Search Results (151)

Utilizing the Kolmogorov–Arnold representation theorem, KANs have emerged as a mathematically rigorous and easily interpretable alternative to traditional neural networks. These networks decompose high-dimensional functions into sums of univariate continuous functions using adaptive activation functions. Compared to MLPs, KANs exhibit superior or comparable performance in accuracy, parameter efficiency, and interpretability. Applications highlight the advantages of KANs in solving complex partial differential equations with enhanced convergence and uncertainty quantification, modeling dynamic systems in a meaningful manner, and making reliable forecasts in the areas of power systems, environmental monitoring, and demand prediction. Based on current research on KANs, they demonstrate a promising frontier in interpretable deep learning, with increasing influence across numerous interdisciplinary scientific and engineering fields. Full article

Many of the most fundamental observables—position, momentum, phase point, and spin direction—cannot be measured by an instrument that obeys the orthogonal projection postulate. Continuous-in-time measurements provide the missing theoretical framework to make physical sense of such observables. The elements of the time-dependent instrument define a group called the instrumental group (IG). Relative to the IG, all of the time dependence is contained in a certain function called the Kraus-operator density (KOD), which evolves according to a classical Kolmogorov equation. Unlike the Lindblad master equation, the KOD Kolmogorov equation is a direct expression of how the elements of the instrument (not just the total quantum channel) evolve. Shifting from continuous measurements to sequential measurements more generally, the structure of combining instruments in sequence is shown to correspond to the convolution of their KODs. This convolution promotes the IG to an involutive Banach algebra (a structure that goes all the way back to the origins of POVM and C*-algebra theory), which will be called the instrumental group algebra (IGA). The IGA is the true home of the KOD, similar to how the dual of a von Neumann algebra is the true home of the density operator. Operators on the IGA, which play the analogous role for KODs as superoperators play for density operators, are called ultraoperators and various important examples are discussed. Certain ultraoperator–superoperator intertwining relationships are also considered throughout, including the relationship between the KOD Kolmogorov equation and the Lindblad master equation. The IGA is also shown to have actually two distinct involutions: one respected by the convolution ultraoperators and the other by the quantum channel superoperators. Finally, the KOD Kolmogorov generators are derived for jump processes and more general diffusive processes. Full article

In this paper, the real-time decentralized integrated sensing, navigation, and communication co-optimization problem is investigated for large-scale mobile wireless sensor networks (MWSN) under limited energy. Compared with traditional sensor network optimization and control problems, large-scale resource-constrained MWSNs are associated with two new challenges, i.e., (1) increased computational and communication complexity due to a large number of mobile wireless sensors and (2) an uncertain environment with limited system resources, e.g., unknown wireless channels, limited transmission power, etc. To overcome these challenges, the Mean Field Game theory is adopted and integrated along with the emerging decentralized multi-agent reinforcement learning algorithm. Specifically, the problem is decomposed into two scenarios, i.e., cost-effective navigation and transmission power allocation optimization. Then, the Actor–Critic–Mass reinforcement learning algorithm is applied to learn the decentralized co-optimal design for both scenarios. To tune the reinforcement-learning-based neural networks, the coupled Hamiltonian–Jacobi–Bellman (HJB) and Fokker–Planck–Kolmogorov (FPK) equations derived from the Mean Field Game formulation are utilized. Finally, numerical simulations are conducted to demonstrate the effectiveness of the developed co-optimal design. Specifically, the optimal navigation algorithm achieved an average accuracy of $2.32\%$ when tracking the given routes. Full article

This work investigates the dynamical properties of the Kolmogorov–Petrovskii–Piskunov (KPP) equation. We begin by establishing its non-integrability through the Painlevé test. Using a traveling wave transformation, we reduce the equation to a planar dynamical system, which we identify as non-conservative. A subsequent bifurcation analysis, supported by Bendixson’s criterion, rules out the existence of periodic orbits and, thus, periodic solutions—a finding further validated by phase portraits. Furthermore, we classify the types and co-dimensions of the bifurcations present in the system. We demonstrate that under certain conditions, the system can exhibit saddle-node, transcritical, and pitchfork bifurcations, while Hopf and Bogdanov–Takens bifurcations cannot occur. This study concludes by systematically deriving a power series solution for the reduced equation. Full article

The method of particular solutions (MPS) has been widely applied for solving various types of partial differential equations. In this paper, the space–time collocation technique is implemented using MPS with polynomial basis functions (MPS-PBF) to solve the nonlinear Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation in both one and two dimensions. The Picard iteration method is used to deal with the nonlinearity of the problem. Four numerical examples are provided, and their results are compared with established methods to demonstrate the effectiveness of the proposed scheme. Full article

