Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for approximations in a much more general setting (e.g. for rough data) than, for instance, classical splitting or exponential integrator methods. However, they lack one important property: the preservation of geometric properties of the flow. This is particularly drastic in the case of the Korteweg--de Vries (KdV) equation and the nonlinear Schr\"odinger equation (NLSE) which are fundamental models in the broad field of dispersive infinite-dimensional Hamiltonian systems, possessing infinitely many conserved quantities, an important property which we wish to capture - at least up to some degree - also on the discrete level. Nowadays, a wide range of structure preserving integrators for Hamiltonian systems are available, however, typically these existing algorithms can only approximate highly regular solutions efficiently. State-of-the-art low-regularity integrators, on the other hand, poorly preserve the geometric structure of the underlying PDE. In this work we introduce a novel framework, so-called Runge-Kutta resonance-based methods, which are able to bridge the gap between low regularity and structure preservation in the KdV and NLSE case. In particular, we are able to characterise a large class of symplectic (in the Hamiltonian picture) resonance-based methods for both equations that allow for low-regularity approximations to the solution while preserving the underlying geometric structure of the continuous problem on the discrete level.
相關內容
A significant portion of research on distributed ledgers has focused on circumventing the limitations of leader-based blockchains mainly in terms of scalability, decentralization and power consumption. Leaderless architectures based on directed acyclic graphs (DAGs) avoid many of these limitations altogether, but their increased flexibility and performance comes at the cost of increased design complexity, so their potential has remained largely unexplored. Management of write access to these ledgers presents a major challenge because ledger updates may be made in parallel, hence transactions cannot simply be serialised and prioritised according to token fees paid to validators. In this work, we propose an access control scheme for leaderless DAG-based ledgers which is based on consuming credits rather than paying fees in the base token. We outline a general model for this new approach and provide some simulation results showing promising performance boosts.
Advances in artificial intelligence (AI) are fueling a new paradigm of discoveries in natural sciences. Today, AI has started to advance natural sciences by improving, accelerating, and enabling our understanding of natural phenomena at a wide range of spatial and temporal scales, giving rise to a new area of research known as AI for science (AI4Science). Being an emerging research paradigm, AI4Science is unique in that it is an enormous and highly interdisciplinary area. Thus, a unified and technical treatment of this field is needed yet challenging. This paper aims to provide a technically thorough account of a subarea of AI4Science; namely, AI for quantum, atomistic, and continuum systems. These areas aim at understanding the physical world from the subatomic (wavefunctions and electron density), atomic (molecules, proteins, materials, and interactions), to macro (fluids, climate, and subsurface) scales and form an important subarea of AI4Science. A unique advantage of focusing on these areas is that they largely share a common set of challenges, thereby allowing a unified and foundational treatment. A key common challenge is how to capture physics first principles, especially symmetries, in natural systems by deep learning methods. We provide an in-depth yet intuitive account of techniques to achieve equivariance to symmetry transformations. We also discuss other common technical challenges, including explainability, out-of-distribution generalization, knowledge transfer with foundation and large language models, and uncertainty quantification. To facilitate learning and education, we provide categorized lists of resources that we found to be useful. We strive to be thorough and unified and hope this initial effort may trigger more community interests and efforts to further advance AI4Science.
Deep reinforcement learning has achieved significant results in low-level controlling tasks. However, for some applications like autonomous driving and drone flying, it is difficult to control behavior stably since the agent may suddenly change its actions which often lowers the controlling system's efficiency, induces excessive mechanical wear, and causes uncontrollable, dangerous behavior to the vehicle. Recently, a method called conditioning for action policy smoothness (CAPS) was proposed to solve the problem of jerkiness in low-dimensional features for applications such as quadrotor drones. To cope with high-dimensional features, this paper proposes image-based regularization for action smoothness (I-RAS) for solving jerky control in autonomous miniature car racing. We also introduce a control based on impact ratio, an adaptive regularization weight to control the smoothness constraint, called IR control. In the experiment, an agent with I-RAS and IR control significantly improves the success rate from 59% to 95%. In the real-world-track experiment, the agent also outperforms other methods, namely reducing the average finish lap time, while also improving the completion rate even without real world training. This is also justified by an agent based on I-RAS winning the 2022 AWS DeepRacer Final Championship Cup.
In this paper, we address two crucial challenges in the design of cell-free (CF) systems: degradation in the performance of CF systems by imperfect channel state information at the transmitter (CSIT) and high computational/signaling loads arising from the increasing number of distributed antennas and parameters to be exchanged. To mitigate the effects of imperfect CSIT, we employ rate-splitting (RS) multiple-access, which separates the messages into common and private streams. Unlike prior works, we present a clustered CF multi-user multiple-antenna framework with RS, which groups the transmit antennas in several clusters to reduce the computational and signaling loads. The proposed RS-CF system employs one common stream per cluster to exploit the network diversity. Furthermore, we propose new cluster-based linear precoders for this framework. We then devise a power allocation strategy for the common and private streams within clusters and derive closed-form expressions for the sum-rate performance of the proposed cluster-based RS-CF system. Numerical results show that the proposed clustered RS-CF system and algorithms outperform existing approaches. % in terms of the sum-rate.
