Numerical Mathematics: Theory, Methods and Applications
Numerical Mathematics: Theory, Methods and Applications (NMTMA) publishes high-quality papers on the construction, analysis and application of numerical methods for solving scientific and engineering problems. Research and expository papers devoted to the numerical solution of mathematical equations arising in all areas of science and technology are expected. The journal originates from Numerical Mathematics: A Journal of Chinese Universities (English Edition), and has been sponsored by Nanjing University.
Color Image Segmentation Based on Hue-Saturation Similarity
In this paper, we propose and develop a novel variational model based on hue-saturation similarity and fuzzy membership function for color image segmentation. The main contribution of the proposed model is that we determine different segments by using the similarity of hue and saturation information in hue, saturation, and value color space. We first provide specific definitions of the hue/saturation distance to describe hue-saturation similarity, then formulate a novel data fitting term with an adaptive weight coefficient by using hue-saturation similarity in the proposed energy functional. Two efficient iterative algorithms based on coordinate descent method and alternating direction method of multipliers have been proposed to solve the proposed optimization problem. Theoretically we study the existence of the solution of the proposed model and the convergence of the proposed coordinate descent algorithm. Numerical experimental results demonstrate that the segmentation performance of the proposed model is much better than that of other existing color image segmentation methods.
Copyright (c) 2025 Numerical Mathematics: Theory, Methods and Applications
Auto-Adaptive PINNs with Applications to Phase Transitions
We propose an adaptive sampling method for the training of physics informed neural networks (PINNs) which allows for sampling based on an arbitrary problem-specific heuristic which may depend on the network and its gradients. In particular we focus our analysis on the Allen-Cahn equations, attempting to accurately resolve the characteristic interfacial regions using a PINN without any post-hoc resampling. In experiments, we show the effectiveness of these methods over residual-adaptive frameworks.
Sequential Subspace Clustering via Joint Capped $ℓ_2$ and $ℓ_{2,p}$ Norm Minimization with Convergence Guarantee
Subspace clustering is a fundamental problem in machine learning that has attracted considerable attention in recent years. Most existing methods focus on designing effective models to regularize the coefficient matrix, often neglecting the impact of noise on subspace structures. However, real-world data are typically corrupted by noise, which can distort the underlying subspace structure. Additionally, for sequential data, a key challenge is the effective exploitation of temporal information. To address these issues, we propose a novel and robust sequential subspace clustering method, termed joint capped $ℓ_2$ and $ℓ_{2,p}$ norm minimization (JCLLM). The capped $ℓ_2$ norm-based loss function mitigates the influence of noise and outliers in regression, while the $ℓ_{2,p}$ norm regularization captures the temporal dependencies inherent in sequential data. We develop an iteratively reweighted optimization algorithm to solve the JCLLM model and prove its convergence to a stationary point. Extensive experiments on both synthetic and real-world datasets demonstrate that our method consistently outperforms several state-of-the-art subspace clustering approaches.
This paper investigates the length of the repeating decimal part when a fraction is expressed in decimal form. First, it provides a detailed explanation of how to calculate the length of the repeating decimal when the denominator of the fraction is a power of a prime number. Then, by factorizing the denominator into its prime factors and determining the repeating decimal length for each prime factor, the paper concludes that the overall repeating decimal length is the least common multiple of these lengths. Furthermore, it examines the conditions under which the repeating decimal length equals the denominator minus 1 and discusses whether such fractions exist in infinite quantity. This topic is connected to an unsolved problem posed by Gauss in the 18th century and is also closely related to the important question of whether cyclic numbers exist in infinite quantity.
A Proximal-Perturbed Bregman ADMM for Solving Nonsmooth and Nonconvex Composite Optimization
In this paper, we focus on a linearly constrained composite minimization problem involving a possibly nonsmooth and nonconvex objective function. Unlike the traditional construction of the augmented Lagrangian function, we design a proximal-perturbed augmented Lagrangian to develop a new Bregman-type alternating direction method of multipliers. Under mild assumptions, we prove that the augmented Lagrangian sequence converges to the limit of the objective function sequence, and the iterative sequence generated by our method converges to a stationary point of the problem. The sublinear convergence rate of the primal residuals is also analyzed. Comparative experiments on testing the linear equation problem, graph-guided fused lasso problem and robust principal component analysis problem demonstrate the efficiency and flexibility of the proposed method.
