arXiv subjects

Ilya Timofeyev

Publications and source records attributed to Ilya Timofeyev.

At least 19 recordsLinked to original sources

arXiv2026

Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem

We develop two parametric data-driven reduced models: a physics-informed neural network (PINN) and a non-intrusive tensorial reduced-order model (TROM), and apply both approaches to the parametrized one-dimensional shallow-water dam-break problem. Both reduced models do not require time integration and learn a direct solution map from space, time, and dam-break parameters to the physical state. We present a detailed comparison for out-of-sample and extrapolated parameter values. In addition, we demonstrate that it is essential to introduce shock-aware collocation to improve the robustness of the PINN model.

math.NA↗

arXiv2026

Subgrid-Scale Parameterization in Burgers' Equation Using Structure-Preserving Neural Networks and Entropy Variables

We present a machine learning approach for developing subgrid-scale (SGS) parametrizations in coarse simulations of partial differential equations. We utilize structure-preserving neural networks and entropy variables to learn subgrid fluxes in coarse simulations of the Burgers' equation. In particular, we employ a decoupled neural network architecture explicitly separating the subgrid corrections into two distinct components: a conservative Flux Potential network and an Eddy Viscosity network. We demonstrate that this reduced-order framework maintains high physical fidelity, accurately reproducing the energy spectrum, spatial and temporal correlation functions, and dynamical characteristics of the full-scale system. Furthermore, we show that our approach is robust and applicable to parameters outside the training regime.

math.NA↗

arXiv2026

Reduced-Order Modeling of Parameterized Visco-Plastic Shallow Flows

We propose a non-intrusive reduced-order modeling framework for parametrized visco-plastic free-surface flows governed by a shallow-water formulation of Herschel--Bulkley fluids. These flows exhibit strong nonlinearities, non-smooth rheology, moving fronts, and yield surfaces, making efficient surrogate modeling particularly challenging. To address this challenge, we employ a tensor-based approach in which the solution manifold is approximated using a low-rank representation obtained via higher-order singular value decomposition of snapshot data over a structured parameter space. The resulting tensorial reduced-order model (TROM) enables rapid online evaluation by directly reconstructing solution trajectories from the compressed representation, thereby avoiding the need to perform time integration of a reduced dynamical system. The proposed non-intrusive framework can be interpreted as an encoder--decoder architecture with a compressed latent representation and efficient multilinear decoding. Numerical experiments demonstrate that the proposed approach accurately captures key flow features, including front propagation, plug and shear regions, and near-stopping dynamics, while achieving substantial computational speedups relative to full-order simulations.

physics.flu-dyn↗

arXiv2026

Parametrization of subgrid scales in long-term simulations of the shallow-water equations using machine learning and convex limiting

We present a method for parametrizing sub-grid processes in the Shallow Water equations. We define coarse variables and local spatial averages and use a feed-forward neural network to learn sub-grid fluxes. Our method results in a local parametrization that uses a four-point computational stencil, which has several advantages over globally coupled parametrizations. We demonstrate numerically that our method improves energy balance in long-term turbulent simulations and also accurately reproduces individual solutions. The long-term simulations refer to numerical studies where a fluid flow is simulated over a duration long enough to reach a statistical steady state. The neural network parametrization can be easily combined with flux limiting to reduce oscillations near shocks. More importantly, our method provides reliable parametrizations, even in dynamical regimes that are not included in the training data.

physics.flu-dyn↗

arXiv2025

A parametric tensor ROM for the shallow water dam break problem

We develop a variant of a tensor reduced-order model (tROM) for the parameterized shallow-water dam-break problem. This hyperbolic system presents multiple challenges for model reduction, including a slow decay of the Kolmogorov $N$-width of the solution manifold, shock formation, and the loss of smooth solution dependence on parameters. These issues limit the performance of traditional Proper Orthogonal Decomposition based ROMs. Our tROM approach, based on a low-rank tensor decomposition, builds a parameter-to-solution map from high-fidelity snapshots and constructs localized reduced bases via a local POD procedure. We apply this method to 1D dry-bed and wet-bed problems and 2D wet-bed problem with topography and bottom friction, showing that the non-interpolatory variant of the tROM, combined with Chebyshev sampling near critical parameter values, effectively captures parameter-dependent behavior and significantly outperforms standard POD-ROMs. This is especially evident in the wet-bed case, where POD-ROMs exhibit poor resolution of shock waves and spurious oscillations.

math.NA↗

arXiv2025

Attractor learning for spatiotemporally chaotic dynamical systems using echo state networks with transfer learning

