First-order Derivatives Research Articles

Selection of optimal spectral features for leaf chlorophyll content estimation

Leaf chlorophyll content (LCC) is crucial for monitoring the physiological processes of crops. Many studies have utilized spectral features to develop regression models for accurate LCC estimation, enabling the quantitative assessment and evaluation of crop growth status. The selection of optimal spectral features and regression algorithms significantly affects the precision of LCC estimation. In this study, we compared and analyzed the optimal spectral features for LCC estimation, as well as the consistency of machine learning methods across different crop types, phenology periods, and sensors. First, we extracted various spectral features, including the original spectral features (OS), first-order derivative spectral features (FDS), original continuum-removed spectra (CR) along with their four related derivative spectral features, principal component variables derived from different spectral features, and highly correlated spectral features with LCC. These extracted spectral features were then employed to construct the LCC models using six common regression algorithms on different datasets. Finally, we analyzed the optimal combination of spectral features and regression algorithms for accurate LCC estimation considering various dimensions, such as crop type, phenological period, and sensor used in observation conditions. The results demonstrate that the combinations of the principal component variables of continuum-removed derivative reflectance with the top 10 correlations with LCC (PCA_CRDR_R) combined with Gaussian process regression (GPR) can be considered as the optimal choice for estimating LCC under diverse observation conditions at a canopy scale, and its R2 can reach 0.62 for sugar beet LCC estimation; thus providing valuable theoretical guidance for selecting appropriate spectral features for LCC estimation.

Pricing policies for printed and electronic books in a triple-channel supply chain

Purpose This study aims to investigate the pricing dynamics within a triple-channel supply chain. The publisher can sell printed books (p-books) through bookstores or online direct sales, and electronic books (e-books) are sold directly through the internet. The primary objectives include determining optimal wholesale and final prices for p-books, assessing the profitability of introducing e-books, comparing profits across channels and supply chain modes and identifying optimal demand volumes. Design/methodology/approach The research uses first-order derivatives and the Stackelberg game to analyze the pricing strategies. Two supply chain modes, centralized and decentralized, are considered, and various parameters are examined to understand their impact on prices, demand volumes and final sales profit. Findings The results indicate that the e-book is either not published or is introduced simultaneously with the printed version in both modes. In the decentralized mode, the wholesale price of a p-book is equivalent to the final price in the bookstore channel in the centralized mode. One channel among the three selling channels is used to maximize the total profit in the centralized supply chain, whereas all demand should be fulfilled through either online direct sales or e-book channels in the decentralized mode. Originality/value This paper introduces a comprehensive triple-channel book supply chain model, considering cross-price sensitivities and lag times for e-books. The study provides insights into the dynamics of the book industry and compares them with existing literature, contributing to a broader understanding of the pricing strategies in a triple-channel context.

Measurement of plasma electron energy distribution functions by Langmuir probe using a modified AC modulation method

Abstract In the diagnosis of vacuum discharge plasma using the Langmuir probe, the electron energy distribution function(EEDF) is closely related to the second derivative of current to voltage (d2I/dV2) on the Langmuir probe. While d2I/dV2 is very sensitive to probe noise, its accuracy directly affects the measurement results of the electron energy distribution function. In this paper, a modified AC modulation method is proposed to improve the measurement accuracy of the electron energy distribution function. First, two small AC signals of different amplitudes are modulated on the measured DC scanning voltage of the Langmuir probe, and the probe current signal is measured. Then, the fundamental amplitudes of the two AC signals are corrected by spectral analysis and using the all-phase Fast Fourier Transformation - Fast Fourier Transformation(apFFT-FFT) amplitude correction algorithm. Then the accurate first-order derivative of the probe current to voltage (dI/dV) is obtained using the AC modulation derivative correction algorithm, and finally the electron energy distribution function is obtained by numerically differentiating dI/dV to obtain d2I/dV2. The method of this paper is compared with the traditional data processing method through experiments, and the results show the superiority and effectiveness of the method.

