Function Approximation Research Articles

Accelerated Gradient Approach For Deep Neural Network-Based Adaptive Control of Unknown Nonlinear Systems.

Recent connections in the adaptive control literature to continuous-time analogs of Nesterov's accelerated gradient method have led to the development of new real-time adaptation laws based on accelerated gradient methods. However, previous results assume that the system's uncertainties are linear-in-the-parameters (LIP). To compensate for non-LIP uncertainties, our preliminary results developed a neural network (NN)-based accelerated gradient adaptive controller to achieve trajectory tracking for nonlinear systems; however, the development and analysis only considered single-hidden-layer NNs. In this article, a generalized deep NN (DNN) architecture with an arbitrary number of hidden layers is considered, and a new DNN-based accelerated gradient adaptation scheme is developed to generate estimates of all the DNN weights in real-time. A nonsmooth Lyapunov-based analysis is used to guarantee the developed accelerated gradient-based DNN adaptation design achieves global asymptotic tracking error convergence for general nonlinear control affine systems subject to unknown (non-LIP) drift dynamics and exogenous disturbances. A comprehensive set of simulation studies are conducted on a two-state nonlinear system, a robotic manipulator, and a complex 20-D nonlinear system to demonstrate the improved performance of the developed method. Our simulation studies demonstrate enhanced tracking and function approximation performance from both DNN architectures and accelerated gradient adaptation.

Point Spread Function Approximation of High-Rank Hessians with Locally Supported Nonnegative Integral Kernels

Point Spread Function Approximation of High-Rank Hessians with Locally Supported Nonnegative Integral Kernels

Efficient function approximation in enriched approximation spaces

Abstract An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can still provide highly accurate numerical approximations under reasonable conditions. In this paper we propose an efficient algorithm to compute such approximations. It is based on the AZ algorithm for overcomplete sets and frames, which simplifies in the case of an enriched basis. In addition, analysis of the original AZ algorithm and of the proposed variant gives constructive insights on how to achieve optimal and stable discretizations using enriched bases. We apply the algorithm to examples of enriched approximation spaces in literature, including a few nonstandard approximation problems and an enriched spectral method for a 2D boundary value problem, and show that the simplified AZ algorithm is indeed stable, accurate and efficient.

FAST: Filter-Adapted Spatio-Temporal Sampling for Real-Time Rendering

Stochastic sampling techniques are ubiquitous in real-time rendering, where performance constraints force the use of low sample counts, leading to noisy intermediate results. To remove this noise, the post-processing step of temporal and spatial denoising is an integral part of the real-time graphics pipeline. The main insight presented in this paper is that we can optimize the samples used in stochastic sampling such that the post-processing error is minimized. The core of our method is an analytical loss function which measures post-filtering error for a class of integrands --- multidimensional Heaviside functions. These integrands are an approximation of the discontinuous functions commonly found in rendering. Our analysis applies to arbitrary spatial and spatiotemporal filters, scalar and vector sample values, and uniform and non-uniform probability distributions. We show that the spectrum of Monte Carlo noise resulting from our sampling method is adapted to the shape of the filter, resulting in less noisy final images. We demonstrate improvements over state-of-the-art sampling methods in three representative rendering tasks: ambient occlusion, volumetric ray-marching, and color image dithering. Common use noise textures, and noise generation code is available at https://github.com/electronicarts/fastnoise.

Electronic heat conductivity in a two-temperature state

Heat transport in solids is governed by two fundamental contributions, atomic and electronic. The electronic energy transport in transient excited states is a defining factor in the problem of ultrafast material irradiation. Here, we calculate the electronic heat conductivity at elevated electron temperatures up to 40,000 K. We apply the novel combined method of tight binding formalism to calculate the electron-phonon contribution to the electronic heat conductivity, and the linear response theory (in the single-pole Ritchie-Howie loss function approximation) for its electron-electron counterpart, implemented in the hybrid code XTANT-3. It allows us to evaluate the electronic heat conductivity in a wide range of materials – fcc metals: Al, Ca, Ni, Cu, Sr, Y, Zr, Rh, Pd, Ag, Ir, Pt, Au, and Pb; hcp metals: Mg, Sc, Ti, Co, Zn, Tc, Ru, Cd, Hf, Re, and Os; bcc metals: V, Cr, Fe, Nb, Mo, Ba, Ta, and W; other metals: Sn, Ga, In, Mn, Te, and Se; semimetal graphite; semiconductors – group IV: Si, Ge, and SiC; group III-V: AlAs, AlP, GaP, GaAs, and GaSb; oxides: ZnO, TiO2, and Cu2O; and others: PbI2, ZnS, and B4C.