In this paper, we explore the predictive capabilities of echo state networks (ESNs) for the generalized Kuramoto-Sivashinsky (gKS) equation, an archetypal nonlinear PDE that exhibits spatiotemporal chaos. Our research focuses on predicting changes in long-term statistical patterns of the gKS model that result from varying the dispersion relation or the length of the spatial domain. We use transfer learning to adapt ESNs to different parameter settings and successfully capture changes in the underlying chaotic attractor. Previous work has shown that transfer learning can be used effectively with ESNs for single-orbit prediction. The novelty of our paper lies in our use of this pairing to predict the long-term statistical properties of spatiotemporally chaotic PDEs. We also show that transfer learning nontrivially improves the length of time that predictions of individual gKS trajectories remain accurate.

math.DS↗

arXiv2025

Using Echo-State Networks to Reproduce Rare Events in Chaotic Systems

We apply Echo-State Networks to predict time series and statistical properties of the competitive Lotka-Volterra model in the chaotic regime. In particular, we demonstrate that Echo-State Networks successfully learn the chaotic attractor of the competitive Lotka-Volterra model and reproduce histograms of dependent variables, including tails and rare events. We also demonstrate that the Echo-State Networks reproduce rare events in the non-equilibrium simulations of the Lotka-Volterra system. We use the Generalized Extreme Value distribution to quantify the tail behavior.

nlin.CD↗

arXiv2025

Parametric Reduced Order Models for the Generalized Kuramoto--Sivashinsky Equations

The paper studies parametric Reduced Order Models (ROMs) for the Kuramoto--Sivashinsky (KS) and generalized Kuramoto--Sivashinsky (gKS) equations. We consider several POD and POD-DEIM projection ROMs with various strategies for parameter sampling and snapshot collection. The aim is to identify an approach for constructing a ROM that is efficient across a range of parameters, encompassing several regimes exhibited by the KS and gKS solutions: weakly chaotic, transitional, and quasi-periodic dynamics. We describe such an approach and demonstrate that it is essential to develop ROMs that adequately represent the short-time transient behavior of the gKS model.

math.NA↗

arXiv2024

Role of Non-Exponential Reversal times in Aggregation Models of Bacterial Populations

In this paper, we consider 1D agent-based and kinetic models of aggregation with reversals. In particular, we fit a Gamma distribution to represent the run times in myxobacteria and analyze numerically the importance of non-exponential reversal times. We demonstrate that non-exponential reversal times aid aggregation and result in tighter aggregates. We compare and contrast the behavior of agent-based and kinetic models, and also consider kinetic models with aggregation driven by chemotaxis. Thus, incorporating non-exponential reversal times into models of aggregation can be particularly important for reproducing experimental data, such as aggregate persistence and dispersal.

physics.bio-ph↗

arXiv2024

Application of Machine Learning and Convex Limiting to Subgrid Flux Modeling in the Shallow-Water Equations

We propose a combination of machine learning and flux limiting for property-preserving subgrid scale modeling in the context of flux-limited finite volume methods for the one-dimensional shallow-water equations. The numerical fluxes of a conservative target scheme are fitted to the coarse-mesh averages of a monotone fine-grid discretization using a neural network to parametrize the subgrid scale components. To ensure positivity preservation and the validity of local maximum principles, we use a flux limiter that constrains the intermediate states of an equivalent fluctuation form to stay in a convex admissible set. The results of our numerical studies confirm that the proposed combination of machine learning with monolithic convex limiting produces meaningful closures even in scenarios for which the network was not trained.

physics.comp-ph↗

arXiv2024

Existence of weak solutions for the kinetic models of motion of myxobacteria with alignment and reversals

In this paper, we consider three non-linear kinetic partial differential equations that emerge in the modeling of motion of rod-shaped cells such as myxobacteria. This motion is characterized by nematic alignment with neighboring cells, orientation reversals from cell polarity switching, orientation diffusion, and transport driven by chemotaxis. Our primary contribution lies in establishing the existence of weak solutions for these equations. Our analytical approach is based on the application of the classical averaging lemma from the kinetic theory, augmented by a novel version where the transport operator is substituted with a uni-directional diffusion operator.

math.AP↗

arXiv2023

Rare Events Analysis and Computation for Stochastic Evolution of Bacterial Populations

In this paper, we develop a computational approach for computing most likely trajectories describing rare events that correspond to the emergence of non-dominant genotypes. This work is based on the large deviations approach for discrete Markov chains describing the genetic evolution of large bacterial populations. We demonstrate that a gradient descent algorithm developed in this paper results in the fast and accurate computation of most-likely trajectories for a large number of bacterial genotypes. We supplement our analysis with extensive numerical simulations demonstrating the computational advantage of the designed gradient descent algorithm over other, more simplified, approaches.

