Search Results (210)

Investigating soil seepage considering free surface conditions under complex geological conditions is of great significance to ensure the safety of dams. In recent years, physics-informed deep learning (PINN) has become a cross-disciplinary hotspot for solving forward and inverse problems based on partial differential equations. However, the challenges in free surface simulation have confined the majority of current PINN research to seepage problems under fixed boundary conditions. To address the above issues, we propose a physics-informed deep learning-based approach for steady seepage in dams. In the proposed method, two different free surface approximation strategies are introduced to accommodate varying boundary conditions in the dam seepage problem. The first strategy approximates the free boundary by sampling points, while the second strategy approximates the free boundaries by an additional deep neural network. To validate the proposed methods, three benchmark cases with different boundary conditions have been conducted. The results indicate that the proposed approach effectively simulates steady seepage in dams. Both point-sampling and deep neural network-based free surface approximation strategies demonstrate high accuracy in predicting the location of the phreatic surface and the discharge of the seepage. Specifically, the prediction results are comparable in accuracy to analytical solutions and advanced numerical simulation methods. Full article

This work presents the first systematic development of the inverse scattering transform for matrix nonlinear Schrödinger equations in the case where the discrete spectrum has triple poles, under nonzero boundary conditions at infinity. These systems arise physically as reductions modeling spinor Bose-Einstein condensates with hyperfine spin $F=1$ and find applications in nonlinear optics. A uniformization variable is employed to map the underlying Riemann surface to the complex plane, enabling a complete characterization of the analyticity, symmetries, and asymptotic behaviors of the Jost functions and scattering data. Extending the established framework for simple and double poles, we show that $\mathrm{rank} P(x,t,z_n)=3$ requires a third-order zero of $\det a\left(z\right)$ at $z=z_n$, while $\mathrm{rank} P(x,t,z_n)=2$ necessitates a fourth-order zero—a nontrivial feature absent in lower-order cases. The discrete spectrum for both rank configurations is fully characterized, and the full singular behavior near a triple pole is derived, respecting the quartet symmetry $z_n$, $z_n^{*}$, $v{k}_0^2/z_n$, $v{k}_0^2/z_n^{*}$ imposed by the nonzero boundary conditions. Solving the resulting matrix Riemann-Hilbert problem with triple poles yields the potential reconstruction formula and, in the reflectionless case, explicit expressions for general N-triple-pole soliton solutions, with a detailed example for $N=1$ presented to illustrate the construction. Full article

A symbolic method is presented for examining the solvability and constructing the exact solution to boundary value problems for systems of linear ordinary loaded differential equations and loaded integro-differential equations with nonlocal boundary conditions. The method uses the inverse of the differential operator involved in the system of loaded differential or integro-differential equations. A solvability criterion based on the determinant of a matrix and an exact analytical matrix-form solution formula are presented. For the implementation of the method into computer algebra system software, two algorithms are provided. The effectiveness of the method is demonstrated by solving several problems. The theoretical and practical results obtained complement the existing literature on the subject. Full article

Traditional methods for geostressfield inversion face issues such as weak physical interpretability and insufficient generalization ability. This study pioneers the application of Physics-Informed Neural Network (PINN) to this problem, developing a data- and physics-driven inversion algorithm. The framework incorporates a constitutive-equation-based regularized loss function as a hard constraint during training to ensure physical consistency. To address boundary load uncertainty, two quantification approaches—Bayesian linear regression and surrogate model optimization—are proposed to establish 95% confidence intervals for boundary coefficients. Verification based on simple three-dimensional models and actual geological models of mines shows that PINN inversion achieves a mean absolute relative error as low as 0.0772%, with an error of 15.67% under sparse sampling conditions—significantly lower than the 31.07% error of the traditional Back propagation neural network. This demonstrates excellent robustness and data efficiency. In the practical engineering application of complex geological bodies, the average error of principal stress inversion is 9.35% with a minimum error of 0.137%. All inversion results fall within the permissible accuracy range of engineering, and the stress distribution conforms to basic laws, with an average error of 0.453 in the constitutive relation. Compared with BP neural network and multiple linear regression methods, it shows obvious accuracy advantages. This method provides a new solution for intelligent ground stress prediction with high accuracy, high efficiency, and strong physical interpretability, and also lays the foundation for early identification of geological disasters. Full article