It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to solve mathematical models. This paper introduces a novel methodological framework based on the SO(3) Lie method to estimate time-dependent correlation matrices (correlation flows) among three variables that have chaotic, entropy, and fractal characteristics, from 11 April 2011 to 31 December 2024 for daily data; from 10 April 2011 to 29 December 2024 for weekly data; and from April 2011 to December 2024 for monthly data. So, it develops the stochastic SO(2) Lie method into the SO(3) Lie method that aims to obtain the correlation flow for three variables with chaotic, entropy, and fractal structure. The results were obtained at three stages. Firstly, we applied entropy (Shannon, Rényi, Tsallis, Higuchi) measures, Kolmogorov–Sinai complexity, Hurst exponents, rescaled range tests, and Lyapunov exponent methods. The results of the Lyapunov exponents (Wolf, Rosenstein’s Method, Kantz’s Method) and entropy methods, and KSC found evidence of chaos, entropy, and complexity. Secondly, the stochastic differential equations which depend on S² (SO(3) Lie group) and Lie algebra to obtain the correlation flows are explained. The resulting equation was numerically solved. The correlation flows were obtained by using the defined covariance flow transformation. Finally, we ran the robustness check. Accordingly, our robustness check results showed the SO(3) Lie method produced more effective results than the standard and Spearman correlation and covariance matrix. And, this method found lower RMSE and MAPE values, greater stability, and better forecast accuracy. For daily data, the Lie method found RMSE = 0.63, MAE = 0.43, and MAPE = 5.04, RMSE = 0.78, MAE = 0.56, and MAPE = 70.28 for weekly data, and RMSE = 0.081, MAE = 0.06, and MAPE = 7.39 for monthly data. These findings indicate that the SO(3) framework provides greater robustness, lower errors, and improved forecasting performance, as well as higher sensitivity to nonlinear transitions compared to standard correlation measures. By embedding time-dependent correlation matrix into a Lie group framework inspired by physics, this paper highlights the deep structural parallels between financial markets and complex physical systems. Full article

Physics-Informed Neural Networks (PINNs) represent a transformative approach to solving partial differential equation (PDE)-based boundary value problems by embedding physical laws into the learning process, addressing challenges such as non-physical solutions and data scarcity, which are inherent in traditional neural networks. This review analyzes critical challenges in PINN development, focusing on loss function design, geometric information integration, and their application in engineering modeling. We explore advanced strategies for constructing loss functions—including adaptive weighting, energy-based, and variational formulations—that enhance optimization stability and ensure physical consistency across multiscale and multiphysics problems. We emphasize geometry-aware learning through analytical representations—signed distance functions (SDFs), phi-functions, and R-functions—with complementary strengths: SDFs enable precise local boundary enforcement, whereas phi/R capture global multi-body constraints in irregular domains; in practice, hybrid use is effective for engineering problems. We also examine adaptive collocation sampling, domain decomposition, and hard-constraint mechanisms for boundary conditions to improve convergence and accuracy and discuss integration with commercial CAE via hybrid schemes that couple PINNs with classical solvers (e.g., FEM) to boost efficiency and reliability. Finally, we consider emerging paradigms—Physics-Informed Kolmogorov–Arnold Networks (PIKANs) and operator-learning frameworks (DeepONet, Fourier Neural Operator)—and outline open directions in standardized benchmarks, computational scalability, and multiphysics/multi-fidelity modeling for digital twins and design optimization. Full article

Objective: The aim of this study was to examine how transcranial electrical stimulation (tES) modulates intermuscular coherence (IMC) in sprinters and develop an interpretable neural network model for performance prediction. Methods: Thirty elite sprinters completed a randomized crossover trial involving three tES conditions: motor cortex stimulation (C1/C2), prefrontal stimulation (F3), and sham. Sprint performance metrics (0–100 m phase analysis) and lower-limb sEMG signals were collected. A Kolmogorov–Arnold Network (KAN) was trained to decode neuromuscular coordination–sprint performance relationships using IMC and time–frequency sEMG features. Results: Motor cortex tDCS increased 30–60 m sprint velocity by 2.2% versus sham (p < 0.05, η² = 0.25). γ-band IMC in key muscle pairs (rectus femoris–biceps femoris, tibialis anterior–gastrocnemius) significantly heightened under motor cortex stimulation (F > 4.2, p < 0.03). The KAN model achieved high predictive accuracy (R² = 0.83) through cross-validation, with derived symbolic equations mapping neuromuscular features to performance. Conclusions: Targeted tDCS enhances neuromuscular coordination and sprint velocity, while KAN provides a transparent framework for performance modeling in elite sports. Full article

