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Vyacheslav Kungurtsev

Publications and source records attributed to Vyacheslav Kungurtsev.

At least 19 recordsLinked to original sources
arXiv2026

Fractional Heat Kernel for Semi-Supervised Graph Learning with Small Training Sample Size

We develop a source-driven fractional heat-kernel framework for semi-super\-vised graph learning that combines nonlocal propagation with sustained label information. A fixed nonzero label source compatible with the Laplacian null space prevents asymptotic collapse into that space, providing a mechanism for mitigating oversmoothing at long diffusion times. The fractional order controls the relative modal attenuation and the spectral weighting of the sustained response, while the diffusion time sets the propagation horizon. We characterize conservation laws and equilibria on normalized and disconnected graphs, develop a null-space deflation, and analyze the approximation of the propagators. On Two-Moon, fractional orders improve source-free propagation, while compatible source-driven diffusion exceeds $96\%$ mean accuracy with one training label per class at orders $0.8$ and $1$. On Cora and CiteSeer, the source-driven pipeline improves mean accuracy over GAT by $9.2$ and $8.0$ percentage points at one training label per class, with model selection on $500$ labeled validation nodes and closely comparable classical and fractional pipeline configurations; it matches GAT on PubMed and trails it at ten and twenty labels per class on Cora. Within GraphHeat, validation-based exponent selection at a common diffusion time yields a paired gain of $1.45$ percentage points at one label per class.

cs.LG↗
arXiv2026

Numerical Algorithms for Partially Segregated Elliptic Systems

We develop numerical methods for elliptic systems governed by a ternary partial-segregation constraint, in which three nonnegative components are required to have a vanishing pointwise product throughout the domain. This condition forces at least one component to vanish at every spatial location while still permitting coexistence of the other two components, and therefore produces a highly nonconvex admissible set. In particular, the limiting constrained Dirichlet minimization problem need not have a unique minimizer, as we show by an explicit one-dimensional example with two distinct global minimizers. We propose two complementary computational frameworks. The first is a strong-competition penalty method solved by a relaxed Picard iteration and a sequential semi-implicit variant, combined with continuation in the penalty parameter. For this formulation, we establish existence of penalized minimizers and weak solutions for every fixed penalty parameter, monotone bracketing properties for the fixed-point iteration, geometric convergence under an explicit contraction condition, and interior exponential estimates in regions of uniformly positive competition. The second framework is a discrete projected-gradient method, together with a safeguarded inertial variant, based on an explicit pointwise projection onto the ternary partial-segregation set. Numerical experiments for several discrete boundary configurations show that, when started on a common solution branch, the two methods produce the same interfaces, and that apparent differences between them reflect the selection of different branches by their default initializations.

math.NA↗
arXiv2026

Symplectic Neural Networks for Learning Non-Separable Hamiltonians

Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency. Identifying the system Hamiltonian from noisy observations of state variables is a challenging task. For simulations to faithfully reflect the long-term behavior of Hamiltonian systems, especially energy conservation, it is essential to use symplectic integrators, which preserve the system's geometric structure. This fidelity comes at a cost: implicit symplectic integrators are more computationally intensive and make backpropagation through the ODE solver non-trivial. However, by leveraging the fact that symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation, we obtain an efficient method of training the Neural Network parameters. In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator. Computationally, a predictor-corrector based ODE solver and fixed point iteration help to mitigate the computational cost of the implicit timestepping, resulting in more efficient generation of gradient updates. We showcase the numerical advantage, in experiments, in system identification and energy preservation on a range of non-separable, chaotic systems and the efficient computation and memory complexity of our method. We also observe that the post-processing of the learned Hamiltonian using backward error analysis yields a modified Hamiltonian that is a more accurate approximation of the true Hamiltonian without the need to use more accurate discretizations of the flow map.

cs.LG↗
arXiv2026

Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus

We develop a framework for PDE-constrained optimal control on Hilbert complexes and its structure-preserving discretization by finite element exterior calculus (FEEC), with emphasis on problems where nontrivial topology creates physically meaningful global modes. We separate the gauge-fixed local state from harmonic components representing global circulation or flux, and introduce finite-dimensional topological actuators whose reachable modes are determined jointly by domain topology and actuator rank. We establish well-posedness, weak optimality conditions, reachability results, and FEEC error estimates preserving the harmonic structure. Numerical experiments on contractible and multiply connected domains demonstrate circulation and cavity- flux control, rank-dependent reachability, mesh-independent optimization, and topology-induced singular convergence. We further extend the framework to time-dependent Maxwell control, where harmonic electric circulation and magnetic flux become dynamical states exhibiting explicit conservation and reachability properties.

math.OC↗
arXiv2026

Mission-Aligned Learning-Informed Control of Autonomous Systems: Formulation and Foundations