Characterization of Deformational Isomerization Potential and Interconversion Dynamics with Ultrafast X-ray Solution Scattering.

Dimeric complexes composed of d8 square planar metal centers and rigid bridging ligands provide model systems to understand the interplay between attractive dispersion forces and steric strain in order to assist the development of reliable methods to model metal dimer complexes more broadly. [Ir2 (dimen)4]2+ (dimen = para-diisocyanomenthane) presents a unique case study for such phenomena, as distortions of the optimal structure of a ligand with limited conformational flexibility counteract the attractive dispersive forces from the metal and ligand to yield a complex with two ground state deformational isomers. Here, we use ultrafast X-ray solution scattering (XSS) and optical transient absorption spectroscopy (OTAS) to reveal the nature of the equilibrium distribution and the exchange rate between the deformational isomers. The two ground state isomers have spectrally distinct electronic excitations that enable the selective excitation of one isomer or the other using a femtosecond duration pulse of visible light. We then track the dynamics of the nonequilibrium depletion of the electronic ground state population─often termed the ground state hole─with ultrafast XSS and OTAS, revealing a restoration of the ground state equilibrium in 2.3 ps. This combined experimental and theoretical study provides a critical test of various density functional approximations in the description of bridged d8-d8 metal complexes. The results show that density functional theory calculations can reproduce the primary experimental observations if dispersion interactions are added, and a hybrid functional, which includes exact exchange, is used.

Diffusing Wave Microrheology in Polymeric Fluids

Recently, there has been interest in determining the viscoelastic properties of polymeric liquids and other complex fluids by means of Diffusing Wave Spectroscopy (DWS). In this technique, light-scattering spectroscopy is applied to highly turbid fluids containing optical probe particles. The DWS spectrum is used to infer the time-dependent mean-square displacement and time-dependent diffusion coefficient D of the probes. From D, values for the storage modulus G′(ω) and the loss modulus G′′(ω) are obtained. This paper is primarily concerned with the inference of the mean-square displacement from a DWS spectrum. However, in much of the literature, central to the inference that is said to yield D is an invocation g(1)(t)=exp(−2q2X(t)2¯) of the Gaussian Approximation for the field correlation function g(1)(t) of the scattered light in terms of the mean-square displacement X(t)2¯ of a probe particle during time t. Experiment and simulation both show that the Gaussian approximation is invalid for probes in polymeric liquids and other complex fluids. In this paper, we obtain corrections to the Gaussian approximation that will assist in interpreting DWS spectra of probes in polymeric liquids. The corrections reveal that these DWS spectra receive contributions from higher moments X(t)2n¯, n>1, of the probe displacement distribution function.

Sliding Mode Speed Control in Synchronous Motors for Agriculture Machinery: A Chattering Suppression Approach

Synchronous motors have extended their presence in different applications, specifically in high-demand environments such as agronomy. These uses need advanced and better control strategies to improve energy efficiency. Within this context, sliding mode control has demonstrated effectiveness in electric machine control due to its advantages in robustness and quick adaptation to uncertain dynamic system disturbances. Nevertheless, this control technique presents the undesirable chattering phenomenon due to the discontinuous control action. This paper introduces a novel speed integral control scheme based on sliding modes for synchronous motors. This approach is designed to track smooth speed profiles and is evaluated through several numeric simulations to verify its robustness against variable torque loads. This approach addresses using electric motors for different applications such as irrigation systems, greenhouses, pumps, and others. Moreover, to address the chattering problem, different sign function approximations are evaluated in the control scheme. Then, the most effective functions for suppressing the chattering phenomenon through extensive comparative analysis are identified. Integral compensation in this technique demonstrates improvement in motor performance, while sign function approximations show a chattering reduction. Different study cases prove the robustness of this control scheme for large-scale synchronous motors. The simulation results validate the proposed control scheme based on sliding modes with integral compensation, by achieving chattering reduction and obtaining an efficient control scheme against uncertain disturbances in synchronous motors for agronomy applications.

Current density functional framework for spin-orbit coupling: Extension to periodic systems.

Spin-orbit coupling induces a current density in the ground state, which consequently requires a generalization for meta-generalized gradient approximations. That is, the exchange-correlation energy has to be constructed as an explicit functional of the current density, and a generalized kinetic energy density has to be formed to satisfy theoretical constraints. Herein, we generalize our previously presented formalism of spin-orbit current density functional theory [Holzer et al., J. Chem. Phys. 157, 204102 (2022)] to non-magnetic and magnetic periodic systems of arbitrary dimension. In addition to the ground-state exchange-correlation potential, analytical derivatives such as geometry gradients and stress tensors are implemented. The importance of the current density is assessed for band gaps, lattice constants, magnetic transitions, and Rashba splittings. In the latter, the impact of the current density may be larger than the deviation between different density functional approximations.