Spectral Characteristics and Identification of Degraded Alpine Meadow in Qinghai–Tibetan Plateau Based on Hyperspectral Data

Grassland degradation poses a significant challenge to achieving the Sustainable Development Goals (SDGs) on the Qinghai–Tibetan Plateau (QTP). Effective monitoring of grassland degradation is essential for ecological restoration. Hyperspectral technology offers efficient and accurate identification of degradation. However, the influence of observation time, data analysis methods and classification techniques on the accuracy of identifying alpine grasslands remains unclear. In this study, the spectral reflectance of degraded alpine meadow, alpine meadow, alpine shrub and Tibetan barley was measured from May to September 2023 using a ground spectrometer in the northeastern QTP. First-order derivatives (FDR) and continuum removal were applied to the spectra, and characteristic parameters and vegetation indices were calculated. Support vector machine (SVM), random forest (RF), artificial neural network (ANN) and decision tree (DT) were then used to compare the classification accuracy between different months, transformation methods and characteristic parameters. The results showed that the spectral reflectance peaked in July, with significant differences in the near infrared (NIR) bands between alpine meadow and degraded alpine meadow. Alpine shrub and Tibetan barley showed greater differences in reflectance compared to other vegetation types, especially in the NIR bands. Data transformations improved reflectance and absorption characteristics in the NIR and visible bands. Indices such as DVI, RVI and NDGI effectively differentiated vegetation types. Optimal accuracy for the identification of degraded alpine meadow in July was achieved using FDR transformations and ANN or SVM for classification. This study provides methodological insights for monitoring grassland degradation on the QTP.

ZEROES: Robust Derivative-Based Demodulation Method for Optical Camera Communication

Most of Optical Camera Communication (OCC) systems benefit from the rolling shutter mechanism of Complementary Metal-Oxide Semiconductor (CMOS) cameras to record the brightness evolution of the Light-Emitting Diode (LED) through dark and bright strips within images. While this technique enhances the maximum achievable data rate, the main difficulty lies in the demodulation of the signal extracted from images, subject to blooming effect. Thus, two main approaches were proposed to deal with this issue, using adaptive thresholds whose value evolves according to amplitude changes or detecting signal variations with the first-order derivative. As the second method is more robust, a new demodulation method based on the detection of the zeros of the first-order derivative of the extracted signal was proposed in this paper. Obtained results clearly show an improvement in the extracted signal demodulation compared to other methods, achieving a raw Bit Error Rate (BER) of 10−3 around 50 cm in a Line-Of-Sight scenario, and increasing the maximum communication distance by 43.5%, reaching 330 cm in the case of a Non-Line-Of-Sight transmission.

Coal-gangue sound recognition using hybrid multi-branch CNN based on attention mechanism fusion in noisy environments

The coal-gangue recognition technology plays an important role in the intelligent realization of fully mechanized caving face and the improvement of coal quality. Although great progress has been made for the coal-gangue recognition in recent years, most of them have not taken into account the impact of the complex environment of top coal caving on recognition performance. Herein, a hybrid multi-branch convolutional neural network (HMBCNN) is proposed for coal-gangue recognition, which based on improved Mel Frequency Cepstral Coefficient (MFCC) as well as Mel spectrogram, and attention mechanism. Firstly, the MFCC and its smooth feature matrix are input into each branch of one-dimensional multi-branch convolutional neural network, and the spliced features are extracted adaptively through multi-head attention mechanism. Secondly, the Mel spectrogram and its first-order derivative are input into each branch of the two-dimensional multi-branch convolutional neural network respectively, and the effective time-frequency information is paid attention to through the soft attention mechanism. Finally, at the decision-making level, the two networks are fused to establish a model for feature fusion and classification, obtaining optimal fusion strategies for different features and networks. A database of sound pressure signals under different signal-to-noise ratios and equipment operations is constructed based on a large amount of data collected in the laboratory and on-site. Comparative experiments and discussions are conducted on this database with advanced algorithms and different neural network structures. The results show that the proposed method achieves higher recognition accuracy and better robustness in noisy environments.

