Provable Convergence Research Articles

Guidance Strategy and Fast Robustness Analysis for Space Debris Flyby

Envisioning the scenario of space debris inspection by orbital flyby, this paper investigates the guidance strategy and fast robustness analysis method considering the relative state constraints. First, the corrected state transition matrix is derived to predict and correct the perturbed relative state at the flyby moment. Based on the state transition matrix, a free-flight corridor model is established to characterize the maximum allowed magnitude of the deviation, which can quickly estimate the robustness of the trajectory under uncertainties and circumvent intensive numerical sampling. The corresponding optimization, with four norm-extreme constraints, can be solved by a proposed double-layer algorithm with proven convergence. Numerical simulations demonstrate that the guidance algorithm is effective, and the free-flight corridor can accurately and efficiently reflect the robustness of the nominal trajectory.

An Improved Penalty-Based Excited-State Variational Monte Carlo Approach with Deep-Learning Ansatzes.

We introduce several improvements to the penalty-based variational quantum Monte Carlo (VMC) algorithm for computing electronic excited states of Entwistle et al. [Nat. Commun. 14, 274 (2023)] and demonstrate that the accuracy of the updated method is competitive with other available excited-state VMC approaches. A theoretical comparison of the computational aspects of these algorithms is presented, where several benefits of the penalty-based method are identified. Our main contributions include an automatic mechanism for tuning the scale of the penalty terms, an updated form of the overlap penalty with proven convergence properties, and a new term that penalizes the spin of the wave function, enabling the selective computation of states with a given spin. With these improvements, along with the use of the latest self-attention-based ansatz, the penalty-based method achieves a mean absolute error below 1 kcal/mol for the vertical excitation energies of a set of 26 atoms and molecules, without relying on variance matching schemes. Considering excited states along the dissociation of the carbon dimer, the accuracy of the penalty-based method is on par with that of natural-excited-state (NES) VMC, while also providing results for additional sections of the potential energy surface, which were inaccessible with the NES method. Additionally, the accuracy of the penalty-based method is improved for a conical intersection of ethylene, with the predicted angle of the intersection agreeing well with both NES-VMC and multireference configuration interaction.

UPrime: Unrolled Phase Retrieval Iterative Method with provable convergence

Phase Retrieval (PR) is an ill-posed inverse problem which arises in various science and engineering applications. Recently, it has been empirically shown that unrolled iterative methods or model-driven deep learning methods are effective for solving this problem. However, the prior modules in these model-driven networks lack model interpretability, leading to a lack of rigorous analysis about the convergence behaviors of these re-implemented iterations, and thus the significance of such PR methods is a little bit vague. For this issue, this paper proposes an effective and provable Unrolled Phase Retrieval Iterative MEthod (UPrime) for the PR problem. Our theoretical analysis demonstrates that UPrime using an elaborated bounded prior module can generate fixed-point convergent trajectories. Meanwhile, the proposed prior module, a flexible and interpretable module, is beneficial for the convergence analysis of regularized imaging methods in the non-convex scenario. Experiments on coded diffraction imaging applications verify the superiority of UPrime.

A convergent finite-volume scheme for the diffusive compressible Euler model—resolved and under-resolved flows

The purpose of this paper is to demonstrate the robustness and accuracy of a provably convergent finite-volume scheme for the diffusive compressible Euler model. To this end, we consider sub- and supersonic flows as well as well-resolved smooth flows and under-resolved turbulent flows. The scheme is formulated for Voronoi meshes to allow complex geometries. It is completely defined without any parameters that require case-by-case tuning. Provable convergence guarantees both stability and admissibility of the numerical solutions, which is confirmed in our computations for both strong shocks and turbulence. For the turbulent Taylor-Green vortex, the scheme performs as well as a linearly-stable minimally-diffusive second-order scheme and for vortex shedding behind a cylinder, it even outperforms energy-stable second-order schemes.

Gradient-Robust Hybrid DG Discretizations for the Compressible Stokes Equations

This paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an H(div)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(\ extrm{div})$$\\end{document}-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier–Stokes equations.