A Novel Consolidation Analysis Framework: Universal Function Approximators Regularized by Physical Principles

Analytical and numerical techniques are widely used to analyse and interpret soil consolidation problems. An important limitation is the requirement for significant geotechnical knowledge and expertise to derive ‘true’ solutions. This study proposes an alternative: a universal function approximator regularized with known physical principles. The proposed approach here advances previous work to solve one-dimensional (1D) consolidation considering both self-weight and large strains, as well as two-dimensional (2D) consolidation by vertical drains. Initial and boundary conditions of the studied consolidation problems are first strongly and weakly enforced for comparison. A neural network is adopted here as the function approximator. To boost prediction accuracy, a novel strategy is proposed to adaptively sample data points for training. An estimate of epistemic uncertainty is achieved using the confidence interval of ensembled multiple outputs. The results show that the proposed approach accurately predicts the behaviours of complex consolidation processes. Results also indicate that regularization using weak physical constraints can alleviate the imbalance of back-propagated gradients of different loss terms and, in turn, achieve higher accuracy. The proposed method is generic, mesh-free, more robust and can be applied to a wide range of geotechnical problems.

Thermodynamics of the gas-phase dimerization of formic acid: Fully anharmonic finite temperature calculations at the CCSD(T) and many DFT levels.

A proof-of-concept study is undertaken to demonstrate the utility of the machine learning combined with the thermodynamic perturbation theory (MLPT) to test the accuracy of electronic structure methods in finite-temperature thermodynamic calculations. As a test example, formic acid dimer is chosen, which is one of the systems included in the popular benchmark set S22 [Jurečka et al., Phys. Chem. Chem. Phys. 8, 1985-1993 (2006)]. Starting from the explicit molecular dynamics and thermodynamic integration performed at the PBE + D2 level, the MLPT is used to obtain fully anharmonic dimerization free and internal energies at the reference quality CCSD(T) level and 19 different density functional approximations, including GGA, meta-GGA, non-local, and hybrid functionals with and without dispersion corrections. Our finite-temperature results are shown to be both qualitatively and quantitatively different from those obtained using the conventional benchmarking strategy based on fixed structures. The hybrid functional HSE06 is identified as the best performing approximate method tested, with the errors in free and internal energies of dimerization being 36 and 41meV, respectively.

Multi-Year Cropland Mapping Based on Remote Sensing Data: A Case Study for the Khabarovsk Territory, Russia

Cropland mapping using remote sensing data is the basis for effective crop monitoring, crop rotation control, and the detection of irrational land use. Classification using Normalized Difference Vegetation Index (NDVI) time series from multi-year data requires additional time costs, especially when sentinel data are sparse. Approximation by nonlinear functions was proposed to solve this problem. Time series of weekly NDVI composites were plotted using multispectral Sentinel-2 (Level-2A) images at a resolution of 10 m for sites in Khabarovsk District from April to October in the years 2021 and 2022. Missing values due to the lack of suitable images for analysis were recovered using cubic polynomial, Fourier series, and double sinusoidal function approximation. The classes that were considered included crops, namely, soybean, buckwheat, oat, and perennial grasses, and fallow. The mean absolute percentage error (MAPE) of each class fitting was calculated. It was found that Fourier series fitting showed the highest accuracy, with a mean error of 8.2%. Different classifiers, such as the support vector machine (SVM), random forest (RF), and gradient boosting (GB), were comparatively evaluated. The overall accuracy (OA) for the site pixels during the cross-validation (Fourier series restored) was 67.3%, 87.2%, and 85.9% for the SVM, RF, and GB classifiers, respectively. Thus, it was established that the best result in terms of combined accuracy, performance, and limitations in cropland mapping was achieved by composite construction using Fourier series and machine learning using GB. Similar results should be expected in regions with similar cropland structures and crop phenological cycles, including other regions of the Far East.

Reinforcement Learning for Efficient Power Systems Planning: A Review of Operational and Expansion Strategies

The efficient planning of electric power systems is essential to meet both the current and future energy demands. In this context, reinforcement learning (RL) has emerged as a promising tool for control problems modeled as Markov decision processes (MDPs). Recently, its application has been extended to the planning and operation of power systems. This study provides a systematic review of advances in the application of RL and deep reinforcement learning (DRL) in this field. The problems are classified into two main categories: Operation planning including optimal power flow (OPF), economic dispatch (ED), and unit commitment (UC) and expansion planning, focusing on transmission network expansion planning (TNEP) and distribution network expansion planning (DNEP). The theoretical foundations of RL and DRL are explored, followed by a detailed analysis of their implementation in each planning area. This includes the identification of learning algorithms, function approximators, action policies, agent types, performance metrics, reward functions, and pertinent case studies. Our review reveals that RL and DRL algorithms outperform conventional methods, especially in terms of efficiency in computational time. These results highlight the transformative potential of RL and DRL in addressing complex challenges within power systems.

