High-dimensional Equations Research Articles

PINN for solving forward and inverse problems involving integrable two-dimensional nonlocal equations

In this paper, the physics informed neural network (PINN) deep learning method is applied to solve two-dimensional nonlocal equations, including the partial reverse space y-nonlocal Mel’nikov equation, the partial reverse space-time nonlocal Mel’nikov equation and the nonlocal two-dimensional nonlinear Schrödinger (NLS) equation. By the PINN method, we successfully derive a data-driven two soliton solution, lump solution and rogue wave solution. Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small, which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations. Moreover, the parameter discovery of the partial reverse space-time nonlocal Mel’nikov equation is analysed in terms of its soliton solution for the first time.

An implicit unified gas-kinetic particle method with large time steps for gray radiation transport

For a long time, efficient algorithms for high-dimensional equations, represented by photon radiation transport, have been one important topic in the development of computational methods for particle transport processes. In this paper, we present an implicit unified gas-kinetic particle (IUGKP) method for multiscale gray radiative transfer. Based on the integral solution of the radiative transfer equation, the photon transport processes are categorized into non-equilibrium transport processes with a large photon free path and equilibrium transport processes with a small photon free path. The long-path processes are solved by an implicit Monte Carlo (IMC) method, and the short-path processes are solved by an implicit diffusion system. The closure formulation of photon distribution is derived from the local integral solution of the radiative transfer equation to couple the IMC and diffusion system. The improvement of the proposed IUGKP method over UGKP method is that particles can be tracked continuously instead of just until the first collision, making simulation with large time steps possible. The IUGKP method has the properties of asymptotic-preserving (AP) and regime-adaptive (RA). The AP property states that the IUGKP method converges to the consistent numerical methods for the asymptotic limiting equations of RTE in the limiting regimes. The RA property states that the computational accuracy of the IUGKP method adapts to the regimes. In this paper, the mathematical proof of the AP and RA properties is presented, and the multiscale numerical tests are performed to demonstrate the accuracy and efficiency of the IUGKP method.

Singular Abreu equations and linearized Monge–Ampère equations with drifts

We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher-dimensional case by transforming singular Abreu equations into linearized Monge–Ampère equations with drifts. We establish global Hölder estimates for linearized Monge–Ampère equations with drifts under suitable hypotheses, and then apply them to prove the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side are also discussed.

Data-Driven Modeling of Tire-Soil Interaction with POD-Based Model Order Reduction

Abstract A data-driven model capable of predicting time-domain solutions of a high-fidelity tire-soil interaction model is developed to enable quick prediction of mobility capabilities on deformable terrain. The adaptive model order reduction based on the proper orthogonal decomposition (POD), for which the high-dimensional equations are projected onto the reduced subspace, is utilized as the basis for predicting the time-domain tire-soil interaction behavior. The projection-based model order reduction, however, requires many online matrix operations due to the successive updates of the nonlinear functions and Jacobians at every time step, thereby hindering the computational improvement. Therefore, a data-driven approach using a Long Short Term Memory (LSTM) neural network is introduced to predict the reduced order coordinates without the projection and time integration processes for computational speedup. With this model, a hybrid data-driven/physics-based off-road mobility model is proposed, where four separate LSTM-POD data-driven tire-soil interaction models are integrated into the physics-based multibody dynamics (MBD) vehicle model through a force-displacement coupling algorithm. By doing so, the individual data-driven tire-soil interaction model can be constructed efficiently, and the MBD and LSTM models are assembled as a single off-road mobility model and analyzed with existing off-road mobility solvers. The predictive ability and computational benefit of the proposed data-driven tire-soil interaction model with the POD-based model order reduction are examined with several numerical examples.

The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates

This is the second part of a two-part series devoted to an in depth understanding of the dynamical behaviour of the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary |x|=h(t). In Part 1 [19], we have shown that the long-time dynamics of this problem is characterised by a spreading-vanishing dichotomy, and there exists a threshold condition on the diffusion kernel J(|x|) such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed c0 is determined by an associated semi-wave problem established in [15]. In Part 2 here, we obtain more precise description of the spreading profile by focusing on some natural classes of kernel functions, including those satisfying J(|x|)∼|x|−β for |x|≫1 in RN. Our results for such kernels reveal a striking difference of behaviour from the pattern exhibited in the one dimension case [18] when β crosses the value N+2. More precisely, (a) when β∈(N,N+1], we show that for t≫1, h(t)∼t1/(β−N) if β∈(N,N+1), and h(t)∼tln⁡t if β=N+1, which is of the same pattern as in dimension one, namely we recover the result in [18] by letting N=1 in the above statements; (b) when β∈(N+1,N+2], the front has a finite spreading speed c0=c0(β) in the sense that limt→∞⁡h(t)/t=c0, and our results here on the order of shift c0t−h(t) are again of the same pattern as in dimension one; (c) when β>N+2, the front still has a finite spreading speed c0, but a significant change happens to the order of shift c0t−h(t) between N≥2 and N=1: for t≫1, c0t−h(t)∼ln⁡t when N≥2, but c0t−h(t)∼1 when N=1.

