To investigate the state-to-state dynamics of the P(D2) + H2(XΣg+1)(v = 0, j = 0) reaction, we reconstructed the PH2(X2B1) potential energy surface (PES) using the permutation invariant polynomial neural network (PIP-NN) method based on 40 595 abinitio points. The aug-cc-pVQZ basis sets with Davidson correction were employed throughout the calculations. The reference wave function for the multi-reference configuration interaction calculations was constructed from a full valence complete-active-space self-consistent field wave function. To achieve higher PES accuracy, the double many-body expansion-scaled external correlation (DMBE-SEC) method was applied to extrapolate to the one-electron complete basis set limit, yielding a total root-mean square deviation of 2.4meV for the final NN-PES. Based on the refined PH2(X2B1) NN-PES, geometries, energies, and harmonic frequencies of stationary points were obtained and analyzed in detail, showing excellent agreement with other theoretical results. Subsequently, quantum time-dependent wave packet(TDWP) and quasi-classical trajectory (QCT) methods were utilized to compute reaction probability, integral cross section (ICS), differential cross section (DCS), product rovibrational distribution, and rate constant on the developed NN-PES. The TDWP results reveal rich resonant structures, while the QCT calculations provide a qualitatively correct description of the coarse-grained reaction cross sections. These results also demonstrate a threshold-type microscopic reaction mechanism characterized by dominant forward-backward scattering, attributable to long-lived collision complex formation. Collectively, these findings provide fundamental mechanistic insights into the microscopic reaction mechanism and dynamics of phosphorus chemistry in interstellar environments.

Effective pest management relies on early and accurate forecasting, yet current models struggle to capture regional specific complex relationship between weather conditions and pest incidence. This study addresses this gap by developing a robust crop pest forecasting model using the Group Method of Data Handling (GMDH) regression. We employed three decomposition techniques like Empirical Mode Decomposition (EMD), Ensemble Empirical Mode Decomposition (EEMD), and Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) to break down nonlinear data into Intrinsic Mode Functions (IMFs). These IMFs were then predicted using GMDH regression, incorporating weather variables as independent factors. The ensemble forecasts were constructed by aggregating the predicted IMFs. The study utilized pest incidence data from 2015 to 2023 for aphid, jassid, thrips, and whitefly pests. Findings indicated that the CEEMDAN-GMDH model outperformed others for forecasting the incidence of aphid, thrips, and whitefly pests, with improvements of 16.3%, 4.3%, and 13.6% over the univariate GMDH model, respectively. For jassid, the EEMD-GMDH model provided the best forecasts, despite CEEMDAN’s superior decomposition capabilities. The study concludes that integrating decomposition methods, with GMDH regression provides a more reliable tool for predicting pest incidences in cotton crops, thereby aiding in better pest management strategies.

Polynomial neural networks for biomedical data: Learnable orthogonal polynomials in image analysis and neural signal interpolation

In this study, interpretable and semi-interpretable soft computing techniques, including Group Method of Data Handling (GMDH), Gene Expression Programming (GEP), and Response Surface Methodology (RSM), were employed to develop predictive relationships for estimating the elastic modulus (E) and splitting tensile strength (STS) of concrete containing waste foundry sand (WFS). A sensitivity analysis was subsequently conducted to evaluate the influence of various parameters on these mechanical properties. The input variables considered were the ratio of waste foundry sand to cement (WFS/C), the ratio of waste foundry sand to fine aggregate (WFS/FA), the ratio of fine aggregate to total aggregate (FA/TA), the ratio of water to cement (W/C), the ratio of coarse aggregate to cement (CA/C), the ratio of superplasticizer to cement (1000SP/C) and the age of the sample. The results revealed that the GMDH model achieved the highest correlation coefficient (R) among all methods in predicting STS, exhibiting the lowest root mean square error (RMSE = 0.533) and mean absolute error (MAE = 0.434). Thus, the GMDH model demonstrated superior performance in predicting STS based on all statistical indicators. For predicting E, the RSM method provided the most accurate results, with the highest R = 0.978 and the lowest errors, RMSE = 1.372 and MAE = 1.088. Sensitivity analysis indicated that CA/C had the most significant effect on STS, while W/C had the greatest influence on E. Other parameters, WFS/C, CA/C, and FA/TA, showed relatively minor impacts on E.