Predicting leaf nitrogen content of coffee trees using the canopy hyperspectral reflectance feature bands, vegetation index and machine learning

ABSTRACT Leaf nitrogen content (LNC) is an essential indicator of crop nitrogen status. To rapidly and correctly estimate the LNC using hyperspectral remote sensing, the canopy hyperspectral reflectance of coffee trees treated with five levels of nitrogen fertilization in a greenhouse was obtained in this study. Five methods were used for hyperspectral data preprocessing, namely, Savitzky–Golay (SG) smoothing, a combination of SG and standard normal variate transformation (SG-SNV), a combination of SG and first-order derivative (SG-FD), a combination of SG and second-order derivative (SG-SD), and a combination of SG and multiplicative scatter correction (SG-MSC). Feature wavelengths were extracted using variables combination population analysis (VCPA), competitive adaptive reweighted sampling (CARS), and the combination (CARS-SPA) of CARS and successive projections algorithm (SPA). Vegetation indexes (VIs) were constructed and subjected to correlation analysis and variance inflation factor (VIF) analysis. Linear and nonlinear models including partial least squares regression (PLSR), back propagation neural network (BPNN), extreme learning machine (ELM), random forest regression (RFR), and support vector regression (SVR), were adopted to construct LNC retrieval models for coffee trees. The results indicated that SG-MSC could increase the signal-to-noise ratio of hyperspectral data well. The wavelengths selected by CARS-SPA were more relevant to LNC, and combined with ELM resulted in the best performance of LNC prediction (R2 P = 0.901, RMSEP = 0.825 g·kg−1, RPD = 3.229). Ten VIs were obtained through correlation analysis and VIF, and the VIs-based ELM prediction model also performed moderately well (R2 P = 0.814, RMSEP = 1.131 g·kg−1, RPD = 2.354). By comparing the coffee LNC prediction models established by different methods, two coffee LNC inversion models with better prediction accuracy were obtained, which provide a scientific basis for accurate diagnosis of coffee trees LNC, and are of great significance for optimizing the field management.

The structural elements controlling the subsurface mineralization as deduced from the integration of magnetic and self-potential methods, Wadi Khosh El-Daba area, southern Sinai, Egypt

Significant copper mineralization are observed to be associated, in most cases, with some radioactive anomalies related to increased uranium and thorium contents in Southern Sinai, Egypt. These mineralizations are related to intense post-magmatic, structurally controlled hydrothermal activities in many localities. Wadi Khosh El-Daba area is one of these locations which is mainly composed of monzogranites dissected by andesite dyke and pegmatite vein. This work deals with the application of ground magnetic and self-potential methods to establish the magnetic sources governing the mineralization, to interpret the significant trends that affected the study area, and to delineate the mineralized zones. The magnetic data were processed, enhanced and interpreted using advanced techniques, such as the first-order vertical derivative (FVD), the tilt derivative (TD), Euler deconvolution (ED), and source parameter image (SPI). The main structural lineaments/contacts that control the distribution of hydrothermal alterations and mineral distributions are oriented to the NW–SE, NE–SW, and NNW–SSE directions. The self-potential (SP) data show that there are numerous mineralized zones; most of them are distributed mainly along the structural lineaments deduced from the magnetic study. The quantitative interpretation revealed that the mineralized bodies are at shallow depths ranging from 5.6 m to 12.7 m, with half-width from 5 m to 13 m. These bodies are shallow to moderate dipping towards the south and north directions. Based on this study an anomaly map is originated to locate the preferred structural elements that may be controlling the mineralization in the area for further future exploration.

Feature Selection and Spectral Indices for Identifying Maize Stress Types

This study aims to identify different types of stress on maize leaves using feature selection and spectral index methods. Spectral data were collected from leaves under heavy metal, water, fertilizer stress, as well as under normal healthy conditions. Preprocessing steps such as continuum removal (CR), standard normal variable (SNV) transformation, multiple scattering correction (MSC), detrend correction (DT), and first-order derivative (FOD) were applied to the raw spectra. Various feature selection methods including ReliefF, chi-square test, recursive feature elimination (FRE), mutual information (MI), random forest (RF), and gradient boosting tree (GBT) were employed to determine the importance scores of different spectral bands, thus identifying sensitive spectral features capable of distinguishing various stress types. Spectral indices for stress type differentiation were constructed using label correlation method. Classification models were built using support vector machine (SVM), K-nearest neighbors (KNN), Gaussian naive Bayes (GNB), extreme gradient boosting (XGBoost), RF, and adaptive boosting (AdaBoost) algorithms. Results indicate that the characteristic spectral bands for differentiating stress types are primarily distributed around the red edge (near 700–800 nm) and water absorption valley (near 1900 nm). Spectral indices constructed using combinations of spectral bands around the near-infrared plateau absorption valley (near 1185 nm) and water absorption valley (near 1460 nm) effectively differentiate maize stress types. Among the modeling classification algorithms, RF and AdaBoost algorithms exhibited optimal performance, demonstrating high classification accuracy on both training and validation sets. These findings hold promise for providing new technical support for maize stress monitoring and diagnosis in agricultural production.

