Zhonghua Qiao

Qiang Du, Lili Ju, Xiao Li et al.

The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete maximum principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadrature-based finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.

Dong Li, Zhonghua Qiao, Tao Tang

Recent results in the literature provide computational evidence that stabilized semi-implicit time-stepping method can efficiently simulate phase field problems involving fourth-order nonlinear dif- fusion, with typical examples like the Cahn-Hilliard equation and the thin film type equation. The up-to-date theoretical explanation of the numerical stability relies on the assumption that the deriva- tive of the nonlinear potential function satisfies a Lipschitz type condition, which in a rigorous sense, implies the boundedness of the numerical solution. In this work we remove the Lipschitz assumption on the nonlinearity and prove unconditional energy stability for the stabilized semi-implicit time-stepping methods. It is shown that the size of stabilization term depends on the initial energy and the perturba- tion parameter but is independent of the time step. The corresponding error analysis is also established under minimal nonlinearity and regularity assumptions.

Wangbo Luo, Zhonghua Qiao, Yanxiang Zhao

In this paper, we develop and analyze exponential time differencing (ETD) schemes for a phase-field model of multicomponent membranes proposed in our previous work \cite{luo2025ohta}, in which membrane deformation is governed by a force-balance phase-field equation and protein segregation is described by a membrane-associated Ohta-Kawasaki (OK) dynamics. For a fixed phase-field membrane, we introduce a geometry-adapted operator splitting method based on the localization function, which reformulates the surface OK dynamics into a form suitable for ETD integration. The resulting first- and second-order ETD schemes, combined with finite-difference spatial discretization, are rigorously proved to satisfy a discrete maximum-bound principle and unconditional energy stability. For the coupled system, we construct stabilized ETD schemes in an FFT-based spectral framework, treating stiff linear terms exactly and nonlinear mechanochemical couplings explicitly. A narrow-band implementation further reduces the computational cost by restricting surface calculations to the diffuse membrane region. Numerical experiments confirm the predicted temporal accuracy, maximum-bound preservation, and energy decay for the fixed-membrane OK problem, and demonstrate stable and efficient three-dimensional simulations of protein-driven pattern formation and membrane deformation.

Kelong Cheng, Zhonghua Qiao, Cheng Wang

In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to the physical model, and an ETD-based multistep approximation is used for time integration of the corresponding equation. In addition, a third order accurate Douglas-Dupont regularization term, in the form of $-A \dt^2 ϕ_0 (L_N) Δ_N^2 ( u^{n+1} - u^n)$, is added in the numerical scheme. A careful Fourier eigenvalue analysis results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the $\ell^\infty (0,T; H_h^1) \cap \ell^2 (0,T; H_h^3)$ norm, with the help of a careful eigenvalue bound estimate, combined with the nonlinear analysis for the NSS model. This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for $\varepsilon=0.02$ (up to $T=3 \times 10^5$) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

Jingwen Dai, Zhonghua Qiao, Dong Wang

This paper introduces a novel, robust, and computationally efficient framework for high-quality quadrilateral mesh generation on general two-dimensional domains. The core of the proposed approach is a novel method for computing cross fields by minimizing a modified and relaxed Ginzburg--Landau-type energy functional. A key innovation is the extension of the problem from the original, potentially complex domain to a larger regular computational domain. This extension transforms the central computational procedure into an iterative scheme that requires only two straightforward and efficient operations: linear diffusion solved globally via the Fast Fourier Transform (FFT) and point-wise normalization. Notably, our method eliminates the conventional need for generating an intermediate triangular mesh or solving complex nonlinear optimization problems on the irregular domain. We provide a rigorous theoretical analysis, proving that the proposed iterative algorithm guarantees unconditional monotonic decay of the objective functional. Comprehensive numerical experiments demonstrate the method's robustness across a wide range of complex geometries, its significant computational efficiency afforded by the FFT-based diffusion, and its consistent generation of high-quality quadrilateral meshes. This work presents a reliable and theoretically sound alternative to existing mesh generation techniques, with strong potential for practical applications in scientific computing.

Xiao Li, Zhonghua Qiao, Hui Zhang

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.

Chaoyu Liu, Zhonghua Qiao, Chao Li et al.