Finite-difference approaches are widely employed to solve partial differential equations in numerous practical applications. However, their computational efficiency is often limited by the need to solve linear systems through matrix inversion or iterative solvers, a challenge that is particularly acute in high-dimensional problems. Consequently, there is a growing demand for methods that ensure both high accuracy and computational efficiency. To address the two-dimensional (2D) heat conduction problem, we propose a novel hybrid technique that integrates a fourth-order implicit compact finite-difference approach with the discrete sine transform (DST). The incorporation of the DST significantly reduces the computational burden associated with solving the heat conduction equation on large grids. Detailed numerical experiments were conducted to evaluate this solver for 2D heat conduction equations subject to homogeneous Dirichlet boundary conditions (DBCs). The results demonstrate that the proposed method not only achieves substantial reductions in computational cost but also maintains a high level of numerical accuracy. All numerical experiments were performed on a computer running MATLAB R2024b. Full article

Electrical impedance tomography (EIT) provides noninvasive, high-temporal-resolution imaging for medical and industrial applications. However, accurate image reconstruction remains challenging due to the severe ill-posedness and nonlinearity of the inverse problem, as well as the limited robustness of existing single-source learning-based methods in real measurement scenarios. To address these limitations, a data-constrained and physics-guided Multi-Source Conditional Diffusion Model (MS-CDM) is proposed for EIT image reconstruction. Unlike conventional conditional diffusion methods that rely on a single measurement or an image prior, MS-CDM utilizes boundary voltage measurements as data-driven constraints and incorporates coarse reconstructions as physics-guided structural priors. This multi-source conditioning strategy provides complementary guidance during the reverse diffusion process, enabling balanced recovery of fine boundary details and global topological consistency. To support this framework, a Hybrid Swin–Mamba Denoising U-Net is developed, combining hierarchical window-based self-attention for local spatial modeling with bidirectional state-space modeling for efficient global dependency capture. Extensive experiments on simulated datasets and three real EIT experimental platforms demonstrate that MS-CDM consistently outperforms state-of-the-art numerical, supervised, and diffusion-based methods in terms of reconstruction accuracy, structural consistency, and noise robustness. Moreover, the proposed model exhibits robust cross-system applicability without system-specific retraining under multi-protocol training, highlighting its practical applicability in diverse real-world EIT scenarios. Full article

This paper presents methods and applications of inverse heat conduction problems (IHCPs) that are ill-posed in the Hadamard sense. The IHCP solution allows for the determination of boundary conditions in the form of heat flux or temperature in places where measurement is impossible or difficult to perform. The applications of IHCP solutions to energy-intensive industrial processes, such as heat treatment and thermochemical treatment, are described. Examples are given of determining boundary conditions on the inner surface of the wall of a power boiler and piston machine, as well as on the surface of a gas turbine blade. It is noted that the application of IHCP solutions to the above-mentioned issues often requires simplification of the computational model, in particular, the method of stabilising the inverse problem (IP). For this purpose, quasi-regularisation of IP and machine learning are currently used. Methods with stabilising properties and neural networks were identified as a challenging and interesting direction for the development of IHCP solutions. Full article

As a mesh-free solving paradigm, Physics-Informed Neural Networks (PINNs) demonstrate potential in both forward and inverse problems by embedding physical equations into the loss function. However, they still face challenges in capturing the spatiotemporal evolution of complex physical processes. When applied to time-dependent complex flows, such as high-Reynolds-number cylinder flow, they often rely on supervised data, which is frequently difficult to obtain accurately in practice. To address these issues, this paper proposes a novel unsupervised solving framework—the Adaptive Hard-Constraint Physics-Informed Neural Network (AHC-PINN). This method integrates an adaptive sampling mechanism based on partial differential equation residuals with a hard-constraint strategy. By dynamically evaluating the contribution of collocation points to the loss and incorporating analytically embedded boundary constraints, it directs the network training entirely toward solving the governing equations. Using two-dimensional unsteady cylinder flow as a validation case, experimental results show that AHC-PINN significantly improves the prediction accuracy of wake evolution under unsupervised conditions. Its performance surpasses that of traditional soft-constraint PINNs by an order of magnitude and is even superior to methods using sparse supervised data. Furthermore, through analysis of the PDE loss and gradient distribution, the study explicitly identifies the impact of large-gradient regions on PINN training stability and prediction accuracy, providing a basis for subsequent optimization. Full article