Vehicle, industrial, and urban emissions remain major contributors to air quality degradation, affecting public health and the level of environmental cleanliness. Cost-effective specific pollutant estimation models, i.e., for carbon monoxide CO, carbon dioxide CO2, and ammonia NH3, are essential to tackle the practical challenge of high-resolution monitoring for reducing vehicle emissions in traffic. Existing model design methods, however, may be insufficient, particularly for peak time estimations, since such models are typically designed using gridding-based vehicle-specific power polynomial and non-optimized artificial neural networks. In this paper, we propose vehicle emission models of pollutants based on a Bayesian Monte Carlo (MC) Dropout-based robust data-driven gated recurrent unit (BMC-GRU) method to enhance estimation robustness and mitigate the overfitting problem in the deep learning network. Bayesian optimization determines the optimal architecture by efficiently and probabilistically searching the hyperparameters of the network, while MC-Dropout quantifies epistemic uncertainty through multiple stochastic forward passes during testing. Therefore, the proposed method improves the models’ calibrations and robustness to distribution shifts. For benchmarking, least squares-based first- and fourth-order polynomials, conventional long-short term memory (LSTM), and bidirectional LSTM (BiLSTM)-based estimation models are designed. The proposed method outperforms the mentioned state-of-the-art methods with strong robust estimation performance. The experimental results on multiple real-world vehicle datasets demonstrate that the proposed method significantly outperforms state-of-the-art approaches. The method presents a promising solution for uncertainty-aware vehicle emission modeling that is applicable to transportation systems.

In recent years, high-order neural networks (HONNs) have gained widespread usage due to their excellent approximation ability, incorporating both ∑ neurons and ∏ neurons. The dynamic ridge polynomial neural network (DRPNN), as a type of HONN, has been highly valued. However, during the training process of the DRPNN, slow convergence occurs due to the presence of unknown factors. Given this, we propose a method to enhance the DRPNN convergence speed by incorporating a penalty term into the error function and training it using the gradient method. The proposed method significantly improves the convergence speed during network training. To explain the training characteristics of the proposed method, three theorems, namely stability, monotonicity, and necessity, are theoretically proven. More precisely, the whole network is stable over time based on the ultimate training results. The loss function monotonically decreases. Regarding the necessity, the ultimate training results must be an optimal solution. Furthermore, under the condition of the same iteration steps and error accuracy, experiments concerning function simulation and image simulation are advanced to verify the theoretical effectiveness of the proposed method, with the advanced analysis of the corresponding generalizability being discussed.

In recent years, the H + F2 reaction has attracted much attention because of its important role in theory and in chemical lasers. The aim of this study was to report a highly accurate potential energy surface (PES) for this reaction and carry out a product-state resolved reaction dynamics study of the H + F2 (v0 = 0, j0 = 0, 1, 2) → HF + F reaction in a collision energy range [0.0, 1.0] eV with the time-dependent wave packet method. The HF2 PES was constructed using the permutation invariant polynomial neural network method with thousands of energy points calculated by the MRCI-F12+Q method with the AVTZ basis sets. The calculated results suggest that the rotational excitation of low-lying states of reactant F2 has little effect on the reaction. The vibrational level population inversion of the product HF is significant, and the HF product is most to be produced in the v' = 4-7 vibrational states. At lower collision energy, the product HF preferred to be populated in highly excited rotational states, but at higher collision energies, the rotational distributions roughly exhibit Gaussian function distributions. The calculated reaction rate constants agree with the experiments well but with a little underestimation. This study suggests that the reaction H + F2 is a wonderful prototype for chemical lasers, just like the more famous H2 + F reaction, agreeing well with the previous findings.

ABSTRACTIn recent years, the concentration of air pollutants has risen significantly due to urbanization and increased transportation. Erzincan province, due to its geographical location, is one of the region's most severely impacted by air pollution. Consequently, it is essential to accurately estimate air quality parameters to effectively analyze and manage the associated health risks. This study utilized a time series methodology to examine hourly air quality parameters, including particulate matter (PM10), sulfur dioxide (SO2), nitrogen dioxide (NO2), nitrogen oxides (NOx), and ozone (O3) throughout the year 2022 in Erzincan province. Various predictive techniques, such as feed‐forward neural networks, group method of data handling, long short‐term memory (LSTM), and least squares boosting tree methods, were employed for the analysis. For estimating air quality parameters, values delayed by up to 4 h—determined as the effective time period through correlation analysis—were utilized. Prediction performance was evaluated using eight different statistical parameters and visualization techniques. Notably, air quality parameters can be effectively estimated using historical data, with the LSTM model demonstrating superior performance compared to other models in this estimation. Furthermore, the most accurate predictions for O3 values were achieved using the LSTM algorithm, yielding an R2 of 0.93 and a root mean square error (RMSE) of 6.02. The findings of this study can aid policymakers in developing water resource management strategies, pollution control policies, and measures to combat climate change.

