Papers
Machine learning for physics, as it is published.
New work from arXiv on CFD surrogates, neural operators and physics-informed learning, refreshed every hour. Search it, filter by topic, or start with the papers we keep coming back to.
Papers
New work from arXiv on CFD surrogates, neural operators and physics-informed learning, refreshed every hour. Search it, filter by topic, or start with the papers we keep coming back to.
564 papers in Turbulence · newest first
physics.flu-dyn
Reynolds-averaged Navier-Stokes (RANS) turbulence models are known to perform poorly in predicting the dynamics of Rayleigh-Taylor mixing when turbulence is not fully developed, particularly during the transition from an initially perturbed interface. In this work, we investigate the use of data-driven strategies to enhance a simple -- model for this transitional regime. The turbulence model is first embedded within a surrogate physics-informed neural network (PINN), enabling the calibration of coefficients that account for parametric errors and the identification of corrective terms representing structural errors associated with missing physical processes. The learned corrections are then re-expressed onto the model state variables and relevant flow indicators, leading to explicit analytical modifications of the closure. The resulting fully interpretable corrected model is assessed against an extensive database of direct numerical simulations (DNS) of Rayleigh-Taylor flows. This framework enables improved predictions of the mixing-layer growth during the transition to turbulence.
physics.flu-dyn
Data-driven turbulence closures are commonly calibrated by embedding a CFD solver within an inverse procedure that requires a converged flow solution at every optimization iteration, which is fragile and costly. We instead formulate closure identification as a PDE-constrained, data-driven inverse problem within a physics-informed neural network (PINN), and subsequently stabilize the inferred closure for deployment in a numerical solver. This separation eliminates the training cost of repeated CFD solves and associated adjoint or ensemble evaluations, providing substantial flexibility to investigate alternative closure forms. Using this flexibility, we develop four closures: three model the Reynolds stress on a realizable tensor basis—a local map, a non-local model transporting the turbulent kinetic energy and recovering the out-of-plane normal stress, and the same with a learned length scale —and a fourth models the Reynolds force directly, free of the realizability constraint. All four are trained across six two-dimensional bluff-body wakes at and deployed frozen in a standard finite-element solver, stabilized primarily by input-gradient smoothing. Under a strict leave-one-shape-out (LOSO) protocol, all four reduce the field errors of a steady SST - baseline, most strongly on the mean velocity. The learned-length-scale closure is most accurate on the stress fields, while the force model generalizes best on the mean velocity and drag (LOSO drag error ). The closures also train efficiently on Particle Image Velocimetry data, enabling geometries intractable for DNS.
cs.CE
Numerical simulations in industrial applications often require performing numerous high-precision computations parameterized by specific experimental conditions. For instance, in vehicle body design, aerodynamic simulations are essential for evaluating the aerodynamic characteristics of various proposed body geometries. However, computational resource constraints often become a bottleneck. Therefore, achieving the desired accuracy while minimizing computational cost is crucial. To address this challenge, model reduction methods have been developed to decrease the degrees of freedom by constraining the possible states of a physical system to a lower-dimensional subspace. In particular, reduction techniques that project the system onto a nonlinear subspace using neural networks have been actively studied. Our previous research developed a reduced-order model that integrates neural-network-based model reduction with a time-evolution method, implemented as a distributed parallel training framework to process high-resolution flow field data efficiently. In this study, we extend this reduction approach by incorporating a variational autoencoder to assess its robustness in high-Reynolds-number flows around multiple vehicle bodies with varying geometries. Specifically, we evaluate the reconstruction accuracy of vortex generation across different spatial and temporal scales using a compact latent representation, with a particular focus on the flow behavior near the rear end of the vehicle body.
physics.flu-dyn
For centuries, the intrinsic chaos of unsteady fluid motion has stood as a formidable barrier to long-term forecasting. While machine learning (ML) has recently emerged as a transformative paradigm for predicting flow evolution, it encounters a pervasive yet unexplained "performance wall": an inevitable deterioration in accuracy as the forecast horizon extends. Here, we demonstrate that this deterioration is not a deficiency of model architecture, no matter how state-of-the-art, but a fundamental constraint imposed by the underlying system, which can be understood through turbulence theory established decades ago. In the setting of bluff body flow, a canonical phenomenon for spatiotemporal complexity in fluid mechanics, we reveal a scaling law governing the deterioration of ML predictability, derived from a Kolmogorov-inspired framework and validated through high-fidelity simulations. Our findings establish a closed loop between the predictability ceiling and its interpretation, bridging the gap between transparent physical theories and modern black-box inference. More broadly, this work provides a theoretical compass for constructing trustworthy ML in complex dynamical systems across the physical sciences.