Gravitational collapse of relativistic compact objects in higher dimension

In this work, we present an iterative method for gravitational collapse in higher dimensions. A framework is developed in a post-quasi-static regime with non-comoving coordinates. The internal five-dimensional system is smoothly matched with the corresponding outer Vaidya space-time over the boundary surface (BS). This correspondence provides a set of dimensionless surface equations in higher dimensions. The physical quantities such as Doppler shift, luminosity, and redshift at the BS of gravitating systems can be described through these surface equations. This procedure offers valuable insights that facilitate comprehension of the behavior of compact objects.

Operator-splitting finite element method for solving the radiative transfer equation

AbstractAn operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

Efficient and scalable path-planning algorithms for curvature constrained motion in the Hamilton-Jacobi formulation

We present a partial-differential-equation-based optimal path-planning framework for curvature constrained motion, with application to vehicles in 2- and 3-spatial-dimensions. This formulation relies on optimal control theory, dynamic programming, and Hamilton-Jacobi-Bellman equations. We develop efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations which can solve these types of path-planning problems efficiently, even in high dimensions, while maintaining the Hamilton-Jacobi formulation. Because our method is rooted in optimal control theory and has no black box components, it has solid interpretability, and thus averts the tradeoff between interpretability and efficiency for high-dimensional path-planning problems. We demonstrate our method with several examples.

A family of linearly weighted-$$\theta $$ compact ADI schemes for sine-Gordon equations in high dimensions

A family of linearly weighted-$$\theta $$ compact ADI schemes for sine-Gordon equations in high dimensions

Dynamical analysis and explicit traveling wave solutions to the higher-dimensional generalized nonlinear wave system

For dealing with exact solutions and properties of PDEs, the higher-dimensional equations are far more complicated than the lower dimensional ones. In the current paper, by using the dynamical system method, we study the (n+1)-dimensional generalized nonlinear coupled KdV (c-KdV) system, which includes a lot of important c-KdV types of systems as its special case. The bifurcations and phase portraits of the nonlinear PDE system are investigated, then we give the explicit parametric representations of all possible solitary wave solutions and periodic wave solutions under different parametric conditions.

Investigation of the Time Fractional Higher-Dimensional Nonlinear Modified Equation of Wave Propagation

In this article, we analyzed the time fractional higher-dimensional nonlinear modified model of wave propagation, namely the (3 + 1)-dimensional Benjamin–Bona–Mahony-type equation. The fractional sense was defined by the classical Riemann–Liouville fractional derivative. We derived firstly the existence of symmetry of the time fractional higher-dimensional equation. Next, we constructed the one-dimensional optimal system to the time fractional higher-dimensional nonlinear modified model of wave propagation. Subsequently, it was reduced into the lower-dimensional fractional differential equation. Meanwhile, on the basis of the reduced equation, we obtained its similarity solution. Through a series of analyses of the time fractional high-dimensional model and the results of the above obtained, we can gain a further understanding of its essence.

Computational dissection of genetic variation modulating the response of multiple photosynthetic phenotypes to the light environment

BackgroundThe expression of biological traits is modulated by genetics as well as the environment, and the level of influence exerted by the latter may vary across characteristics. Photosynthetic traits in plants are complex quantitative traits that are regulated by both endogenous genetic factors and external environmental factors such as light intensity and CO2 concentration. The specific processes impacted occur dynamically and continuously as the growth of plants changes. Although studies have been conducted to explore the genetic regulatory mechanisms of individual photosynthetic traits or to evaluate the effects of certain environmental variables on photosynthetic traits, the systematic impact of environmental variables on the dynamic process of integrated plant growth and development has not been fully elucidated.ResultsIn this paper, we proposed a research framework to investigate the genetic mechanism of high-dimensional complex photosynthetic traits in response to the light environment at the genome level. We established a set of high-dimensional equations incorporating environmental regulators to integrate functional mapping and dynamic screening of gene‒environment complex systems to elucidate the process and pattern of intrinsic genetic regulatory mechanisms of three types of photosynthetic phenotypes of Populus simonii that varied with light intensity. Furthermore, a network structure was established to elucidate the crosstalk among significant QTLs that regulate photosynthetic phenotypic systems. Additionally, the detection of key QTLs governing the response of multiple phenotypes to the light environment, coupled with the intrinsic differences in genotype expression, provides valuable insights into the regulatory mechanisms that drive the transition of photosynthetic activity and photoprotection in the face of varying light intensity gradients.ConclusionsThis paper offers a comprehensive approach to unraveling the genetic architecture of multidimensional variations in photosynthetic phenotypes, considering the combined impact of integrated environmental factors from multiple perspectives.