Maximum Principle-Preserving Explicit Scheme for the Ternary Allen-Cahn Equation
We investigate the stability of the ternary Allen-Cahn (tAC) equation solved by the fully explicit Euler method. The tAC equation models interface dynamics and phase separation in three-component systems, and is formulated as an $L^2$-gradient flow of the Ginzburg-Landau energy. We derive a sufficient condition on the time step size that satisfies the discrete maximum principle and guarantees the numerical stability of the fully explicit Euler method. Through detailed analysis, we confirm that the derived condition provides the largest possible time step size among those that preserve stability. Numerical experiments are conducted to verify the theoretical results, and the numerical solution violates the maximum principle and becomes unstable when the time step size exceeds the derived bound. Conversely, when the time step condition is satisfied, the computational solution preserves the maximum principle and maintains the energy dissipation property of the tAC equation. These results provide a rigorous theoretical foundation for determining the stability limit of the explicit Euler method applied to the tAC equation.
Locally Optimal Preconditioned Conjugate Gradient Method for Computing Ground State of Space-Fractional Nonlinear Schrödinger Equation
In this paper, we investigate numerical methods for computing the ground state of space-fractional nonlinear Schrödinger equation. We focus on extending the locally optimal preconditioned conjugate gradient method, originally designed for linear eigenvalue problems, to address this nonlinear fractional framework. Through comparative numerical experiments with the discrete gradient flow method, we demonstrate that the locally optimal preconditioned conjugate gradient method achieves significantly superior efficiency in many scenarios. This advantage is particularly pronounced in multidimensional problems and cases involving lower fractional derivative orders. The results highlight the adaptability of the locally optimal preconditioned conjugate gradient approach to eigenproblem of fractional nonlinear systems, offering a robust and high efficient computational alternative for high-dimensional or low-order fractional derivative settings.
locally optimal preconditioned conjugate gradient method
A Novel Semi-Analytical Multiple Invariants-Preserving Integrator for Conservative PDEs
Many conservative partial differential equations such as the Korteweg-de Vries (KdV) equation, the nonlinear Schrödinger equation, and the Klein-Gordon equation have more than one invariant functionals. In this paper, we propose the definition of the discrete variational derivative, based on which, a novel semi-analytical multiple invariants-preserving integrator for the conservative partial differential equations is constructed by projection technique. The proposed integrators, constructed by applying a projection technique to existing numerical methods, are shown to preserve the same order of accuracy as their underlying base integrators. For applications, some concrete mass-momentum-energy-preserving integrators are derived for the KdV equation.
Conservative partial differential equation
A Resampling-Free Stochastic Projection Contraction Algorithm for Solving Stochastic Variational Inequalities
Sampling is a major computational bottleneck in stochastic algorithms. This paper proposes a stochastic projection contraction algorithm for stochastic variational inequality problems, significantly reducing runtime by eliminating resampling in the correction step. We introduce an adjustable offset weight to optimize search direction, along with different adaptive step size strategies in prediction and correction steps. We further present discrete differential equation interpretations for specific offset weight values. To address bias due to the absence of resampling in the correction step, we develop an error control scheme and provide convergence guarantees. Numerical experiments demonstrate the algorithm’s efficiency.
Stochastic projection contraction algorithm
Density Estimation via Mixture Discrepancy and Moments
With the aim of generalizing histogram statistics to higher dimensional cases, density estimation via discrepancy based sequential partition (DSP) has been proposed to learn an adaptive piecewise constant approximation defined on a binary sequential partition of the underlying domain, where the star discrepancy is adopted to measure the uniformity of particle distribution. However, the calculation of the star discrepancy is NP-hard and it does not satisfy the reflection invariance and rotation invariance either. To this end, we use the mixture discrepancy and the comparison of moments as a replacement of the star discrepancy, leading to the density estimation via mixture discrepancy based sequential partition (DSP-mix) and density estimation via moment-based sequential partition (MSP), respectively. Both DSP-mix and MSP are computationally tractable and exhibit the reflection and rotation invariance. Numerical experiments in reconstructing beta mixtures, Gaussian mixtures and heavy-tailed Cauchy mixtures up to 30 dimension are conducted, demonstrating that MSP can maintain the same accuracy compared with DSP, while gaining an increase in speed by a factor of two to twenty for large sample size. DSP-mix can achieve satisfactory accuracy and boost the efficiency in low-dimensional tests ($d \leq 6$), but might lose accuracy in high-dimensional problems due to a reduction in partition level.
