Differentiable Research Articles

An additive variant of the differential symbol maps

An additive variant of the differential symbol maps

A new filled function method for solving constrained global optimization problems

ABSTRACT Filled function methods have been considered as effective algorithms for solving global optimization problems. However, their effectiveness is greatly affected by the selection of parameters, the noncontinuous or non-differentiable properties of the constructed filled function. In addition, many of the constructed filled functions are only for unconstrained optimization problems, and they are unable to solve constrained optimization problems. In this paper, a new filled function is constructed for solving constrained global optimization problems. The new filled function has only one parameter which needs to be adjusted, and, when the objective functions and constrained functions are all continuously differentiable functions, the constructed filled function is also a continuously differentiable function. Then, the classical local optimization methods can be used to find a better minimum of the proposed filled function and a few parameter adjustments are needed. At last, a new filled function algorithm for constrained global optimization is developed based on the proposed filled function. The new algorithm is applied to several test examples. The results of the numerical experiments show that the new filled function algorithm is effective and efficient.

EPMITS: An efficient prediction method incorporating trends and shapes features for chemical process variables

With the transformation of industrial production digitization and automation, process monitoring has been an indispensable technical method to realize the safe and efficient production of chemical process. Accurate prediction of process variables in chemical process can indicate the possible system change to reduce the probability of abnormal conditions. Current popular deep learning prediction methods trained with MSE or its variants may exhibit limitations in extracting shape features of chemical process data. In this paper, we proposed an efficient prediction method incorporating trends and shapes features (EPMITS) for chemical process variables. Specifically, we introduced a novel differentiable loss function Efficient Shape Error (ESE) to quantify shape differences between two time series of equal length in chemical process data. Then we trained deep learning models with MSE and ESE as loss function by two steps in training stage, to effectively acquire both trend and shape features of chemical process data. The proposed method was evaluated by the Tennessee Eastman process datasets and a real fluid catalytic cracking dataset from a petrochemical company. The results indicate that EPMITS models exhibit high prediction accuracy and short model training time across various time scales. These findings demonstrate the considerable feasibility and significant potential of EPMITS for future fault prognosis applications.

Nonlinear conjugate gradient methods for unconstrained set optimization problems whose objective functions have finite cardinality

In this paper, we propose nonlinear conjugate gradient methods for unconstrained set optimization problems in which the objective function is given by a finite number of continuously differentiable vector-valued functions. First, we provide a general algorithm for the conjugate gradient method using Wolfe line search but without imposing an explicit restriction on the conjugate parameter. Later, we study two variants of the algorithm, namely Fletcher-Reeves and conjugate descent, using two different choices of the conjugate parameter but with the same line search rule. In the general algorithm, the direction of movement at each iterate is identified by finding out a descent direction of a vector optimization problem. This vector optimization problem is identified with the help of the concept of partition set at the current iterate. As this vector optimization problem is different at different iterations, the conventional conjugate gradient method for vector optimization cannot be straightly extended to solve the set optimization problem under consideration. The well-definedness of the methods is provided. Further, we prove the Zoutendijk-type condition, which assists in proving the global convergence of the methods even without a regular condition of the stationary points. No convexity assumption is assumed on the objective function to prove the convergence. Lastly, some numerical examples are illustrated to exhibit the performance of the proposed method. We compare the performance of the proposed conjugate gradient methods with the existing steepest descent method. It is found that the proposed method commonly outperforms the existing steepest descent method for set optimization.

Impacts of thermal sterilization on volatile substances in xiongzhang tofu seasoning packets according to electronic nose and gas chromatography-mass spectrometry

Seasoning packets, which are essential for pre-cooked Xiongzhang tofu dishes, contain odor substances that are crucial for providing food aromas because of their low molecular weights, boiling points, and volatilities. Identifying these volatile substances is crucial; however, the impact of sterilization on these compounds is not well understood. Therefore, we explored the effect of thermal sterilization on the volatile substance content of Xiongzhang tofu seasoning packets. Three different sterilization treatments were investigated using sensory evaluation, electronic noses, and gas chromatography-mass spectrometry (GC-MS) combined with microbial indicators, orthogonal partial least squares discriminant analysis, the odor activity value (OAV), and differential heat maps. Heat sterilization enhanced flavor richness and preference scores, with the microbial content meeting national standards. GC-MS analysis revealed that pasteurized samples had high similarity with non-sterilized samples, whereas high-temperature/high-pressure sterilized samples showed significant differences. OAV analysis revealed dominant fruity, floral, and spicy aromas, with significant differences in the characteristic compounds among samples. The main characteristic differences among the three treatments were attributed to compounds including (methoxymethyl) ethylene oxide, 2,3-butanediol, epoxypropane, methoxyacetone, methyl thioether, and formic acid. High-temperature/high-pressure sterilization may have facilitated the release of sulfur-containing substances. This study enriches our understanding of the impacts of thermal sterilization of the volatile substance content of seasoning packets in pre-cooked dishes.

