Methods For Partial Differential Equations Research Articles

Cervical cancer incidence rates considering migration status in mainland China using Bayesian model-Estimation based on 2016 cancer registry data.

In mainland China, cancer registration relies on household-registered populations, overlooking migrant populations. Estimating cervical cancer incidence among permanent residents, including migrants, offers a more accurate representation of the true burden. The data from 487 cancer registries across China in 2016 were analyzed using a Bayesian spatial regression model with the integrated nested Laplace approximation-stochastic partial differential equation method. The study estimated cervical cancer incidence among household-registered populations and adjusted for migrant populations using a weighting method based on interprovincial distribution and age stratification to derive the incidence of cervical cancer in the permanent residents. Data from the China Population Census, the China Migrants Dynamic Survey, and the Urban Statistical Yearbook were incorporated. The estimated crude incidence rate of cervical cancer among permanent residents was 17.4/100,000 in mainland China, with an age-standardized incidence rate (ASIR) of 17.2/100,000. The largest disparities in cervical cancer crude incidence rate between permanent residents and household-registered populations were observed in Guizhou (2.4/100,000, 95% CI 1.9-2.9/100,000), Zhejiang (-1.2/100,000, 95% CI -1.8 to -0.6/100,000) and Tianjin (-1.1/100,000, 95% CI -1.5 to -0.7/100,000). The number of the estimated cervical cancer incident cases was 8948. Guangdong saw an increase of 887 cases, while Henan had a decrease of 1430 cases. Guizhou had the highest ASIR (28.1/100,000), and Beijing had the lowest ASIR (11.0/100,000). The significance of this study is that it improves the accuracy of cervical cancer data in China. These findings provide evidence for developing cervical cancer prevention and control strategies, and offer insights for other countries and regions facing migration challenges.

Preface to Focused Section on Efficient High-Order Time Discretization Methods for Partial Differential Equations

Preface to Focused Section on Efficient High-Order Time Discretization Methods for Partial Differential Equations

Dilated convolution neural operator for multiscale partial differential equations

This paper introduces a data-driven operator learning method for multiscale partial differential equations, with a particular emphasis on preserving high-frequency information. Drawing inspiration from the representation of multiscale parameterized solutions as a combination of low-rank global bases (such as low-frequency Fourier modes) and localized bases over coarse patches (analogous to dilated convolution), we propose the Dilated Convolutional Neural Operator (DCNO). The DCNO architecture effectively captures both high-frequency and low-frequency features while maintaining a low computational cost through a combination of convolution and Fourier layers. We conduct experiments to evaluate the performance of DCNO on various datasets, including the multiscale elliptic equation, its inverse problem, Navier–Stokes equation, and Helmholtz equation. Our results demonstrate that DCNO achieves an optimal balance between accuracy and computational efficiency, making it a promising approach for multiscale operator learning.

Transformers as neural operators for solutions of differential equations with finite regularity

Neural operator learning models have emerged as very effective surrogates in data-driven methods for partial differential equations (PDEs) across different applications from computational science and engineering. Such operator learning models not only predict particular instances of a physical or biological system in real-time but also forecast classes of solutions corresponding to a distribution of initial and boundary conditions or forcing terms. DeepONet is the first neural operator model and has been tested extensively for a broad class of solutions, including Riemann problems. Transformers have not been used in that capacity, and specifically, they have not been tested for solutions of PDEs with low regularity. In this work, we first establish the theoretical groundwork that transformers possess the universal approximation property as operator learning models. We then apply transformers to forecast solutions of diverse dynamical systems with solutions of finite regularity for a plurality of initial conditions and forcing terms. In particular, we consider three examples: the Izhikevich neuron model, the tempered fractional-order Leaky Integrate-and-Fire (LIF) model, and the one-dimensional Euler equation Riemann problem. For the latter problem, we also compare with variants of DeepONet, and we find that transformers outperform DeepONet in accuracy but they are computationally more expensive.

Direct estimates for adaptive time-stepping finite element methods

We study direct estimates for adaptive time-stepping finite element methods for time-dependent partial differential equations. Our results generalize previous findings from “On approximation classes for adaptive time-stepping finite element methods” by Actis et al. (2023), where the approximation error was only measured in L2([0,T],L2(Ω)). In particular, we now also cover the error norms L∞([0,T],L2(Ω)) and L2([0,T],H1(Ω)) which are more natural in this context.