The Method of Fundamental Solutions for Optical Fluorescence Tomography
In this paper, the method of fundamental solutions (MFS) is first developed for solving direct problems in bi-layer materials in the biomedical field of optical fluorescence. The governing system of second-order linear partial differential equations (PDEs) for the emission and excitation fluences is transformed into a single fourth-order PDE with appropriate boundary and interface matching conditions. The MFS is subsequently further developed, in conjunction with a constrained minimization regularization procedure, to solve nonlinear inverse optical fluorescence tomography problems. Numerical results confirm the accuracy, stability and versatility of the proposed meshless technique.
Stability and Optimal Error Estimates Analysis of an LDG Method for the Stochastic Nonlinear KdV Equation
To address the computational challenges of stochastic nonlinear partial differential equations with high-order derivatives, a local discontinuous Galerkin method is proposed for the stochastic KdV equation. The method is proven to be $\mathcal{L}^2$-stable and to attain optimal error estimates of order $n+1$ measured in the mean-square norm when degree-$n$ polynomials are used. Temporal integration of the spatial semi-discrete stochastic system in the numerical experiments is carried out by using the implicit midpoint method. The simulation results verify the method's accuracy and its consistency with the theoretical analysis.
Analysis of Dirichlet-Neumann and Neumann-Dirichlet Methods for Time-Periodic Parabolic Optimal Control Problems
Dirichlet-Neumann and Neumann-Neumann methods are not only the parallel strategies in the spatial domain, but also can be used as a class of parallel methods in time. In this paper, we propose the Dirichlet-Neumann and Neumann-Dirichlet algorithms for time-periodic parabolic optimal control problems. By the Lagrange multiplier approach, a coupled system is obtained with a special time-periodic condition. For this coupled system, the Dirichlet-Neumann and Neumann-Dirichlet algorithms and their three variants are derived. We present the convergence analysis for all proposed algorithms. The numerical performance of the convergence factors is shown to illustrate our theoretical analysis. Based on our analysis, there is a class of algorithms with the better convergence compared with the natural Dirichlet-Neumann algorithm. Finally, numerical experiments are provided to illustrate the theoretical results.
Time-periodic parabolic optimal control problems
A Fast Fully Discrete Mixed Finite Element Scheme for Fractional Viscoelastic Models of Wave Propagation
Due to the nonlocal feature of fractional differential operators, the numerical solution to fractional partial differential equations usually requires expensive memory and computation costs. This paper develops a fast scheme for fractional viscoelastic models of wave propagation. We first apply the Laplace transform to convert the time-fractional constitutive equation into an integro-differential form that involves the Mittag-Leffler function as a convolution kernel. Then we construct an efficient sum-of-exponentials (SOE) approximation for the Mittag-Leffler function. We use mixed finite elements for the spatial discretization and the Newmark scheme for the temporal discretization of the second time-derivative of the displacement variable in the kinematical equation and finally obtain the fast algorithm. Compared with the traditional L1 scheme for time fractional derivative, our fast scheme reduces the memory complexity from $\mathcal{O}(N_sN)$ to $\mathcal{O}(N_sN_{\text{exp}})$ and the computation complexity from $\mathcal{O}(N_sN^2)$ to $\mathcal{O}(N_sN_{\text{exp}}N)$, where $N$ denotes the total number of temporal grid points, $N_{\text{exp}}$ the number of exponentials in SOE, and $N_s$ the complexity of memory and computation related to the spatial discretization. Numerical experiments confirm the theoretical results.
An Asymptotic Preserving Method for the Weakly Nonlinear Klein-Gordon Equation
In this paper, we propose an asymptotic preserving (AP) method for the weakly nonlinear Klein-Gordon equation (NKGE). Firstly, we apply a multi-scale expansion for the weakly NKGE and obtain the equation for the leading-order term, for which an error estimate has been provided. Secondly, by solving the equation for the leading-order term numerically, we construct an AP method for the weakly NKGE. Finally, numerical results in one spatial dimension are provided to show that: (i) The method is asymptotic preserving, i.e., the error between the leading-order term and the solution of the weakly NKGE behaves as $\mathcal{O}(\varepsilon)$ as $\varepsilon \to 0$. (ii) It is uniformly accurate since the numerical solution obtained by the method is independent of the small parameter $\varepsilon$. (iii) It can make correct predictions about the solution of the original NKGE. Moreover, extension of the method to the two-dimensional weakly NKGE are also provided.