An Extension of Left Radau Type Inequalities to Fractal Spaces and Applications

In this study, we introduce a novel local fractional integral identity related to the Gaussian two-point left Radau rule. Based on this identity, we establish some new fractal inequalities for functions whose first-order local fractional derivatives are generalized convex and concave. The obtained results not only represent an extension of certain previously established findings to fractal sets but also a refinement of these when the fractal dimension μ is equal to one. Finally, to support our findings, we present a practical application to demonstrate the effectiveness of our results.

A comparative study on numerical methods for Fredholm integro-differential equations of convection-diffusion problem with integral boundary conditions

This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results.

Predicting cognitive load with EEG using Riemannian geometry-based features

Objective. We show that electroencephalography (EEG)-based cognitive load (CL) prediction using Riemannian geometry features outperforms existing models. The performance is estimated using Riemannian Procrustes Analysis (RPA) with a test set of subjects unseen during training.Approach. Performance is evaluated by using the Minimum Distance to Riemannian Mean model trained on CL classification. The baseline performance is established using spatial covariance matrices of the signal as features. Various novel features are explored and analyzed in depth, including spatial covariance and correlation matrices computed on the EEG signal and its first-order derivative. Furthermore, each RPA step effect on the performance is investigated, and the generalization performance of RPA is compared against a few different generalization methods.Main results. Performances are greatly improved by using the spatial covariance matrix of the first-order derivative of the signal as features. Furthermore, this work highlights both the importance and efficiency of RPA for CL prediction: it achieves good generalizability with little amounts of calibration data and largely outperforms all the comparison methods.Significance. CL prediction using RPA for generalizability across subjects is an approach worth exploring further, especially for real-world applications where calibration time is limited. Furthermore, the feature exploration uncovers new, promising features that can be used and further experimented within any Riemannian geometry setting.

Three-dimensional compact multi-resolution weighted essentially non-oscillatory reconstruction under the Arbitrary Lagrange–Euler framework

A third-order compact multi-resolution weighted essentially non-oscillatory (CMR-WENO) reconstruction method for three-dimensional (3D) hybrid unstructured grids is developed using the Arbitrary Lagrange–Euler framework. The finite volume method is used to discretize the governing equations, and some turbulent and moving boundary problems are simulated. Only one compact center stencil comprising the neighboring cells of each control cell is required to construct the polynomials in the algorithm. As a result, the number of stencils and stencil cells is significantly reduced when compared with the traditional WENO scheme. This simplifies the code and improves the robustness of the algorithm. By ensuring the cell average and first-order derivatives are consistent with that in stencil cells an over-determined system of equations can be used to reconstruct the polynomials. This system can then be solved using the compact least squares method to avoid an ill-conditioned coefficient matrix. Furthermore, a coupled implicit iteration strategy is used to solve for the unknown coefficients, so no extra determination is required for the derivatives of each control cell. The final interpolation function for discontinuities in the flow field is obtained using CMR-WENO to nonlinearly combine polynomials of different orders, which further improves the stability of the algorithm. The CMR-WENO can be implemented on 3D hybrid unstructured grids and can be used to simulate complex problems such as those involving turbulence and moving boundaries. Finally, the algorithm presented here is verified to be third-order accurate and to exhibit good robustness when used on several representative numerical examples.