Bifurcation analysis and exact solutions of the conformable time fractional Symmetric Regularized Long Wave equation

This paper investigates exact solutions for the conformable time fractional Symmetric Regularized Long Wave equation by applying the bifurcation analysis method and the exp(−Φ(ξ))-expansion method. By analyzing the long behaviors of the exact solutions plotted in 3D and 2D figures, we can model weakly nonlinear ion acoustic and space-charge waves. The bifurcation of the equation is analyzed based on the condition where the first integral constant is zero and the second integral constant is not zero. Based on different parameter conditions, many phase portraits and exact solutions including dark soliton, bright soliton, breaking wave, periodic and singular solutions for the equation are obtained. It has been proven that bifurcation method provides a wider range of solutions compared with other methods. Then the exp(−Φ(ξ))−expansion method is utilized to get the more solutions. Next graphical representations are presented that show physical characteristics of the solutions and the significance of the methods for fractional partial differential equations. Finally, we make a comprehensive comparison with other literatures.

Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows

We develop a registration-based nonlinear model-order reduction (MOR) method for partial differential equations (PDEs) with applications to transonic Euler and Reynolds-averaged Navier–Stokes (RANS) equations in aerodynamics. These PDEs exhibit discontinuous features, namely shocks, whose location depends on problem configuration parameters, and the associated parametric solution manifold exhibits a slowly decaying Kolmogorov N-width. As a result, conventional linear MOR methods, which use linear reduced approximation spaces, do not yield accurate low-dimensional approximations. We present a registration-based nonlinear MOR method to overcome this challenge. Our formulation builds on the following key ingredients: (i) a geometrically transformable parametrized PDE discretization; (ii) localized spline-based parametrized transformations which warp the domain to align discontinuities; (iii) an efficient dilation-based shock sensor and metric to compute optimal transformation parameters; (iv) hyperreduction and online-efficient output-based error estimates; and (v) simultaneous transformation and adaptive finite element training. Compared to existing methods in the literature, our formulation is efficiently scalable to larger problems and is equipped with error estimates and hyperreduction. We demonstrate the effectiveness of the method on two-dimensional inviscid and turbulent flows modeled by the Euler and RANS equations, respectively.

The natural extension to PDEs of Lie’s reduction of order algorithm for ODEs

In this paper, we further consider the symmetry-based method for seeking nonlocally related systems for partial differential equations. In particular, we show that the symmetry-based method for partial differential equations is the natural extension of Lie’s reduction of order algorithm for ordinary differential equations by looking at this algorithm from a different point of view. Many examples exhibit various situations that can arise.

Accelerating the convergence of Newton’s method for nonlinear elliptic PDEs using Fourier neural operators

It is well known that Newton’s method can have trouble converging if the initial guess is too far from the solution. Such a problem particularly occurs when this method is used to solve nonlinear elliptic partial differential equations (PDEs) discretized via finite differences. This work focuses on accelerating Newton’s method convergence in this context. We seek to construct a mapping from the parameters of the nonlinear PDE to an approximation of its discrete solution, independently of the mesh resolution. This approximation is then used as an initial guess for Newton’s method. To achieve these objectives, we elect to use a Fourier neural operator (FNO). The loss function is the sum of a data term (i.e., the comparison between known solutions and outputs of the FNO) and a physical term (i.e., the residual of the PDE discretization). Numerical results, in one and two dimensions, show that the proposed initial guess accelerates the convergence of Newton’s method by a large margin compared to a naive initial guess, especially for highly nonlinear and anisotropic problems, with larger gains on coarse grids.

Conformable bilinear neural network method: a novel method for time-fractional nonlinear partial differential equations in the sense of conformable derivative

Conformable bilinear neural network method: a novel method for time-fractional nonlinear partial differential equations in the sense of conformable derivative

Multigrid Reduction‐In‐Time Convergence for Advection Problems: A Fourier Analysis Perspective

ABSTRACTA long‐standing issue in the parallel‐in‐time community is the poor convergence of standard iterative parallel‐in‐time methods for hyperbolic partial differential equations (PDEs), and for advection‐dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two‐level variant of the iterative parallel‐in‐time method of multigrid reduction‐in‐time (MGRIT). This closed‐form theory allows for new insights into the poor convergence of MGRIT for advection‐dominated PDEs when using the standard approach of rediscretizing the fine‐grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse‐grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady‐state advection‐dominated PDEs. We apply this convergence theory to show that, for certain semi‐Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse‐grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re‐purposed in the MGRIT context to develop more robust parallel‐in‐time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions ( d = 2 ) (d=2) and surfaces in three dimensions ( d = 3 ) (d=3) with an arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) , where n ∈ S d − 1 \boldsymbol {n}\in \mathbb {S}^{d-1} represents the outward unit vector. By introducing a novel unified surface energy matrix G k ( n ) \boldsymbol {G}_k(\boldsymbol {n}) depending on γ ( n ) \gamma (\boldsymbol {n}) , the Cahn-Hoffman ξ \boldsymbol {\xi } -vector and a stabilizing function k ( n ) : S d − 1 → R k(\boldsymbol {n}):\ \mathbb {S}^{d-1}\to {\mathbb R} , we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators, including the surface gradient operator, the surface divergence operator and the surface Laplace-Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on γ ( n ) \gamma (\boldsymbol {n}) , we propose a new framework via local energy estimate for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) arising from different applications.