Research, innovation and practical capital investment have been increasing rapidly toward the realization of autonomous physical agents. This includes industrial and service robots, unmanned aerial vehicles, embedded control devices, and a number of other realizations of cybernetic/mechatronic implementations of intelligent autonomous devices. In this paper, we consider a stylized version of robotic care, which would normally involve a two-level Reinforcement Learning procedure that trains a policy for both lower level physical movement decisions as well as higher level conceptual tasks and their sub-components. In order to deliver greater safety and reliability in the system, we present the general formulation of this as a two-level optimization scheme which incorporates control at the lower level, and classical planning at the higher level, integrated with a capacity for learning. This synergistic integration of multiple methodologies -- control, classical planning, and RL -- presents an opportunity for greater insight for algorithm development, leading to more efficient and reliable performance. Here, the notion of reliability pertains to physical safety and interpretability into an otherwise black box operation of autonomous agents, concerning users and regulators. This work presents the necessary background and general formulation of the optimization framework, detailing each component and its integration with the others.

math.OC↗
arXiv2026

"Cause" is Mechanistic Narrative within Scientific Domains: An Ordinary Language Philosophical Critique of "Causal Machine Learning"

Causal Learning has emerged as a major theme of research in statistics and machine learning in recent years, promising computational techniques to reveal ``true'' causality. In this paper, we critique the premise of causal learning by considering the epistemology of causality across disciplines, applying the Ordinary Language method of an anthropological investigation of customary word use in reasoning about cause and effect in the real world. We observe that although cause-and-effect semantics vary between scientific domains, they maintain a consistent central function of describing the mechanisms underlying forces most salient to the systems under research. A critical distinction is the degree to which the mathematical representations of the models used for statistical analysis exhibit a faithful correspondence to expert knowledge in mechanism. We demarcate 1) physics and engineering as domains wherein mathematical models are sufficient to comprehensively describe causality, in contrast to 2) biology, which studies open and irreducible systems with mechanisms crossing scales through emergence, and 3) the social sciences as suffering from compounding difficulties for precision but providing, through Hermeneutics, the potential for subjective phenomenology yielding findings instrumentally useful to individuals. We posit the greater the discrepancy between expert-defined models and complete characterization of mechanism, the more that epistemic virtue requires that definitive causal claims regarding phenomena can only come through an agglomeration of consistent evidence across multiple domains. Exercising greater caution in communicating the degree of certainty evidence provides, especially demanding restraint in the face of incentives to overstate research conclusions, is the only durable solution to modern science's collective action problems.

cs.LG↗
arXiv2026

Towards Tsallis Fully Probabilistic Design

Fully Probabilistic design (FPD) is a powerful framework offering an elegant and unifying account of stochastic control, learning and decision-making. Here we introduce a generalized FPD framework, which we term as Tsallis FPD. Tsallis FPD uses Tsallis divergence in place of the Kullback-Leibler divergence that defines the standard FPD cost term. Tsallis divergence is a natural generalization of the KL divergence, rooted in non-extensive statistical mechanics and providing flexibility towards modeling stochastic processes with non-Gaussian tail behavior. After formulating Tsallis FPD, we present a double iteration scheme that performs a sequence of backwards inductions, rather than a single pass down the stages that constitutes the proven approach for classical FPD.

math.OC↗
arXiv2026

A Survey of Quantum Alternatives to Randomized Algorithms: Monte Carlo Integration and Beyond

Monte Carlo sampling is a powerful toolbox of algorithmic techniques widely used for a number of applications wherein some noisy quantity, or summary statistic thereof, is sought to be estimated. In this paper, we survey the literature for implementing Monte Carlo procedures using quantum circuits, focusing on the potential to obtain a quantum advantage in the computational speed of these procedures. We revisit the quantum algorithms that could replace classical Monte Carlo and then consider both the existing quantum algorithms and the potential quantum realizations that include adaptive enhancements as alternatives to the classical procedure.

quant-ph↗
arXiv2026

Structural Assessment for Understanding and Guiding Dataset Distillation in Discrete Token Space

Dataset distillation (DD) has proven to reduce training cost while preserving accuracy. While promising, the factors that make one distilled dataset more effective than another remain poorly understood. In this work, we investigate this question through the lens of discrete visual tokenizers. Whereas many prior DD efforts emphasize matching global data distributions, we suggest that the effectiveness depends on which semantic concepts are captured and how they are composed. Discrete visual tokenizers provide a finite vocabulary that enables direct statistical analysis of such compositional structure. Through quantitative analysis of token-level statistics, we introduce the structural score to measure the adequacy of token compositions. We observe that distilled datasets with balanced token composition yield higher validation performance. On the other hand, divergence from the original data does not necessarily harm performance. We further show that samples with high structural scores in the discrete token space can effectively guide diffusion-based DD. Our findings highlight the importance of token composition in dataset effectiveness, offering a principled complement to distributional similarity considerations in DD.