Sensitivity analysis of frequency response functions with imaginary parts decoupling based on multicomplex-step perturbation

Imaginary perturbation is used in the complex step differentiation method to compute first-order derivatives, widely known as an effective approach for sensitivity analysis in structural dynamics. However, coupling of imaginary parts occurs in the damped frequency response functions when employing this method. To mitigate this coupling, a novel approach for sensitivity analysis based on multicomplex-step perturbation is proposed in this paper, for sensitivity analysis of Frequency Response Functions in structural dynamics. The structural parameters are perturbed in multicomplex domain, the dimensions of structural matrices are expanded using the Cauchy Riemann matrix representation, the equation of motion for sensitivity analysis in frequency domain is transformed to matrix operation in field of real numbers, imaginary term will not exist in the equation of motion for sensitivity analysis, the imaginary part of the frequency response function and the imaginary part of the perturbation are decoupled, the structural frequency response functions and the corresponding sensitivities are obtained from the dimension-expanded equation of motion. A truss structure and a solar wing are adopted to verify the accuracy of the proposed method. Results show that the sensitivity of FRFs can be effectively calculated using the proposed method. Compare to the finite difference method, the proposed method is not depended on the step-size selection procedure. The multi-order and mixed-order sensitivity matrices, especially Hessian matrix can also be obtained using the proposed method.

A detection method of oil content for maize kernels based on CARS feature selection and deep sparse autoencoder feature extraction

To achieve accurate, rapid, and non-destructive oil content determination in maize, a neural fitting model that combines feature selection and feature extraction is proposed. Competitive adaptive re-weighted sampling (CARS) is used to choose features, while the deep sparse autoencoder (DSAE) network is utilized to extract features. The 160 spectral data from the near-infrared spectra public datasets of maize served as experimental samples. Several preprocessing approaches were tested, and the first-order derivatives fared the best. To capture the various degrees of abstract features in spectral data, SSAE networks with two or three hidden layers are developed. The number of nodes in the network's hidden layer (30, 50, 70, 100, 200, 300, respectively) and the network's sparsity parameter (0.001, 0.004, 0.007, 0.01, 0.04, 0.07, 0.1, 0.2, …, 1) are tuned to generate partial least squares regression (PLSR) models for testing the feature learning effect. The highest performance comes from a three-layer network with 200 hidden layer nodes and a sparsity parameter of 0.04. This network extracts three levels of features from the preprocessed spectral data: F1, F2, and F3. Full_cars, F1_cars, F2_cars, and F3_cars are obtained using the CARS algorithm for variable selection on the preprocessed spectral data, F1, F2, and F3, respectively. They are then integrated into a final feature set to build a neural-fitting network (FNN). After training and testing, the test set has a correlation coefficient of 0.964205 and an mean square error (MSE) of 0.00636809. Comparing this model to support vector machine regression (SVR), PLSR, and gaussian process regression (GPR), the findings demonstrate that the model described in this study outperforms the others in terms of fitting accuracy and generalization performance. To summarize, this paper introduces a new regression analysis method based on near-infrared spectra.

基于透射光谱技术的马铃薯种薯黑心病检测研究

Black heart disease is one of the screening indicators of seed potatoes, which has a serious impact on the quality and yield of potato, and at present there are fewer non-destructive testing methods for internal defects of seed potatoes. This paper aims to utilize non-destructive testing technology to quickly identify qualified and black hearted seed potatoes, and then to protect yield and food security. In this paper, transmission spectroscopy system was utilized to collect the spectral data of 104 qualified seed potatoes and 104 black hearted seed potatoes in 450~940 nm band. Subsequently, four algorithms, namely Savitzky-Golay (SG), Standard Normal Variate (SNV), Multiplicative Scatter Correction (MSC) and First-order Derivative (FD), were utilized to pre-process the seed potatoes spectral data to improve the data noise ratio. Feature wavelength extraction was made using Competitive Adaptive Reweighted Sampling (CARS) and Successive Projections Algorithm (SPA) to enhance the sample data characteristics and improve the model interpretability. The construction of classification models for qualified and black hearted seed potatoes relied on two deep learning techniques, Convolutional Neural Networks (CNN) and Recurrent Neural Network (RNN), which were trained and tested for the feature bands respectively. The experimental results showed that SG-CARS-CNN was the optimal combination of classification algorithms, and the classification accuracies of both the training set and the test set reached 100%, which improved the accuracy of the test set by 3.85% compared with that of the traditional machine learning algorithms, and provided an accurate method for the rapid screening of qualified seed potatoes.

Extension of Milne-type inequalities to Katugampola fractional integrals

This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences.