The evaluation of building energy performance under dynamic conditions requires reliable estimates of the thermophysical properties of envelope components. In existing buildings, however, the properties of multilayer walls are often unknown or uncertain, limiting the applicability of detailed physical models. To address this issue, this study proposes an inverse modeling framework for identifying the equivalent thermophysical parameters of a multilayer wall through a simplified homogeneous one-dimensional conduction model. The equivalent parameters are determined by matching the inner-side dynamic thermal response of the homogeneous model to that of the actual multilayer structure under the same external excitation. The approach explicitly accounts for the role of inner boundary conditions, which govern both the identifiability of the equivalent parameters and the formulation of the inverse problem. Adiabatic, isothermal, and more general inner boundary conditions are analyzed to determine how many independent parameters can be reliably identified and which response variables should be used in the objective function. Synthetic datasets, generated via numerical simulations driven by real weather data, are first employed to assess the method and to quantify the effect of transient initialization. The framework is then applied to experimental measurements collected from a full-scale test room. The results show that, under adiabatic conditions, the wall dynamics can be accurately reproduced by identifying a single equivalent thermal diffusivity, whereas isothermal and near-isothermal conditions require the simultaneous estimation of thermal conductivity and volumetric heat capacity. Moreover, the analysis demonstrates that inverse formulations based on inner heat flux are significantly more robust than temperature-based formulations, particularly when the inner-surface temperature is weakly varying or tightly controlled, as commonly occurs in real buildings. In a nearly isothermal experimental case, the inverse identification failed (${EF}_T=-5.76$) when based on the inner-surface temperature, while it resulted in a better match (${EF}_q=0.63$) when based on the inner heat flux. Overall, the proposed framework provides a physically consistent and practically robust methodology for the dynamic thermal characterization of multilayer building walls using equivalent homogeneous models. Full article

Accurate inversion of the deep initial in situ stress field is a fundamental prerequisite for stability analysis of surrounding rock in underground engineering, roadway support design, and prevention and control of dynamic disasters. To address the problems of scarce in situ stress measurements in deep mining areas, the inability of conventional regression methods to capture the nonlinear characteristics of complex tectonic stress fields, and the tendency of traditional inversion algorithms to fall into local optima and overfitting, this paper proposes an intelligent inversion method based on support vector regression optimized by the artificial bee colony algorithm (ABC-SVR). The artificial bee colony algorithm is employed to adaptively optimize the core parameters of the SVR model, thereby enabling high-precision inversion of complex deep stress fields. Comparing the results with acoustic emission tests demonstrated that the ABC-SVR model significantly outperforms conventional SVR and backpropagation neural networks across various performance metrics. The inversion results show high consistency with the measured data, achieving a root mean square error (RMSE) of 1.25, a mean absolute percentage error (MAPE) of 4.16%, and a coefficient of determination (R²) of 0.908. This method can rapidly reconstruct high-precision initial in situ stress fields in deep unmined regions, providing highly reliable boundary conditions for numerical simulations and demonstrating significant engineering application potential. Full article

We study an inverse source problem for a semilinear diffusion equation involving a Caputo-type time-fractional derivative whose order is a function of time. The equation is considered in a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$, $d\ge 1$, and is supplemented with homogeneous Dirichlet boundary conditions. The source term is taken to be separable, $h\left(t\right)f\left(x\right)$, where the temporal component $h\left(t\right)$ is unknown. This quantity is to be identified from spatially localized measurements $m\left(t\right)$ of the solution. In this setting, we establish existence and uniqueness results in suitable function spaces, thereby demonstrating the well-posedness of the corresponding inverse source problem. Full article