physics.flu-dyn
Turbulence closure models are essential for solving the equations of motion in realistic systems, where fully resolving all relevant scales of motion is computationally infeasible. Developing turbulence closures remains one of the most challenging problems in fluid dynamics. Specifically, the Navier-Stokes equations, when filtered to isolate large-scale motions, introduce new terms representing the influence of subgrid-scale turbulent stresses. These terms, which can only be computed directly by resolving the turbulence itself, therefore lead to the closure problem: we must add new equations or introduce assumptions to relate the unresolved scales of motions to the resolved flow. Here we consider the closure problem for oceanic flows, i.e., stratified, Boussinesq, incompressible, in a rotating frame of reference. In particular, we focus on a closure for ocean mesoscale eddies, which have horizontal scales of 10-100km and are key to the redistribution of momentum, energy, and tracers in the ocean. In particular, mesoscale eddies can reinject energy and momentum into the large-scale flow through an inverse energy cascade. Here, we explore a range of theoretical and data-driven ocean mesoscale closures and examine their connections using analytical and data-driven methods. This note aims to bridge the gap between novel methods from artificial intelligence (AI) and machine learning and theoretical fluid dynamics to address significant challenges in the physics of turbulence.
physics.flu-dyn
Free-stream-turbulence-induced transition under streamwise-varying pressure gradients, as encountered on turbine blades, has received much less attention in high-fidelity simulations than its zero-pressure-gradient counterpart. The ERCOFTAC T3 experiments provide a carefully documented reference for this problem and remain central to transition-model calibration. Complementing them with full-field, three-dimensional data opens new opportunities for the development of physics-based and machine-learning transition models. To this end, wall-resolved implicit large-eddy simulations of the T3A, T3B, and T3C2–C5 flat-plate cases are performed with a high-order compressible solver, using an upstream slip-wall block that lets the synthetic free-stream turbulence develop up to the leading edge. The simulations reproduce well the measured transition locations, integral quantities, and velocity profiles, and predict a separation bubble near the end of the plate for the lowest-Reynolds-number case T3C4. As the Reynolds number decreases and transition moves into the adverse-pressure-gradient region, the mechanism evolves continuously from classical bypass transition to transition governed by the tendency of the laminar boundary layer to separate, up to separation-induced transition. Intermittency, conditioned statistics, and Reynolds-stress budgets document this evolution and the associated non-equilibrium effects, which challenge current RANS transition models.
physics.ao-ph
High-altitude Clear Air Turbulence (CAT) poses significant risks to aviation safety due to its unpredictability and challenges in detection. This study leverages machine learning models to improve CAT prediction within U.S. airspace at 200-350 hPa pressure levels, utilizing Pilot Reports (PIREPs), ERA5 reanalysis data, and aircraft aerodynamic parameters from the BADA database. Gradient boosting algorithms, particularly XGBoost, achieved the highest performance with an AUC of 0.904, demonstrating superior capability in capturing non-linear atmospheric dynamics. Key findings highlight the dominance of geographic coordinates (17.5% feature importance) and turbulence indices like TI3 in prediction, emphasizing the role of regional topography and upper-tropospheric instability. The integration of aerodynamic features such as drag force and wing loading improved the detection of moderate-to-severe perceived turbulence intensity (POD improved from 0.845 to 0.866), providing additional value to traditional aircraft-independent methods. Seasonal analysis revealed winter months as peak periods for CAT incidents, correlating with jet stream activity. While results align with global studies, limitations include geographic scope and aircraft-type diversity. This research underscores the potential of machine learning for operational CAT forecasting, with recommendations for future work focusing on global data integration and real-time telemetry to address climate-driven turbulence trends.
