arXiv subjects

Toby van Gastelen

Publications and source records attributed to Toby van Gastelen.

4 recordsLinked to original sources

arXiv2025

A new data-driven energy-stable Evolve-Filter-Relax model for turbulent flow simulation

We present a novel approach to define the filter and relax steps in the evolve-filter-relax (EFR) framework for simulating turbulent flows. The EFR main advantages are its ease of implementation and computational efficiency. However, as it only contains two parameters (one for the filter step and one for the relax step) its flexibility is rather limited. In this work, we propose a data-driven approach in which the optimal filter is found based on DNS data in the frequency domain. The optimization step is computationally efficient and only involves one-dimensional least-squares problems for each wavenumber. Across both decaying turbulence and Kolmogorov flow, our learned filter decisively outperforms the standard differential filter and the Smagorinsky model, yielding significantly improved accuracy in energy spectra and in the temporal evolution of both energy and enstrophy. In addition, the relax parameter is determined by requiring energy and/or enstrophy conservation, which enforces stability of the method and reduces the appearance of numerical wiggles, especially when the filter is built in scarce data regimes. Applying the learned filter is also more computationally efficient compared to traditional differential filters, as it circumvents solving a linear system.

math.NA↗

arXiv2025

Energy-Conserving Neural Network Closure Model for Long-Time Accurate and Stable LES

Machine learning-based closure models for LES have shown promise in capturing complex turbulence dynamics but often suffer from instabilities and physical inconsistencies. In this work, we develop a novel skew-symmetric neural architecture as closure model that enforces stability while preserving key physical conservation laws. Our approach leverages a discretization that ensures mass, momentum, and energy conservation, along with a face-averaging filter to maintain mass conservation in coarse-grained velocity fields. We compare our model against several conventional data-driven closures (including unconstrained convolutional neural networks), and the physics-based Smagorinsky model. Performance is evaluated on decaying turbulence and Kolmogorov flow for multiple coarse-graining factors. In these test cases we observe that unconstrained machine learning models suffer from numerical instabilities. In contrast, our skew-symmetric model remains stable across all tests, though at the cost of increased dissipation. Despite this trade-off, we demonstrate that our model still outperforms the Smagorinsky model in unseen scenarios. These findings highlight the potential of structure-preserving machine learning closures for reliable long-time LES.

cs.LG↗

arXiv2025

Modeling Advection-Dominated Flows with Space-Local Reduced-Order Models

Reduced-order models (ROMs) are often used to accelerate the simulation of large physical systems. However, traditional ROM techniques, such as those based on proper orthogonal decomposition (POD), often struggle with advection-dominated flows due to the slow singular value decay. This results in high computational costs and potential instabilities. This paper proposes a novel approach using space-local POD to address the challenges arising from the slow singular value decay. Instead of global basis functions, our method employs local basis functions that are applied across the domain, analogous to the finite element method, but with a data-driven basis. By dividing the domain into subdomains and applying the space-local POD, we achieve a representation that is sparse and that generalizes better outside the training regime. This allows the use of a larger number of basis functions compared to standard POD, without prohibitive computational costs. To ensure smoothness across subdomain boundaries, we introduce overlapping subdomains inspired by the partition of unity method. Our approach is validated through simulations of the 1D and 2D advection equation. We demonstrate that using our space-local approach we obtain a ROM that generalizes better to flow conditions which are not part of the training data. In addition, we show that the constructed ROM inherits the energy conservation and non-linear stability properties from the full-order model. Finally, we find that using a space-local ROM allows for larger time steps.

math.NA↗

arXiv2024

Energy-Conserving Neural Network for Turbulence Closure Modeling

In turbulence modeling, we are concerned with finding closure models that represent the effect of the subgrid scales on the resolved scales. Recent approaches gravitate towards machine learning techniques to construct such models. However, the stability of machine-learned closure models and their abidance by physical structure (e.g. symmetries, conservation laws) are still open problems. To tackle both issues, we take the `discretize first, filter next' approach. In this approach we apply a spatial averaging filter to existing fine-grid discretizations. The main novelty is that we introduce an additional set of equations which dynamically model the energy of the subgrid scales. Having an estimate of the energy of the subgrid scales, we can use the concept of energy conservation to derive stability. The subgrid energy containing variables are determined via a data-driven technique. The closure model is used to model the interaction between the filtered quantities and the subgrid energy. Therefore the total energy should be conserved. Abiding by this conservation law yields guaranteed stability of the system. In this work, we propose a novel skew-symmetric convolutional neural network architecture that satisfies this law. The result is that stability is guaranteed, independent of the weights and biases of the network. Importantly, as our framework allows for energy exchange between resolved and subgrid scales it can model backscatter. To model dissipative systems (e.g. viscous flows), the framework is extended with a diffusive component. The introduced neural network architecture is constructed such that it also satisfies momentum conservation. We apply the new methodology to both the viscous Burgers' equation and the Korteweg-De Vries equation in 1D. The novel architecture displays superior stability properties when compared to a vanilla convolutional neural network.

math.NA↗