In this work, we propose a numerical method to compute the Wasserstein Hamiltonian flow (WHF), which is a Hamiltonian system on the probability density manifold. Many well-known PDE systems can be reformulated as WHFs. We use parameterized function as push-forward map to characterize the solution of WHF, and convert the PDE to a finite-dimensional ODE system, which is a Hamiltonian system in the phase space of the parameter manifold. We establish error analysis results for the continuous time approximation scheme in Wasserstein metric. For the numerical implementation, we use neural networks as push-forward maps. We apply an effective symplectic scheme to solve the derived Hamiltonian ODE system so that the method preserves some important quantities such as total energy. The computation is done by fully deterministic symplectic integrator without any neural network training. Thus, our method does not involve direct optimization over network parameters and hence can avoid the error introduced by stochastic gradient descent (SGD) methods, which is usually hard to quantify and measure. The proposed algorithm is a sampling-based approach that scales well to higher dimensional problems. In addition, the method also provides an alternative connection between the Lagrangian and Eulerian perspectives of the original WHF through the parameterized ODE dynamics.
Motivated by bid recommendation in online ad auctions, this paper considers a general class of multi-level and multi-agent games, with two major characteristics: one is a large number of anonymous agents, and the other is the intricate interplay between competition and cooperation. To model such complex systems, we propose a novel and tractable bi-objective optimization formulation with mean-field approximation, called MESOB (Mean-field Equilibria & Social Optimality Balancing), as well as an associated occupation measure optimization (OMO) method called MESOB-OMO to solve it. MESOB-OMO enables obtaining approximately Pareto efficient solutions in terms of the dual objectives of competition and cooperation in MESOB, and in particular allows for Nash equilibrium selection and social equalization in an asymptotic manner. We apply MESOB-OMO to bid recommendation in a simulated pay-per-click ad auction. Experiments demonstrate its efficacy in balancing the interests of different parties and in handling the competitive nature of bidders, as well as its advantages over baselines that only consider either the competitive or the cooperative aspects.
The criticality problem in nuclear engineering asks for the principal eigen-pair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step withing judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigen-pair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus.
We consider a cooperative multi-agent system consisting of a team of agents with decentralized information. Our focus is on the design of symmetric (i.e. identical) strategies for the agents in order to optimize a finite horizon team objective. We start with a general information structure and then consider some special cases. The constraint of using symmetric strategies introduces new features and complications in the team problem. For example, we show in a simple example that randomized symmetric strategies may outperform deterministic symmetric strategies. We also discuss why some of the known approaches for reducing agents' private information in teams may not work under the constraint of symmetric strategies. We then adopt the common information approach for our problem and modify it to accommodate the use of symmetric strategies. This results in a common information based dynamic program where each step involves minimization over a single function from the space of an agent's private information to the space of probability distributions over actions. We present specialized models where private information can be reduced using simple dynamic program based arguments.
In this paper we develop a new well-balanced discontinuous Galerkin (DG) finite element scheme with subcell finite volume (FV) limiter for the numerical solution of the Einstein--Euler equations of general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Z4 system, which is composed of 59 equations, is analyzed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows. Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching three achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum ($\rho=p=0$) up to $t=1000$ with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered un--tractable within the Z4 formalism.
Scientific workflow management systems (SWMSs) and resource managers together ensure that tasks are scheduled on provisioned resources so that all dependencies are obeyed, and some optimization goal, such as makespan minimization, is achieved. In practice, however, there is no clear separation of scheduling responsibilities between an SWMS and a resource manager because there exists no agreed-upon separation of concerns between their different components. This has two consequences. First, the lack of a standardized API to exchange scheduling information between SWMSs and resource managers hinders portability. It incurs costly adaptations when a component should be replaced by a different one (e.g., an SWMS with another SWMS on the same resource manager). Second, due to overlapping functionalities, current installations often actually have two schedulers, both making partial scheduling decisions under incomplete information, leading to suboptimal workflow scheduling. In this paper, we propose a simple REST interface between SWMSs and resource managers, which allows any SWMS to pass dynamic workflow information to a resource manager, enabling maximally informed scheduling decisions. We provide an implementation of this API as an example, using Nextflow as an SWMS and Kubernetes as a resource manager. Our experiments with nine real-world workflows show that this strategy reduces makespan by up to 25.1% and 10.8% on average compared to the standard Nextflow/Kubernetes configuration. Furthermore, a more widespread implementation of this API would enable leaner code bases, a simpler exchange of components of workflow systems, and a unified place to implement new scheduling algorithms.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191