Explore the fascinating realm of comparing metal melting curves by applying the equation of state and Lindemann's law

Our research, which is not just theoretical but also has practical implications, is a comprehensive study that aims to find the equation of state needed to calculate the pressure-dependent melting curves of metals. We have introduced a unique model for the melting curve, utilizing equations of state such as those proposed by Srivastava-Pandey and Kholiya. This model has been compared with Lindemann's model and available experimental data. It establishes the relationship between pressure, bulk modulus, pressure derivative of bulk modulus, and volume compression. Our findings show that a significant increase in melting temperature is directly linked to a substantial rise in bulk modulus and a gradual decrease in the first-order pressure derivative of bulk modulus. This study provides insights into understanding the effect of pressure on melting temperature. The present study finds that the equation of state of Srivastava-Pandey is more suitable for describing the melting temperature of metals, which could have significant implications for material science and engineering.

Aggregation and assessment of grape quality parameters with visible-near-infrared spectroscopy: Introducing a novel quantitative index

Grape quality is generally assessed by measuring individual indicators rather than utilizing a comprehensive metric. A flexible approach to aggregating multiple quality-related parameters of grapes is introduced in this study, and an aggregative index was proposed for the synthetical evaluation of grape quality. First, visible-near-infrared (Vis-NIR) spectral data of two grape varieties were obtained, and the physical and chemical quality indicators were measured as reference standards. The variation trends and correlations of the indicators were analyzed, and an aggregative quality indicator (AQI) was proposed using the min-max normalization and factor analysis methods. The maturity stages of grapes were categorized by analyzing the AQI values. The spectral data underwent preprocessing using multiplicative scatter correction (MSC), Savitzky-Golay smoothing (SG), first-order derivative (FD), SG+FD and de-trending (DT) methods. Feature wavelengths were extracted using uninformative variable elimination (UVE), competitive adaptive reweighted sampling (CARS), and the UVE-CARS algorithms. Three regression models for the AQI were established based on partial least squares regression (PLSR), support vector machine regression (SVR) and convolutional neural network (CNN). Results indicated that the PLSR combined with the FD+SG preprocessing and CARS-selected feature wavelengths achieved optimal performance for Cabernet Sauvignon grapes, with the coefficient of determination for the prediction set (Rp2)=0.972, for the calibration set (Rc2)=0.995, the root mean square error of prediction (RMSEP)=0.040 and residual predictive deviation (RPD)=6.064. In addition, for the Muscat Kyoho grape, CNN combined with DT preprocessed full-spectrum data yielded the best prediction for the AQI, with Rp2=0.989, Rc2=0.981, RMSEP=0.103 and RPD=9.482. This research demonstrated that the non-destructive prediction of grape aggregative qualities can be achieved through the combination of Vis-NIR spectroscopy and a chemometric analysis, offering a practical and comprehensive metric for determining grape maturity stages and optimal harvest times. This approach is particularly valuable for the wine industry, as it facilitates better decision-making and quality control.

Deep finite volume method for partial differential equations

In this paper, we introduce the Deep Finite Volume Method (DFVM), an innovative deep learning framework tailored for solving high-order (order ≥2) partial differential equations (PDEs). Our approach centers on a novel loss function crafted from local conservation laws derived from the original PDE, distinguishing DFVM from traditional deep learning methods. By formulating DFVM in the weak form of the PDE rather than the strong form, we enhance accuracy, particularly beneficial for PDEs with less smooth solutions compared to strong-form-based methods like Physics-Informed Neural Networks (PINNs). A key technique of DFVM lies in its transformation of all second-order or higher derivatives of neural networks into first-order derivatives which can be computed directly using Automatic Differentiation (AD). This adaptation significantly reduces computational overhead, particularly advantageous for solving high-dimensional PDEs. Numerical experiments demonstrate that DFVM achieves equal or superior solution accuracy compared to existing deep learning methods such as PINN, Deep Ritz Method (DRM), and Weak Adversarial Networks (WAN), while drastically reducing computational costs. Notably, for PDEs with nonsmooth solutions, DFVM yields approximate solutions with relative errors up to two orders of magnitude lower than those obtained by PINN. The implementation of DFVM is available on GitHub at https://github.com/Sysuzqs/DFVM.