We study a stochastic differential model for the dynamics of institutional corruption, extending a deterministic three-variable system—corruption perception, proportion of sanctioned acts, and policy laxity—by incorporating Gaussian perturbations into key parameters. We prove global existence and uniqueness of solutions in the physically relevant domain, and we analyze the linearization around the asymptotically stable equilibrium of the deterministic system. Explicit mean square bounds for the linearized process are derived in terms of the spectral properties of a symmetric matrix, providing insight into the temporal validity of the linear approximation. To investigate global behavior, we relate the first exit time from the domain of interest to backward Kolmogorov equations and numerically solve the associated elliptic and parabolic PDEs with FreeFEM, obtaining estimates of expectations and survival probabilities. An application to the case of Mexico highlights nontrivial effects: while the spectral structure governs local stability, institutional volatility can non-monotonically accelerate global exit, showing that highly reactive interventions without effective sanctions increase uncertainty. Policy implications and possible extensions are discussed. Full article

This study examines the spatio-temporal evolution of precipitation in the State of Guanajuato, Mexico, from 1981 to 2016 by analyzing monthly series from 65 meteorological stations. A rigorous data quality protocol was implemented, selecting stations with more than 30 years of continuous data and less than 10% missing values. Multiple Imputation by Chained Equations (MICE) with Predictive Mean Matching was applied to handle missing data, preserving the statistical properties of the time series as validated by Kolmogorov–Smirnov tests ($p=1.000$ for all stations). Homogeneity was assessed using Pettitt, SNHT, Buishand, and von Neumann tests, classifying 60 stations (93.8%) as useful, 3 (4.7%) as doubtful, and 2 (3.1%) as suspicious for monthly analysis. Breakpoints were predominantly clustered around periods of instrumental changes (2000–2003 and 2011–2014), underscoring the necessity of homogenization prior to trend analysis. The Trend-Free Pre-Whitening Mann–Kendall (TFPW-MK) test was applied to account for significant first-order autocorrelation (${\rho }_1 > 0.3$) present in all series. The analysis revealed no statistically significant monotonic trends in monthly precipitation at any of the 65 stations ($\alpha =0.05$). While 75.4% of the stations showed slight non-significant increasing tendencies (Kendall’s $\tau$ range: 0.0016 to 0.0520) and 24.6% showed non-significant decreasing tendencies ($\tau$ range: −0.0377 to −0.0008), Sen’s slope estimates were negligible (range: −0.0029 to 0.0111 mm/year) and statistically indistinguishable from zero. No discernible spatial patterns or correlation between trend magnitude and altitude ($\rho =-0.022$, $p>0.05$) were found, indicating region-wide precipitation stability during the study period. The integration of advanced imputation, multi-test homogenization, and robust trend detection provides a comprehensive framework for hydroclimatic analysis in semi-arid regions. These findings suggest that Guanajuato’s severe water crisis cannot be attributed to declining precipitation but rather to anthropogenic factors, primarily unsustainable groundwater extraction for agriculture. Full article

In this study, a new approach was developed for the estimation of optimum parameters (ODP), in terms of materials and design in livestock barns, and for optimal design. For this purpose, two thousand simulations were run using Monte Carlo (MC) techniques and Latin hypercube methods using the Energy Plus program on a 50-head closed dairy farm. In this study, the heat balance in the barn was adapted to Energy Plus using an innovative approach, using heat balance equations according to the ASHRAE Standard. First, data normality was determined using the Shapiro–Wilk (SW) and Kolmogorov–Smirnov (KS) tests. Data on thermal stress duration and energy consumption for dairy cattle welfare were estimated directly from the simulations, and sensitivity (SA) and uncertainty (UA) analyses were conducted. Furthermore, the statistical relationship between thermal comfort and energy consumption was determined using Pearson correlation. The predicted values obtained from the simulations were validated with barn values, and time-series overlay plots and histograms were generated. Furthermore, interpretations of the validation processes were made based on MBE, RSME, and R² statistical values. The study estimated an indoor thermal comfort temperature of 12 °C, and this value was taken into account in the innovatively developed simulations. The estimated optimum design parameters in the study resulted in energy reductions of 25% and 41% for walls and roofs, 48% and 19% for cooling and heating setpoint temperatures, 43% and 37% for window areas, and 75% and 40% for natural and mechanical ventilation, respectively. When the design parameters were evaluated holistically and analyzed in terms of average values, the new simulation model achieved approximately 50% energy savings. We believe that the newly developed approach will guide future planning for countries, the public, and private sectors to ensure animal welfare and reduce energy consumption. Full article