Life-Cycle Gradient-Based Production Optimization Including Well-Shutoff Option with Least-Squares Support Vector Regression

Summary In this work, we focus on well-shutoff/well-control optimization, which enables shutting off a production well or injection well to be part of the well-control optimization within a net present value (NPV) formulation that includes the operation expenditures (OPEX) if the well is not economical to produce or inject. For this purpose, we formulate an objective function for the NPV by introducing a production time fraction as the design variable to shut off a production well over its total production life, divided into a fixed number of cycles. Unlike the previous studies, we use a shutoff period within each cycle instead of shutting it off with a bottomhole pressure (BHP) control, where the BHPs for each producer are part of the optimization variables. When BHPs are considered control optimization variables with an NPV having OPEX, the NPV could be discontinuous for BHPs. To avoid this problem, we built a continuously differentiable proxy function for NPV using least-square support vector regression (LS-SVR). We use linear equality constraints so that the sums of the lengths of the cycles at each producer and injector are equal to the life of the production. Thus, we do not need to truncate the size of the last cycle, as in the previous studies, which may lead to suboptimal solutions. We use a simulator-based optimization method with stochastic simplex approximate gradient (StoSAG) and a machine learning-based (LS-SVR) optimization method to solve such an optimization problem. We update the LS-SVR proxy during optimization so that the updated proxy remains predictive toward promising regions of search space during the optimization. We compare the performance of the proposed LS-SVR-based iterative sampling refinement (ISR) method with the StoSAG-based and the finite difference (FD)-based optimization methods. To demonstrate the applicability of our proposed methodologies, we consider a synthetic example of a waterflooding process in a tight oil reservoir with two water injectors and four producers. Results show that the LS-SVR-based optimization method is at least three to seven times more computationally efficient than the StoSAG-based optimization method using a high-fidelity numerical simulator. However, we observe that the size and sampling of the training data, as well as the selection of bound constraints for the well controls, influence the performance of the LS-SVR-based optimization method. Designing multiple shutoffs and making cycle lengths unknown are found to be ineffective compared to single shutoff cases, as they yield lower NPVs.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Addressing bias in bagging and boosting regression models.

As artificial intelligence (AI) becomes widespread, there is increasing attention on investigating bias in machine learning (ML) models. Previous research concentrated on classification problems, with little emphasis on regression models. This paper presents an easy-to-apply and effective methodology for mitigating bias in bagging and boosting regression models, that is also applicable to any model trained through minimizing a differentiable loss function. Our methodology measures bias rigorously and extends the ML model's loss function with a regularization term to penalize high correlations between model errors and protected attributes. We applied our approach to three popular tree-based ensemble models: a random forest model (RF), a gradient-boosted model (GBT), and an extreme gradient boosting model (XGBoost). We implemented our methodology on a case study for predicting road-level traffic volume, where RF, GBT, and XGBoost models were shown to have high accuracy. Despite high accuracy, the ML models were shown to perform poorly on roads in minority-populated areas. Our bias mitigation approach reduced minority-related bias by over 50%.

Effect of Ethanol on Brain Electrical Tissue Conductivity in Social Drinkers.

How the biophysics of electrical conductivity measures relate to brain activity is poorly understood. The sedative, ethanol, reduces metabolic activity but its impact on brain electrical conductivity is unknown. To investigate whether ethanol reduces brain electrical tissue conductivity. Prospective. Fifty-two healthy volunteers (aged 18-37 years, 22 females, 30 males). 3 T, T1-weighted, multi-shot, turbo-field echo (TFE); 3D balanced fast-field echo (bFFE). Brain gray and white matter tissue conductivity measured with phase-based magnetic resonance electrical properties tomography (MREPT) compared before and 20 minutes after ethanol consumption (0.7 g/kg body weight). Differential conductivity whole brain maps were generated for three subgroups: those with strong ( > 0.1 S/m; N = 33), weak (0.02 S/m ≤ ≤ 0.1 S/m; N = 9) conductivity decrease, and no significant response ( < 0.02 S/m, N = 10). Maps were compared in the strong response group where breath alcohol rose between scans, vs. those where it fell. Average breath alcohol levels were compared to the differential conductivity maps using linear regression. T-maps were generated (threshold P < 0.05 and P < 0.001; minimum cluster 48 mm3). Differential conductivity maps were compared with ANOVA. Whole-group analysis showed decreased conductivity that did not survive statistical thresholding. Strong responders (N = 33) showed a consistent pattern of significantly decreased conductivity ( > 0.1 S/m) in frontal/occipital and cerebellar white matter. The weak response group (N = 9) showed a similar pattern of conductivity decrease (0.02 S/m ≤ ≤ 0.1 S/m). There was no significant relationship with breath alcohol levels, alcohol use, age, ethnicity, or sex. The strong responders' regional response was different between ascending (N = 12) or descending (N = 20) alcohol during the scan. Ethanol reduces brain tissue conductivity in a participant-dependent and spatially dependent fashion. 1 TECHNICAL EFFICACY: Stage 2.