Infinite-dimensional reservoir computing

Reservoir computing approximation and generalization bounds are proved for a new concept class of input/output systems that extends the so-called generalized Barron functionals to a dynamic context. This new class is characterized by the readouts with a certain integral representation built on infinite-dimensional state–space systems. It is shown that this class is very rich and possesses useful features and universal approximation properties. The reservoir architectures used for the approximation and estimation of elements in the new class are randomly generated echo state networks with either linear or ReLU activation functions. Their readouts are built using randomly generated neural networks in which only the output layer is trained (extreme learning machines or random feature neural networks). The results in the paper yield a recurrent neural network-based learning algorithm with provable convergence guarantees that do not suffer from the curse of dimensionality when learning input/output systems in the class of generalized Barron functionals and measuring the error in a mean-squared sense.

A peer‐to‐peer joint energy and reserve market considering renewable generation uncertainty: A generalized Nash equilibrium approach

AbstractPeer‐to‐peer (P2P) energy trading enhances distribution network resilience by reducing energy demand from central power plants and enabling distributed energy resources to support critical loads after extreme events. However, adequate reserves from main grids are still required to ensure real‐time energy balance in distribution networks due to the uncertainty in renewable generation. This paper introduces a novel two‐stage joint energy and reserve market for prosumers, wherein local flexible resources are fully utilized to manage renewable generation uncertainty. In contrast to cooperative optimization methods, the interactions between prosumers are modelled as a generalized Nash game (GNG), considering that prosumers are self‐interested and should follow distribution network constraints. Then, linear decision rules are employed to ensure a feasible market equilibrium and develop a privacy‐preserving algorithm to guide prosumers toward the market equilibrium with a proven convergence. Finally, the numerical study on a modified IEEE 33‐power system demonstrates that the designed market effectively manages renewable generation uncertainty, and that the algorithm converges to the market equilibrium.

Contingency detection in modern power systems: A stochastic hybrid system method

This paper introduces a new stochastic hybrid system (SHS) framework for contingency detection in modern power systems (MPS). The framework uses stochastic hybrid system representations in state space models to expand and facilitate capability of contingency detection. In typical microgrids (MGs), buses may contain various synchronous generators, renewable generators, controllable loads, battery systems, regular loads, etc. For development of SHS models in power systems, this paper introduces the concept of dynamic and non-dynamic buses. By converting a physical power grid into a virtual linearized state space model and representing contingencies as random switching of system structures and parameters, this paper formulates the contingency detection problem as a joint estimation problem of discrete events and continuous states in stochastic hybrid systems. This method offers unique advantages, including using common measurement signals on voltage and current synchrophasors to detect different types and locations of contingencies, avoiding expensive local direct fault measurements and detecting certain contingencies that cannot be directly measured. The method employs a small and suitably-designed probing signal to sustain the ability of persistent contingency detection. Joint estimation algorithms are presented with their proven convergence and reliability properties. Examples that use an IEEE 5-Bus system demonstrate the main ideas and derivation steps. Simulation case studies on an IEEE 33-Bus system are used for detecting transmission line faults and sensor interruptions.

Constrained Bayesian Optimization under Partial Observations: Balanced Improvements and Provable Convergence

The partially observable constrained optimization problems (POCOPs) impede data-driven optimization techniques since an infeasible solution of POCOPs can provide little information about the objective as well as the constraints. We endeavor to design an efficient and provable method for expensive POCOPs under the framework of constrained Bayesian optimization. Our method consists of two key components. Firstly, we present an improved design of the acquisition functions that introduce balanced exploration during optimization. We rigorously study the convergence properties of this design to demonstrate its effectiveness. Secondly, we propose Gaussian processes embedding different likelihoods as the surrogate model for partially observable constraints. This model leads to a more accurate representation of the feasible regions compared to traditional classification-based models. Our proposed method is empirically studied on both synthetic and real-world problems. The results demonstrate the competitiveness of our method for solving POCOPs.