A learning based numerical method for Helmholtz equations with high frequency

High-frequency issues have been remarkable challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validate the error analysis and demonstrate the high-precision and high-efficiency features.

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

GPU-enabled extreme-scale turbulence simulations: Fourier pseudo-spectral algorithms at the exascale using OpenMP offloading

Fourier pseudo-spectral methods for nonlinear partial differential equations are of wide interest in many areas of advanced computational science, including direct numerical simulation of three-dimensional (3-D) turbulence governed by the Navier-Stokes equations in fluid dynamics. This paper presents a new capability for simulating turbulence at a new record resolution up to 35 trillion grid points, on the world's first exascale computer, Frontier, comprising AMD MI250x GPUs with HPE's Slingshot interconnect and operated by the US Department of Energy's Oak Ridge Leadership Computing Facility (OLCF). Key programming strategies designed to take maximum advantage of the machine architecture involve performing almost all computations on the GPU which has the same memory capacity as the CPU, performing all-to-all communication among sets of parallel processes directly on the GPU, and targeting GPUs efficiently using OpenMP offloading for intensive number-crunching including 1-D Fast Fourier Transforms (FFT) performed using AMD ROCm library calls. With 99% of computing power on Frontier being on the GPU, leaving the CPU idle leads to a net performance gain via avoiding the overhead of data movement between host and device except when needed for some I/O purposes. Memory footprint including the size of communication buffers for MPI_ALLTOALL is managed carefully to maximize the largest problem size possible for a given node count.Detailed performance data including separate contributions from different categories of operations to the elapsed wall time per step are reported for five grid resolutions, from 20483 on a single node to 327683 on 4096 or 8192 nodes out of 9408 on the system. Both 1D and 2D domain decompositions which divide a 3D periodic domain into slabs and pencils respectively are implemented. The present code suite (labeled by the acronym GESTS, GPUs for Extreme Scale Turbulence Simulations) achieves a figure of merit (in grid points per second) exceeding goals set in the Center for Accelerated Application Readiness (CAAR) program for Frontier. The performance attained is highly favorable in both weak scaling and strong scaling, with notable departures only for 20483 where communication is entirely intra-node, and for 327683, where a challenge due to small message sizes does arise. Communication performance is addressed further using a lightweight test code that performs all-to-all communication in a manner matching the full turbulence simulation code. Performance at large problem sizes is affected by both small message size due to high node counts as well as dragonfly network topology features on the machine, but is consistent with official expectations of sustained performance on Frontier. Overall, although not perfect, the scalability achieved at the extreme problem size of 327683 (and up to 8192 nodes — which corresponds to hardware rated at just under 1 exaflop/sec of theoretical peak computational performance) is arguably better than the scalability observed using prior state-of-the-art algorithms on Frontier's predecessor machine (Summit) at OLCF. New science results for the study of intermittency in turbulence enabled by this code and its extensions are to be reported separately in the near future.

Prescribed-Time Network-Based Deployment of Nonlinear Multiagent Systems: A Discrete-Space PDE Method.

This article attempts to design the prescribed-time time-varying deployment schemes for first-order and second-order nonlinear multiagent systems (MASs). We assume that all agents can obtain the information of their current and final relative positions with their neighbors, and the final absolute velocities (as well as their current and final relative velocities, the final absolute accelerations for the second-order MASs) through a communication network, whereas two boundary agents are able to obtain their current and final absolute positions (as well as their current and final absolute velocities for the second-order MASs). The neighbor relationship of all agents is described by a spatial variable and two static-feedback controllers are introduced, which can be expressed as a second-order space difference of the spatial variable. Then, the deployment of MASs can be transformed into the stabilization of discrete-space partial differential equation (PDE) systems. Three virtual agents are introduced to constitute the Dirchlet and Neumann boundary conditions. Several algebraic inequality criteria are derived to guarantee that the prescribed-time time-varying deployment can be achieved within a prescribed time under the Dirchlet and mixed boundary conditions. Unlike the published results, our results are derived based on the discrete-space PDE systems instead of continuous-space PDE systems, which is consistent with the discrete spatial distribution of agents. Finally, two numerical examples are given to illustrate the effectiveness of our results.