physics.flu-dyn
We introduce a Mori–Zwanzig graph neural network (MZ–GNN) framework for learning reduced-order Lagrangian dynamics of tracer particles in homogeneous isotropic turbulence. The model represents particle acceleration as a finite-memory expansion over present and delayed particle-neighborhood graphs, with each memory contribution parameterized by an equivariant message-passing graph neural network. By construction, the architecture respects the relevant physical symmetries of the problem, including permutation equivariance, Galilean invariance, and equivariance under rotations and reflections. Trained on direct numerical simulation data, the model is rolled out autoregressively and evaluated on observables that are not imposed during training. We show that memory is essential for recovering the intermittent, heavy-tailed acceleration statistics, and that the learned dynamics accurately reproduce single-particle dispersion, pair-dispersion statistics, and four-particle tetrad geometry. Our results establish a physically structured, scalable route to data-driven multi-particle simulation of turbulent transport, and a template for learning reduced dynamics of correlated, symmetry-rich particle systems.
physics.flu-dyn
This study presents a hybrid experimental and computational framework that couples in vitro 4D phase-contrast magnetic resonance imaging (4D-flow MRI) measurements with data assimilation and linear modeling to characterize the flow linear amplification mechanisms. We manufacture an idealized stenosis phantom with a cosine-shaped contraction and acquire three-dimensional (3D) mean velocity measurements at Reynolds number 3960 using 4D-flow MRI. To overcome the inherent displacement artifact, we perform data assimilation via a two-step optimization strategy using physics-informed neural network (PINN). This approach first corrects measurement artifacts before extracting the unknown mean pressure and eddy viscosity fields. The RANS-compatible mean flow then serves as the base state for global linear stability analysis (LSA) and resolvent analysis. The global LSA reveals stationary eigenmodes located in the recirculation bubble that exhibit a positive growth rate for azimuthal wavenumbers m=2 and m=3. The forced dynamics of this eigenmode dominates the low-frequency dynamics. Resolvent analysis identifies a broadband pseudo-resonance associated with the convective instability of the separated shear-layer, with maximal amplification for m=0. This methodology demonstrates how integrating sparse experimental MRI data with physics-based modeling enables the identification of mean fields and coherent structures. By leveraging the capabilities of 4D-flow MRI to non-invasively measure 3D velocity fields without requiring physical or optical access, this approach is a first step in the application of linear analysis to cardiovascular flows.
physics.flu-dyn
Whether physics foundation models can be usefully deployed on laboratory experiments remains an open question for scientific machine learning (ML). We test this question on the Rayleigh-Taylor instability (RTI), a ubiquitous and demanding fluid instability seen from tabletop flows to supernova explosions, in which small perturbations at a density interface grow into chaotic, multiscale mixing as a lighter fluid accelerates into a heavier one. Standard ML models struggle with RTI, and despite over a century of theoretical, numerical, and experimental work, it carries an unresolved discrepancy between simulation and experiment: the late-time mixing growth rate, , measured in most laboratory experiments ( 0.06-0.07), is roughly three times the value from idealized direct numerical simulations (DNS, 0.02). The gap's origin remains debated. These properties make RTI a stringent test for a question that matters well beyond RTI: can foundation models trained only on simulations generalise to sparse, messy, and noisy laboratory settings? We finetune Walrus, a foundation model for continuum dynamics, on three or fewer DNS realizations and recover key RTI physics over long rollouts. Applied zero-shot to sliding-barrier laboratory data, the finetuned model leaves the DNS-like regime and enters the observed growth band, having never seen a single experimental sample. These results provide independent, data-driven evidence that initial conditions play a crucial role in the longstanding sim-experiment gap in . The model also generalises zero-shot to stable stratification, a buoyancy regime absent from training, correctly slowing mixing-layer growth. Together, our results show that foundation models can generalise well beyond their training data, predicting laboratory behavior and unseen physical regimes, opening new ways to probe longstanding simulation-experiment gaps.