(3+1)-Dimensional Gardner Equation Deformed from (1+1)-Dimensional Gardner Equation and its Conservation Laws

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to beyorzindependent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.

Fisher information and shape-morphing modes for solving the Fokker–Planck equation in higher dimensions

The Fokker–Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional numerical methods becomes computationally prohibitive. Here, we introduce a fast, scalable, and interpretable method for solving the Fokker–Planck equation which is applicable in higher dimensions. This method approximates the solution as a linear combination of shape-morphing Gaussians with time-dependent means and covariances. These parameters evolve according to the method of reduced-order nonlinear solutions (RONS) which ensures that the approximate solution stays close to the true solution of the PDE for all times. As such, the proposed method approximates the transient dynamics as well as the equilibrium density, when the latter exists. Our approximate solutions can be viewed as an evolution on a finite-dimensional statistical manifold embedded in the space of probability densities. We show that the metric tensor in RONS coincides with the Fisher information matrix on this manifold. We also discuss the interpretation of our method as a shallow neural network with Gaussian activation functions and time-varying parameters. In contrast to existing deep learning methods, our method is interpretable, requires no training, and automatically ensures that the approximate solution satisfies all properties of a probability density.

Toward null-state equations in d > 2

In two-dimensional CFTs with a large central charge, the level-two BPZ equation governs the heavy-light scalar four-point correlator when the light probe scalar has dimension h = −12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ -\\frac{1}{2} $$\\end{document} ; the corresponding linear ordinary differential equation can be recast into a schematic form x2uxx + u = 0. In this paper, we make an observation that in a class of four-dimensional CFTs with a large central charge, the heavy-light scalar correlator in the near-lightcone limit obeys a similar equation, x3uxxxy + u = 0, when the light scalar has dimension ∆ = –1. We focus on the multi-stress tensor sector of the theory and also discuss the corresponding equations for the cases with ∆ = –2, –3. The solutions to these linear partial differential equations in higher dimensions are shown, after a suitable change of variables, to reproduce the near-lightcone correlators previously obtained via holography and the conformal bootstrap.

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker–Planck equation in high dimensions and the Kuramoto–Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training.

Long-time dynamics of classical Keller–Segel equation

In this article, we obtain global dynamics of 2D Keller–Segel equation (KS for short) in subcritical regime with no additional assumptions. With initial data small in some Triebel-Lizorkin space, we also characterize the long-time asymptotics of global mild solutions to the KS equation in higher dimensions where the spatial dimension n≥3.

On the propagation of regularity for solutions of the Zakharov–Kuznetsov equation

In this paper, we focus on the Zakharov–Kuznetsov (ZK) equation in the [Formula: see text]-dimensional setting with [Formula: see text] and investigate its smoothness properties. We extend the well-known regularity propagation phenomenon observed in the 2D and 3D cases, where the regularity of the initial data on certain half-spaces propagates with infinite speed, to the case where the regularity of the initial data is measured on a fractional scale. To achieve this, we introduce new localization formulas that enable us to describe the regularity of the solution on a specific class of subsets in Euclidean space. This work provides insights into the regularity behavior of solutions of the ZK equation in higher dimensions and with more general initial data.

Nonlocal symmetries and solutions of the (2+1) dimension integrable Burgers equation

The conservation laws of low-dimensional partial differential equations have been employed by Lou et al. to construct various high-dimensional equations, particularly focusing on high-dimensional integrable equations. However, solving these high-dimensional equations poses significant challenges. In this paper, the (2+1)-dimensional Burgers equation is studied by means of nonlocal symmetry method for the first time. For this high-dimensional equation, we establish a link with the exact solution of the (1+1)-dimensional Burgers equation through nonlocal symmetry. Furthermore, we successfully construct multiple exact solutions for the high-dimensional Burgers equation by leveraging the exact solution of the low-dimensional counterpart. We also give the corresponding images of several solutions to study the dynamic behavior of the equations.

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.