Accurate modeling of event sequences is valuable in domains like electronic health records, financial risk management, and social networks. Random time intervals in these sequences contain key dynamic information, and temporal point processes (TPPs) are widely used to analyze event triggering mechanisms and probability evolution patterns in asynchronous sequences. Neural TPPs (NTPPs) enhanced by deep learning improve modeling capabilities, but most suffer from limited interpretability due to predefined functional structures. This study proposes KANJDP (Kolmogorov–Arnold Neural Jump-Diffusion Process), a novel event sequence modeling method: it decomposes the intensity function via stochastic differential equations (SDEs), with each component parameterized by learnable spline functions. By analyzing each component’s contribution to event occurrence, KANJDP quantitatively reveals core event generation mechanisms. Experiments on real-world and synthetic datasets show that KANJDP achieves higher prediction accuracy with fewer trainable parameters. Full article

The soft-reset process, or a sequence of charge emissions from a floating storage node through a transistor biased in a subthreshold bias condition, is modeled by a master (Kolmogorov–Bateman) equation. The Coulomb interaction energy after each one-charge emission leads to a stepwise potential increase, giving correlated emission rates represented by Boltzmann factors. The governing probability distribution function is a hypoexponential type, and its cumulants describe characteristics of the single-charge Coulomb interaction at room temperature on a mesoscopic scale. The cumulants are further extended into a complex domain. Starting from three fundamental assumptions, i.e., the generation of non-degenerated states due to single-charge Coulomb energy, the Markovian property of each emission event, and the independence of each state, a moment function is identified as a product of mutually prime elements (algebraically termed as prime ideals) comprising the eigenvalues or the lifetimes of the emission states. Then, the algebraic structure of the moment function is found to be highly analogous to that of an integer uniquely factored into prime numbers. Treating the lifetimes as analogs of the prime numbers, two types of zeta functions are constructed. Standard analyses of the zeta functions analogous to the prime number problem or the Riemann Hypothesis are performed. For the zeta functions, the analyticity and poles are specified, and the functional equations are derived. Also, the zeta functions are found to be equivalent to the analytic extension of the cumulants. Finally, between the number of emitted charges and the lifetime, a logarithmic relation analogous to the prime number theorem is derived. Full article

Research on reliability control for enhancing power systems under random loads holds significant and undeniable importance in maintaining system stability, performance, and safety. The primary challenge lies in determining the reliability index while optimizing system parameters. To effectively address this challenge, we developed a novel intelligent algorithm and conducted an optimal reliability assessment for a Negative Stiffness Device (NSD) seismic isolation structure incorporating fractional-order damping. This algorithm combines the Gaussian Radial Basis Function Neural Network (GRBFNN) with the Particle Swarm Optimization (PSO) algorithm. It takes the reliability function with unknown parameters as the objective function, while using the Backward Kolmogorov (BK) equation, which governs the reliability function and is accompanied by boundary and initial conditions, as the constraint condition. During the operation of this algorithm, the neural network is employed to solve the BK equation, thereby deriving the fitness function in each iteration of the PSO algorithm. Then the PSO algorithm is utilized to obtain the optimal parameters. The unique advantage of this algorithm is its ability to simultaneously achieve the optimization of implicit objectives and the solution of time-dependent BK equations.To evaluate the performance of the proposed algorithm, this study compared it with the algorithm combines the GRBFNN with Genetic Algorithm (GA-GRBFNN)across multiple dimensions, including performance and operational efficiency. The effectiveness of the proposed algorithm has been validated through numerical comparisons and Monte Carlo simulations. The control strategy presented in this paper provides a solid theoretical foundation for improving the reliability performance of mechanical engineering systems and demonstrates significant potential for practical applications. Full article

The paper oresents the analytical construction of approximate solutions to the generalized Fisher–Kolmogorov equation in the complex domain. The existence and uniqueness of such solutions are established within an analytic domanin of the complex plane. The study employs techniques from complex function theory and introduces a modified version of the Cauchy majorant method. The principal innovation of the proposed approach, as opposed to the classical method, lies in constructing the majorant for the solution of the equation rather than for its right-hand side. A formula for calculating the analyticity radius is derived, which guarantees the absence of a movable singular point of algebraic type for the solutions under consideration. Special exact periodic solutions are found in elementary functions. Theoretical results are verified by numerical study. Full article