q-bio.PE↗

arXiv2023

Modeling Information Flow with a Multi-Stage Queuing Mode

In this paper, we introduce a nonlinear stochastic model to describe the propagation of information inside a computer processor. In this model, a computational task is divided into stages, and information can flow from one stage to another. The model is formulated as a spatially-extended, continuous-time Markov chain where space represents different stages. This model is equivalent to a spatially-extended version of the M/M/s queue. The main modeling feature is the throttling function which describes the processor slowdown when the amount of information falls below a certain threshold. We derive the stationary distribution for this stochastic model and develop a closure for a deterministic ODE system that approximates the evolution of the mean and variance of the stochastic model. We demonstrate the validity of the closure with numerical simulations.

math.PR↗

arXiv2023

Breakdown of Boltzmann-type Models for the Alignment of Self-propelled Rods

Studies in the collective motility of organisms use a range of analytical approaches to formulate continuous kinetic models of collective dynamics from rules or equations describing agent interactions. However, the derivation of these kinetic models often relies on Boltzmann's hypothesis of "molecular chaos", that correlations between individuals are short-lived. While this assumption is often the simplest way to derive tractable models, it is often not valid in practice due to the high levels of cooperation and self-organization present in biological systems. In this work, we illustrated this point by considering a general Boltzmann-type kinetic model for the alignment of self-propelled rods where rod reorientation occurs upon binary collisions. We examine the accuracy of the kinetic model by comparing numerical solutions of the continuous equations to an agent-based model that implements the underlying rules governing microscopic alignment. Even for the simplest case considered, our comparison demonstrates that the kinetic model fails to replicate the discrete dynamics due to the formation of rod clusters that violate statistical independence. Additionally, we show that introducing noise to limit cluster formation helps improve the agreement between the analytical model and agent simulations but does not restore agreement completely. These results highlight the need to both develop and disseminate improved moment-closure methods for modeling biological and active matter systems.

physics.bio-ph↗

arXiv2022

A mean-field model for nematic alignment of self-propelled rods

In this paper we develop a model for nematic alignment of self-propelled rods interacting through binary collisions. We avoid phenomenological descriptions of rod interaction in favor of rigorously using a set of microscopic-level rules. Under the assumption that each collision results in a small change to a rod's orientation, we derive the Fokker-Planck equation for the evolution of the kinetic density function. Using analytical and numerical methods, we study the emergence of the nematic order from a homogeneous, uniform steady-state of the mean-field equation.

physics.bio-ph↗

arXiv2021

Predicting Shallow Water Dynamics using Echo-State Networks with Transfer Learning

In this paper we demonstrate that reservoir computing can be used to learn the dynamics of the shallow-water equations. In particular, while most previous applications of reservoir computing have required training on a particular trajectory to further predict the evolution along that trajectory alone, we show the capability of reservoir computing to predict trajectories of the shallow-water equations with initial conditions not seen in the training process. However, in this setting, we find that the performance of the network deteriorates for initial conditions with ambient conditions (such as total water height and average velocity) that are different from those in the training dataset. To circumvent this deficiency, we introduce a transfer learning approach wherein a small additional training step with the relevant ambient conditions is used to improve the predictions.

cs.LG↗

arXiv2021

Non-parametric estimation of Stochastic Differential Equations from stationary time-series

We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients. These estimators involve time- and space-discretization parameters for computing expected values from discretely-sampled stationary data. Next, we analyze consistency and mean squared error of these estimators depending on computational parameters. We derive relationships between the number of observational points, time- and space-discretization parameters in order to achieve the optimal speed of convergence and minimize computational complexity. We illustrate our approach with numerical simulations.

math.PR↗

arXiv2021

Reaction-diffusion models for morphological patterning of hESCs

In this paper we consider mathematical modeling of the dynamics of self-organized patterning of spatially confined human embryonic stem cells (hESCs) treated with BMP4 (gastruloids) described in recent experimental works. In the first part of the paper we use the activator-inhibitor equations of Gierer and Meinhardt to identify 3 reaction-diffusion regimes for each of the three morphogenic proteins, BMP4, Wnt and Nodal, based on the characteristic features of the dynamic patterning. We identify appropriate boundary conditions which correspond to the experimental setup and perform numerical simulations of the reaction-diffusion (RD) systems, using the finite element approximation, to confirm that the RD systems in these regimes produce realistic dynamics of the protein concentrations. In the second part of the paper we use analytic tools to address the questions of the existence and stability of non-homogeneous steady states for the reaction-diffusion systems of the type considered in the first part of the paper. We find sufficient conditions on the data of the problem under which the system has an universal attractor.

math.AP↗