PDE Problems Research Articles

An implicit GNN solver for Poisson-like problems

This paper presents Ψ-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems on general unstructured meshes with mixed boundary conditions. By leveraging the Implicit Layer Theory, Ψ-GNN models an “infinitely” deep network, thus avoiding the empirical tuning of the number of required Message Passing layers to attain the solution. Its original architecture explicitly takes into account the boundary conditions, a critical pre-requisite for physical applications, and is able to adapt to any initially provided solution. Ψ-GNN is trained using a physics-informed loss, and the training process is stable by design. Furthermore, the consistency of the approach is theoretically proven, and its flexibility and generalization efficiency are experimentally demonstrated: the same learned model can accurately handle unstructured meshes of various sizes, as well as different boundary conditions. To the best of our knowledge, Ψ-GNN is the first physics-informed GNN-based method that can handle various unstructured domains, boundary conditions and initial solutions while also providing convergence guarantees.

Weakening the effect of boundaries: “diffusion-free” boundary conditions as a “do least harm” alternative to Neumann

In this note, we discuss a poorly known alternative boundary condition to the usual Neumann or “stress-free” boundary condition typically used to weaken boundary layers when diffusion is present but very small. These “diffusion-free” boundary conditions were first developed (as far as the authors know) in 1995 (Sureshkumar and Beris J. Non-Newtonian Fluid Mech. 60, 53–80, 1995) in viscoelastic flow modelling but are worthy of general consideration in other research areas. To illustrate their use, we solve two simple ODE problems and then treat a PDE problem – the inertial wave eigenvalue problem in a rotating cylinder, sphere and spherical shell for small but non-zero Ekman number E. Where inviscid inertial waves exist (cylinder and sphere), the viscous flows in the Ekman boundary layer are O ( E 1 / 2 ) weaker than for the corresponding stress-free layer and fully O ( E ) weaker than in a non-slip layer. These diffusion-free boundary conditions can also be used with hyperdiffusion and provide a systematic way to generate as many further boundary conditions as required. The weakening effect of this boundary condition could allow precious numerical resources to focus on other areas of the flow and thereby make smaller, more realistic values of diffusion accessible to simulations.

Low-velocity impact behavior analysis on composite double curved-shell reinforced with graphene platelet

ABSTRACT This research investigates the low-velocity impact behavior of a spherical rigid body on a composite shell reinforced with graphene platelets (GPLs), in which GPL have been uniformly dispersed and randomly arranged within each individual layer. The weight proportion of GPL has been altered across the thickness. The modified Halpin Tsai model is employed to analyze the Young's modulus and the effective material characteristics of a composite reinforced with graphene platelets (GPLRC). The mass density and Poisson's ratio of the viscoelastic multilayer composite are determined using the rule of mixture. The material parameters of each layer are defined according to the Kelvin–Voigt model. The governing equations have been derived by applying the first order shear deformation shell theory, Hamilton's principle, and modifying the Hertz contact law. The PDE problems have been transformed into ODE equations through the utilization of the Galerkin method. Subsequently, the ordinary equations have been solved utilizing the Newmark method. A parametric study is conducted to examine the effects of various factors, such as fraction weight, GPL distribution pattern, impactor mass, radius, viscoelastic modulus, and initial velocity, on the low velocity impact response of a composite shell reinforced with GPL.

A frequency-independent bound on trigonometric polynomials of Gaussians and applications

We prove a frequency-independent bound on trigonometric functions of a class of singular Gaussian random fields, which arise naturally from weak universality problems for singular stochastic PDEs. This enables us to reduce the regularity assumption on the nonlinearity of the microscopic models (for pathwise convergence) in KPZ and dynamical Φ34 in the previous works of Hairer-Xu and Furlan-Gubinelli to heuristically optimal thresholds required by PDE structures.

Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control

It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of 4-stage methods with order pairs (3, 3) and (4, 3) in detail and provide three Peer triplets of practical interest. We prove convergence of order s-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s-1$$\\end{document}, at least, for s-stage methods if state, adjoint and control satisfy the corresponding order conditions. Numerical tests show the expected order of convergence for the new Peer triplets.

Travelling waves with continuous profile for hyperbolic Keller-Segel equation

Abstract This work describes a hyperbolic model for cell-cell repulsion with population dynamics. We consider the pressure produced by a population of cells to describe their motion. We assume that cells try to avoid crowded areas and prefer locally empty spaces far away from the carrying capacity. Here, our main goal is to prove the existence of travelling waves with continuous profiles. This article complements our previous results about sharp travelling waves. We conclude the paper with numerical simulations of the PDE problem, illustrating such a result. An application to wound healing also illustrates the importance of travelling waves with a continuous and discontinuous profile.

