Ilya Timofeyev

Md Rezwan Bin Mizan, Maxim Olshanskii, Ilya Timofeyev

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.

Ilya Timofeyev, Alexey Schwarzmann, Dmitri Kuzmin

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.

Md Rezwan Bin Mizan, Ilya Timofeyev, Maxim Olshanskii

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.

Md Amran Hossan Mojamder, Zhihang Xu, Min Wang et al.

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 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.

Mohammad Shah Alam, William Ott, Ilya Timofeyev

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. We introduce a novel methodology that integrates ESNs with transfer learning, aiming to enhance predictive performance across various parameter regimes of the gKS model. 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.

Anton Erofeev, Balasubramanya T. Nadiga, Ilya Timofeyev

We apply the Echo-State Networks to predict the 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 use the Generalized Extreme Value distribution to quantify the tail behavior.

Xiaoqian Chen, Balasubramanya T. Nadiga, Ilya Timofeyev

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.