cs.CV↗
arXiv2026

Multistage Stochastic Programming for Rare Event Risk Mitigation in Power Systems Management

High intermittent renewable penetration in the energy mix presents challenges in robustness for the management of power systems' operation. If a tail realization of the distribution of weather yields a prolonged period of time during which solar irradiation and wind speed are insufficient for satisfying energy demand, then it becomes critical to ramp up the generation of conventional power plants with adequate foresight. This event trigger is costly, and inaccurate forecasting can either be wasteful or yield catastrophic undersupply. This encourages particular attention to accurate modeling of the noise and the resulting dynamics within the aforementioned scenario. In this work we present a method for rare event-aware control of power systems using multi-stage scenario-based stochastic programming. A Fleming-Viot particle approach is used to bias the scenario generation towards rare realizations of very low wind power, in order to obtain a cost-effective control of conventional power plants that is robust under prolonged renewable energy shortfalls.

math.OC↗
arXiv2026

Towards Real Time Control of Water Engineering with Nonlinear Hyperbolic Partial Differential Equations

This paper examines aspirational requirements for software addressing mixed-integer optimization problems constrained by the nonlinear Shallow Water partial differential equations (PDEs), motivated by applications such as river-flow management in hydropower cascades. Realistic deployment of such software would require the simultaneous treatment of nonlinear and potentially non-smooth PDE dynamics, limited theoretical guarantees on the existence and regularity of control-to-state mappings under varying boundary conditions, and computational performance compatible with operational decision-making. In addition, practical settings motivate consideration of uncertainty arising from forecasts of demand, inflows, and environmental conditions. At present, the theoretical foundations, numerical optimization methods, and large-scale scientific computing tools required to address these challenges in a unified and tractable manner remain the subject of ongoing research across the associated research communities. Rather than proposing a complete solution, this work uses the problem as a case study to identify and organize the mathematical, algorithmic, and computational components that would be necessary for its realization. The resulting framework highlights open challenges and intermediate research directions, and may inform both more circumscribed related problems and the design of future large-scale collaborative efforts aimed at addressing such objectives.

math.OC↗
arXiv2026

Direct-search methods for decentralized blackbox optimization

Derivative-free optimization algorithms are particularly useful for tackling blackbox optimization problems where the objective function arises from complex and expensive procedures that preclude the use of classical gradient-based methods. In contemporary decentralized environments, such functions are defined locally on different computational nodes due to technical or privacy constraints, introducing additional challenges within the optimization process. In this paper, we adapt direct-search methods, a classical technique in derivative-free optimization, to the decentralized setting. In contrast with zeroth-order algorithms, our algorithms rely on positive spanning sets to define suitable search directions while still possessing global convergence guaranties, thanks to carefully chosen stepsizes. Numerical experiments highlight the advantages of direct-search techniques over gradient-approximation-based strategies.

math.OC↗
arXiv2026

Noncooperative Human-AI Agent Dynamics

This paper investigates the dynamics of noncooperative interactions between artificial intelligence agents and human decision-makers in strategic environments. In particular, motivated by extensive literature in behavioral Economics, human agents are more faithfully modeled with respect to the state of the art using Prospect Theoretic preferences, while AI agents are modeled with standard expected utility maximization. Prospect Theory incorporates known cognitive heuristics employed by humans, including reference dependence and greater loss aversion relative to utility to relative gains. This paper runs different combinations of expected utility and prospect theoretic agents in a number of classic matrix games as well as examples specialized to tease out distinctions in strategic behavior with respect to preference functions, to explore the emergent behaviors from mixed population (human vs. AI) competition. Extensive numerical simulations are performed across AI, aware humans (those with full knowledge of the game structure and payoffs), and learning Prospect Agents (i.e., for AIs representing humans). A number of interesting observations and patterns show up, spanning barely distinguishable behavior, behavior corroborating Prospect preference anomalies in the theoretical literature, and unexpected surprises. Code can be found at https://github.com/dylanwaldner/noncooperative-human-AI.

cs.GT↗
arXiv2026

Federated Sinkhorn

We study distributed Sinkhorn iterations for entropy-regularized optimal transport when the Gibbs kernel operator is row-partitioned across c workers and cannot be centralized. We present Federated Sinkhorn, two exact synchronous protocols that exchange only scaling-vector slices: (i) an All-to-All scheme implemented by Allgather, and (ii) a Star (parameter-server) scheme implemented by client to server sends and server to client broadcasts. For both, we derive closed-form per-iteration compute, communication, and memory costs under an alpha-beta latency--bandwidth model, and show that the distributed iterates match centralized Sinkhorn under standard positivity assumptions. Multi-node CPU/GPU experiments validate the model and show that repeated global scaling exchange quickly becomes the dominant bottleneck as c increases. We also report an optional bounded-delay asynchronous schedule and an optional privacy measurement layer for communicated log-scalings.

