Search Results (59)

Physics-Informed Neural Networks (PINNs) provide a promising framework for solving partial differential equations (PDEs). By incorporating temporal causality, Causal PINN improves training stability in time-dependent problems. However, applying Causal PINN to higher-order nonlinear PDEs, such as the Cahn–Hilliard equation (CHE), presents notable challenges due to the inefficient utilization of temporal information. This inefficiency often results in numerical instabilities and physically inconsistent solutions. This study systematically analyzes the limitations of Causal PINN in solving the one-dimensional CHE. To resolve these issues, we propose a novel framework called APM (Adaptive Progressive Marching)-PINN that enhances temporal representation and improves model robustness. APM-PINN mainly integrates a progressive temporal marching strategy, a causality-based adaptive sampling algorithm, and a residual-based adaptive loss weighting mechanism (effective with the chemical potential reformulation). Comparative experiments on two one-dimensional CHE test cases show that APM-PINN achieves relative errors consistently near 10⁻³ or even 10⁻⁴. It also preserves mass conservation and energy dissipation better. The promising results highlight APM-PINN’s potential for the accurate, stable modeling of complex high-order dynamic systems. Full article

This paper is focused on developing a Galerkin finite element framework for the fractional Allen–Cahn equation with regularized logarithmic potential over the ${\mathbb{R}}^d$ ($d=1,2,3$) domain, where the regularization of the singular potential extends beyond the classical double-well formulation. A fully discrete finite element scheme is developed using a k-th-order finite element space for spatial approximation and a backward Euler scheme for the temporal discretization of a regularized system. The existence and uniqueness of numerical solutions are rigorously established by applying Brouwer’s fixed-point theorem. Moreover, the proposed numerical framework is shown to preserve the discrete energy dissipation law analytically, while a priori error estimates are derived. Finally, numerical experiments are conducted to verify the theoretical results and the inherent physical property, such as phase separation phenomenon and coarsening processes. The results show that the fractional Allen–Cahn model provides enhanced capability in capturing phase transition characteristics compared to its classical equation. Full article

In this article, we present a nonlocal Cahn–Hilliard (nCH) equation incorporating a space-dependent parameter to model microphase separation phenomena in diblock copolymers. The proposed model introduces a modified formulation that accounts for spatially varying average volume fractions and thus captures nonlocal interactions between distinct subdomains. Such spatial heterogeneity plays a critical role in determining the morphology of the resulting phase-separated structures. To efficiently solve the resulting partial differential equation, a Fourier spectral method is used in conjunction with a linearly stabilized splitting scheme. This numerical approach not only guarantees stability and efficiency but also enables accurate resolution of spatially complex patterns without excessive computational overhead. The spectral representation effectively handles the nonlocal terms, while the stabilization scheme allows for large time steps. Therefore, this method is suitable for long-time simulations of pattern formation processes. Numerical experiments conducted under various initial conditions demonstrate the ability of the proposed method to resolve intricate phase separation behaviors, including coarsening dynamics and interface evolution. The results show that the space-dependent parameters significantly influence the orientation, size, and regularity of the emergent patterns. This suggests that spatial control of average composition could be used to engineer desirable microstructures in polymeric materials. This study provides a robust computational framework for investigating nonlocal pattern formation in heterogeneous systems, enables simulations in complex spatial domains, and contributes to the theoretical understanding of morphology control in polymer science. Full article

We present a reduced-order battery management system (BMS) for lithium-ion cells in electric and hybrid vehicles that couples a physics-based single-particle model (SPM) derived from the Cahn–Hilliard phase-field formulation with a lumped heat-transfer model. A three-dimensional COMSOL^® 5.0 simulation of a LiFePO₄ particle produced voltage and temperature data across ambient temperatures (253–298 K) and discharge rates (1 C–20.5 C). Principal component analysis (PCA) reduced this dataset to five latent variables, which we then mapped to experimental voltage–temperature profiles of an A123 Systems 26650 2.3 Ah cell using a self-normalizing neural network (SNN). The resulting ROM achieves real-time prediction accuracy comparable to detailed models while retaining essential electrothermal dynamics. Full article