A Gradient Recovery Technique for Enhancing the Convergence of Demagnetizing Field Based on PDE Approach
The PDE approach is a popular technique for the demagnetizing field calculation due to the flexibility in handling complex domains. However, it faces a challenge on delivering desired accuracy due to the suboptimal convergence of the function gradient and singularity on the boundary. In this work, a robust gradient recovery technique is applied and analysed for fixing such an issue. An $L^2$ error estimate of the finite element approximation for the demagnetizing field is derived, which consists of two parts, i.e., of finite element discretization error and boundary approximation error. A gradient recovery method based on the polynomial preserving recovery technique is applied to enhance the accuracy of the finite element approximation, and a superconvergence result is established. An idea of locally refined surface meshes is applied to resolve the singularity in the boundary conditions, thereby reducing boundary approximation errors. Extensive numerical tests are provided to verify our theoretical findings and the efficiency of our proposed method. The results indicate that the proposed method achieves second-order convergence for the approximate demagnetizing field, positioning it as a highly competitive technique to develop optimal algorithms in computational micromagnetics.
Theoretical Guarantees of Recovery Algorithms for Ternary Sparse Signals
This paper focuses on the problem of recovering ternary sparse signals with $s$ nonzero entries of 1 and -1. We propose three novel algorithms: ternary matching pursuit (TMP), ternary generalized orthogonal matching pursuit (TGOMP), and piecewise ternary generalized orthogonal matching pursuit (PTGOMP). First, inspired by the binary matching pursuit algorithm, we introduce the TMP algorithm, which assigns values of 1 or -1 based on the most correlated residual, and provide theoretical guarantees based on the mutual coherence, denoted by $\mu$ and the restricted isometry property of the measurement matrix, respectively. Second, we propose the TGOMP algorithm, by selecting multiple ($M$) indices at each iteration in order to improve the performance of the TMP algorithm. We establish a sufficient condition $\mu < 1/(2s-1)$ that ensures the TGOMP algorithm selects $M$ correct indices and corresponding entries of $x$ in each iteration. Especially, all correct indices and entries can be selected in the first iteration. Additionally, we present a sufficient condition based on the restricted isometry property that guarantees all correct indices are selected in at most $s$ iterations. Third, we propose the PTGOMP algorithm, which employs a piecewise selection strategy at each iteration, further improves recovery performance. Theoretical guarantees for the PTGOMP algorithm are derived based on the mutual coherence, showing its advantages in ternary sparse signal recovery. Finally, we validate the effectiveness of our algorithms through simulations and numerical experiments, demonstrating that combining appropriate matrix structures with suitable sparsity patterns can significantly improve recovery performance.
A Space-Time Extrapolation Cascadic Multigrid Method for 2D Linear Parabolic Problems
We present a new space-time extrapolation cascadic multigrid method with SSOR preconditioned GMRES smoother for linear parabolic equations. This method simultaneously solves all time steps of the system using an all-at-once approach, employing Crank-Nicolson discretization in time and central difference discretization in space. The key components of the new algorithm are the Richardson extrapolation and Lagrange interpolation operators. By utilizing these techniques with numerical solutions of current and previous grids, we generate a good initial guess for the iterative solution on the next finer grid, greatly reducing the number of required iterations and computational time. Finally, we explain how to implement the new multigrid method and show its efficiency through numerical experiments.
Splitting Schemes for Second-Order Backward Stochastic Differential Equations
In this work, we focus on splitting methods for solving second-order backward stochastic differential equations (2BSDEs). By splitting the original $d$-dimensional 2BSDEs into $d$ 2BSDEs and approximating these split 2BSDEs, we derive the splitting schemes that only require one-dimensional approximations to evaluate the conditional mathematical expectations. Combining the Sinc quadrature rule to approximate the conditional expectations, we further propose the first-order fully discrete splitting schemes. Numerical tests are presented to show the capacity of the schemes.
Second-order backward stochastic differential equations
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A Biquadratic Characteristic Finite Volume Element Method for Time-Fractional Convection-Dominated Equations
We propose a biquadratic characteristic finite volume element (CFVE) method for two classes of time-fractional convection-dominated equations on quadrilateral meshes. For the mixed time-fractional problem (i.e., involving both time-fractional and integer-order derivatives), optimal convergence rate in $L^2$-norm is established for the CFVE method with temporal discretization based on the $L1$-formula. For the pure time-fractional problem, we decompose the equation by separating an integer-order time derivative term, thereby enabling the application of the method of characteristics. Furthermore, the influence of the decomposition parameter is investigated through numerical experiments, providing practical insights into parameter selection. Extensive numerical experiments validate the efficacy of the proposed CFVE method in simulating both classes of time-fractional convection-dominated equations.