cs.DC↗
arXiv2026

An efficient penalty decomposition algorithm for minimization over sparse symmetric sets

This paper proposes an improved quasi-Newton penalty decomposition algorithm for the minimization of continuously differentiable functions, possibly nonconvex, over sparse symmetric sets. The method solves a sequence of penalty subproblems approximately via a two-block decomposition scheme: the first subproblem admits a closed-form solution without sparsity constraints, while the second subproblem is handled through an efficient sparse projection over the symmetric feasible set. Under a new assumption on the gradient of the objective function, weaker than global Lipschitz continuity from the origin, we establish that accumulation points of the outer iterates are basic feasible and cardinality-constrained Mordukhovich stationarity points. To ensure robustness and efficiency in finite-precision arithmetic, the algorithm incorporates several practical enhancements, including an enhanced line search strategy based on either backtracking or extrapolation, and four inexpensive diagonal Hessian approximations derived from differences of previous iterates and gradients or from eigenvalue-distribution information. Numerical experiments on a diverse benchmark of $30$ synthetic and data-driven test problems, including machine-learning datasets from the UCI repository and sparse symmetric instances with dimensions ranging from $10$ to $500$, demonstrate that the proposed algorithm is competitive with several state-of-the-art methods in terms of efficiency, robustness, and strong stationarity.

math.OC↗
arXiv2026

A Three-Tier Time-Scale Architecture for Controlling Complex Nonlinear Systems

This letter proposes a three-tier computational architecture for the real-time control of nonlinear complex systems, such as time-dependent PDEs. There is an important class of such problems for which existing single- and two-time-scale approaches are fundamentally insufficient due to lack of a priori system knowledge, computational complexity, model fidelity requirements, and uncertainty. The proposed architecture consists of an offline, meso-scale, and real-time layer of computation, with distinct roles for each layer and specific information flow between them. The result is a practical systems-level paradigm that enables real-time operation of complex nonlinear control problems.

math.OC↗
arXiv2025

Survey of Mathematical Models of and Numerical Methods for Fluid Dynamics Water Engineering

Computational fluid dynamics (CFD) has become a cornerstone of modern water engineering, providing quantitative tools for the analysis, prediction, and management of complex hydraulic systems across a wide range of spatial and temporal scales. This survey reviews the mathematical models and numerical methods that underpin CFD applications in water engineering, from depth-averaged formulations such as the shallow water equations to fully three-dimensional Navier-Stokes models, as well as selected alternative modeling approaches. We examine the historical development of these models, their mathematical structure, and the numerical discretization and solution strategies commonly employed in practice, including finite difference, finite volume, and finite element methods. Beyond core solver technology, the survey addresses practical modeling issues such as source-term treatment, wetting and drying, turbulence modeling, free-surface representation, and computational efficiency. The growing role of data integration is also discussed, encompassing data assimilation, uncertainty quantification, and emerging machine-learning-assisted approaches that complement physics-based solvers. To illustrate the impact of modeling and numerical choices on real-world applications, representative case studies from large-scale water management systems are reviewed. By integrating theory, numerical techniques, and applied perspectives, this survey provides a unified reference for researchers and practitioners seeking to understand both the foundational principles and contemporary challenges of CFD in water engineering.

physics.flu-dyn↗
arXiv2025

Learning Generalized Hamiltonians using fully Symplectic Mappings

Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and in particular Hamiltonian Neural Networks have emerged as a mechanism to incorporate structural inductive bias into the NN model. By ensuring physical invariances are conserved, the models exhibit significantly better sample complexity and out-of-distribution accuracy than standard NNs. Learning the Hamiltonian as a function of its canonical variables, typically position and velocity, from sample observations of the system thus becomes a critical task in system identification and long-term prediction of system behavior. However, to truly preserve the long-run physical conservation properties of Hamiltonian systems, one must use symplectic integrators for a forward pass of the system's simulation. While symplectic schemes have been used in the literature, they are thus far limited to situations when they reduce to explicit algorithms, which include the case of separable Hamiltonians or augmented non-separable Hamiltonians. We extend it to generalized non-separable Hamiltonians, and noting the self-adjoint property of symplectic integrators, we bypass computationally intensive backpropagation through an ODE solver. We show that the method is robust to noise and provides a good approximation of the system Hamiltonian when the state variables are sampled from a noisy observation. In the numerical results, we show the performance of the method concerning Hamiltonian reconstruction and conservation, indicating its particular advantage for non-separable systems.

cs.LG↗