arXiv subjects

Benjamin Sanderse

Publications and source records attributed to Benjamin Sanderse.

At least 19 recordsLinked to original sources

arXiv2026

A differentiable software suite for accelerated simulation of turbulent flows

We present IncompressibleNavierStokes.jl, an open-source Julia package for solving the incompressible Navier--Stokes equations on staggered Cartesian grids. The package features matrix-free, hardware-agnostic kernels that are compiled from a single source for multi-threaded CPU or GPU execution, and hand-written adjoint kernels for all discrete operators, enabling efficient reverse-mode automatic differentiation through the entire solver. This differentiability allows neural network closure models to be trained a-posteriori while embedded in a large-eddy simulation. Memory optimizations permit double-precision direct numerical simulations at resolutions up to $840^3$ on a single GPU. The software design, numerical methods, hardware performance, and integration of neural network closure models are described, and results for turbulent channel flow are validated against reference data.

math.NA↗

arXiv2026

Generalized Tadmor Conditions and Structure-Preserving Numerical Fluxes for the Compressible Flow of Real Gases

We generalize Tadmor's algebraic numerical flux condition for entropy-conservative discretizations of conservation laws to a broader class of secondary structures, i.e. possibly non-convex secondary quantities whose evolution can consist of both conservative and non-conservative contributions. The resulting generalized Tadmor condition yields a discrete local balance law for secondary structures alongside the discrete conservation law that is solved. In contrast to the convex entropy setting, non-convex secondary quantities can have singular Hessians and non-injective gradients; this introduces an additional necessary structural requirement, which we term (discrete) null-consistency. Null-consistency constrains admissible numerical work terms and is required for the existence and well-posedness of fluxes satisfying the generalized Tadmor condition. To construct such fluxes in practice, we show how discrete gradient operators provide systematic construction methods even when some of the functions entering the secondary structure are arbitrary, as in compressible flow closed by an arbitrary equation of state. As an application, we derive an entropy-conserving and kinetic-energy-consistent numerical flux for the Euler equations with an arbitrary (non-ideal) equation of state. We demonstrate the performance of the resulting scheme on a set of supercritical/transcritical compressible-flow test cases using several non-ideal equations of state, including a fully turbulent transcritical flow with a state-of-the-art equation of state and models for viscosity and heat conductivity. Computations are performed with our new open-source, flexible, JAX-based, multi-GPU compressible flow solver for Helmholtz-based equations of state available at github.com/rbklein/HelmEOS2.

math.NA↗

arXiv2026

Approaching the optimal closure: equivariance, inductive bias, and Reynolds-number generalization in data-driven LES

Data-driven closures for large-eddy simulation (LES) are commonly built to respect the symmetries of the Navier--Stokes equations, on the premise that this improves accuracy and generalization. We test this premise in a controlled comparison of three data-driven LES closures that share a pointwise, Galilean-invariant velocity-gradient construction but span non-equivariant, octahedral-equivariant, and tensor-basis designs: an unconstrained multi-layer perceptron (MLP), a group-convolutional network whose exactly equivariant weights we synthesize in closed form, and a tensor-basis neural network (TBNN). The designs follow from an analysis of which symmetries survive discretization on a uniform grid, where the continuous orthogonal group reduces to the 48-element octahedral group. Across a range of network sizes the three closures saturate to the same a priori and a posteriori accuracy, and a direct conditional-mean estimate identifies the a priori floor as the one-point optimal closure of Langford and Moser. The equivariant and tensor-basis models reach this floor with $25$ times fewer parameters than the MLP: the inductive bias buys parameter efficiency rather than a lower error floor. Finally, we train the closures across several viscosities and supply the global filter-scale Reynolds number $\operatorname{Re}_\Delta = \Delta^2 \| \nabla \bar{u} \| / \nu$ as an input, a scaling-invariant feature dictated by the same symmetry analysis. The closures then generalize across Reynolds number: they hold their dissipation calibration at held-out viscosities and filter ratios where Reynolds-blind closures mis-dissipate, and partially correct it on an out-of-distribution Taylor--Green flow. Reynolds-number generalization is thus largely a calibration that the right input feature supplies.

math.NA↗

arXiv2025

Stable self-adaptive timestepping for Reduced Order Models for incompressible flows