Comparative analysis of Polynomial Chaos Expansion and Neural Networks for inferring elastic modulus from beam deflection with Matérn covariance

Polynomial neural network for solving Caputo-conformable fractional Volterra–Fredholm integro-differential equation with three-point non-local boundary conditions

In this article, an investigation has been carried out on nonlinear time-space fractional reaction–diffusion equations governed by Caputo derivatives. A deep neural network approach based on shifted Legendre polynomial expansions has been developed to obtain the numerical solution of the considered model. The proposed architecture consists of a four-layer neural network-comprising input, hidden, and output layers—where Legendre polynomials of various degrees are employed as activation functions. The exact solution is encoded into the model through an appropriate forcing term, allowing the network to capture the underlying dynamics of the fractional system effectively. Furthermore, a detailed comparison with the classical finite difference method has been conducted, which highlights the superior accuracy and efficiency of the proposed approach. The high accuracy of the method has been demonstrated through absolute error plots and comparative numerical tables, confirming its effectiveness in solving time–space fractional partial differential equations.

The accumulation of mineral deposits on industrial equipment surfaces poses a major concern in a variety of processes. Gypsum (CaSO4·2H2O) is one of the most widely produced minerals in both natural and industrial environments. Currently, intelligent white-box models can serve as a suitable alternative to time-consuming and high-priced experiments, enabling the identification of possible gypsum scaling issues in the chemical and petroleum industries. In this regard, the current study focused on the development of robust mathematical correlations to estimate the solubility of gypsum in aqueous electrolyte solutions. For this purpose, three rigorous techniques of Genetic Programming (GP), Gene Expression Programming (GEP), and Group Method of Data Handling (GMDH) were implemented on two distinct data banks, including 2288 experimental data-points taken from previously published literature. Solution temperature (T), solution molecular weight (MW), and molal concentrations of monovalent, divalent, and trivalent compounds (mI, mII, and mIII) were the input/independent variables employed in the first data bank, whereas solution temperature (T), solution molecular weight (MW), and solution ionic strength (I) were included in the second data bank. The performance and accuracy of correlations were evaluated using various statistical indicators such as Mean Bias Error (MBE), Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Coefficient of Determination (R2). Following multiple statistical and graphical analyses on the novel correlations' outcomes, it was found that the correlation established by implementing the GMDH technique onto the first data bank (i.e., GMDH-1) performed significantly better than all other correlations, with MAE = 0.01095, RMSE = 0.01482, and R2 = 0.8508. The correlations obtained by applying the GEP and GMDH techniques to the second data bank (i.e., GEP-2 and GMDH-2) also revealed a satisfactory level of performance. By comparing the new correlations developed in this study with models reported in previous studies, a reasonable level of agreement was found.

Activation degree thresholds and expressiveness of polynomial neural networks

Over the past decades, the use of fiber-reinforced polymers (FRPs) for retrofitting existing structures has become a widespread practice. More recently, however, the use of FRP bars as the primary reinforcement in new structures—especially concrete columns—has attracted global attention. Various equations have been proposed to predict the compressive strength of columns reinforced with FRP bars. However, these equations typically estimate axial strength without directly accounting for eccentricity. In this study, additional important parameters are considered: eccentricity of axial load, types of longitudinal and transverse reinforcement, and column height. To this end, a dataset of 525 samples is compiled, and machine learning methods—including artificial neural networks (ANN), gene expression programming (GEP), the group method of data handling (GMDH), and multiple linear regression (MLR)—are employed. Among these, ANN yielded the best predictive performance with an R2 value of 0.974. Using the GEP, GMDH, and MLR methods, three predictive equations are proposed. Of these, the GMDH and GEP approaches demonstrate relatively high accuracy, with R2 values of 0.966 and 0.942, respectively. The proposed equations can be used to predict the strength of reinforced concrete (RC) columns under axial loading with or without eccentricity, for various cross-sectional shapes and different types of longitudinal and transverse reinforcement (steel and FRP).