Enhancing generalized spectral clustering with embedding Laplacian graph regularization

AbstractAn enhanced generalised spectral clustering framework that addresses the limitations of existing methods by incorporating the Laplacian graph and group effect into a regularisation term is presented. By doing so, the framework significantly enhances discrimination power and proves highly effective in handling noisy data. Its versatility enables its application to various clustering problems, making it a valuable contribution to unsupervised learning tasks. To optimise the proposed model, the authors have developed an efficient algorithm that utilises the standard Sylvester equation to compute the coefficient matrix. By setting the derivatives to zero, computational efficiency is maintained without compromising accuracy. Additionally, the authors have introduced smoothing strategies to make the non‐convex and non‐smooth terms differentiable. This enables the use of an alternative iteration re‐weighted procedure (AIwRP), which distinguishes itself from other first‐order optimisation algorithms by introducing auxiliary variables. The authors provide a provable convergence analysis of AIwRP based on the iteration procedures of unconstrained problems to support its effectiveness. Extensive numerical tests have been conducted on synthetic and benchmark databases to validate the superiority of their approaches. The results demonstrate improved clustering performance and computational efficiency compared to several existing spectral clustering methods, further reinforcing the advantages of their proposed framework. The source code is available at https://github.com/ZhangHengMin/LGR_LSRLRR.

Mean-Field Multiagent Reinforcement Learning: A Decentralized Network Approach

One of the challenges for multiagent reinforcement learning (MARL) is designing efficient learning algorithms for a large system in which each agent has only limited or partial information of the entire system. Whereas exciting progress has been made to analyze decentralized MARL with the network of agents for social networks and team video games, little is known theoretically for decentralized MARL with the network of states for modeling self-driving vehicles, ride-sharing, and data and traffic routing. This paper proposes a framework of localized training and decentralized execution to study MARL with the network of states. Localized training means that agents only need to collect local information in their neighboring states during the training phase; decentralized execution implies that agents can execute afterward the learned decentralized policies, which depend only on agents’ current states. The theoretical analysis consists of three key components: the first is the reformulation of the MARL system as a networked Markov decision process with teams of agents, enabling updating the associated team Q-function in a localized fashion; the second is the Bellman equation for the value function and the appropriate Q-function on the probability measure space; and the third is the exponential decay property of the team Q-function, facilitating its approximation with efficient sample efficiency and controllable error. The theoretical analysis paves the way for a new algorithm LTDE-Neural-AC, in which the actor–critic approach with overparameterized neural networks is proposed. The convergence and sample complexity are established and shown to be scalable with respect to the sizes of both agents and states. To the best of our knowledge, this is the first neural network–based MARL algorithm with network structure and provable convergence guarantee. Funding: X. Wei is partially supported by NSFC no. 12201343. R. Xu is partially supported by the NSF CAREER award DMS-2339240.

Online neuro-fuzzy model learning of dynamic systems with measurement noise

Model identification of nonlinear time varying dynamic systems is challenging because the system behaviours may vary significantly in different operational conditions. If the changes are insufficiently captured by training data, the trained model is unable to capture the system response well when the operational condition changes. The model performance may also be deteriorated in real-time implementation due to the noise in sensors or the environment. This paper presents a self-adaptive Neuro-Fuzzy (NF) modelling framework to address these challenges. The NF model, trained offline based on experimental data, combines the Auto-Regressive with eXogenous (ARX) models and Gaussian activation functions to capture the nonlinear system behaviours. During online implementation, the ARX model parameters are updated using new data through a recursive generalised least squares method, which embeds a noise model to eliminate effects of the noise. The online updating algorithm has a provable convergence guarantee and enables the proposed NF model to adapt to changes in system behaviours automatically. Efficacy of the algorithm is verified through two numerical examples and an experiment on a commercial automotive engine.