A Novel Extended Unscented Kalman Filter Is Designed Using the Higher-Order Statistical Property of the Approximate Error of the System Model

A Novel Extended Unscented Kalman Filter Is Designed Using the Higher-Order Statistical Property of the Approximate Error of the System Model

Impact of nonconvergence and various approximations of the partition function on the molecular column densities in the interstellar medium (Corrigendum)

Impact of nonconvergence and various approximations of the partition function on the molecular column densities in the interstellar medium <i>(Corrigendum)</i>

Limits of Bessel functions for root systems as the rank tends to infinity

We study the asymptotic behaviour of Bessel functions associated to root systems of type An−1 and type Bn with positive multiplicities as the rank n tends to infinity. In both cases, we characterize the possible limit functions and the Vershik-Kerov type sequences of spectral parameters for which such limits exist. In the type A case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of β-ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over F=R,ℂ,H (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.

Advanced Fusion of MTM-LSTM and MLP Models for Time Series Forecasting: An Application for Forecasting the Solar Radiation

Accurate time series forecasting has become increasingly important across various domains such as finance, energy, and medicine. This study introduces an innovative hybrid model that leverages the power of neural networks, precisely Many To Many LSTM (MTM LSTM) and Multilayer Perceptron (MLP), to improve time series forecasting accuracy. In this new combination, we trained network MTM LSTM to approximate the target at each step, and finally, we used network MLP to combine these approximations. To perform the evaluation, we made a forecast for the amount of solar energy radiation in the city of Mashhad, Iran. The experiment results concerning MSE and MAE showed that the proposed method with five lags outperforms the standard models.We hypothesize that MTM LSTM can effectively capture solar radiation's intricate temporal dependencies and nonlinearity. At the same time, MLP can enhance function approximation by modeling complex interactions, resulting in improved forecast accuracy. By employing the hybrid MTM LSTM and MLP model, we achieved improved accuracy in predicting solar energy radiation, which has significant implications for the renewable energy sector and its energy management and planning applications. This research advances time series forecasting techniques, highlighting the effectiveness of combining neural networks to address complex and dynamic patterns in time-dependent data.Overall, our findings underscore the potential and efficacy of the proposed hybrid model as a robust tool for accurate time series forecasting in various domains, supporting effective decision-making and planning processes.

Non‐Euclidean Sliced Optimal Transport Sampling

AbstractIn machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well‐dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real‐world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space 𝕊d, the hyperbolic plane ℍd, and the real projective plane ℙd. Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

Strong Converse Inequality for Weighted Approximation of Functions by the Szász–Mirakjan–Kantorovich Operator

Strong Converse Inequality for Weighted Approximation of Functions by the Szász–Mirakjan–Kantorovich Operator

Impact of Cobalt-Doping on the Optical, Magnetic, and Electronic Features of Hexagonal Cadmium Sulfide

Abstract CdS and Cd0.9Co0.1S samples were prepared under an N2 atmosphere. The structural analysis was conducted using X-ray diffraction. The structural and microstructure parameters were determined using Rietveld refinement method. The incorporation of cobalt ions into CdS matrix was confirmed by energy-dispersive spectroscopy and Fourier-transform infrared analysis. CdS sample has a non-magnetic feature while the Co-doped sample exhibited a magnetic behavior. The origin of magnetic property transformation has been investigated, revealing the emergence of ferromagnetic ordering and the conversion to a diluted magnetic semiconductor (DMS) with a calculated magnetic moment of 2.56 μB upon Co doping. We also investigated how this Cobalt-doping-driven transformation affected optical, photoluminescence, and electronic properties. These effects correlated with the emergence of hyper-deep defect states. Electronic properties were calculated using density functional theory (DFT) with the HSE06 hybrid functional approximation. The calculated energy bandgaps for both Co-doped and pure CdS were 2.13 and 2.12 eV, respectively, while experimental measurements from our UV analysis yielded values of 2.26 and 2.15 eV. DFT calculations were employed to explore the magnetic properties, absorption coefficients, refractive indices, real and imaginary dielectric components, and energy loss spectra in both samples.&amp;#xD;&amp;#xD;