Modeling and predicting heavy metal uptake by plants using organic amendments helps reduce metal concentrations in contaminated soils. This study examined the effects of 1% and 2% (W/W) biochar and urban waste compost on the growth and cadmium (Cd) and lead (Pb) uptake by Bromus tomentellus in contaminated soil. The highest plant height (34.0 cm) and biomass (30.0 g) occurred with 2% biochar, compared to 16.0 cm and 9.0 g in control. For Pb, the maximum bioconcentration factor (BCF) was 2.25 with 1% compost, and the highest translocation factor (TF) was 1.4 with 2% biochar. For Cd, both max BCF (3.40) and TF (1.4) were seen at 1% biochar. Metal uptake and transfer significantly correlated with biomass and soil factors such as fertility (N, P, and K), pH, sodium adsorption ratio (SAR), and organic matter (OM) (Mantel test: p = 0.1, r = 0.4). The Group Method of Data Handling (GMDH) model, with high accuracy (R 2 = 0.998), showed compost caused an initial rise then decline in Cd uptake, while biochar had the opposite effect. Pb uptake increased with compost up to 1.052%, peaking at 763.7 ppm, then decreased. The GMDH model can optimize biochar or compost levels to enhance metal uptake by plants in polluted soils.

The study uses the comparison of Group Method of Data Handling (GMDH)-type neural network algorithm with Vector error correction model to predict the value of FDI on the basis of accuracy of prediction and to model the determinants of FDI in Pakistan. It was estimated in the times series data during the time period 1971-2013 and the time period 2014-2018 has been utilized to predict FDI. The Augmented Dickey-Fuller (ADF) test suggests that all variables data are of I (1) and Johansen co-integration test proved nexus of the variables FDI and its determinants in the long run. Nonetheless, the error correction-term of the vector error correction model (VECM) indicates that around 3 % of cumulative disequilibrium was expunged in every year in Pakistan. This paper compares two new methods Group Method of Data Handling (GMDH) Neural Network Algorithm and Vector Error Correction Model (VECM) in an attempt to obtain the more useful one based on accuracy of analysis when making predictions. Whereas, the effectiveness of VECM is then compared with a non parametric GMDH-like type neural network. An evaluation of forecasting model accuracies in terms of FDI was done using Root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE) and Diebold Mariano (DM) test. The empirical outcomes are a clear indication that Group Method of Data Handling GMDH-type of neural network algorithm did very well in comparison with the vector error correction model based on forecasting.

Determination of physical and mechanical properties of agricultural products plays an important role in the usage areas of the products and industrial applications. Correct determination and evaluation of physical and mechanical properties of agricultural products is of critical importance in determining the quality, durability and usage potential of the product. In this study, the relationship between moisture content and friction coefficients of Samsoy variety soybean seed, which is a trial material, was determined in order to contribute to making correct decisions in industrial design and material selection. The central aim of this research is to expose with different moisture contents and friction surfaces well-accepted data-driven models to predict friction coefficients for soybean seed using different soft computing techniques. Determination of friction coefficient of agricultural products is important in terms of design and functionality of equipment used in post-harvest technologies and agricultural applications. In the study, 3 different moisture contents and five different friction surfaces (steel, stainless steel, galvanized sheet, PVC, court fabric) were used. Artificial neural network (ANN), adaptive neuro-fuzzy inference system (ANFIS), group method of data handling (GMDH) are used to predict of friction coefficients. The best accuracy values were recorded as GMDH 7-7-1 for seven input and 7-15-1 model for five input structures for kinetic and static friction that were calculated performance criteria R2 = 0.99-0.98, RMSE =0.00004-0.00006 , MSE = 0.00009 -0.00011, respectively. These selected the best models predicted which can be used in the soft computing techniques determined different conditions to estimating the friction coefficient for soybean seeds.

