Abstract: This paper describes Dedalus, an open-source Python code for simulating partial differential equations from all areas of physics. Dedalus translates plain-text equations into efficient and parallelized solvers using global spectral methods. Here the authors detail the numerical methods enabling this translation and describe the code's design and implementation. They also illustrate its capabilities with diverse examples, including optical network dynamics, magnetized shocks in plasmas, large-scale oceanic flows, low Reynolds number flows, stellar and atmospheric waves, and diamagnetic levitation.

Abstract: Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing theoretical explanation for this success is still lacking. Since these applications revolve around approximating an unknown function from data observations, part of the answer must involve the ability of NNs to produce accurate approximations. This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion, i.e. error versus the number of parameters used to create the approximant. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation, and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward. The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of f into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parametrized nonlinear manifold. It is shown that this manifold has certain space-filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates the challenge to the numerical method of finding best or good parameter choices when trying to approximate.

Abstract: Hybrid High-Order (HHO) methods are new generation numerical methods for models based on Partial Differential Equations with features that set them apart from traditional ones. These include: the support of polytopal meshes including non star-shaped elements and hanging nodes; the possibility to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics. The first part of the monograph lays the foundation of the method considering linear scalar second-order models, including scalar diffusion, possibly heterogeneous and anisotropic, and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity and incompressible fluid flows.

Abstract: In this paper, an extension is paid to an idea of fractal and fractional derivatives which has been applied to a number of ordinary differential equations to model a system of partial differential equations. As a case study, the fractal fractional Schnakenberg system is formulated with the Caputo operator (in terms of the power law), the Caputo-Fabrizio operator (with exponential decay law) and the Atangana-Baleanu fractional derivative (based on the Mittag-Liffler law). We design some algorithms for the Schnakenberg model by using the newly proposed numerical methods. In such schemes, it worth mentioning that the classical cases are recovered whenever α = 1 and β = 1 . Numerical results obtained for different fractal-order ( β ∈ ( 0 , 1 ) ) and fractional-order ( α ∈ ( 0 , 1 ) ) are also given to address any point and query that may arise.

Abstract: Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs it has also been proved mathematically that deep neural networks overcome the curse of dimensionality in the numerical approximation of solutions of such linear PDEs. The key contribution of this article is to rigorously prove this for the first time for a class of nonlinear PDEs. More precisely, we prove in the case of semilinear heat equations with gradient-independent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced full history recursive multilevel Picard approximations for semilinear PDEs.

Abstract: This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modelling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state of the art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modelling and their numerical methods in materials science, fluid mechanics, biology and image science.

Abstract: In this work, we introduce a numerical and analytical study of the Peyrard-Bishop DNA dynamic model equation. This model is studied analytically by hyperbolic and exponential ansatz methods and numerically by finite difference method. A comparison between the results obtained by the analytical methods and the numerical method is investigated. Furthermore, some figures are introduced to show how accurate the solutions will be obtained from the analytical and numerical methods.

Abstract: Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to adequately model observed phenomena or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modeling and algorithmic extensions which serve to show the wide applicability of nonlocal modeling.

Abstract: The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numerical methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one-, two-, and three-dimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Furthermore, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately.

Abstract: Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature. The approaches discussed, in contrast to many other methods, have good mesh properties that avoid mesh coalescence and very nonuniform meshes. Mean curvature flow, surface diffusion, anisotropic geometric flows, solidification, two-phase flow, Willmore and Helfrich flow as well as biomembranes are treated. We show stability results as well as results explaining the good mesh properties.

Abstract: In this article, the effects of swimming gyrotactic microorganisms for magnetohydrodynamics nanofluid using Darcy law are investigated. The numerical results of nonlinear coupled mathematical model are obtained by means of Successive Local Linearization Method. This technique is based on a simple notion of the decoupling systems of equations utilizing the linearization of the unknown functions sequentially according to the order of classifying the system of governing equations. The linearized equations, that developed a sequence of linear differential equations along with variable coefficients, were solved by employing the Chebyshev spectral collocation method. The convergence speed of the SLLM technique can be willingly upgraded by successive applying over relaxation method. The comparison of current study with available published literature has been made for the validation of obtained results. It is found that the reported numerical method is in perfect accord with the said similar methods. The results are displayed through tables and graphs.