Characteristic finite volume element method
Machine Learning-Based Bias Correction for the GEIM Model
This study focuses on real-time reconstruction of the spatial distribution of nuclear power using limited measurement observations. While physical models, such as the generalized empirical interpolation method (GEIM), can reconstruct the spatial field, they often cause bias if the model used to construct it is biased. The parametrized background data weak (PBDW) method attempts to mitigate this model bias, but its effectiveness is limited. To improve model bias correction, this paper proposes leveraging machine learning techniques-specifically, support vector regression, ${\rm K}$-nearest neighbors, and decision trees to enhance the GEIM method. These techniques predict model bias distributions across the entire field based on observed model bias at measurement points. The results demonstrate that Gaussian process based correction performs comparably to PBDW, both offering superior accuracy and robustness against noise, while other machine learning methods exhibit instability under varying parameter settings.
generalized empirical interpolation method
Two Globally Convergent Hybrid CG Methods with Adaptive Restart Strategy for Riemannian Optimization
Hybrid conjugate gradient methods (CGMs) with adaptive restart strategies have been well-researched in Euclidean space. In this paper, we extend two methods of this type to solve optimization problems on Riemannian manifolds. Firstly, we present two Riemannian hybrid CGMs, and their hybrid conjugate parameters are yielded by projection or convex combination of the classical parameters. The first is a Riemannian hybrid CGM that projects between the Dai-Yuan method and another flexible conjugate parameter. The second is a combination of projection and convex combination of the modified Riemannian Liu-Storey method and the modified Riemannian Hestenes-Stiefel method. In the framework of Riemannian CGMs, we apply a uniform adaptive restart strategy to both methods based on the hybrid conjugate parameters we proposed. The search direction of the presented methods satisfies the sufficient descent condition. Under the usual assumption and the weak Wolfe line search, we prove the global convergence of the two proposed methods. Finally, preliminary numerical results are reported and compared with several existing Riemannian CGMs, showing that our methods are effective.
A Moving Mesh Isogeometric Method Based on Harmonic Maps
Although isogeometric analysis has shown great potential in achieving highly accurate numerical solutions of partial differential equations, how to efficiently implement the method is one of the challenges that makes it more competitive in practical simulations. In this paper, an integration of isogeometric analysis and a moving mesh method is proposed, providing a competitive approach to resolve the efficiency issue. Focusing on the Poisson equation, the implementation of the algorithm is presented in detail, including the numerical discretization of the governing equation using isogeometric analysis, and a mesh redistribution technique developed via harmonic maps. It is found that the isogeometric analysis brings attractive features in the realization of moving mesh method, such as providing an accurate expression for moving direction of mesh nodes, and allowing for more choices to construct monitor functions. Through a series of numerical experiments, the efficiency and effectiveness of the proposed method are successfully verified. Moreover, the potential of the method towards the practical applications is also well presented with the simulation of a helium atom in Kohn-Sham density functional theory.
Decoupling Analysis for Solving the Diffusive Peterlin Viscoelastic Model
The paper analyzes a system decoupling approach over the entire time domain to solve a diffusive Peterlin viscoelastic model which is a type of non-Newtonian fluids. The main idea is decoupling this non-Newtonian fluid equations into Newtonian fluid equations and second-order quasilinear parabolic equations entirely in the time domain, which can be solved iteratively. Under this approach, existing excellent computational method interfaces for Newtonian fluid equations and parabolic equations can be directly utilized. By employing the weak form of the Peterlin viscoelastic model, a specific decoupling scheme for the continuous-time case is developed. We prove the convergence of this decoupling scheme. Then, for the discrete-time case, a first-order scheme and its iterative error analysis are provided. Numerical experiments validate the theoretical convergence of the algorithm and its dependence on relative parameters.
Numerical Analysis of a Second Order Crank-Nicolson Mixed Finite Element Method for the Swift-Hohenberg Equation
In this paper, we conduct an error analysis for a mixed finite element method combined with Crank-Nicolson time-stepping for the Swift-Hohenberg (SH) equation. The original fourth-order differential equation from the SH equation is reformulated as a coupled system comprising a nonlinear parabolic equation and an elliptic equation. This transformation allows us to propose a mixed finite element method that utilizes only continuous elements for the approximation of the system. We rigorously demonstrate that our proposed scheme is unconditionally uniquely solvable and maintains unconditional nonlinear energy stability. Additionally, we provide a thorough analysis to determine the convergence rate of the method. To substantiate our theoretical findings, we present numerical tests that confirm both the accuracy and energy stability of the scheme.