cs.LG
We propose a method combining Multi-Agent Deep Reinforcement Learning (MARL) and eXplainable Deep Learning (XDL) to reduce drag in wall-bounded turbulent flows. Taking as a baseline the results of training agents directly targeting wall-shear stress and opposition control, three SHAP-guided approaches are compared. In the first, the reward is computed from SHAP attributions of a U-net predicting the future velocity field; in the second, from SHAP attributions of a U-net predicting the skin-friction coefficient; in the third, from a combination of SHAP attributions of two U-nets predicting the skin-friction coefficient and the wall pressure fluctuations, respectively. The combined SHAP strategy based on skin-friction coefficient and wall-pressure fluctuations achieves the best overall performance, achieving a DR of 34.44% and a NES of 34.01% with only 0.43% normalized input power. Relative to opposition control, drag reduction and net energy saving increase by 49.41% and 48.52%, respectively. Compared with the direct wall-shear-stress baseline, the proposed strategy simultaneously improves performance while reducing the normalized actuation cost from 5.90% to 0.43%. Analysis of the results reveals that the energetically efficient policy is consistent with pressure-gated actuation, activating predominantly at near-zero wall pressure, and operates on a temporal timescale comparable to the lifetime of the near-wall turbulent structures.
physics.flu-dyn
Turbulence is ubiquitous in engineering and science, yet direct simulation is prohibitively expensive. The Reynolds-averaged Navier-Stokes (RANS) equations provide savings exceeding ten orders of magnitude but introduce unclosed terms (the closure problem). Offline-trained machine-learning (ML) closures suffer distribution shift in predictive simulations, while ML methods that bypass the governing equations struggle to generalise from scarce high-fidelity data. We develop a physics-derived deep learning closure model for RANS, the Deep Algebraic Reynolds Stress Model (DARSM), which can be trained on small datasets and accurately generalise across Reynolds numbers, to unseen geometries, and to different flow regimes. A neural network maps flow invariants to empirical parameters in an implicit algebraic Reynolds stress equation, derived from the Reynolds stress transport equations under the weak-equilibrium assumption, imposing physics-based structure on the ML closure. End-to-end optimisation through the governing PDEs and the coupled implicit closure eliminates distribution shift, but both unrolled and implicit automatic differentiation fail on the stiff coupled solver. We derive adjoint equations that exploit the solver's implicit-explicit structure for efficient optimisation. On canonical square-duct and periodic-hill benchmarks, DARSM reduces average test velocity error over baseline RANS by - across Reynolds number, geometries, and flow regimes, with peak case-level reductions of . The model trained on attached, anisotropy-dominated flows (square duct) accurately generalises without retraining to separated flows (periodic hills), a regime change in the underlying physics. DARSM also outperforms five established ML methods: offline training, tensor-basis neural networks, field-inversion machine learning, DeepONets, and physics-informed neural networks.
cs.CE
This work presents an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user-friendly methodology applicable to a wide range of problems, from data assimilation to closure modeling. Our approach combines a baseline differentiable PDE solver, which predicts the state w from the nonlinear system , with a generic additive, parametrized, and differentiable correction , with trainable parameters . We show how to optimize phi within a fully differentiable Python workflow by reformulating the PDE as an implicit layer, enabling its integration into arbitrary objective functions, while leveraging PyTorch's automatic differentiation graph. The method is demonstrated on the Reynolds-Averaged Navier-Stokes equations for compressible flows, where the closure term, or a portion of it, is modeled using trainable parameters or a Neural Network. The first application considers the 2D NASA Wall-Mounted Hump test case, where a production-term parameter is optimized against time-averaged LES data. A second application is carried out on the VKI LS-59 turbine blade, where the Spalart-Allmaras eddy viscosity field is reconstructed through the optimization of a trainable spatial field. A dataset is generated starting from the VKI LS-59 turbine blade geometry using the differentiable BROADCAST solver with the Spalart-Allmaras turbulence model. The results highlight the flexibility of the framework, showing its applicability beyond turbulence modeling to a broader class of physics-informed PDE-constrained problems with data-driven components.
physics.flu-dyn
Long-horizon prediction of three-dimensional (3D) wall-bounded turbulence with machine-learning methods remains a challenging task, due to the rapid accumulation of autoregressive errors and the substantially computational cost. To address these challenges, we present a hybrid machine-learning framework, in which a channel-time-attention Swin-UNet (CTA-Swin-UNet) and a multi-time-scale fusion correction (MTFC) strategy are developed to predict the turbulent flow fields in a wall-parallel plane, with affordable computational cost. Then, 3D flow fields are reconstructed via a resolvent-based spectral linear stochastic estimation (SLSE), rooting from the predicted planar flow. Results show that the CTA-Swin-UNet outperforms the baseline models (LSTM, FNO and traditional Swin-UNet) in both single-step prediction and autoregressive rollouts, indicating the effectiveness of introducing the CTA module into the Swin-UNet architecture. At the same temporal interval, the CTA-Swin-UNet remains stable for approximately 150 rollout steps, while the baseline models fail within 20 to 50 rollout steps. After introducing the MTFC strategy, a longer horizon upto 300 steps is achieved. Using the resolvent-based SLSE reconstruction further recovers the 3D flow structures and energy spectral distributions from the predicted planar inputs, which demonstrates that the proposed framework provides an effective and computationally efficient approach for long-horizon autoregressive prediction of 3D wall-bounded turbulence.