Numerical discretization of initial–boundary value problems for PDEs with integer and fractional order time derivatives

This paper is mainly concerned with introducing a numerical method for solving initial–boundary value problems with integer and fractional order time derivatives. The method is based on discretizing the considered problems with respect to spatial and temporal domains. With the help of finite difference methods, we transformed the studied problem into a set of fractional differential equations. Then, we implemented the fractional Adams method to solve this set in order to provide approximate solutions to the main problem. This combination results in an algorithm that can efficiently and accurately solve a general class of integer and fractional order initial–boundary value problems, such that it does not need to solve large systems of linear equations. In addition, we discussed the stability of the proposed scheme. Three illustrative examples are numerically solved to reveal the effectiveness and validity of the proposed technique.

Artificial neural network prediction on free convection in a triangular chamber with a v-shaped corrugated bottom wall filled with nanofluid

Artificial neural network (ANN) prediction on magnetohydrodynamic (MHD) free convection for complex systems using nanofluid plays a significant role in understanding and optimizing the system’s behaviour. The current research investigates how various governing parameters affect heat transport within a v-shaped corrugated bottom triangular chamber consisting of Al2O3-H2O nanofluid. The vertical wall is kept at a uniform low temperature, the inclined wall is adiabatic, and the v-shaped corrugated bottom wall is exposed to constant high temperature. The Galerkin weighted residual approach, based on the method of finite elements, is used to solve the dimensionless PDE problems with boundary constraints. The isotherms, streamlines, and average Nusselt numbers are displayed for various Hartmann numbers (0 ≤ Ha ≤ 100), Rayleigh numbers (103 ≤ Ra ≤ 106), and nanoparticles volume fractions (0.01 ≤ ϕ ≤ 0.05). It demonstrates how the average Nu is affected by Ha, Ra, and ϕ. Furthermore, it has been discovered that the average Nu increases with Ra and ϕ but decreases with Ha. However, the Levenberg-Marquardt back propagation method is employed for the artificial neural network (ANN) approach on 120 datasets to estimate the average Nu, demonstrating an excellent correlation with simulated data and an average R2 of 0.99689. MATLAB software is used to calculate the ANN predictions.

PDE generalization of in-context operator networks: A study on 1D scalar nonlinear conservation laws

Can we build a single large model for a wide range of PDE-related scientific learning tasks? Can this model generalize to new PDEs without any fine-tuning? In-context operator learning and the corresponding model In-Context Operator Networks (ICON) represent an initial exploration of these questions. The capability of ICON regarding the first question has been demonstrated previously. In this paper, we present a detailed methodology for solving PDE problems with ICON, and show how a single ICON model can make forward and reverse predictions for different equations with different strides, provided with appropriately designed data prompts. We show positive evidence for the second question above, through a study on 1D scalar nonlinear conservation laws, a family of PDEs with temporal evolution. In particular, we show that an ICON model trained on conservation laws with cubic flux functions can generalize well to some other flux functions of more general forms, without fine-tuning. We also show how to broaden the range of problems that an ICON model can address, by transforming functions and equations to ICON's capability scope. We believe that the progress in this paper is a significant step towards the goal of training a foundation model for PDE-related tasks under the in-context operator learning framework.

Remarks on control and inverse problems for PDEs

Remarks on control and inverse problems for PDEs

Meshfree RBF–FD methods for numerical simulation of PDE problems

The meshfree RBF–FD method is becoming an increasingly popular discretization method for challenging PDE problems due to its flexibility with respect to geometry and its ease of implementation. Using a combination of polyharmonic spline basis functions and polynomial basis functions has proven to be particularly successful with a convergence rate depending on the polynomial degree and good behavior at boundaries due to the contribution of the spline part. A challenge that has been observed is that derivative boundary conditions can give rise to large errors. A known remedy for this problem is to introduce one or more layers of ghost points, which improves the boundary approximations. In a recent paper, Tominec, Larsson, Heryudono (2021), it was shown that using oversampling is another effective way to handle this problem. Oversampling also allows for theoretical analysis of the method, which can then be viewed as a discretization of a continuous least-squares projection. By looking deeper into the error estimates we gain an understanding about how to best implement the method.

Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem

An effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders for approximating the first and second derivatives of sufficiently differentiable functions using the radial basis function-generated Hermite finite-difference (RBF-HFD) scheme. By utilizing only three equally spaced points in 1D, we are able to boost the convergence rate to four. Extensive tests have been conducted to demonstrate the effectiveness of the proposed theoretical weighting coefficients in solving interpolation and PDE problems.