Crank-Nicolson unconditionally energy stable
Parallel and Scalable Two-Grid Nonlinear Preconditioning Technique for Fully Coupled Solution of Thermal Convective Flow Problems
The focus of this paper is on the numerical solution of large sparse nonlinear systems of algebraic equations, which arise from the discretization of the incompressible Navier-Stokes equations for fluid flows. The inexact Newton algorithm is an efficient and popular technique for solving this kind of nonlinear systems. However, the method may not be suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the number of processors, the grid size, and robust with respect to some physical parameters such as the Reynolds number and the Grashof number. In this paper, we investigate some fully coupled parallel two-grid nonlinear preconditioning algorithm, including a grid sequencing method for the Newton iteration and a two-level overlapping Schwarz preconditioner for the linear iteration. We show numerically that it performs well for solving the thermal convective flow problems on parallel computers.
Solving the Helmholtz Equation by Sparse Fundamental Solution Neural Network
A sparse fundamental solution neural network (SFSNN) for solving the Helmholtz equation with constant coefficients and relatively large wave numbers $k$ is proposed. The method combines the strengths of fundamental solution techniques and neural networks by employing a radial basis function neural network, where fundamental solution functions serve as activation functions. Since these functions inherently satisfy the homogeneous Helmholtz equation, SFSNN only requires boundary sampling, significantly accelerating training. To enhance sparsity and generalization, an $ℓ_1$ regularization term of the weights is introduced into the loss function, reformulating the weight optimization as a least absolute shrinkage and selection operator (Lasso) problem. This not only reduces the number of basis functions but also improves the network’s generalization capability. Numerical experiments validate the method’s effectiveness for high-wavenumber isotropic Helmholtz equations in two dimensions and three dimensions. The results reveal that when the analytical solution is a linear combination of fundamental solutions, SFSNN accurately identifies their centers. Otherwise, the number of required basis functions scales as $N =\mathcal{O}(k^{(τ(d−1))}),$ where $τ < 1$ and $d$ is the problem dimension. Moreover, SFSNN has been successfully extended to non-homogeneous and semi-infinite Helmholtz equations, achieving high accuracy. Codes of the examples in this paper are available at https://github.com/wangzhiwensuda/SFSNN-Helmholtz-problem.
A Highly Efficient Adaptive Mesh Refinement Algorithm for the 1D Schrödinger-Poisson Problem
The one-dimensional stationary Schrödinger equation, coupled with the transparent boundary conditions and self-consistently linked to the Poisson equation, is a well-established model for describing quantum effects. In this paper, we introduce a general framework for constructing arbitrarily high-order finite difference schemes on arbitrary grids, whether they are uniform or nonuniform, inspired by the analytic discrete transparent boundary conditions [M. Guo et al., arXiv:2411.13175]. To enhance the accuracy of approximations while keeping computational costs low, we develop an optimal mesh refinement strategy that balances the need to resolve intervals with large gradient and high curvature of the potential function. We further propose an adaptive mesh refinement algorithm to solve the 1D Schrödinger-Poisson problem, incorporating a third-order compact finite difference discretization of the Schrödinger-Poisson system based on nonuniform grids, and a given mesh refinement strategy. Numerical experiments on a resonant tunneling diode are conducted to validate the algorithm’s high efficiency and to study the I-V characteristic curve of the device.
Uncertainty Quantification with Physics-Informed Generative Process Distributions for the Linear Diffusion Equation
We propose a novel diffusion-based generative model for solving linear stochastic diffusion equations, while enabling uncertainty quantification (UQ). By embedding the governing physical laws into the generative process, our approach establishes a mapping between control parameters and solutions across the latent space of the diffusion model. This ensures that the generated solutions satisfy the underlying physical constraints. Additionally, our method overcomes the limitation of conventional diffusion models, which struggle to generate accurate solutions for new control terms, and achieves superior accuracy compared to traditional data-driven operator learning techniques. Furthermore, by sampling different noise realizations and analyzing variations in the generated solutions, we efficiently capture solution diversity, enabling simultaneous prediction and comprehensive UQ. Experimental results demonstrate that our method outperforms deep operator networks and variational inference-based deep operator networks in both accuracy and confidence estimation.