arXiv subjects

Konstantin M. Zuev

Publications and source records attributed to Konstantin M. Zuev.

7 recordsLinked to original sources

arXiv2026

Upper Generalization Bounds for Neural Oscillators

Neural oscillators that originate from second-order ordinary differential equations (ODEs) have shown competitive performance in learning mappings between dynamic loads and responses of complex nonlinear structural systems. Despite this empirical success, theoretically quantifying the generalization capacities of their neural network architectures remains undeveloped. In this study, the neural oscillator consisting of a second-order ODE followed by a multilayer perceptron (MLP) is considered. Its upper probably approximately correct (PAC) generalization bound for approximating causal and uniformly continuous operators between continuous temporal function spaces and that for approximating the uniformly asymptotically incrementally stable second-order dynamical systems are derived by leveraging the Rademacher complexity framework. These bounds are further extended to the squared Wasserstein-1 distances between the probability measures of quantities of interest calculated from target causal operators and the corresponding learned neural oscillators. The theoretical results show that the estimation errors grow polynomially with respect to both MLP sizes and the time length, thereby avoiding the curse of parametric complexity. Furthermore, the derived error bounds demonstrate that constraining the Lipschitz constants of the MLPs via loss function regularization can improve the generalization ability of the neural oscillator. Numerical studies considering a Bouc-Wen nonlinear system under stochastic seismic excitation validates the theoretically predicted power laws of the estimation errors with respect to the sample size and time length, and confirms the effectiveness of constraining MLPs' matrix and vector norms in enhancing the performance of the neural oscillator under limited training data.

cs.LG↗

arXiv2026

Upper Approximation Bounds for Neural Oscillators

Neural oscillators, originating from second-order ordinary differential equations (ODEs), have demonstrated strong performance in stably learning causal mappings between long-term sequences or continuous temporal functions, as well as in accurately approximating physical systems. However, theoretically quantifying the capacities of their neural network architectures remains a significant challenge. In this study, the neural oscillator consisting of a second-order ODE followed by a multilayer perceptron (MLP) is considered. Its upper approximation bound for approximating causal and uniformly continuous operators between continuous temporal function spaces and that for approximating uniformly asymptotically incrementally stable second-order dynamical systems are derived. The established proof method of the approximation bound for approximating the causal continuous operators can also be directly applied to state-space models consisting of a linear time-continuous complex recurrent neural network followed by an MLP. Theoretical results reveal that the approximation error of the neural oscillator for approximating the second-order dynamical systems scales polynomially with the reciprocals of the widths of two utilized MLPs, thus overcoming the curse of parametric complexity. The convergence rates of two established approximation error bounds are validated through four numerical cases. These results provide a robust theoretical foundation for the effective application of the neural oscillator in science and engineering.

cs.LG↗

arXiv2020

History matching with probabilistic emulators and active learning

The scientific understanding of real-world processes has dramatically improved over the years through computer simulations. Such simulators represent complex mathematical models that are implemented as computer codes which are often expensive. The validity of using a particular simulator to draw accurate conclusions relies on the assumption that the computer code is correctly calibrated. This calibration procedure is often pursued under extensive experimentation and comparison with data from a real-world process. The problem is that the data collection may be so expensive that only a handful of experiments are feasible. History matching is a calibration technique that, given a simulator, it iteratively discards regions of the input space using an implausibility measure. When the simulator is computationally expensive, an emulator is used to explore the input space. In this paper, a Gaussian process provides a complete probabilistic output that is incorporated into the implausibility measure. The identification of regions of interest is accomplished with recently developed annealing sampling techniques. Active learning functions are incorporated into the history matching procedure to refocus on the input space and improve the emulator. The efficiency of the proposed framework is tested in well-known examples from the history matching literature, as well as in a proposed testbed of functions of higher dimensions.

stat.CO↗

arXiv2018

Efficient Pricing of Barrier Options on High Volatility Assets using Subset Simulation

Barrier options are one of the most widely traded exotic options on stock exchanges. In this paper, we develop a new stochastic simulation method for pricing barrier options and estimating the corresponding execution probabilities. We show that the proposed method always outperforms the standard Monte Carlo approach and becomes substantially more efficient when the underlying asset has high volatility, while it performs better than multilevel Monte Carlo for special cases of barrier options and underlying assets. These theoretical findings are confirmed by numerous simulation results.

q-fin.PR↗

arXiv2015

Rare Event Simulation

Rare events are events that are expected to occur infrequently, or more technically, those that have low probabilities (say, order of $10^{-3}$ or less) of occurring according to a probability model. In the context of uncertainty quantification, the rare events often correspond to failure of systems designed for high reliability, meaning that the system performance fails to meet some design or operation specifications. As reviewed in this section, computation of such rare-event probabilities is challenging. Analytical solutions are usually not available for non-trivial problems and standard Monte Carlo simulation is computationally inefficient. Therefore, much research effort has focused on developing advanced stochastic simulation methods that are more efficient. In this section, we address the problem of estimating rare-event probabilities by Monte Carlo simulation, Importance Sampling and Subset Simulation for highly reliable dynamic systems.

stat.CO↗

arXiv2011

Bayesian Post-Processor and other Enhancements of Subset Simulation for Estimating Failure Probabilities in High Dimensions

Estimation of small failure probabilities is one of the most important and challenging computational problems in reliability engineering. The failure probability is usually given by an integral over a high-dimensional uncertain parameter space that is difficult to evaluate numerically. This paper focuses on enhancements to Subset Simulation (SS), proposed by Au and Beck, which provides an efficient algorithm based on MCMC (Markov chain Monte Carlo) simulation for computing small failure probabilities for general high-dimensional reliability problems. First, we analyze the Modified Metropolis algorithm (MMA), an MCMC technique, which is used in SS for sampling from high-dimensional conditional distributions. We present some observations on the optimal scaling of MMA, and develop an optimal scaling strategy for this algorithm when it is employed within SS. Next, we provide a theoretical basis for the optimal value of the conditional failure probability $p_0$, an important parameter one has to choose when using SS. Finally, a Bayesian post-processor SS+ for the original SS method is developed where the uncertain failure probability that one is estimating is modeled as a stochastic variable whose possible values belong to the unit interval. Simulated samples from SS are viewed as informative data relevant to the system's reliability. Instead of a single real number as an estimate, SS+ produces the posterior PDF of the failure probability, which takes into account both prior information and the information in the sampled data. This PDF quantifies the uncertainty in the value of the failure probability and it may be further used in risk analyses to incorporate this uncertainty. The relationship between the original SS and SS+ is also discussed

stat.CO↗

arXiv2011

Asymptotically Independent Markov Sampling: a new MCMC scheme for Bayesian Inference

In Bayesian statistics, many problems can be expressed as the evaluation of the expectation of a quantity of interest with respect to the posterior distribution. Standard Monte Carlo method is often not applicable because the encountered posterior distributions cannot be sampled directly. In this case, the most popular strategies are the importance sampling method, Markov chain Monte Carlo, and annealing. In this paper, we introduce a new scheme for Bayesian inference, called Asymptotically Independent Markov Sampling (AIMS), which is based on the above methods. We derive important ergodic properties of AIMS. In particular, it is shown that, under certain conditions, the AIMS algorithm produces a uniformly ergodic Markov chain. The choice of the free parameters of the algorithm is discussed and recommendations are provided for this choice, both theoretically and heuristically based. The efficiency of AIMS is demonstrated with three numerical examples, which include both multi-modal and higher-dimensional target posterior distributions.

stat.CO↗