This work introduces RedEigCD, the first self-adaptive timestepping technique specifically tailored for reduced-order models (ROMs) of the incompressible Navier-Stokes equations. Building upon linear stability concepts, the method adapts the timestep by directly bounding the stability function of the employed time integration scheme using exact spectral information of matrices related to the reduced operators. Unlike traditional error-based adaptive methods, RedEigCD relies on the eigenbounds of the convective and diffusive ROM operators, whose computation is feasible at reduced scale and fully preserves the online efficiency of the ROM. A central theoretical contribution of this work is the proof, based on the combined theorems of Bendixson and Rao, that, under linearized assumptions, the maximum stable timestep for projection-based ROMs is shown to be larger than or equal to that of their corresponding full-order models (FOMs). Numerical experiments for both periodic and non-homogeneous boundary conditions demonstrate that RedEigCD yields stable timestep increases up to a factor 40 compared to the FOM, without compromising accuracy. The methodology thus establishes a new link between linear stability theory and reduced-order modeling, offering a systematic path towards efficient, self-regulating ROM integration in incompressible flow simulations.

math.NA↗

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

Exact expressions for the unresolved stress in a finite-volume based large-eddy simulation

In this article we propose new discretization-informed expressions for the residual stress tensor (RST) in a finite-volume based large-eddy simulation (LES-FVM). In addition to the classical RST $\overline{u u} - \bar{u} \bar{u}$ resulting from the non-commutation between filtering and the nonlinear stress, our RST also contains contributions from the numerical flux, discrete divergence, and pressure terms. Unlike the classical RST, our proposed RST is non-symmetric and non-local. The proposed form of the RST is important for generating appropriate reference data for LES closure modeling. Based on DNS results of the 1D Burgers and 3D incompressible Navier-Stokes equations, we show that the discretization-induced parts of the RST play an important role in the LES-FVM equation for common LES filter widths. When the discrete contribution is included, our RST expression gives zero a-posteriori error in LES, while existing RST expressions give errors that increase over time. For a Smagorinsky model, we show that the Smagorinsky coefficient is higher when fitted to our new RST than when fitted to the classical RST and gives improved results.

math.NA↗

arXiv2025

Exact operator inference with minimal data

This work introduces a novel method to generate snapshot data for operator inference that guarantees the exact reconstruction of intrusive projection-based reduced-order models (ROMs). To ensure exact reconstruction, the operator inference least squares matrix must have full rank, without regularization. Existing works have achieved this full rank using heuristic strategies to generate snapshot data and a-posteriori checks on full rank, but without a guarantee of success. Our novel snapshot data generation method provides this guarantee thanks to two key ingredients: first we identify ROM states that induce full rank, then we generate snapshots corresponding to exactly these states by simulating multiple trajectories for only a single time step. This way, the number of required snapshots is minimal and orders of magnitude lower than typically reported with existing methods. The method avoids non-Markovian terms and does not require re-projection. Since the number of snapshots is minimal, the least squares problem simplifies to a linear system that is numerically more stable. In addition, because the inferred operators are exact, properties of the intrusive ROM operators such as symmetry or skew-symmetry are preserved. Numerical results for differential equations involving 2nd, 3rd and 8th order polynomials demonstrate that the novel snapshot data generation method leads to exact reconstruction of the intrusive reduced order models.

math.NA↗

arXiv2025

Harnessing Equivariance: Modeling Turbulence with Graph Neural Networks

This work proposes a novel methodology for turbulence modeling in Large Eddy Simulation (LES) based on Graph Neural Networks (GNNs), which embeds the discrete rotational, reflectional and translational symmetries of the Navier-Stokes equations into the model architecture. In addition, suitable invariant input and output spaces are derived that allow the GNN models to be embedded seamlessly into the LES framework to obtain a symmetry-preserving simulation setup. The suitability of the proposed approach is investigated for two canonical test cases: Homogeneous Isotropic Turbulence (HIT) and turbulent channel flow. For both cases, GNN models are trained successfully in actual simulations using Reinforcement Learning (RL) to ensure that the models are consistent with the underlying LES formulation and discretization. It is demonstrated for the HIT case that the resulting GNN-based LES scheme recovers rotational and reflectional equivariance up to machine precision in actual simulations. At the same time, the stability and accuracy remain on par with non-symmetry-preserving machine learning models that fail to obey these properties. The same modeling strategy translates well to turbulent channel flow, where the GNN model successfully learns the more complex flow physics and is able to recover the turbulent statistics and Reynolds stresses. It is shown that the GNN model learns a zonal modeling strategy with distinct behaviors in the near-wall and outer regions. The proposed approach thus demonstrates the potential of GNNs for turbulence modeling, especially in the context of LES and RL.

