Diffusion-Guided graph generation for multi-view semi-Supervised classification.

PIT-Net: Physics-informed transformer with differentiable diffusion constraints for quantitative photoacoustic tomography of deep tissues.

A Crank-Nicolson ADI compact difference scheme for the two-dimensional tempered space-fractional diffusion equation

Modeling anomalous transport and pattern formation using coupled fractional reaction–diffusion equations

Inverse reconstruction of a spatial source term in a space-fractional diffusion equation via Nyström discretization

Under the microscope: microbial life at pore scale under extreme deep-environment conditions.

Graph adiabatic diffusion neural networks for distribution-shift breast tumor image classification.

Moiré superlattices have emerged as a highly tunable platform for exploring correlated bosonic states, such as the recently discovered dipole ladder arising from strong on-site exciton-exciton interactions. Although spectroscopic signatures of these dipole ladders are established, their influence on exciton transport has remained unknown. Here, we employ hyperspectral transient photoluminescence microscopy to directly visualize exciton dynamics in a WS2/WSe2 moiré superlattice and demonstrate that dipole ladder mediates highly efficient exciton drift. At high excitation densities, the formation of a transient dipolar ladder generates a steep internal potential gradient. This gradient triggers a transition from localized, diffusion-limited transport (with a diffusion coefficient of D ≈ 0.01 cm2 s-1) to rapid, drift-dominated flow, yielding an equivalent diffusion coefficient of 0.75 cm2 s-1. These results establish dipole ladders as an interaction-driven mechanism for controlling exciton flow in moiré superlattices.

In this paper, we solve the Cauchy problem for a loaded fractional diffusion equation in an infinite strip. The loaded term is defined as the trace of the fractional derivative of the desired solution on a continuous curve lying inside the domain. We consider all three cases of possible distribution of the order of differentiation in the loaded term (µ) and the order of the time-fractional derivative in the principal differential part of the equation (α). In the first case considered (α > µ), the problem under study is reduced to an integral equation. In the second case (α = µ), we obtain a functional equation. In the third case (α < µ), we are dealing with a differential equation. We show that the condition α > µ ensures the unique solvability of the problem under consideration. In the case of an essentially loaded equation (α ≤ µ), the problem may lose both uniqueness and solvability. In particular, it is shown that if α < µ, then the problem under consideration ceases to be uniquely solvable, and the corresponding homogeneous problem has infinitely many nontrivial solutions. Moreover, in this case, the solvability requires additional conditions that narrow the set of admissible input data.

Comparative numerical study of caputo time-fractional reaction–diffusion equations with applications in physical phenomena

This paper introduces a novel approach using physics-informed neural networks (PINNs) to simultaneously solve variable-order time fractional diffusion equations and infer the time-dependent fractional order from data. By embedding the governing equations into the neural network’s loss function, our method achieves high accuracy and flexibility, even with sparse or noisy data. We present a dual-network architecture where one network approximates the solution u(x,t) while another learns the fractional order . Numerical experiments demonstrate the effectiveness of our approach, achieving mean squared errors below 10−4 for solutions and 10−3 for fractional orders in smooth cases, while also handling noisy data and non-smooth orders robustly.

Coal seam gas content is a crucial parameter for ensuring coal mine safety and accurately assessing coalbed methane reserves. With the increasing demand for advancing gas control in mining faces, the depth of boreholes for gas content determination continues to extend, making it difficult to guarantee short sampling times. Consequently, existing gas loss compensation methods struggle to apply to prolonged exposure loss estimation. To address the limitations of current methods, which rely heavily on initial desorption data and are inadequate for long-duration loss estimation, a transient gas diffusion equation incorporating gas adsorption effects was established. The relationship between the dimensionless time number T and the dimensionless desorption quantity number Y was analyzed. Furthermore, novel loss compensation methods based on the tangent at the initial measurement point and the tangent at a dynamic point were proposed. The results indicate that the developed diffusion model can effectively characterize long-term gas diffusion behavior. The T-Y relationship conforms to a power-law function, but its exponent varies with the selected time interval; fixing the exponent leads to significant errors. Directly extrapolating the loss amount by fitting field desorption data to a power function introduces substantial inaccuracies. Therefore, the tangent at the initial point of the field desorption curve was used to substitute for the desorption curve during the exposure period for loss estimation (i.e., the Initial Measurement Point Tangent Method, CQSB). Comparative analysis revealed that the calculation error increases with longer coal sample exposure time, making this method suitable only for scenarios with short exposure times (less than 2 min). An improved method was proposed, which involves using the fitted power function to predict the slope at a dynamic point within the exposure period and then utilizing the tangent at this point to calculate the gas loss (i.e., the Dynamic Point Tangent Method, DQSB). This method enables accurate determination of gas loss under prolonged exposure conditions (3-30 min). This research outcome provides a new approach for estimating gas loss during deep hole drilling and sampling processes using either open or sealed core barrels. However, how to theoretically determine the location of the dynamic point without relying on experimental data, and whether the method proposed in this paper is applicable to lump coal samples containing fractures, require further research and experimental verification.