Traditional image processing methods employing partial differential equations (PDEs) offer a multitude of meaningful regularizers, along with valuable theoretical foundations for a wide range of image-related tasks. This makes their integration into neural networks a promising avenue. In this paper, we introduce a novel regularization approach inspired by the reverse process of PDE-based evolution models. Specifically, we propose inverse evolution layers (IELs), which serve as bad property amplifiers to penalize neural networks of which outputs have undesired characteristics. Using IELs, one can achieve specific regularization objectives and endow neural networks' outputs with corresponding properties of the PDE models. Our experiments, focusing on semantic segmentation tasks using heat-diffusion IELs, demonstrate their effectiveness in mitigating noisy label effects. Additionally, we develop curve-motion IELs to enforce convex shape regularization in neural network-based segmentation models for preventing the generation of concave outputs. Theoretical analysis confirms the efficacy of IELs as an effective regularization mechanism, particularly in handling training with label issues.

Yuwei Fan, Jun Li, Ruo Li et al.

We model the Knudsen layer in Kramers' problem by linearized high order hyperbolic moment system. Due to the hyperbolicity, the boundary conditions of the moment system is properly reduced from the kinetic boundary condition. For Kramers' problem, we give the analytical solutions of moment systems. With the order increasing of the moment model, the solutions are approaching to the solution of the linearized BGK kinetic equation. The velocity profile in the Knudsen layer is captured with improved accuracy for a wide range of accommodation coefficients.

Xiao Li, Zhonghua Qiao, Hui Zhang

A fast explicit operator splitting (FEOS) method for the molecular beam epitaxy model has been presented in [Cheng, et al., Fast and stable explicit operator splitting methods for phase-field models, J. Comput. Phys., submitted]. The original problem is split into linear and nonlinear subproblems. For the linear part, the pseudo-spectral method is adopted; for the nonlinear part, a 33-point difference scheme is constructed. Here, we give a compact center-difference scheme involving fewer points for the nonlinear subproblem. Besides, we analyze the convergence rate of the algorithm. The global error order $\mathcal{O}(τ^2+h^4)$ in discrete $L^2$-norm is proved theoretically and verified numerically. Some numerical experiments show the robustness of the algorithm for small coefficients of the fourth-order term for the one-dimensional case. Besides, coarsening dynamics are simulated in large domains and the $1/3$ power laws are observed for the two-dimensional case.

Chaoyu Liu, Zhonghua Qiao, Qian Zhang

In this paper, we propose an active contour model with a local variance force (LVF) term that can be applied to multi-phase image segmentation problems. With the LVF, the proposed model is very effective in the segmentation of images with noise. To solve this model efficiently, we represent the regularization term by characteristic functions and then design a minimization algorithm based on a modification of the iterative convolution-thresholding method (ICTM), namely ICTM-LVF. This minimization algorithm enjoys the energy-decaying property under some conditions and has highly efficient performance in the segmentation. To overcome the initialization issue of active contour models, we generalize the inhomogeneous graph Laplacian initialization method (IGLIM) to the multi-phase case and then apply it to give the initial contour of the ICTM-LVF solver. Numerical experiments are conducted on synthetic images and real images to demonstrate the capability of our initialization method, and the effectiveness of the local variance force for noise robustness in the multi-phase image segmentation.

Yongqiang Cai, Gaohang Chen, Zhonghua Qiao

The universal approximation property is fundamental to the success of neural networks, and has traditionally been achieved by training networks without any constraints on their parameters. However, recent experimental research proposed a novel permutation-based training method, which exhibited a desired classification performance without modifying the exact weight values. In this paper, we provide a theoretical guarantee of this permutation training method by proving its ability to guide a ReLU network to approximate one-dimensional continuous functions. Our numerical results further validate this method's efficiency in regression tasks with various initializations. The notable observations during weight permutation suggest that permutation training can provide an innovative tool for describing network learning behavior.

Chaoyu Liu, Zhonghao Li, Gaohang Chen et al.

Neural operators have achieved promising performance on partial differential equations (PDEs), but most existing models are built on fixed Eulerian coordinates. This mismatch between evolving physical structures and static coordinates creates spatial misalignment, leading to unnecessarily non-local operator mappings and reinforcing a smoothness preference near sharp transitions. Inspired by adaptive coordinate transformations in classical PDE analysis, we propose the Adaptive Coordinate Transform (ACT) block, a plug-and-play module for data-driven geometric adaptation in neural operators. ACT blocks resolve this structural limitation by learning adaptive coordinate systems within the operator learning pipeline. Specifically, given an input feature, the ACT block learns a coordinate transformation and represents the same feature under the transformed coordinates via differentiable sampling. This operation preserves the underlying signal while changing its spatial representation, equivalent to expressing the same physical quantity in different coordinate systems. By adapting the coordinate system to the data, ACT allows the network to better track evolving structures, reduce operator complexity, and dynamically focus on critical features to improve learning. We evaluate the proposed approach across diverse PDE benchmarks and multiple neural operator architectures. Experimental results demonstrate consistent and significant improvements in predictive accuracy, indicating that learning coordinate systems provides a powerful mechanism for enhancing operator learning.

