Methods For Partial Differential Equations Research Articles

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.

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.

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.

Image inpainting algorithm based on double curvature-driven diffusion model with P-Laplace operator.

The method of partial differential equations for image inpainting achieves better repair results and is economically feasible with fast repair time. Addresses the inability of Curvature-Driven Diffusion (CDD) models to repair complex textures or edges when the input image is affected by severe noise or distortion, resulting in discontinuous repair features, blurred detail textures, and an inability to deal with the consistency of global image content, In this paper, we have the CDD model of P-Laplace operator term to image inpainting. In this method, the P-Laplace operator is firstly introduced into the diffusion term of CDD model to regulate the diffusion speed; then the improved CDD model is discretized, and the known information around the broken region is divided into two weighted average iterations to get the inpainting image; finally, the final inpainting image is obtained by weighted averaging the two image inpainting images according to the distancing. Experiments show that the model restoration results in this paper are more rational in terms of texture structure and outperform other models in terms of visualization and objective data. Comparing the inpainting images with 150, 1000 and 100 iterations respectively, Total Variation(TV) model and the CDD model inpainting algorithm always has inpainting traces in details, and TV model can't meet the visual connectivity, but the algorithm in this paper can remove the inpainting traces well, TV model and the CDD model inpainting algorithm always have inpainting traces in details, and TV model can't meet the visual connectivity, but the algorithm in this paper can remove the inpainting traces well. Of the images used for testing, the highest PSNR reached 38.7982, SSIM reached 0.9407, and FSIM reached 0.9781, the algorithm not only inpainting the effect and, but also has fewer iterations.

RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects

This article discusses standard Radial Basis Function (RBF) partition of unity method and an enhanced version known as the direct RBF partition of unity method. We introduce the framework of these methods for the effective and accurate numerical analysis of certain two-dimensional time-dependent Partial Differential Equations (PDEs). Additionally, we conduct a convergence analysis of these methods and establish error bounds for local approximations. These error bounds are determined based on conditions associated with the eigenvalues of the Laplacian operator matrices generated by the proposed methods. To validate the accuracy and reliability of these approaches, we provide several numerical examples. We assess the consistency between theoretical and numerical results. The localized techniques introduced in this research demonstrate a significant reduction in computational cost and algorithmic complexity, achieved through the direct local procedure. Despite this computational advantage, both versions of the RBF partition of unity method exhibit a similar level of accuracy. Finally, we conduct a comparative analysis between the RBF partition of unity local methods and the RBF-generated finite differences method, offering valuable insights into the strengths and limitations of each approach.

An augmented subspace based adaptive proper orthogonal decomposition method for time dependent partial differential equations

In this paper, we propose an augmented subspace based adaptive proper orthogonal decomposition (POD) method for solving the time dependent partial differential equations. We use the difference between the approximation obtained in the augmented subspace and that obtained in the original POD subspace to construct an error indicator, by which we obtain a general framework for augmented subspace based adaptive POD method. We then provide two strategies to construct the augmented subspaces, the residual type augmented subspace and the coarse-grid approximation type augmented subspace. We apply our new methods to two typical 3D advection-diffusion equations with the advection being the Kolmogorov flow and the ABC flow. Numerical results show that both the residual type augmented subspace based adaptive POD method and the coarse-grid approximation type augmented subspace based adaptive POD method are more efficient than the existing adaptive POD methods, especially for the advection dominated models.

A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in \(\mathbb {R}^2 \) or \(\mathbb {R}^3 \) , such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold’s interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM’s finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver that can scale to tens of millions of degrees of freedom, allowing PDEs to be solved on more complex manifolds. We demonstrate our method’s convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

A hierarchical Bayesian model to monitor pelagic larvae in response to environmental changes

European anchovies and round sardinella play a crucial role, both ecological and commercial, in the Mediterranean Sea. In this paper, we investigate the distribution of their larval stages by analyzing a dataset collected over time (1998–2016) and spaced along the area of the Strait of Sicily. Environmental factors are also integrated. We employ a hierarchical spatio-temporal Bayesian model and approximate the spatial field by a Gaussian Markov Random Field to reduce the computation effort using the Stochastic Partial Differential Equation method. Furthermore, the Integrated Nested Laplace Approximation is used for the posterior distributions of model parameters. Moreover, we propose an index that enables the temporal evaluation of species abundance by using an abundance aggregation within a spatially confined area. This index is derived through Monte Carlo sampling from the approximate posterior distribution of the fitted models. Model results suggest a strong relationship between sea currents’ directions and the distribution of larval European anchovies. For round sardinella, the analysis indicates increased sensitivity to warmer ocean conditions. The index suggests no clear overall trend over the years.