This paper explores the application of kernel learning methods for parameter prediction and evaluation in the algebraic multigrid method (AMG), focusing on several partial differential equation (PDE) problems. AMG is an efficient iterative solver for large-scale sparse linear systems, particularly those derived from elliptic and parabolic PDE discretizations. However, its performance heavily relies on numerous parameters, which are often set empirically, and different parameter settings can significantly impact the effectiveness of AMG. Traditional parameter optimization methods are either computationally expensive or lack theoretical support. To address this, we propose a Gaussian process regression (GPR)-based strategy to optimize AMG parameters and introduce evaluation metrics to assess their effectiveness. Trained on small-scale datasets, GPR predicts nearly optimal parameters, bypassing the time-consuming parameter sweeping process. We also use kernel learning techniques to build a kernel function library and determine the optimal kernel function through linear combination, enhancing prediction accuracy. In numerical experiments, we tested typical PDEs such as Poisson equation, Parabolic equation, Diffusion equation, and Helmholtz equation. Results show that GPR-predicted parameters match grid search results in iteration steps while significantly reducing computational time. A comprehensive analysis using metrics like mean squared error, prediction interval coverage, and Bayesian information criterion confirms GPR's efficiency and reliability. These findings validate GPR's effectiveness in AMG parameter optimization and provide theoretical support for AMG's practical application.

We analyze the existence of traveling waves for a degenerate reaction diffusion equation featuring flux limitation effects together with time delay. The approach adopted is the upper and lower solutions method. The main technical issue for the proof is to overcome the obstacle caused by the flux-limited nonlinear degenerate diffusion and the time delay.

Magnetic targeting of drug carriers is commonly studied at macroscopic scales, while its impact on drug transport across individual cell membranes remains poorly quantified. Here, we present a theoretical and numerical model of magnetically assisted drug transport across the membrane of a single tumor cell exposed to magnetic field gradients. Extracellular transport is described by an advection–diffusion equation that couples passive diffusion with magnetophoretic drift, whereas intracellular transport is governed by diffusion and first-order uptake kinetics. The cell membrane is modeled as a semi-permeable interface with finite permeability, providing explicit coupling between extracellular and intracellular domains. Assuming spherical symmetry, the coupled transport equations are solved using finite-difference schemes, with magnetic forcing represented through an effective drift velocity and interpreted using the magnetic Peclet number. To enable a controlled comparison between healthy and tumor cells, identical geometric, diffusive, and magnetic parameters are used, while biological differences are introduced solely through membrane permeability and intracellular uptake rates. By separating cumulative membrane delivery from cumulative intracellular uptake, the model resolves ambiguities arising from heterogeneous uptake kinetics. The results show that magnetophoretic drift enhances near-membrane drug accumulation and effective transmembrane flux without modifying intrinsic membrane properties. Magnetic targeting therefore acts as a transport amplifier, magnifying pre-existing biological differences and producing a larger model-predicted delivery advantage in tumor cells. Overall, the framework identifies the magnetic Peclet number as the key parameter governing the transition from diffusion-dominated to drift-enhanced cellular drug transport.

Analysis of power law fluid for nutritional transport in the human capillary.

Dust concentration distribution and variation in multi-level high ore passes during ore discharge

Numerical study on characteristics of discrete motion and equivalent heat diffusion in granular shear flow

Inverse source problem in a distributed-order time-space fractional diffusion equation with a variable sign time source

A high-order hybrid spectral-Isogeometric solver for time-fractional neutron diffusion equations