Some Classical Inequalities Associated with Generic Identity and Applications

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of γ and parameter ξ.

Comparison of Brain Function and Structure in Patients with Major Depression: A Systematic Review and Meta-Analysis of MRI-Based Data.

Depression is a common mental illness worldwide. Neuroimaging techniques, such as magnetic resonance imaging and functional magnetic resonance imaging, play an essential role in diagnosing and evaluating depression. This study is based on magnetic resonance imaging (MRI)-related research to explore the comparison of brain function and structure between patients with severe depression and normal individuals, and to conduct meta-analysis. We conducted searches in various databases such as PubMed, Web of Science, Embase, and Cochrane Library to obtain research data on comparing brain function and structure between patients with severe depression and healthy individuals. The search keywords included "Major Depressive Disorder", "Brain Function", "Brain Structure", "Depression", "MRI", and "Magnetic Resonance". The quality assessment was conducted using the bias risk assessment tool recommended by the Cochrane Collaborative Network. Literature was screened following the predetermined inclusion and exclusion criteria, and Anisotropic Effect-Size Seed-Based Differential Mapping (AES-SDM) was used for systematic meta-analysis. Regression analysis was performed on age, gender, disease duration, years of education, and treatment status. After a thorough screening process, 10 documents were selected for subsequent analysis. These studies consisted of 477 study subjects, including 231 depression patients and 246 healthy individuals. The proportion of women was 36%-75%, and the disease duration was 3-60 months. The patients in 4 documents had first attacks, and the patients in the other 6 documents had multiple attacks. The baseline conditions of the 10 included documents were consistent and comparable. None of the studies reported blinding methods, and none of the results had incomplete data. The Regional homogeneity (ReHo) levels in the left precuneus (BA7), lentiform nucleus (BA48), and left prefrontal lobe (BA32) were significantly increased in the depression group, with voxel numbers of 358, 116, and 181, respectively. Conversely, the left postcentral gyrus (BA4), left cerebellar area (hemispheric lobule I, IV/V, lingual gyrus, fusiform gyrus), left fusiform gyrus (BA30), and right cingulate gyrus (BA23) were significantly reduced, with voxel numbers of 17, 50, and 124, respectively. Furthermore, regression analysis showed that gender, age, disease duration, years of education, and disease severity were potential influencing factors, and the disease duration demonstrated the most significant impact on the left cingulate gyrus (SDM = 2.777). There are significant differences in brain function and structure between patients with major depression and healthy individuals. Furthermore, our findings reveal a substantial correlation between the severity of depressive symptoms and brain function and structure indicators. These findings provide novel research directions and ideas for the diagnosis and treatment of depression.

The Widom–Sobolev formula for discontinuous matrix-valued symbols

We prove the Widom–Sobolev formula for the asymptotic behaviour of truncated Wiener–Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum except when closing the asymptotics for twice differentiable test functions with Hölder singularities. The cut-off domains are allowed to have piecewise differentiable boundaries. In contrast to the case where the symbol is smooth in one variable, the resulting coefficient in the enhanced area law we obtain here remains as explicit for matrix-valued symbols as it is for scalar-valued symbols.