Data-Driven Distributionally Robust Optimization for Railway Timetabling Problem

Unpredictable disturbances that occur when the train is running often make the actual timetable deviate from the planned timetable and deteriorate service quality for passengers. To provide a robust planned timetable that is reliable in operation, this paper presents a two-stage distributionally robust railway timetabling (DRRT) model, where the decisions in the first and second stages are the planned and actual timetables, respectively. To deal with the uncertain travel time, the proposed DRRT model assumes the true distribution belongs to a Wasserstein ambiguity set centered at the empirical distribution constructed from finite samples and aims to minimize the sum of the total idle time, travel time, and worst-case expected delay over the ambiguity set. To solve this model, we design a decomposition algorithm with provable convergence, which iteratively solves a master problem and a set of subproblems. To enhance the algorithm efficiency, a blend of two popular decomposition methods is proposed to integrate their advantages, and the solution of the subproblems is accelerated by using an uncapacitated minimum-cost flow interpretation of the second-stage optimization problem. Experiments on real-world instances demonstrate the effectiveness of these enhancements. The results also show that our model prominently outperforms the sample average approximation model in terms of the out-of-sample performance, especially when only a small sample set is available. Moreover, at the expense of equal traffic efficiency, our model is more robust than those models that do not utilize distribution information. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper was motivated by the problem of using finite historical data to provide a robust planned railway timetable, aiming at minimizing the traffic efficiency losses and expected delays under the worst-case distribution. Existing studies on dealing with the uncertain travel time usually assumed that the distribution is exactly known, or the distribution is completely unknown but the travel time belongs to an uncertainty set. However, in practice, the exact distribution is hard to know but can only be coarsely estimated from data. This paper proposes a data-driven two-stage model by assuming that the true distribution belongs to a 1-Wasserstein ball centered at the empirical distribution with a bounded support set, which is a middle ground between the above two distribution assumptions. To solve this problem efficiently, we propose a customized algorithm with provable convergence and then enhance it by exploiting problem-specific properties. This distributionally robust model can be easily shifted to other timetabling problems, and the algorithm enhancement techniques can also be applied to efficiently solve other two-stage recourse models with similar properties.

Single-Loop Deep Actor-Critic for Constrained Reinforcement Learning With Provable Convergence

Single-Loop Deep Actor-Critic for Constrained Reinforcement Learning With Provable Convergence

One-Step Multiview Clustering via Adaptive Graph Learning and Spectral Rotation.

In graph based multiview clustering methods, the ultimate partition result is usually achieved by spectral embedding of the consistent graph using some traditional clustering methods, such as k -means. However, optimal performance will be reduced by this multistep procedure since it cannot unify graph learning with partition generation closely. In this article, we propose a one-step multiview clustering method through adaptive graph learning and spectral rotation (AGLSR). For every view, AGLSR adaptively learns affinity graphs to capture similar relationships of samples. Then, a spectral embedding is designed to take advantage of the potential feature space shared by different views. In addition, AGLSR utilizes a spectral rotation strategy to obtain the discrete clustering labels from the learned spectral embeddings directly. An effective updating algorithm with proven convergence is derived to optimize the optimization problem. Sufficient experiments on benchmark datasets have clearly demonstrated the effectiveness of the proposed method in six metrics. The code of AGLSR is uploaded at https://github.com/tangchuan2000/AGLSR.

Fast and accurate nonadiabatic molecular dynamics enabled through variational interpolation of correlated electron wavefunctions.

We build on the concept of eigenvector continuation to develop an efficient multi-state method for the rigorous and smooth interpolation of a small training set of many-body wavefunctions through chemical space at mean-field cost. The inferred states are represented as variationally optimal linear combinations of the training states transferred between the many-body bases of different nuclear geometries. We show that analytic multi-state forces and nonadiabatic couplings from the model enable application to nonadiabatic molecular dynamics, developing an active learning scheme to ensure a compact and systematically improvable training set. This culminates in application to the nonadiabatic molecular dynamics of a photoexcited 28-atom hydrogen chain, with surprising complexity in the resulting nuclear motion. With just 22 DMRG calculations of training states from the low-energy correlated electronic structure at different geometries, we infer the multi-state energies, forces and nonadiabatic coupling vectors at 12 000 geometries with provable convergence to high accuracy along an ensemble of molecular trajectories, which would not be feasible with a brute force approach. This opens up a route to bridge the timescales between accurate single-point correlated electronic structure methods and timescales of relevance for photo-induced molecular dynamics.

Regularized Instance Weighting Multiview Clustering via Late Fusion Alignment.