Robust Variational Physics-Informed Neural Networks

We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov–Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN’s loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm of the residual. The main advantage of such a loss definition is that it provides a reliable and efficient estimator of the true error in the energy norm under the assumption of the existence of a local Fortin operator. We test the performance and robustness of our algorithm in several advection–diffusion problems. These numerical results perfectly align with our theoretical findings, showing that our estimates are sharp.

Incorporating Lasso Regression to Physics-Informed Neural Network for Inverse PDE Problem

Incorporating Lasso Regression to Physics-Informed Neural Network for Inverse PDE Problem

Second-Order Hamilton–Jacobi PDE Problems and Certain Related First-Order Problems, Part 1: Approximation

Second-Order Hamilton–Jacobi PDE Problems and Certain Related First-Order Problems, Part 1: Approximation

Adaptive task decomposition physics-informed neural networks

As a new approach for solving partial differential equations, physics-informed neural networks (PINNs) have received extensive attention in recent years and have made breakthroughs in various fields. Serving as a neural network-based universal solver, they possess advantages such as high continuity and being meshless. However, an increasing number of studies have pointed out that PINNs may fail to converge to accurate results when dealing with complex tasks. In this paper, we highlight the differences between training PINNs and regular neural network tasks, summarizing three key characteristics of training PINNs. Based on these findings, we propose a strategy of task decomposition and progressive learning, which successfully explains the effectiveness of existing methods for time-dependent problems and reveals their limitations. For time-dependent and time-independent problems, we partition the tasks into physically complete and progressively challenging sub-tasks. The division is based on the resolution requirements for time-independent problems and the coverage of the time domain for time-dependent problems. Through task parameters and the decomposition of the loss terms, we adaptively control the progressive increase in task complexity during the training process. The proposed method overcomes the limitations of existing approaches, incurs no additional computational cost, and is applicable to both time-dependent and time-independent problems, demonstrating higher generality. Furthermore, building upon the concepts of task decomposition and progressive learning, we introduce a stable and efficient minibatch training method for the first time. In a series of numerical experiments on baseline PDE problems, our approaches outperformed other state-of-the-art algorithms.

A large deviation principle for reflected SPDEs on infinite spatial domain

In this paper, we study a large deviation principle for a reflected stochastic partial differential equation on infinite spatial domain. A new sufficient condition for the weak convergence criterion proposed by Matoussi, Sabbagh and Zhang [A. Matoussi, W. Sabbagh and T.-S. Zhang, Large deviation principles of obstacle problems for quasilinear stochastic PDEs, Appl. Math. Optim. 83(2) (2021) 849–879] plays an important role in the proof.

Boundary corrections on multi-dimensional PDEs

Two new boundary correction techniques are proposed in order to mitigate the order reduction phenomenon associated with the numerical solution of initial boundary value problems for parabolic partial differential equations in arbitrary spatial dimensions with time-dependent Dirichlet boundary conditions. The new techniques are based on the idea of discretizing the PDE problem at the boundary points as similar as possible to that of the interior points of the domain. These new techniques are considered for the time integration with W-methods based on approximate matrix factorization. By suitably modifying the internal stages of the methods on the boundary points, it is illustrated by numerical testing with time-dependent boundary conditions that the new boundary correction techniques are able to keep the same accuracy and order of convergence that the method reaches in the case of homogeneous boundary conditions.

M-WDRNNs: Mixed-Weighted Deep Residual Neural Networks for Forward and Inverse PDE Problems

Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations in recent years. But studies have shown that there is a gradient pathology in PINNs. That is, there is an imbalance gradient problem in each regularization term during back-propagation, which makes it difficult for neural network models to accurately approximate partial differential equations. Based on the depth-weighted residual neural network and neural attention mechanism, we propose a new mixed-weighted residual block in which the weighted coefficients are chosen autonomously by the optimization algorithm, and one of the transformer networks is replaced by a skip connection. Finally, we test our algorithms with some partial differential equations, such as the non-homogeneous Klein–Gordon equation, the (1+1) advection–diffusion equation, and the Helmholtz equation. Experimental results show that the proposed algorithm significantly improves the numerical accuracy.

Regularity of time-periodic solutions to autonomous semilinear hyperbolic PDEs

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable x and C∞-smooth with respect to the unknown function u, then the solution is C∞-smooth with respect to the time variable t, and if the PDEs are C∞-smooth with respect to x and u, then the solution is C∞-smooth with respect to t and x. The same is true for appropriate weak solutions. Moreover, we show examples of time-periodic functions, which do not satisfy the non-resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not C∞-smooth, neither with respect to t nor with respect to x, even if the PDEs are C∞-smooth with respect to x and u. For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant equations.