The Clausius–Mossotti (CM) factor underpins the theoretical description of dielectrophoresis (DEP) and is widely used in micro- and nano-scale systems for frequency-dependent particle and cell manipulation. It has further been proposed as an “electrophysiology Rosetta Stone” capable of linking DEP spectra to intrinsic cellular electrical properties. In this paper, the mathematical foundations and interpretive limits of this proposal are critically examined. By analyzing contrast factors derived from Laplace’s equation across multiple physical domains, it is shown that the CM functional form is a universal consequence of geometry, material contrast, and boundary conditions in linear Laplacian fields, rather than a feature unique to biological systems. Key modelling assumptions relevant to DEP are reassessed. Deviations from spherical symmetry lead naturally to tensorial contrast factors through geometry-dependent depolarisation coefficients. Complex, frequency-dependent CM factors and associated relaxation times are shown to inevitably arise from the coexistence of dissipative and storage mechanisms under time-varying forcing, independent of particle composition. Membrane surface charge influences DEP response through modified interfacial boundary conditions and effective transport parameters, rather than by introducing an independent driving mechanism. These results indicate that DEP spectra primarily reflect boundary-controlled field–particle coupling. From an inverse-problem perspective, this places fundamental constraints on parameter identifiability in DEP-based characterization. The CM factor remains a powerful and general modelling tool for micromachines and microfluidic systems, but its interpretive scope must be understood within the limits imposed by Laplacian field theory. Full article

This article presents an algorithm for solving the direct and inverse problem for a model consisting of a fractional differential equation with non-integer order derivatives with respect to time and space. The Caputo derivative was taken as the fractional derivative with respect to time, and the Riemann–Liouville derivative in the case of space. On one of the boundaries of the considered domain, a fractional boundary condition of the third kind was adopted. In the case of the direct problem, a differential scheme was presented, and a metaheuristic optimization algorithm, namely the Group Teaching Optimization Algorithm (GTOA), was used to solve the inverse problem. The article presents numerical examples illustrating the operation of the proposed methods. In the case of inverse problem, a function occurring in the fractional boundary condition was identified. The presented approach can be an effective tool for modeling the anomalous diffusion phenomenon. Full article

Over the last decade, numerical simulations and experiments have confirmed the existence of a novel class of vibrationally excited solid-particle attractors in cubic cavities containing a fluid in non-isothermal conditions. The diversity of emerging particle structures, in both morphology and multiplicity, depends strongly on the uni- or multi-directional nature of the imposed temperature gradients. The present study seeks to broaden this theoretical framework by further increasing the complexity of the thermal “information” coded along the external boundary of the fluid container. In particular, in place of the thermal inhomogeneities located in the center of otherwise uniformly cooled or heated walls, here, a cubic cavity with temperature boundary conditions satisfying the D₂h (in Schoenflies notation) or “mmm” (in Hermann–Mauguin notation) symmetry is considered. This configuration, equivalent to a bipartite vertex coloring of a cube leading to a total of 24 thermally controlled planar surfaces, possesses three mutually perpendicular twofold rotation axes and inversion symmetry through the cube’s center. To reduce the problem complexity by suppressing potential asymmetries due to fluid-dynamic instabilities of inertial nature, the numerical analysis is carried out under the assumption of dilute particle suspension and one-way solid–liquid phase coupling. The results show that a kaleidoscope of new particle structures is enabled, whose main distinguishing mark is the essentially one-dimensional (filamentary) nature. These show up as physically disjoint or intertwined particle circuits in striking contrast to the single-curvature or double-curvature spatially extended accumulation surfaces reported in earlier investigations. Full article

This work introduces a new category of proportional fractional integro-differential equations (PFIDEs) governed by functional boundary conditions. We derive verifiable sufficient criteria that guarantee the Ulam–Hyers Stability, existence and uniqueness of solutions to this problem. Our analytical approach leverages Babenko’s method to construct an inverse operator, which allows us to reformulate the differential problem into an equivalent integral equation. The analysis is then conducted using key mathematical tools, including contraction mapping principle of Banach, the Leray–Schauder alternative, and properties of multivariate Mittag–Leffler functions. The Ulam–Hyers Stability is rigorously examined to assess the system’s resilience to small perturbations. The applicability and effectiveness of the established theoretical results are demonstrated through two illustrative examples. This research provides a unified and adaptable framework that advances the analysis of complex fractional-order dynamical systems subject to nonlocal constraints. Full article