Abstract: The various studies of partial differential equations (PDEs) are hot topics of mathematical research. Among them, solving PDEs is a very important and difficult task. Since many partial differential equations do not have analytical solutions, numerical methods are widely used to solve PDEs. Although numerical methods have been widely used with good performance, researchers are still searching for new methods for solving partial differential equations. In recent years, deep learning has achieved great success in many fields, such as image classification and natural language processing. Studies have shown that deep neural networks have powerful function-fitting capabilities and have great potential in the study of partial differential equations. In this paper, we introduce an improved Physics Informed Neural Network (PINN) for solving partial differential equations. PINN takes the physical information that is contained in partial differential equations as a regularization term, which improves the performance of neural networks. In this study, we use the method to study the wave equation, the KdV–Burgers equation, and the KdV equation. The experimental results show that PINN is effective in solving partial differential equations and deserves further research.

Abstract: Mechanical behavior of Lithium-ion batteries under dynamic impact loading is crucial in assessing and improving the crash safety of batteries. To understand the possible causes of internal short circuit (ISC) in the impacted batteries, both experimental and numerical methods are necessary. Quasi-static and dynamic tests for a type of pouch-type battery cell are performed according to a UN standard. The mechanical characterization of all the component materials in the battery is carefully carried out under different strain rate and loading conditions. Then a detailed 3D Finite Element (FE) Model of the pouch cell with all the components is established. Combining the experimental and numerical approaches, the dynamic deformation process under the UN 38.3 Impact tests and the corresponding failure mechanism of pouch cells are firstly captured. Meanwhile, the conditions for the occurrence of thermal runaway are investigated with the comparison of quasi-static and dynamic tests. It is believed that the rough fracture surface and the crushed powdery residue are responsible for the continuous internal short circuits and thermal runaway of the broken batteries. To improve the crash safety performance of the pouch cell, some safety suggestions are proposed.

Abstract: The ultraslow-roll (USR) inflationary models predict large-amplitude scalar perturbations at small scales which can lead to the primordial black hole production and scalar-induced gravitational waves. In general, scalar perturbations in the USR models can only be obtained using a numerical method because the usual slow-roll approximation breaks. In this work, we propose an analytical approach to estimate the scalar spectrum which is consistent with the numerical result. We find that the USR inflationary models predict a peak with power-law slopes in the scalar spectrum and energy spectrum of gravitational waves, and we derive the expression of the spectral indexes in terms of the inflationary potential. In turn, the inflationary potential near the USR regime can be reconstructed from the negative spectral index of the gravitational wave energy spectrum.

Abstract: In some previous works, the authors have introduced a strategy to develop well-balanced high-order numerical methods for nonconservative hyperbolic systems in the framework of path-conservative numerical methods. The key ingredient of these methods is a well-balanced reconstruction operator, i.e. an operator that preserves the stationary solutions in some sense. A strategy has been also introduced to modify any standard reconstruction operator like MUSCL, ENO, CWENO, etc. in order to be well-balanced. In this article, the specific case of 1d systems of balance laws is addressed and difficulties are gradually introduced: the methods are presented in the simpler case in which the source term does not involve Dirac masses. Next, systems whose source term involves the derivative of discontinuous functions are considered. In this case, the notion of weak solution is discussed and the Generalized Hydrostatic Reconstruction technique is used for the treatment of singular source terms. A technique to preserve the well-balancedness of the methods in the presence of numerical integration is introduced. The strategy is applied to derive first, second and third order well-balanced methods for Burgers’ equation with a nonlinear source term and for the Euler equations with gravity.

Abstract: The Iterative Filtering method is a technique developed recently for the decomposition and analysis of nonstationary and nonlinear signals. In this work, we propose two alternative formulations of the original algorithm which allows to transform the iterative filtering method into a direct technique, making the algorithm closer to an online algorithm. We present a few numerical examples to show the effectiveness of the proposed approaches.

Abstract: A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method co...

Abstract: In this work, optimal control for a fractional-order nonlinear mathematical model of cancer treatment is presented. The suggested model is determined by a system of eighteen fractional differential equations. The fractional derivative is defined in the Atangana–Baleanu Caputo sense. Necessary conditions for the control problem are derived. Two control variables are suggested to minimize the number of cancer cells. Two numerical methods are used for simulating the proposed optimal system. The methods are the iterative optimal control method and the nonstandard two-step Lagrange interpolation method. In order to validate the theoretical results, numerical simulations and comparative studies are given.

Abstract: We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory predicts that the performance of the model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation. We use various methods to establish comparability between test-cases by minimizing the effect of the choice of test-cases on the optimization and sampling aspects of the learning problem. We find strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis.