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1. Identifying Arguments of Space-Time Fractional Diffusion: Data-Driven Approach

   https://doi.org/https://doi.org/10.3389/fams.2020.00014  

   Znaidi Mohamed Ridha, Gupta Gaurav ( May 2020 , Frontiers in applied mathematics and statistics)

   A plethora of complex dynamical systems from disordered media to biological systems exhibit mathematical characteristics (e.g., long-range dependence, self-similar and power law magnitude increments) that are well-fitted by fractional partial differential equations (PDEs). For instance, some biological systems displaying an anomalous diffusion behavior, which is characterized by a non-linear mean-square displacement relation, can be mathematically described by fractional PDEs. In general, the PDEs represent various physical laws or rules governing complex dynamical systems. Since prior knowledge about the mathematical equations describing complex dynamical systems in biology, healthcare, disaster mitigation, transportation, or environmental sciences may not be available, we aim to provide algorithmic strategies to discover the integer or fractional PDEs and their parameters from system's evolution data. Toward deciphering non-trivial mechanisms driving a complex system, we propose a data-driven approach that estimates the parameters of a fractional PDE model. We study the space-time fractional diffusion model that describes a complex stochastic process, where the magnitude and the time increments are stable processes. Starting from limited time-series data recorded while the system is evolving, we develop a fractional-order moments-based approach to determine the parameters of a generalized fractional PDE. We formulate two optimization problems to allow us to estimate the argumentsmore »of the fractional PDE. Employing extensive simulation studies, we show that the proposed approach is effective at retrieving the relevant parameters of the space-time fractional PDE. The presented mathematical approach can be further enhanced and generalized to include additional operators that may help to identify the dominant rule governing the measurements or to determine the degree to which multiple physical laws contribute to the observed dynamics.« less
2. A Large-Scale Comparison of Tetrahedral and Hexahedral Elements for Solving Elliptic PDEs with the Finite Element Method

   https://doi.org/10.1145/3508372  

   Schneider, Teseo ; Hu, Yixin ; Gao, Xifeng ; Dumas, Jérémie ; Zorin, Denis ; Panozzo, Daniele ( June 2022 , ACM Transactions on Graphics)

   The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of this benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance. Asmore »a specific application of our geometry and benchmark dataset, we explore the question of relative advantages of unstructured (triangular/ tetrahedral) and structured (quadrilateral/hexahedral) discretizations. We observe that for Lagrange-type elements, while linear tetrahedral elements perform poorly, quadratic tetrahedral elements perform equally well or outperform hexahedral elements for our set of problems and currently available mesh generation algorithms. This observation suggests that for common problems in structural analysis, thermal analysis, and low Reynolds number flows, high-quality results can be obtained with unstructured tetrahedral meshes, which can be created robustly and automatically. We release the description of the benchmark problems, meshes, and reference implementation of our testing infrastructure to enable statistically significant comparisons between different FE methods, which we hope will be helpful in the development of new meshing and FEA techniques.« less
3. Delayed Hopf Bifurcation and Space–Time Buffer Curves in the Complex Ginzburg–Landau Equation

   https://doi.org/10.1093/imamat/hxac001  

   Goh, Ryan ; Kaper, Tasso J ; Vo, Theodore ( March 2022 , IMA Journal of Applied Mathematics)

   Abstract In this article, the recently discovered phenomenon of delayed Hopf bifurcations (DHB) in reaction–diffusion partial differential equations (PDEs) is analysed in the cubic Complex Ginzburg–Landau equation, as an equation in its own right, with a slowly varying parameter. We begin by using the classical asymptotic methods of stationary phase and steepest descents on the linearized PDE to show that solutions, which have approached the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the instantaneous Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linearized PDE has a saddle point, and the Stokes and anti-Stokes lines are central to the asymptotics. The non-linear terms are treated by applying an iterative method to the mild form of the PDE given by perturbations about the linear particular solution. This tracks the closeness of solutions near the attracting and repelling QSS in the full, non-linear PDE. Next, we show that beyond a key Stokes line through the saddle there is a curve in the space-time plane along which the particular solution of the linear PDE ceases to be exponentially small, causing the solution of the non-linearmore »PDE to diverge from the repelling QSS and exhibit large-amplitude oscillations. This curve is called the space–time buffer curve. The homogeneous solution also stops being exponentially small in a spatially dependent manner, as determined also by the initial data and time. Hence, a competition arises between these two solutions, as to which one ceases to be exponentially small first, and this competition governs spatial dependence of the DHB. We find four different cases of DHB, depending on the outcomes of the competition, and we quantify to leading order how these depend on the main system parameters, including the Hopf frequency, initial time, initial data, source terms, and diffusivity. Examples are presented for each case, with source terms that are a uni-modal function, a smooth step function, a spatially periodic function and an algebraically growing function. Also, rich spatio-temporal dynamics are observed in the post-DHB oscillations. Finally, it is shown that large-amplitude source terms can be designed so that solutions spend substantially longer times near the repelling QSS, and hence, region-specific control over the delayed onset of oscillations can be achieved.« less
4. Spectrally adapted physics-informed neural networks for solving unbounded domain problems

   https://doi.org/10.1088/2632-2153/acd0a1  

   Xia, Mingtao ; Böttcher, Lucas ; Chou, Tom ( May 2023 , Machine Learning: Science and Technology)

   Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient numerical methods that can resolve the dependence of the PDE on the unbounded variable over at least several orders of magnitude. We propose a solution to such problems by combining two classes of numerical methods: (i) adaptive spectral methods and (ii) physics-informed neural networks (PINNs). The numerical approach that we develop takes advantage of the ability of PINNs to easily implement high-order numerical schemes to efficiently solve PDEs and extrapolate numerical solutions at any point in space and time. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs. Through a number of examples, we demonstrate the advantages of the proposed spectrally adapted PINNs in solving PDEs and estimating model parameters from noisy observations in unbounded domains.
5. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness

   https://doi.org/10.1007/s10884-021-10013-5  

   Hupkes, H. J. ; Van Vleck, E. S. ( June 2022 , Journal of Dynamics and Differential Equations)

   Abstract In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations.