High-order Integration Research Articles

Learning dynamical systems from noisy data with inverse-explicit integrators

We introduce the mean inverse integrator (MII), a novel approach that improves accuracy when training neural networks to approximate vector fields of dynamical systems using noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge–Kutta methods. We show that the class of mono-implicit Runge–Kutta methods (MIRK) offers significant advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained by inserting the training data in the MIRK formulae. This allows the use of symmetric and high-order integrators without requiring the solution of non-linear systems of equations. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with extensive numerical experiments using data from chaotic and high-dimensional Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, providing theoretical support for the favorable performance of MII and symmetric MIRK methods.

Implicit Frictional Dynamics With Soft Constraints.

Dynamics simulation with frictional contacts is important for a wide range of applications, from cloth simulation to object manipulation. Recent methods using smoothed lagged friction forces have enabled robust and differentiable simulation of elastodynamics with friction. However, the resulting frictional behavior can be inaccurate and may not converge to analytic solutions. Here we evaluate the accuracy of lagged friction models in comparison with implicit frictional contact systems. We show that major inaccuracies near the stick-slip threshold in such systems are caused by lagging of friction forces rather than by smoothing the Coulomb friction curve. Furthermore, we demonstrate how systems involving implicit or lagged friction can be correctly used with higher-order time integration and highlight limitations in earlier attempts. We demonstrate how to exploit forward-mode automatic differentiation to simplify and, in some cases, improve the performance of the inexact Newton method. Finally, we show that other complex phenomena can also be simulated effectively while maintaining smoothness of the entire system. We extend our method to exhibit stick-slip frictional behavior and preserve volume on compressible and nearly-incompressible media using soft constraints.

Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence

We study the ODE/IM correspondence between two-dimensional WAr/WDr-type conformal field theories and the higher-order ordinary differential equations (ODEs) obtained from the affine Toda field theories associated with Ar(1)/Dr(1)-type affine Lie algebras. We calculate the period integrals of the WKB solution to the ODE along the Pochhammer contour, where the WKB expansions correspond to the classical conserved currents of the Drinfeld-Sokolov integrable hierarchies. We also compute the integrals of motion for WAr(WDr) algebras on a cylinder. Their eigenvalues on the vacuum state are confirmed to agree with the period integrals up to the sixth order. These results generalize the ODE/IM correspondence to higher-order ODEs and can be used to predict higher-order integrals of motion.

A novel high-order explicit exponential integrator scheme for the space–time fractional Bloch–Torrey equation

A novel high-order explicit exponential integrator scheme for the space–time fractional Bloch–Torrey equation

Disturbances of Interoception in Schizophrenia – Focus on Neuroimaging

Abstract: The current paper presents the most recent theoretical understanding of interoception (INTC) in light of the predictive processing framework and the Embodied Predictive Interoceptive Coding model and reviews recent experimental data on different interoceptive disturbances in schizophrenia (SCZ) such as lower interoceptive accuracy, nociceptive dysfunction reflected by reduced pain sensitivity, disturbed thermoregulation, impaired gastrointestinal regulation, cardiovascular and respiratory dysfunction, and immune disturbances. The immense clinical implications of these disturbances are discussed in the context of the high rates of morbidity and mortality in SCZ related to suicide and cardiac death. Next, the potential of INTC-related treatments is reviewed, and conclusions are made on the evidence for the use of vagus nerve stimulation, mindfulness-based interventions, and exercise therapy. The neuroimaging perspective is given in a brief account of the INTC-related findings in healthy individuals suggesting the right anterior insula and anterior cingulate cortex as key brain areas for interoceptive awareness, accuracy, and sensibility, by supporting higher-order representation and integration of visceral, and somatosensory interoception with exteroceptive information. In SCZ, on the other hand, neuroimaging research on specific interoceptive abilities is lacking but indirect evidence points to disrupted activation of major INTC network hubs. There is converging evidence of changes in these hubs in terms of both structure and function (volume reductions, aberrant activation, and connectivity). This narrative review identifies several gaps in contemporary INTC-related research in SCZ that need to be addressed in future experimental and clinical studies to elucidate the exact nature of the disturbances of interoceptive abilities in modalities other than cardioception; the interactions between different interoceptive modalities; the association between interoceptive abilities and other mental functions (such as emotion recognition and regulation, decision making, learning, memory, etc.); the efficacy of nonpharmacological and pharmacological INTC-based interventions; and the relation of disturbed interoceptive abilities with brain structure and function.

Multiscale Analysis of Sandwich Beams with Polyurethane Foam Core: A Comparative Study of Finite Element Methods and Radial Point Interpolation Method.

This study presents a comprehensive multiscale analysis of sandwich beams with a polyurethane foam (PUF) core, delivering a numerical comparison between finite element methods (FEMs) and a meshless method: the radial point interpolation method (RPIM). This work aims to combine RPIM with homogenisation techniques for multiscale analysis, being divided in two phases. In the first phase, bulk PUF material was modified by incorporating circular holes to create PUFs with varying volume fractions. Then, using a homogenisation technique coupled with FEM and four versions of RPIM, the homogenised mechanical properties of distinct PUF with different volume fractions were determined. It was observed that RPIM formulations, with higher-order integration schemes, are capable of approximating the solution and field smoothness of high-order FEM formulations. However, seeking a comparable field smoothness represents prohibitive computational costs for RPIM formulations. In a second phase, the obtained homogenised mechanical properties were applied to large-scale sandwich beam problems with homogeneous and approximately functionally graded cores, showing RPIM's capability to closely approximate FEM results. The analysis of stress distributions along the thickness of the beam highlighted RPIM's tendency to yield lower stress values near domain edges, albeit with convergence towards agreement among different formulations. It was found that RPIM formulations with lower nodal connectivity are very efficient, balancing computational cost and accuracy. Overall, this study shows RPIM's viability as an alternative to FEM for addressing practical elasticity applications.

High-order finite volume method for solving compressible multicomponent flows with Mie–Grüneisen equation of state

In this paper, we propose a new high-order finite volume method for solving the multicomponent fluids problem with Mie–Grüneisen EOS. Firstly, based on the cell averages of conservative variables, we develop a procedure to reconstruct the cell averages of the primitive variables in a high-order manner. Secondly, the high-order reconstructions employed in computing numerical fluxes are implemented in a characteristic-wise manner to reduce numerical oscillations as much as possible and obtain high-resolution results. Thirdly, advection equation within the governing system is rewritten in a conservative form with a source term to enhance the scheme’s performance. We utilize integration by parts and high-order numerical integration techniques to handle the source terms. Finally, all variables are evolved by using Runge–Kutta time discretization. All steps are carefully designed to maintain the equilibrium of pressure and velocity for the interface-only problem, which is crucial in designing a high-resolution scheme and adapting to more complex multicomponent problems. We have performed extensive numerical tests for both one- and two-dimensional problems to verify our scheme’s high resolution and accuracy.

A very fast high-order flux reconstruction for Finite Volume schemes for Computational Aeroacoustics

AbstractGiven the small wavelengths and wide range of frequencies of the acoustic waves involved in Aeroacoustics problems, the use of very accurate, low-dissipative numerical schemes is the only valid option to accurately capture these phenomena. However, as the order of the scheme increases, the computational time also increases. In this work, we propose a new high-order flux reconstruction in the framework of finite volume (FV) schemes for linear problems. In particular, it is applied to solve the Linearized Euler Equations, which are widely used in the field of Computational Aeroacoustics. This new reconstruction is very efficient and well suited in the context of very high-order FV schemes, where the computation of high-order flux integrals are needed at cell edges/faces. Different benchmark test cases are carried out to analyze the accuracy and the efficiency of the proposed flux reconstruction. The proposed methodology preserves the accuracy while the computational time relatively reduces drastically as the order increases.

Towards modelling AR Sco: generalized particle dynamics and strong radiation-reaction regimes

ABSTRACT Numerical simulations of relativistic plasmas have become more feasible, popular, and crucial for various astrophysical sources with the availability of computational resources. The necessity for high-accuracy particle dynamics is especially highlighted in pulsar modelling due to the extreme associated electromagnetic fields and particle Lorentz factors. Including the radiation-reaction force in the particle dynamics adds even more complexity to the problem, but is crucial for such extreme astrophysical sources. We have also realized the need for such modelling concerning magnetic mirroring and particle injection models proposed for AR Sco, the first white dwarf pulsar. This paper demonstrates the benefits of using higher-order explicit numerical integrators with adaptive time-step methods to solve the full particle dynamics with radiation-reaction forces included. We show that for standard test scenarios, namely various combinations of uniform E- and B-fields and a static dipole B-field, the schemes we use are equivalent to and in extreme field cases outperform standard symplectic integrators in accuracy. We show that the higher-order schemes have massive computational time improvements due to the adaptive time-steps we implement, especially in non-uniform field scenarios and included radiation reaction where the particle gyro-radius rapidly changes. When balancing accuracy and computational time, we identified the adaptive Dormand–Prince eighth-order scheme to be ideal for our use cases. The schemes we use maintain accuracy and stability in describing the particle dynamics and we indicate how a charged particle enters radiation-reaction equilibrium and conforms to the analytical Aristotelian Electrodynamics expectations.

Perturbation-theory informed integrators for cosmological simulations

Large-scale cosmological simulations are an indispensable tool for modern cosmology. To enable model-space exploration, fast and accurate predictions are critical. In this paper, we show that the performance of such simulations can be further improved with time-stepping schemes that use input from cosmological perturbation theory. Specifically, we introduce a class of time-stepping schemes derived by matching the particle trajectories in a single leapfrog/Verlet drift-kick-drift step to those predicted by Lagrangian perturbation theory (LPT). As a corollary, these schemes exactly yield the analytic Zel'dovich solution in 1D in the pre-shell-crossing regime (i.e. before particle trajectories cross). One representative of this class is the popular ‘FastPM’ scheme by Feng et al. 2016[1], which we take as our baseline. We then construct more powerful LPT-inspired integrators and show that they outperform FastPM and standard integrators in fast simulations in two and three dimensions with O(1−100) timesteps, requiring fewer steps to accurately reproduce the power spectrum and bispectrum of the density field. Furthermore, we demonstrate analytically and numerically that, for any integrator, convergence is limited in the post-shell-crossing regime (to order 32 for planar-wave collapse), owing to the lacking regularity of the acceleration field, which makes the use of high-order integrators in this regime futile. Also, we study the impact of the timestep spacing and of a decaying mode present in the initial conditions. Importantly, we find that symplecticity of the integrator plays a minor role for fast approximate simulations with a small number of timesteps.

Gauss–Newton Runge–Kutta integration for efficient discretization of optimal control problems with long horizons and least-squares costs

This work proposes an efficient treatment of continuous-time optimal control problems with long horizons and nonlinear least-squares costs. In particular, we present the Gauss–Newton Runge–Kutta (GNRK) integrator which provides a high-order cost integration. Crucially, the Hessian of the cost terms required within an SQP-type algorithm is approximated with a Gauss–Newton Hessian. Moreover, L2 penalty formulations for constraints are shown to be particularly effective for optimization with GNRK. An efficient implementation of GNRK is provided in the open-source software framework acados. We demonstrate the effectiveness of the proposed approach and its implementation on an illustrative example showing a reduction of relative suboptimality by a factor greater than 10 while increasing the runtime by only 10%.

Variational Integrators on Manifolds for Constrained Mechanical Systems

Abstract Variational integrators play a pivotal role in the simulation and control of constrained mechanical systems. Current Lagrange multiplier-free variational integrators for such systems are concise but inevitably face issues with parameterization singularities, hindering global integration. To tackle this problem, this paper proposes a novel method for constructing variational integrators on manifolds without introducing Lagrange multipliers, offering the benefit of avoiding singularities. Our approach unfolds in three key steps: (1) the local parameterization of configuration space; (2) the formulation of forced discrete Euler–Lagrange equations on manifolds; and (3) the construction and implementation of high-order variational integrators. Numerical tests are conducted for both conservative and forced mechanical systems, demonstrating the excellent global energy behavior of the proposed variational integrators.

Response and reliability analysis of a nonlinear VEH systems with FOPID controller by improved stochastic averaging method and LBFNN algorithm

In engineering, it is an important issue to guarantee the normal working of a vibration energy harvesting (VEH) device to harvest electricity. However, the determination of the system dynamics is a huge challenge due to random excitation, nonlinear structure, and control force. This paper aims to study the response and reliability of a kind of nonlinear and coupled VEH system with an FOPID controller. The stationary response is analyzed by an improved stochastic averaging method, in which an amplitude-dependent frequency is taken into account during the averaging procedure in the case of the strongly nonlinear system. Time-varying reliability in the VEH system is analyzed by using a new LBFNN algorithm, and weight coefficients at each time in this algorithm is obtained through an iteration procedure. Numerical results show that there are essential differences between the weakly nonlinear system and the strongly nonlinear one in terms of averaging procedure and response probability. The fractional orders in the FOPID controller are critical parameters to bring stochastic bifurcations. In addition, strongly nonlinear structure together with lower-order fractional derivative and higher-order fractional integration in FOPID controller are helpful to enhance the reliability performance of the VEH system.

Distributed Byzantine-Resilient Observer for High-Order Integrator Multiagent Systems on Directed Graphs: An Edge-Based Approach

Distributed Byzantine-Resilient Observer for High-Order Integrator Multiagent Systems on Directed Graphs: An Edge-Based Approach

Consensus of Multiple High-Order Integrator Agents With Time-Varying Connectivity and Delays: Protocols Using Only Relative States

Consensus of Multiple High-Order Integrator Agents With Time-Varying Connectivity and Delays: Protocols Using Only Relative States

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

Differentiable solver for time-dependent deformation problems with contact

We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed incremental potential contact method for handling contact and friction forces to solve ODE- and PDE-constrained optimization problems on scenes with complex geometry. It supports static and dynamic problems and differentiation with respect to all physical parameters involved in the physical problem description, which include shape, material parameters, friction parameters, and initial conditions. Our analytically derived adjoint formulation is efficient, with a small overhead (typically less than 10% for nonlinear problems) over the forward simulation, and shares many similarities with the forward problem, allowing the reuse of large parts of existing forward simulator code. We implement our approach on top of the open-source PolyFEM library and demonstrate the applicability of our solver to shape design, initial condition optimization, and material estimation on both simulated results and physical validations.

Algebraically stable SDIRK methods with controllable numerical dissipation for first/second-order time-dependent problems

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions.

Adaptive scaled boundary finite element method for two/three-dimensional structural topology optimization based on dynamic responses

This paper presents an efficient solution for designing the topology of structures that can withstand dynamic loads. The method, called image/stereolithography (STL)-based adaptive scaled boundary finite element (SBFE), is a novel approach to topology optimization (TO) that is particularly effective for designing two/three-dimensional structures. The SBFE-based TO algorithm adopts a bi-evolutionary structural optimization (BESO) approach, which utilizes image/STL-based procedures to automatically adapt meshes and minimize mechanical compliance under varying volume fractions in the time domain. A key advantage of this approach is that it uses images of TO responses, rather than specific error functions, to guide the automatic adaptive mesh schemes. The method also incorporates a quadtree/octree mesh refinement to address the problem of hanging nodes, ensuring fast mesh convergence during the optimization process. Furthermore, a high-order implicit time integration technique provides a close approximation of equivalent static loads, allowing for the accurate mapping of displacement fields under actual dynamic responses at each time step. The effectiveness of this method is demonstrated through a variety of numerical examples and benchmarks.

Distributed Nash Equilibrium Seeking and Disturbance Rejection for High-Order Integrators Over Jointly Strongly Connected Switching Networks.

In this article, we study the problem of Nash equilibrium (NE) seeking of N -player games with high-order integrator agents over jointly strongly connected networks. The same problem was studied recently over static, undirected, and connected networks. Since a jointly strongly connected network can be directed and disconnected at every time instant, our result strictly includes the existing results as special cases. Moreover, in addition to some analytic conditions on the payoff functions, the existing results also rely on the satisfaction of an inequality involving some Lipschitz constant and the smallest nonzero eigenvalue of the Laplacian of the network graph. By adopting a different approach, our result does not rely on the satisfaction of this inequality. Furthermore, our result can also be extended to the case where the additive disturbances are imposed on the input of each agent. Our design is illustrated by two numerical examples.