribution Function Using Particle Swarm Optimization based Finite Mixtures of Logistic Distribution

This paper proposes a data-driven approximation of the Cumulative Distribution Function using the Finite Mixtures of the Cumulative Distribution Function of Logistic distribution. Since it is not possible to solve the logistic mixture model using the Maximum likelihood method, the mixture model is modeled to approximate the empirical cumulative distribution function using the computational intelligence algorithms. The Probability Density Function is obtained by differentiating the estimate of the Cumulative Distribution Function. The proposed technique estimates the Cumulative Distribution Function of different benchmark distributions. Also, the performance of the proposed technique is compared with the state-of-the-art kernel density estimator and the Gaussian Mixture Model. Experimental results on κ−μ distribution show that the proposed technique performs equally well in estimating the probability density function. In contrast, the proposed technique outperforms in estimating the cumulative distribution function. Also, it is evident from the experimental results that the proposed technique outperforms the state-of-the-art Gaussian Mixture model and kernel density estimation techniques with less training data.

Area-driven Boolean bi-decomposition by function approximation

Bi-decomposition rewrites logic functions as the composition of simpler components. It is related to Boolean division, where a given function is rewritten as the product of a divisor and a quotient, but bi-decomposition can be defined for any Boolean operation of two operands. The key questions are how to find a good divisor and then how to compute the quotient. In this paper we select the divisor by approximation of the original function and then characterize by an incompletely specified function the full flexibility of the quotient for each binary operator. We target area-driven exact bi-decomposition and we apply it to the bi-decomposition of SOP forms. We report experiments that exhibit significant gains in literals of SOP forms when rewritten as bi-decompositions with respect to the product operator. This suggests the application of this framework to other logic forms and binary operations, both for exact and approximate implementations.

Analysis of in-store crowdshipping in a stochastic dynamic pickup-and-delivery system

AbstractTo meet the increasing demands of home delivery resulting from the proliferation of internet shopping and compounded by the rising expectation of fast fulfillment (often within hours of request), companies seek new delivery methods supported by information and communication technologies. In this study, we consider a dispatching platform with delivery capacity consisting of a dedicated fleet of vehicles complemented by crowdsourced couriers. We consider the crowdsourced couriers to be in-store customers who, upon checking out of the store, declare themselves available to deliver one or more requests from e-shoppers. The role of the collaborative platform is to aggregate e-shopper orders from the participating businesses and then manage the routing for the pickup of the corresponding products at the physical stores and the subsequent deliveries to the e-shoppers’ locations. We model this dynamic stochastic pickup-and-delivery problem as a Markov decision process to represent the uncertainty in the e-shopper requests and in-store crowdshipper appearances. We adapt a real-time insertion method enhanced with a cost function approximation to account for differences in the temporal availability of the dedicated vehicles and in-store crowdshippers. We conduct computational experiments to demonstrate the conditions under which in-store crowdshippers provide a cost benefit.

Bridging electronic and classical density-functional theory using universal machine-learned functional approximations.

The accuracy of density-functional theory (DFT) calculations is ultimately determined by the quality of the underlying approximate functionals, namely the exchange-correlation functional in electronic DFT and the excess functional in the classical DFT formalism of fluids. For both electrons and fluids, the exact functional is highly nonlocal, yet most calculations employ approximate functionals that are semi-local or nonlocal in a limited weighted-density form. Machine-learned (ML) nonlocal density-functional approximations show promise in advancing applications of both electronic and classical DFTs, but so far these two distinct research areas have implemented disparate approaches with limited generality. Here, we formulate a universal ML framework and training protocol to learn nonlocal functionals that combine features of equivariant convolutional neural networks and the weighted-density approximation. We prototype this new approach for several 1D and quasi-1D problems and demonstrate that functionals with exactly the same hyperparameters achieve excellent accuracy for a diverse set of systems, including the hard-rod fluid, the inhomogeneous Ising model, the exact exchange energy of electrons, the electron kinetic energy for orbital-free DFT, as well as for liquid water with 1D inhomogeneities. These results lay the foundation for a universal ML approach to approximate exact 3D functionals spanning electronic and classical DFTs.

Phase transition in the shape memory alloy NiTi described by the SCAN meta-GGA functional.

Climbing the ladder of density functional approximations has long been proposed to systematically improve the accuracy of first-principles calculations employing the density functional theory (DFT); however, up until now, the Perdew-Burke-Ernzerhof (PBE) functional at the second rung of the ladder, has dominated. Here, we present a study of the martensitic phase transition in NiTi based on abinitio molecular dynamics simulations and thermodynamic integration using the third-rung approximation of the strongly constrained and appropriately normalized (SCAN) meta-generalized gradient approximation (GGA). Although the predicted equilibrium lattice constants and formation enthalpy agree well with experimental data, the martensitic transition temperature (MTT) is overestimated by 94% (or 324K too high), compared with only 22% (77K) overestimation by PBE. The latent heat (q) is severely overestimated by SCAN as well. This deteriorated performance originates from the enlarged energy difference (ΔE) between the austenite and martensite phases, compared with the PBE result. Furthermore, a large variation (over 50meV/atom) in ΔE using different meta-GGAs indicates large variations in computed MTTs (∼400-500 K) and q, i.e., the predicted thermodynamic properties depend sensitively on the choice of meta-GGA. This would pose a serious problem when upgrading DFT calculations to the third rung. One possible solution is to add NiTi as a normsystem so that the revised SCAN meta-GGA could reproduce the PBE results of the relevant energy difference.

A Wasserstein perspective of Vanilla GANs

The empirical success of Generative Adversarial Networks (GANs) caused an increasing interest in theoretical research. The statistical literature is mainly focused on Wasserstein GANs and generalizations thereof, which especially allow for good dimension reduction properties. Statistical results for Vanilla GANs, the original optimization problem, are still rather limited and require assumptions such as smooth activation functions and equal dimensions of the latent space and the ambient space. To bridge this gap, we draw a connection from Vanilla GANs to the Wasserstein distance. By doing so, existing results for Wasserstein GANs can be extended to Vanilla GANs. In particular, we obtain an oracle inequality for Vanilla GANs in Wasserstein distance. The assumptions of this oracle inequality are designed to be satisfied by network architectures commonly used in practice, such as feedforward ReLU networks. By providing a quantitative result for the approximation of a Lipschitz function by a feedforward ReLU network with bounded Hölder norm, we conclude a rate of convergence for Vanilla GANs as well as Wasserstein GANs as estimators of the unknown probability distribution.

Quasi-interpolation for high-dimensional function approximation

Quasi-interpolation for high-dimensional function approximation