We propose an unconditionally stable computational algorithm that preserves the maximum principle for the three-dimensional (3D) high-order Allen–Cahn (AC) equation. The presented algorithm applies an operator-splitting technique that decomposes the original equation into nonlinear and linear diffusion equations. To guarantee the unconditional stability of the numerical solution, we solve the nonlinear equation using the frozen coefficient technique, which simplifies computations by approximating variable coefficients by constants within small regions. For the linear equation, we use an implicit finite difference scheme under the operator-splitting method. To validate the efficiency of the proposed algorithm, we conducted several computational tests. The numerical results confirm that the scheme achieves unconditional stability even for large time step sizes and high-order polynomial potential. In addition, we analyze motion by mean curvature in three-dimensional space and show that the numerical solutions closely match the analytical solutions. Finally, the robustness of the method is evaluated under noisy data conditions, and its ability to accurately classify complex data structures is demonstrated. These results confirm the efficiency and reliability of the proposed computational algorithm for simulating phase-field models with a high-order polynomial potential. Full article

The increasing interest in metamaterials stems from the ability to expand the design space for material properties by tailoring the material architecture. Spinodal decomposition-inspired topologies are an emerging class of minimal surface-based metamaterials with promising properties. The diffusion process of the binary mixture is modeled using the Cahn–Hilliard equation, which is typically approached with a statistical method (superimposition of Gaussian random fields) that is well founded only for its initial stages. In this work, a fast and efficient computational method based on the finite difference method is employed to simulate the entire 2D dynamic evolution. Next, the solution field is converted into a CAD model forming the metamaterial. The elastic properties are assessed using a computational homogenization in which both homogeneous displacement and periodic boundary conditions have been applied for comparison. Finally, results from experimental testing confirm the accuracy of the FE homogenization procedure developed here. Full article

Periodic pattern formation is a prominent phenomenon in chemical, physical, and geochemical systems. This phenomenon can arise from various processes, such as the reaction and mass transport of chemical species, solidification, or solvent evaporation. We investigated the formation of ring-banded spherulites of l-menthol using a thin liquid film in a Petri dish. We found that the film thickness and cooling rate strongly influence the generation of crystallization patterns. We performed two-dimensional numerical simulations using the Cahn–Hilliard model to support the experimentally observed trend on the dependence of the layer thickness on the periodicity of the generated macroscopic patterns. In a specific scenario, we observed the formation of rings consisting of needle-like crystals on the cover of the Petri dish. This phenomenon was due to the evaporation of the menthol and its subsequent crystallization. In addition to these findings, we created crystallization patterns by solvent evaporation (using tert-butyl alcohol, methyl alcohol, and acetone). Full article

In this paper, we conduct a numerical investigation into the influence of polynomial order on wave-front propagation in the Allen–Cahn (AC) equations with high-order polynomial potentials. The conventional double-well potential in these equations is typically a fourth-order polynomial. However, higher-order double-well potentials, such as sixth, eighth, or any even order greater than four, can model more complex dynamics in phase transition problems. Our study aims to explore how the order of these polynomial potentials affects the speed and behavior of front propagation in the AC framework. By systematically varying the polynomial order, we observe significant changes in front dynamics. Higher-order polynomials tend to influence the sharpness and speed of moving fronts, leading to modifications in the overall pattern formation process. These results have implications for understanding the role of polynomial potentials in phase transition phenomena and offer insights into the broader application of AC equations for modeling complex systems. This work demonstrates the importance of considering higher-order polynomial potentials when analyzing front propagation and phase transitions, as the choice of polynomial order can dramatically alter system behavior. Full article