physics.flu-dyn↗

arXiv2025

Physics-aware generative models for turbulent fluid flows through energy-consistent stochastic interpolants

Generative models have demonstrated remarkable success in domains such as text, image, and video synthesis. In this work, we explore the application of generative models to fluid dynamics, specifically for turbulence simulation, where classical numerical solvers are computationally expensive. We propose a novel stochastic generative model based on stochastic interpolants, which enables probabilistic forecasting while incorporating physical constraints such as energy stability and divergence-freeness. Unlike conventional stochastic generative models, which are often agnostic to underlying physical laws, our approach embeds energy consistency by making the parameters of the stochastic interpolant learnable coefficients. We evaluate our method on a benchmark turbulence problem - Kolmogorov flow - demonstrating superior accuracy and stability over state-of-the-art alternatives such as autoregressive conditional diffusion models (ACDMs) and PDE-Refiner. Furthermore, we achieve stable results for significantly longer roll-outs than standard stochastic interpolants. Our results highlight the potential of physics-aware generative models in accelerating and enhancing turbulence simulations while preserving fundamental conservation properties.

cs.CE↗

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↗

arXiv2024

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

A new temperature evolution equation that enforces thermodynamic vapour-liquid equilibrium in multiphase flows -- application to CO2 modeling

This work presents a novel framework for numerically simulating the depressurization of tanks and pipelines containing carbon dioxide (CO2). The framework focuses on efficient solution strategies for the coupled system of fluid flow equations and thermodynamic constraints. A key contribution lies in proposing a new set of equations for phase equilibrium calculations which simplifies the traditional vapor-liquid equilibrium (VLE) calculations for two-phase CO2 mixtures. The first major novelty resides in the reduction of the conventional four-equation VLE system to a single equation, enabling efficient solution using a non-linear solver. This significantly reduces computational cost compared to traditional methods. Furthermore, a second novelty is introduced by deriving an ordinary differential equation (ODE) directly from the UV-Flash equation. This ODE can be integrated alongside the governing fluid flow equations, offering a computationally efficient approach for simulating depressurization processes.

physics.flu-dyn↗

arXiv2024

Entropy-Stable Model Reduction of One-Dimensional Hyperbolic Systems using Rational Quadratic Manifolds

In this work we propose a novel method to ensure important entropy inequalities are satisfied semi-discretely when constructing reduced order models (ROMs) on nonlinear reduced manifolds. We are in particular interested in ROMs of systems of nonlinear hyperbolic conservation laws. The so-called entropy stability property endows the semi-discrete ROMs with physically admissible behaviour. The method generalizes earlier results on entropy-stable ROMs constructed on linear spaces. The ROM works by evaluating the projected system on a well-chosen approximation of the state that ensures entropy stability. To ensure accuracy of the ROM after this approximation we locally enrich the tangent space of the reduced manifold with important quantities. Using numerical experiments on some well-known equations (the inviscid Burgers equation, shallow water equations and compressible Euler equations) we show the improved structure-preserving properties of our ROM compared to standard approaches and that our approximations have minimal impact on the accuracy of the ROM. We additionally generalize the recently proposed polynomial reduced manifolds to rational polynomial manifolds and show that this leads to an increase in accuracy for our experiments.

math.NA↗

arXiv2024

Discretize first, filter next: learning divergence-consistent closure models for large-eddy simulation

We propose a new neural network based large eddy simulation framework for the incompressible Navier-Stokes equations based on the paradigm "discretize first, filter and close next". This leads to full model-data consistency and allows for employing neural closure models in the same environment as where they have been trained. Since the LES discretization error is included in the learning process, the closure models can learn to account for the discretization. Furthermore, we employ a divergence-consistent discrete filter defined through face-averaging and provide novel theoretical and numerical filter analysis. This filter preserves the discrete divergence-free constraint by construction, unlike general discrete filters such as volume-averaging filters. We show that using a divergence-consistent LES formulation coupled with a convolutional neural closure model produces stable and accurate results for both a-priori and a-posteriori training, while a general (divergence-inconsistent) LES model requires a-posteriori training or other stability-enforcing measures.