Multiview clustering has become a prominent research topic in data analysis, with wide-ranging applications across various fields. However, the existing late fusion multiview clustering (LFMVC) methods still exhibit some limitations, including variable importance and contributions and a heightened sensitivity to noise and outliers during the alignment process. To tackle these challenges, we propose a novel regularized instance weighting multiview clustering via late fusion alignment (R-IWLF-MVC), which considers the instance importance from various views, enabling information integration to be more effective. Specifically, we assign each sample an importance attribute to enable the learning process to focus more on the key sample nodes and avoid being influenced by noise or outliers, while laying the groundwork for the fusion of different views. In addition, we continue to employ late fusion alignment to integrate base clustering from various views and introduce a new regularization term with prior knowledge to ensure that the learning process does not deviate too much from the expected results. After that, we design a three-step alternating optimization strategy with proven convergence for the resultant problem. Our proposed approach has been extensively evaluated on multiple real-world datasets, demonstrating its superiority to state-of-the-art methods.

Beyond Battery State of Charge Estimation: Observer for Electrode-Level State and Cyclable Lithium With Electrolyte Dynamics

This paper presents a provably convergent battery estimation scheme based on a single particle model with electrolyte dynamics (SPMe), by proposing a systematic methodology to estimate critical information such as electrode-level states, electrolyte dynamics, and cyclable lithium. Electrode-level state estimation suffers from weak observability originating from two standalone electrode dynamics, which is then aggravated by the addition of electrolyte dynamics. This lack of observability can be alleviated by exploiting lithium inventory conservation enabled by Kalman decomposition, allowing one to separate out the unobservable subspace. Assuming the knowledge of cyclable lithium, a nonlinear state observer with provable convergence can be constructed for the SPMe model, using voltage and current measurements. To relax this strong assumption, a sensitivity-based parameter estimation scheme is also deployed to track cyclable lithium &#x2013; a crucial physical variable for capacity fade. Ultimately, the estimation framework can perform finer monitoring and diagnosis of battery charge and health down to the level of individual electrode and the electrolyte. Experimental validation demonstrates &#x003C;1% estimation error for cyclable lithium inventory. Solid phase lithium concentration estimates, especially in the negative electrode, can be sensitive to disturbances in cyclable lithium.

Posture Adjustment for a Wheel-Legged Robotic System Via Leg Force Control With Prescribed Transient Performance

This work proposes a force control strategy with prescribed transient performance for the legs of a wheel-legged robotic system to realize the posture adjustment on uneven roads. A dynamic model of the robotic system is established with the body posture references and feedback to calculate the desired leg forces, which are the tracking references for the wheel-legs. Based on the funnel control scheme, the legs realize force tracking with prescribed transient performance. To improve the robustness of the force control system, an event-triggering condition is designed for the online segment of the funnel function. As a result, the force tracking error of the wheel-leg evolves inside the performance funnel with proven convergence. The absence of Zeno behavior for the event-based mechanism is also guaranteed. The proposed control scheme is applied to the wheel-legged physical prototype for the performance of force tracking and posture adjustment. Multiple comparative experimental results are presented to validate the stability and effectiveness of the proposed methodology.

Constrained Network Slicing Games: Achieving Service Guarantees and Network Efficiency

Network slicing is a key capability for next generation mobile networks. It enables infrastructure providers to cost effectively customize logical networks over a shared infrastructure. A critical component of network slicing is resource allocation, which needs to ensure that slices receive the resources needed to support their services while optimizing network efficiency. In this paper, we propose a novel approach to slice-based resource allocation named Guaranteed seRvice Efficient nETwork slicing (GREET). The underlying concept is to set up a constrained resource allocation game, where ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$i$</tex-math> </inline-formula> ) slices unilaterally optimize their allocations to best meet their (dynamic) customer loads, while ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ii$</tex-math> </inline-formula> ) constraints are imposed to guarantee that, if they wish so, slices receive a pre-agreed share of the network resources. The resulting game is a variation of the well-known Fisher market, where slices are provided a budget to contend for network resources (as in a traditional Fisher market), but (unlike a Fisher market) prices are constrained for some resources to ensure that the pre-agreed guarantees are met for each slice. In this way, GREET combines the advantages of a share-based approach (high efficiency by flexible sharing) and reservation-based ones (which provide guarantees by assigning a fixed amount of resources). We characterize the Nash equilibrium, best response dynamics, and propose a practical slice strategy with provable convergence properties. Extensive simulations exhibit substantial improvements over network slicing state-of-the-art benchmarks.