Different researchers have analyzed effective computational methods that maintain the precision of Allen–Cahn (AC) equations and their constant security. This article presents a method known as the reduced-order model technique by utilizing kernel principle component analysis (KPCA), a nonlinear variation of traditional principal component analysis (PCA). KPCA is utilized on the data matrix created using discrete solution vectors of the AC equation. In order to achieve discrete solutions, small variations are applied for dividing up extraterrestrial elements, while Kahan’s method is used for temporal calculations. Handling the process of backmapping from small-scale space involves utilizing a non-iterative formula rooted in the concept of the multidimensional scaling (MDS) method. Using KPCA, we show that simplified sorting methods preserve the dissipation of the energy structure. The effectiveness of simplified solutions from linear PCA and KPCA, the retention of invariants, and computational speeds are shown through one-, two-, and three-dimensional AC equations. Full article

This paper aims to present a robust computational technique utilizing finite difference schemes for accurately solving time fractional reaction–diffusion models, which are prevalent in chemical and biological phenomena. The time-fractional derivative is treated in the Caputo sense, addressing both linear and nonlinear scenarios. The proposed schemes were rigorously evaluated for stability and convergence. Additionally, the effectiveness of the developed schemes was validated through various linear and nonlinear models, including the Allen–Cahn equation, the KPP–Fisher equation, and the Complex Ginzburg–Landau oscillatory problem. These models were tested in one-, two-, and three-dimensional spaces to investigate the diverse patterns and dynamics that emerge. Comprehensive numerical results were provided, showcasing different cases of the fractional order parameter, highlighting the schemes’ versatility and reliability in capturing complex behaviors in fractional reaction–diffusion dynamics. Full article

Within recent years, considerable progress has been made regarding high-performance solvers for partial differential equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely remains the status quo for scientists and engineers focusing on applying simulation tools to specific problems in practice. We attribute this growing technical gap to the increasing complexity and knowledge required to pick and assemble state-of-the-art methods. Thus, with this work, we initiate an effort to build a common taxonomy for the most popular grid-based approximation schemes to draw comparisons regarding accuracy and computational efficiency. We then build upon this foundation and introduce a method to systematically guide an application expert through classifying a given PDE problem setting and identifying a suitable numerical scheme. Great care is taken to ensure that making a choice this way is unambiguous, i.e., the goal is to obtain a clear and reproducible recommendation. Our method not only helps to identify and assemble suitable schemes but enables the unique combination of multiple methods on a per-field basis. We demonstrate this process and its effectiveness using different model problems, each comparing the resulting numerical scheme from our method with the next best choice. For both the Allen–Cahn and advection equations, we show that substantial computational gains can be attained for the recommended numerical methods regarding accuracy and efficiency. Lastly, we outline how one can systematically analyze and classify a coupled multiphysics problem of considerable complexity with six different unknown quantities, yielding an efficient, mixed discretization that in configuration compares well to high-performance implementations from the literature. Full article

In the fields of physics and engineering, it is crucial to understand phase transition dynamics. This field involves fundamental partial differential equations (PDEs) such as the Allen–Cahn, Burgers, and two-dimensional (2D) wave equations. In alloys, the evolution of the phase transition interface is described by the Allen–Cahn equation. Vibrational and wave phenomena during phase transitions are modeled using the Burgers and 2D wave equations. The combination of these equations gives comprehensive information about the dynamic behavior during a phase transition. Numerical modeling methods such as finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) are often applied to solve phase transition problems that involve many partial differential equations (PDEs). However, physical problems can lead to computational complexity, increasing computational costs dramatically. Physics-informed neural networks (PINNs), as new neural network algorithms, can integrate physical law constraints with neural network algorithms to solve partial differential equations (PDEs), providing a new way to solve PDEs in addition to the traditional numerical modeling methods. In this paper, a hard-constraint wide-body PINN (HWPINN) model based on PINN is proposed. This model improves the effectiveness of the approximation by adding a wide-body structure to the approximation neural network part of the PINN architecture. A hard constraint is used in the physically driven part instead of the traditional practice of PINN constituting a residual network with boundary or initial conditions. The high accuracy of HWPINN for solving PDEs is verified through numerical experiments. One-dimensional (1D) Allen–Cahn, one-dimensional Burgers, and two-dimensional wave equation cases are set up for numerical experiments. The properties of the HWPINN model are inferred from the experimental data. The solution predicted by the model is compared with the FDM solution for evaluating the experimental error in the numerical experiments. HWPINN shows great potential for solving the PDE forward problem and provides a new approach for solving PDEs. Full article