math.NA↗

arXiv2024

Scientific machine learning for closure models in multiscale problems: a review

Closure problems are omnipresent when simulating multiscale systems, where some quantities and processes cannot be fully prescribed despite their effects on the simulation's accuracy. Recently, scientific machine learning approaches have been proposed as a way to tackle the closure problem, combining traditional (physics-based) modeling with data-driven (machine-learned) techniques, typically through enriching differential equations with neural networks. This paper reviews the different reduced model forms, distinguished by the degree to which they include known physics, and the different objectives of a priori and a posteriori learning. The importance of adhering to physical laws (such as symmetries and conservation laws) in choosing the reduced model form and choosing the learning method is discussed. The effect of spatial and temporal discretization and recent trends toward discretization-invariant models are reviewed. In addition, we make the connections between closure problems and several other research disciplines: inverse problems, Mori-Zwanzig theory, and multi-fidelity methods. In conclusion, much progress has been made with scientific machine learning approaches for solving closure problems, but many challenges remain. In particular, the generalizability and interpretability of learned models is a major issue that needs to be addressed further.

math.NA↗

arXiv2023

A pressure-free long-time stable reduced-order model for two-dimensional Rayleigh-B\'enard convection

The present work presents a stable POD-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh-B\'enard convection in a square geometry for three Rayleigh numbers: $10^4$ (steady state), $3\times 10^5$ (periodic), and $6 \times 10^6$ (chaotic). Stability is obtained through a particular (staggered-grid) full-order model (FOM) discretization that leads to a ROM that is pressure-free and has skew-symmetric (energy-conserving) convective terms. This yields long-time stable solutions without requiring stabilizing mechanisms, even outside the training data range. The ROM's stability is validated for the different test cases by investigating the Nusselt and Reynolds number time series and the mean and variance of the vertical temperature profile. In general, these quantities converge to the FOM when increasing the number of modes, and turn out to be a good measure of accuracy. However, for the chaotic case, convergence with increasing numbers of modes is relatively difficult and a high number of modes is required to resolve the low-energy structures that are important for the global dynamics.

physics.flu-dyn↗

arXiv2023

Energy-consistent discretization of viscous dissipation with application to natural convection flow

A new energy-consistent discretization of the viscous dissipation function in incompressible flows is proposed. It is implied by choosing a discretization of the diffusive terms and a discretization of the local kinetic energy equation and by requiring that continuous identities like the product rule are mimicked discretely. The proposed viscous dissipation function has a quadratic, strictly dissipative form, for both simplified (constant viscosity) stress tensors and general stress tensors. The proposed expression is not only useful in evaluating energy budgets in turbulent flows, but also in natural convection flows, where it appears in the internal energy equation and is responsible for viscous heating. The viscous dissipation function is such that a consistent total energy balance is obtained: the 'implied' presence as sink in the kinetic energy equation is exactly balanced by explicitly adding it as source term in the internal energy equation. Numerical experiments of Rayleigh-B\'enard convection (RBC) and Rayleigh-Taylor instabilities confirm that with the proposed dissipation function, the energy exchange between kinetic and internal energy is exactly preserved. The experiments show furthermore that viscous dissipation does not affect the critical Rayleigh number at which instabilities form, but it does significantly impact the development of instabilities once they occur. Consequently, the value of the Nusselt number on the cold plate becomes larger than on the hot plate, with the difference increasing with increasing Gebhart number. Finally, 3D simulations of turbulent RBC show that energy balances are exactly satisfied even for very coarse grids; therefore, we consider that the proposed discretization forms an excellent starting point for testing sub-grid scale models.

physics.flu-dyn↗

arXiv2023

Structure-Preserving Hyper-Reduction and Temporal Localization for Reduced Order Models of Incompressible Flows

A novel hyper-reduction method is proposed that conserves kinetic energy and momentum for reduced order models of the incompressible Navier-Stokes equations. The main advantage of conservation of kinetic energy is that it endows the hyper-reduced order model (hROM) with a nonlinear stability property. The new method poses the discrete empirical interpolation method (DEIM) as a minimization problem and subsequently imposes constraints to conserve kinetic energy. Two methods are proposed to improve the robustness of the new method against error accumulation: oversampling and Mahalanobis regularization. Mahalanobis regularization has the benefit of not requiring additional measurement points. Furthermore, a novel method is proposed to perform structure-preserving temporal localization with the principle interval decomposition: new interface conditions are derived such that energy and momentum are conserved for a full time-integration instead of only during separate intervals. The performance of the new structure-preserving hyper-reduction methods and the structure-preserving temporal localization method is analysed using two convection-dominated test cases; a shear-layer roll-up and two-dimensional homogeneous isotropic turbulence. It is found that both Mahalanobis regularization and oversampling allow hyper-reduction of these test cases. Moreover, the Mahalanobis regularization provides comparable robustness while being more efficient than oversampling.

physics.flu-dyn↗