Introduction. Cryptocurrencies are gaining popularity among individuals, businesses, and financial institutions. They are used for various purposes, particularly to pay for goods and services. Selling goods and services for cryptocurrencies can help companies attract new customers, increase sales, expand market share, and more. This article explores whether cryptocurrencies today function as a means of payment similar to fiat money, and examines the risks faced by companies that accept cryptocurrencies for goods and services. While cryptocurrencies are approaching the fulfillment of the economic functions of money, they have not yet fully reached this level. Nonetheless, in many countries, cryptocurrencies can be used to pay for goods or services, exchanged for other currencies, and more. In Ukraine, some companies sell household appliances, tickets, fuel, and other goods and services for cryptocurrencies. Problem Statement. Cryptocurrency developers emphasize that it is an alternative, private form of digital money that is not issued by national governments or controlled by financial intermediaries such as banks. The National Bank of Ukraine notes that the complex legal nature of cryptocurrencies prevents them from being recognized as cash, foreign currency, electronic money, securities, or a monetary surrogate. Cryptocurrencies offer certain advantages over traditional money, such as reducing transaction costs. However, transactions involving cryptocurrencies also carry inherent risks. Purpose. The study identifies the main approaches to organizing the sale of goods and services for cryptocurrencies. Additionally, the article aims to identify the risks associated with the sale of goods and services for cryptocurrencies and outline ways to minimize these risks. Materials and methods. We employed various research methods, including historical and legal methods, which involve the study of the legislative framework surrounding cryptocurrency transactions, as well as the empirical method, which investigates different practices of selling goods and services for cryptocurrencies. This approach also helps in identifying the risks companies face when engaging in such activities. One of the primary risks associated with cryptocurrency transactions is significant fluctuations in their exchange rates, which can result in economic losses in the event of a sharp devaluation. To better understand the nature of the risks associated with using cryptocurrencies, we conducted a statistical analysis of fluctuations in the Bitcoin exchange rate and built a correlation model with other market indicators, such as the Nasdaq Composite index and the exchange price of silver, for the period from March 1, 2012, to February 29, 2024. Using the Group Method of Data Handling (GMDH), we identified a connection between Bitcoin’s value fluctuations and market indicators that differ in terms of technological orientation (Nasdaq Composite) and investment risk (silver). Results. More and more countries are legalizing cryptocurrencies. In Ukraine, however, legislation regarding cryptocurrencies is still in development, and the sale of goods and services for cryptocurrencies is treated similarly to barter agreements. Depending on market characteristics and the specifics of their business, sellers of goods and services choose between directly selling for cryptocurrencies or using third-party intermediaries. This raises the question of what risks sellers face when accepting cryptocurrencies and how to mitigate or reduce those risks, such as the risk of sharp devaluation. Our model reveals a connection between Bitcoin’s exchange rate and other market indicators, such as the Nasdaq Composite index and the price of silver. However, the potential risks associated with using cryptocurrencies as a means of payment warrant further exploration. The lack of a clear regulatory framework and consistent definitions also introduces uncertainty in cryptocurrency operations. In practice, varying definitions of cryptocurrencies can create additional risks, particularly regarding the taxation of income received in cryptocurrency. Therefore, selling goods and services via intermediaries and converting cryptocurrency into fiat money can help mitigate legal, financial, and tax risks for companies. Additionally, gove

This study utilizes a hybrid model that combines generalized autoregressive conditional heteroskedasticity (GARCH) with the group method of data handling (GMDH) model to forecast returns and volatility. We focus on the Korean real estate investment trust (REIT) market and compare our hybrid model with the traditional GARCH and GMDH single models. The GARCH (1,1) model effectively captures the volatility clustering phenomenon but cannot fully reflect nonlinear patterns. In contrast, the GMDH model is stronger in handling nonlinearities; therefore, the hybrid model combining GARCH and GMDH is expected to have a higher forecasting performance. This study empirically analyzes Korean REIT market data from the listing date of each stock to the end of December 2024. The results show that our GARCH-GMDH hybrid model outperforms the GARCH and GMDH single models in forecasting returns and volatility, especially in short-term forecasting; the hybrid model can also stably capture extreme volatility and volatility clustering. Analyzing the differences in forecasting performance by market capitalization, trading volume, and time horizon of REITs reveals that REITs with larger market capitalizations and higher trading volumes had higher forecasting accuracy. Conversely, newly listed REITs had higher forecasting accuracy than long-term REITs.

Developing an accurate model to estimate the Interfacial Tension (IFT) of CO2-brine mixtures is a challenging task because of the varying operating conditions. Although different physics-based and data-driven methods have been proposed to estimate the CO2-brine IFT, the estimation error is still high. This study aimed to employ the Group Method of Data Handling (GMDH) to develop a deep artificial neural network to accurately estimate the CO2-brine IFT including Na2SO4 salt within a wide range of temperature, pressure, and salt concentration conditions. The advantage of the proposed method is that it can automatically optimize the structure and parameters of the model during the training phase. Therefore, there is no need to do hyperparameter optimization before deploying the model. A databank, consisting of experimentally measured IFT samples by an in-house high-temperature, high pressure IFT measurement device and samples collected from the literature, was used to develop and test the proposed method. Results revealed the strong ability of the proposed method to estimate the CO2-brine IFT with a coefficient of determination of 0.95 on all samples. The proposed method was compared with four IFT estimation methods from the literature. It was shown that the proposed method significantly outperformed the other methods. The contribution of this work is to show the ability of the GMDH-based ANN to estimate the CO2-brine IFT that can be used to assess the success of CO2 capture and storage projects.