The formed morphology during phase separation is crucial for determining the properties of the resulting product, e.g., a functional membrane. However, an accurate morphology prediction is challenging due to the inherent complexity of molecular interactions. In this study, the phase separation of a two-dimensional model polymer solution is investigated. The spinodal decomposition during the formation of polymer-rich domains is described by the Cahn–Hilliard equation incorporating the Flory–Huggins free energy description between the polymer and solvent. We circumvent the heavy burden of precise morphology prediction through two aspects. First, we systematically analyze the degree of impact of the parameters (initial polymer volume fraction, polymer mobility, degree of polymerization, surface tension parameter, and Flory–Huggins interaction parameter) in a phase-separating system on morphological evolution characterized by geometrical fingerprints to determine the most influential factor. The sensitivity analysis provides an estimate for the error tolerance of each parameter in determining the transition time, the spinodal decomposition length, and the domain growth rate. Secondly, we devise a set of physics-informed neural networks (PINN) comprising two coupled feedforward neural networks to represent the phase-field equations and inversely discover the value of the embedded parameter for a given morphological evolution. Among the five parameters considered, the polymer–solvent affinity is key in determining the phase transition time and the growth law of the polymer-rich domains. We demonstrate that the unknown parameter can be accurately determined by renormalizing the PINN-predicted parameter by the change of characteristic domain size in time. Our results suggest that certain degrees of error are tolerable and do not significantly affect the morphology properties during the domain growth. Moreover, reliable inverse prediction of the unknown parameter can be pursued by merely two separate snapshots during morphological evolution. The latter largely reduces the computational load in the standard data-driven predictive methods, and the approach may prove beneficial to the inverse design for specific needs. Full article

Due to the significant value that hotel reviews hold for both consumers and businesses, the development of an accurate sentiment classification method is crucial. By effectively distinguishing the authenticity of reviews, consumers can make informed decisions, and businesses can gain insights into customer feedback to improve their services and enhance overall competitiveness. In this paper, we propose a partial differential equation model based on phase-field for sentiment analysis in the field of hotel comment texts. The comment texts are converted into word vectors using the Word2Vec tool, and then we utilize the multifractal detrended fluctuation analysis (MF-DFA) model to extract the generalized Hurst exponent of the word vector time series to achieve dimensionality reduction of the word vector data. The dimensionality reduced data are represented in a two-dimensional computational domain, and the modified Allen–Cahn (AC) function is used to evolve the phase values of the data to obtain a stable nonlinear boundary, thereby achieving automatic classification of hotel comment texts. The experimental results show that the proposed method can effectively classify positive and negative samples and achieve excellent results in classification indicators. We compared our proposed classifier with traditional machine learning models and the results indicate that our method possesses a better performance. Full article

Percolative memristive networks based on self-organized ensembles of silver and gold nanoparticles are synthesized and investigated. Using cyclic voltammetry, pulse and step voltage excitations, we study switching between memristive and capacitive states below the percolation threshold. The resulting systems demonstrate scale-free (self-similar) temporal dynamics, long-term correlations, and synaptic plasticity. The observed plasticity can be manipulated in a controlled manner. The simplified stochastic model of resistance dynamics in memristive networks is testified. A phase field model based on the Cahn–Hilliard and Ginzburg–Landau equations is proposed to describe the dynamics of a self-organized network during the dissolution of filaments. Full article