physics.flu-dyn
Computational fluid dynamics (CFD)-driven machine learning frameworks based on symbolic regression offer a promising pathway for turbulence model discovery, but are often hindered by numerical instability, residual stagnation, and non-physical model behavior during training. In particular, realizability, which is rarely enforced explicitly during model development, remains a critical yet overlooked requirement, especially for accurate wake prediction. In this work, a residual- and realizability-filtered CFD-driven framework is proposed to enhance both efficiency and robustness within a gene expression programming (GEP) paradigm. The method integrates two residual-based filtering criteria along with a barycentric-map-based realizability constraint directly into the CFD solution loop, enabling early identification and rejection of unstable and non-realizable candidate models. This reduces unnecessary computational effort while guiding the search toward physically admissible solutions. The proposed approach achieves a 42.3% reduction in computational cost relative to the baseline CFD-driven GEP framework and reduces non-realizable models at convergence from 58.4% to 1.7%. The framework is trained on a canonical cylinder wake. The resulting models enhance mean wake prediction and remain realizable across training and test cases, with robust generalization to diverse geometries and operating conditions, including a rectangular cylinder, an airfoil, and an axisymmetric body. The study further provides insights into realizable model statistics, coefficient trends, and conditions governing physically consistent wake behavior. These results demonstrate that incorporating realizability and stability constraints within CFD-driven learning enables efficient and physically consistent turbulence model discovery, offering a scalable pathway toward reliable data-driven closure development.
physics.comp-ph
Manifold-based combustion models decrease the cost of turbulent combustion simulations by projecting the thermochemical state onto a lower-dimensional manifold, allowing the thermochemical state to be computed separately from the flow solver. The solutions to the manifold equations have traditionally been precomputed and pretabulated, but this results in large memory requirements and significant precomputation cost even for simple models. One approach to alleviate the memory requirements is to use In-Situ Adaptive Manifolds (ISAM), which only stores solutions that are encountered during a simulation in a database built with In-Situ Adaptive Tabulation (ISAT). Even with ISAM, as the manifold complexity increases, the memory requirements can still grow too large. Another approach to reduce memory of these databases are machine learning methods, for they represent functions in a highly memory-compact manner. However, current implementations of these methods require the pregeneration of training datasets with little knowledge of the states present in a simulation. This work develops the Neural In-Situ Adaptive Manifolds (Neural-ISAM) method, which is designed to address the drawbacks of both adaptive tabulation and machine learning methods, and leverage their benefits by coupling neural networks to manifold databases on-the-fly. ISAM databases are built via ISAT, which stores the manifold solutions in a binary tree, and Neural-ISAM periodically searches this tree to identify regions that can be pruned. Neural networks are trained on the candidate regions, and these portions of the binary tree are then replaced by the trained neural network, reducing the memory requirements of the database. Neural-ISAM memory usage, computational performance, and accuracy is evaluated in LES of two turbulent flames with increasing manifold model complexity: Sandia Flame D and the Sandia Sooting flame.