Z-scheme γ-Fe2O3/g-C3N4 in Photo-Fenton reaction for oxytetracycline degradation: Mechanism study and DFT calculation

With the urbanization and industrialization of the world, the pollution of water due to the misuse of antibiotics is increasing. In this study, a Z-scheme γ-Fe2O3/g-C3N4 composite photocatalyst was prepared, and the removal efficiency of oxytetracycline (OTC) reached 85.7% within 60 min under the Photo-Fenton reaction system higher than that of g-C3N4 and γ-Fe2O3. The composite has great anti-interference capability and sustainability, and the recyclability was demonstrated using hysteresis return lines data. The photoelectrochemical characterization was analyzed to investigate the effect of the building of Z-scheme heterojunction on the photogenerated carrier separation efficiency. The charge behavior of heterojunction interfaces was probed in conjunction with DFT calculations, and differential charge maps were used to visualize charge distributions to analyze possible charge transfer pathways. In addition, the active substances in the system were identified by ESR data and capture experiments. The results showed that ·OH was the dominant active substance for OTC degradation. Meanwhile, LC-MS data combined with Fukui function calculations led to the derivation of a possible degradation pathway dominated by ·OH. This research provides a reference for subsequent studies of non-homogeneous Photo-Fenton systems and offers a viable option for mitigating OTC pollution in water bodies.

On Some Positive Linear Operators Preserving the B^{-1}-Convexity of Functions

In the study, we first give an inequality that non-negative differentiable functions must satisfy to be B^{-1}-convex. Then, using the inequality, we show that the B^{-1}-convexity property of functions is preserved by Bernstein-Stancu operators, Sz'asz-Mirakjan operators and Baskakov operators. In addition, we compare the concepts B^{-1}-convexity and B-concavity of functions.

X-SLAM: Scalable Dense SLAM for Task-aware Optimization using CSFD

We present X-SLAM, a real-time dense differentiable SLAM system that leverages the complex-step finite difference (CSFD) method for efficient calculation of numerical derivatives, bypassing the need for a large-scale computational graph. The key to our approach is treating the SLAM process as a differentiable function, enabling the calculation of the derivatives of important SLAM parameters through Taylor series expansion within the complex domain. Our system allows for the real-time calculation of not just the gradient, but also higher-order differentiation. This facilitates the use of high-order optimizers to achieve better accuracy and faster convergence. Building on X-SLAM, we implemented end-to-end optimization frameworks for two important tasks: camera relocalization in wide outdoor scenes and active robotic scanning in complex indoor environments. Comprehensive evaluations on public benchmarks and intricate real scenes underscore the improvements in the accuracy of camera relocalization and the efficiency of robotic navigation achieved through our task-aware optimization. The code and data are available at https://gapszju.github.io/X-SLAM.

Rings of differentiable semialgebraic functions

Rings of differentiable semialgebraic functions

The multi-parameterized integral inequalities for multiplicative Riemann–Liouville fractional integrals

Using the multiplicative Riemann–Liouville fractional integrals, for ⁎⁎differentiable functions, we present a fractional integral identity together with multi-parameters. Relying on it, and the fact that the function f⁎⁎ is multiplicatively convex or (lnf⁎⁎)q is convex for q>1, in particular considering the case 0<q≤1, a series of three-point Newton–Cotes type inequalities are obtained in this article. To enable the readers to comprehend the results better, we provide three examples to support the findings obtained here. Some applications in special means are offered as well.

Differentiable, holomorphic, and analytic functions on complex Φ-algebras

Using the notion of order convergent nets, we develop an order-theoretic approach to differentiable functions on Archimedean complex Φ-algebras. Most notably, we improve the Cauchy-Hadamard formulas for universally complete complex vector lattices given by both authors in a previous paper in order to prove that analytic functions are holomorphic in this abstract setting.

New cyclic Jensen type bounds via weighted HH inequalities and Montgomery identity with application to information inequalities

In this article a new method is introduced to give weighted estimations of the differences of cyclic refinements of Jensen functionals. We utilize weighted form of Hermite–Hadamard (HH) inequality along with the approximations of Montgomery two- and one-point formulae with Peano type kernel along with the consequences of the n-times differentiable convex functions. As a result, we present new upper and lower bounds that are also verified with concrete examples to show the correctness of the bounds obtained for special cases. As a result, we provide estimations in terms of cyclic power means. We also improvise our results to give new uniform estimations for useful distance functions in information theory. Finally, some estimations of Zipf and Hybrid Zipf law are provided as well.

Truncated Taylor Solvers for Simple First-Order Ordinary Differential Equations

Certain exact solutions of first-order ordinary differential equations naturally become corresponding solvers for simple firstorder ordinary differential equations whose form is the equality of a twice continuously differentiable function of the dependent variable to the derivative with respect the independent variable of the dependent variable. When the twice continuously differentiable function of the dependent variable is replaced by its truncated Taylor expansions through second order about its initial value, the resulting first-order ordinary differential equations have exact solutions that naturally become corresponding solvers for those simple first-order ordinary differential equations