Discrete variational method for partial differential equations of functionally gradient beam vibration

The field of engineering is becoming increasingly complex. In order to adapt to the numerical simulation of solving the partial differential equation of functionally graded beam vibration, a higher order stable numerical algorithm has been constructed. Differential quadrature method is used in discrete space domain. The discrete variational method is constructed in the time domain. The index differential Algebraic equation are obtained by combining the two methods. The discrete variational scheme is constructed for simulation. The results indicate that under long-term simulation, both the velocity and displacement constraints of the Runge Kutta method have defaulted. Displacement constraint values differ by 5 × 10 - 10. The velocity, displacement and acceleration constraints of the discrete variational method are stable. Compared with the Runge Kutta method, the constraint magnitude is reduced. The speed constraint is maintained at within 2.5 × 10 - 15. The displacement constraint level is maintained at within 1 × 10 - 16. This indicates that the discrete variational method has high accuracy and good stability when solving problems such as the vibration equation of functionally graded beams. When the step sizes are h= 0.1 m and h= 0.01 m, the accuracy of the discrete variational method is close. The larger the step size h, the higher the computational efficiency of the discrete variational method. The discrete variational method can maintain structural and energy conservation, making it suitable for long-term simulations. This has a good effect on solving complex problems in the field of partial differential equations.

Deep finite volume method for partial differential equations

In this paper, we introduce the Deep Finite Volume Method (DFVM), an innovative deep learning framework tailored for solving high-order (order ≥2) partial differential equations (PDEs). Our approach centers on a novel loss function crafted from local conservation laws derived from the original PDE, distinguishing DFVM from traditional deep learning methods. By formulating DFVM in the weak form of the PDE rather than the strong form, we enhance accuracy, particularly beneficial for PDEs with less smooth solutions compared to strong-form-based methods like Physics-Informed Neural Networks (PINNs). A key technique of DFVM lies in its transformation of all second-order or higher derivatives of neural networks into first-order derivatives which can be computed directly using Automatic Differentiation (AD). This adaptation significantly reduces computational overhead, particularly advantageous for solving high-dimensional PDEs. Numerical experiments demonstrate that DFVM achieves equal or superior solution accuracy compared to existing deep learning methods such as PINN, Deep Ritz Method (DRM), and Weak Adversarial Networks (WAN), while drastically reducing computational costs. Notably, for PDEs with nonsmooth solutions, DFVM yields approximate solutions with relative errors up to two orders of magnitude lower than those obtained by PINN. The implementation of DFVM is available on GitHub at https://github.com/Sysuzqs/DFVM.

Control Methods in Hyperbolic PDEs

Control of hyperbolic partial differential equations (PDEs) is a truly interdisciplinary area of research in applied mathematics nurtured by challenging problems arising in most modern applications ranging from road traffic, gas pipeline management, blood circulation, to opinion dynamics and socio-economical models, as well as in environmental and biological issues, or more recently in the analysis of deep learning and machine learning methods. The topic has gained an increasing attraction of researchers in the last twenty years due to fundamental theoretical as well as numerical advances achieved in the field of nonlinear hyperbolic PDEs. The hyperbolic and the control of PDEs communities, while pursuing separate interests in their respective range of action with a different focus and, often, with a different array of technical tools, do share a substantial body of common knowledge and background. We think the time is right and the momentum is propitious to bring those communities together at a joint workshop, to mutually stimulate each other and inter- act with one another, for a marked advancement of this area of research on a broad spectrum of control ranging from theoretical to numerical problems and covering also the emerging challenges involving the interplay between (topics of) control and learning. In order to also attract young scientists to this striving field we focus on selected lectures, in-depth discussions, transfer of information from senior to young researchers, and vice versa, and invited plenary talks.

Spatio-temporal ecological models via physics-informed neural networks for studying chronic wasting disease

To mitigate the negative effects of emerging wildlife diseases in biodiversity and public health it is critical to accurately forecast pathogen dissemination while incorporating relevant spatio-temporal covariates. Forecasting spatio-temporal processes can often be improved by incorporating scientific knowledge about the dynamics of the process using physical models. Ecological diffusion equations are often used to model epidemiological processes of wildlife diseases where environmental factors play a role in disease spread. Physics-informed neural networks (PINNs) are deep learning algorithms that constrain neural network predictions based on physical laws and therefore are powerful forecasting models useful even in cases of limited and imperfect training data. In this paper, we develop a novel ecological modeling tool using PINNs, which fits a feedforward neural network and simultaneously performs parameter identification in a partial differential equation (PDE) with varying coefficients. We demonstrate the applicability of our model by comparing it with the commonly used Bayesian stochastic partial differential equation method and traditional machine learning approaches, showing that our proposed model exhibits superior prediction and forecasting performance when modeling chronic wasting disease in deer in Wisconsin. Furthermore, our model provides the opportunity to obtain scientific insights into spatio-temporal covariates affecting spread and growth of diseases. This work contributes to future machine learning and statistical methodology development by studying spatio-temporal processes enhanced by prior physical knowledge.