cs.LG
High-fidelity experimental characterization of turbulent premixed flames remains limited by the cost and complexity of advanced diagnostics, particularly under elevated pressures and intense turbulence where measurements of coupled flame morphology and burning dynamics are sparse. Here, we develop a hierarchical multi-fidelity neural network framework (MuFiNNs) to address this challenge by integrating sparse high-fidelity experimental data with structured low-fidelity representations encoding dominant physical trends. The framework combines hierarchical low-fidelity construction with nonlinear multi-fidelity correction to learn coupled geometric and reactive flame behavior while recovering discrepancies that simplified models alone cannot capture. The methodology is applied to expanding turbulent premixed flames to predict three-dimensional flame wrinkling dynamics and turbulent mass burning velocity across varying fuels, pressures, and turbulence intensities. Using experimentally informed low-fidelity trend models with sparse high-fidelity measurements, MuFiNNs accurately reconstruct observed flame behavior, enable interpolation across unseen operating conditions, and demonstrate robust extrapolation beyond the training domain. Importantly, the framework remains effective in noisy, weakly structured, or experimentally inaccessible regimes where conventional data-driven approaches often fail. These results show that hierarchical multi-fidelity learning provides a scalable and physically grounded strategy for predictive combustion modeling in data-limited regimes. More broadly, this work establishes multi-fidelity scientific machine learning as a practical framework for extracting physically meaningful predictive models from sparse experiments, particularly for instability-dominated and turbulence-sensitive reactive flows where high-fidelity data acquisition is demanding.
physics.flu-dyn
If sufficient training data are available, neural networks are attractive for representing missing physics in simulations, such as sub-grid scales in the coarse-mesh particle-turbulence system we consider. Physical constraints are known to both increase performance and reduce the need for data; we use the complete physics represented in the discretized governing equations as a constraint. Two-way coupled particles in two-dimensional turbulence provide a sufficiently complex system to assess effectiveness for various training data, all constructed from well-resolved simulations, in cases intentionally degraded to assess robustness. Surprisingly, using the full space-time data actually hinders model effectiveness. Instead, training that targets only spectra – hence, neglecting phase information – provides better closures, which is related to the well-known success of non-dissipative discretizations for simulating turbulence. It is found that some of the missing physics that lead to preferential particle concentration errors are fundamentally stochastic on the coarse mesh and therefore uncorrectable by the basic approach; a learning formulation is introduced for a Langevin-type closure to correct this. Most importantly, training just for particle kinetic energy – without any direct input from the flow field – also yields effective sub-grid-scale stress models. This holds true even if noise is added to the particle data, if only a sub-sample of particles are used, or if only one component of the particle velocity is used. In sum, these results show a path for inferring sub-grid-scale physics based just on particle data from experiments.
physics.flu-dyn
Adding flexible polymers to a Newtonian solvent confers complex properties to the resulting solution. The additional complexity substantially increases the computational cost of numerical simulations, which often makes them prohibitively expensive. Here, we propose hybrid reduced-order models to accelerate simulations of viscoelastic turbulent jets. The model combines modal decompositions with deep networks: we use proper orthogonal decomposition to obtain a compact representation of the data, and a neural network is trained to predict the mode coefficients in the low-dimensional space. Results show that the hybrid model effectively captures the long-term behavior of the viscoelastic jet, that we demonstrate by computing relevant statistics of the jet. While small models are capable of predicting large-scale dynamics more than one-step at a time, thus facilitating greater accelerations, larger models are mandatory for forecasting smaller-scale dynamics, with skip connections the most effective strategy for deeper and generalizable models. The proposed methodology underpins the potential of hybrid approaches for compact and robust reduced-order models of viscoelastic turbulent jets.
physics.flu-dyn
The differentiable physics paradigm may be leveraged as an a-posteriori approach for discovering turbulence closure models by embedding a neural network parameterization directly inside the solver and optimizing it given potentially sparse target data. This addresses a key limitation of a-priori learning where direct numerical simulation (DNS) data is used to approximate the subgrid stress with the assumption of a low-pass filter. Closures trained in this a-priori manner frequently lead to unstable deployments due to the mismatch between the assumed filter and the effect of numerical discretizations and coarse-graining. In comparison, while typically stable during deployment, a-posteriori learning incurs high computational costs due to the need to backpropagate through a large eddy simulation (LES) solver. Furthermore, a-posteriori methods are challenging to apply broadly since they require significant modification of existing solvers. Finally, both approaches are limited when generalization is desired across different numerical schemes with their implicit filtering characteristics. In this work, we present a deep-learning approach for turbulence closure modeling built on the continuous data assimilation framework. Our approach enables the a-priori training of closures using sparsely observed DNS data without modifying or differentiating through the LES solver, while preserving stability during deployment for the recovery of invariant statistics. We focus on the model's ability to adapt to different discretizations by explicitly conditioning it on the numerical scheme. We use two- and three-dimensional canonical cases to test our framework and show that the learned correction systematically tracks the discretization error of the coarse solver.