Tao Cheng, Lili Ju, Zhonghua Qiao et al.

Deep learning has emerged as a compelling framework for scientific and engineering computing, motivating growing interest in neural network-based solvers for partial differential equations (PDEs). Within this landscape, network architectures with deterministic feature construction have become an appealing approach, offering both high accuracy and computational efficiency in practice. Among them, the transferable neural network (TransNet) is a special class of shallow neural networks (i.e., single-hidden-layer architectures), whose hidden-layer parameters are predetermined according to the principle of uniformly distributed partition hyperplanes. Although TransNet has demonstrated strong performance in solving PDEs with relatively smooth solutions, its accuracy and stability may deteriorate in the presence of highly oscillatory solution structures, where activation saturation and system conditioning issues become limiting factors. In this paper, we propose a generalized transferable neural network (GTransNet) for solving steady-state PDEs, which augments the original TransNet design with additional hidden layers while preserving its interpretable feature-generation mechanism. In particular, the first hidden layer of GTransNet retains TransNet's parameter sampling strategy but incorporates an additional symmetry constraint on the neuron biases, while the subsequent hidden layers omit bias terms and employ a variance-controlled sampling strategy for selecting neuron weights.

Qizhe Fan, Chaoyu Liu, Zhonghua Qiao et al.

Domain Generalized Semantic Segmentation (DGSS) focuses on training a model using labeled data from a source domain, with the goal of achieving robust generalization to unseen target domains during inference. A common approach to improve generalization is to augment the source domain with synthetic data generated by diffusion models (DMs). However, the generated images often contain structural or semantic defects due to training imperfections. Training segmentation models with such flawed data can lead to performance degradation and error accumulation. To address this issue, we propose to integrate inverse evolution layers (IELs) into the generative process. IELs are designed to highlight spatial discontinuities and semantic inconsistencies using Laplacian-based priors, enabling more effective filtering of undesirable generative patterns. Based on this mechanism, we introduce IELDM, an enhanced diffusion-based data augmentation framework that can produce higher-quality images. Furthermore, we observe that the defect-suppression capability of IELs can also benefit the segmentation network by suppressing artifact propagation. Based on this insight, we embed IELs into the decoder of the DGSS model and propose IELFormer to strengthen generalization capability in cross-domain scenarios. To further strengthen the model's semantic consistency across scales, IELFormer incorporates a multi-scale frequency fusion (MFF) module, which performs frequency-domain analysis to achieve structured integration of multi-resolution features, thereby improving cross-scale coherence. Extensive experiments on benchmark datasets demonstrate that our approach achieves superior generalization performance compared to existing methods.

Gaohang Chen, Lili Ju, Zhonghua Qiao

Solving partial differential equations (PDEs) with neural networks (NNs) has shown great potential in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDE formulas in the strong or weak sense, without explicitly considering some intrinsic physical properties, such as mass and momentum conservation, or energy dissipation. This limitation may result in nonphysical or unstable numerical solutions, particularly in long-term simulations. To address this issue, we propose ``Sidecar'', a novel framework that enhances the physical consistency of existing NN solvers for solving parabolic PDEs. Inspired by the time-dependent spectral renormalization approach, our Sidecar framework introduces a small network as a copilot, guiding the primary function-learning NN solver to respect the structure-preserving properties. Our framework is highly flexible, allowing the preservation of various physical quantities for different PDEs to be incorporated into a wide range of NN solvers. Experimental results on some benchmark problems demonstrate significant improvements brought by the proposed framework to both accuracy and structure preservation of existing NN solvers.

Zhicheng Hu, Ruo Li, Zhonghua Qiao

We study the acceleration of steady-state computation for microflow, which is modeled by the high-order moment models derived recently from the steady-state Boltzmann equation with BGK-type collision term. By using the lower-order model correction, a novel nonlinear multi-level moment solver is developed. Numerical examples verify that the resulting solver improves the convergence significantly thus is able to accelerate the steady-state computation greatly. The behavior of the solver is also numerically investigated. It is shown that the convergence rate increases, indicating the solver would be more efficient, as the total levels increases. Three order reduction strategies of the solver are considered. Numerical results show that the most efficient order reduction strategy would be $m_{l-1} = \lceil m_{l} / 2 \rceil$.