Subgrid Selection: Methods and Applications

• Subgrid selection is the process of identifying and modeling fine-scale phenomena below a simulation’s resolution by leveraging combinatorial, physical, and data-driven methods.
• Mathematical frameworks and physical parameterizations enable precise estimation of unresolved interactions, as seen in grid-based combinatorial optimization and turbulent flow simulations.
• Data-driven and algorithmic techniques, including deep learning and importance sampling, enhance simulation fidelity by adapting subgrid models to varying scales and constraints.

Subgrid selection refers to the mathematical, algorithmic, or physical strategies by which unresolved or fine-scale features—below the resolution of a computational grid, model, or measurement device—are identified, modeled, or incorporated into numerical simulations and data-driven systems. The motivation for subgrid selection arises in diverse domains as a consequence of scale separation: many phenomena (astrophysics, fluid dynamics, machine learning, combinatorics) exhibit dynamics or structures much smaller than the chosen resolution, yet these subgrid effects can have non-negligible feedback on the resolved scales. Approaches to subgrid selection vary according to discipline, ranging from combinatorial optimization to physical parameterization, stochastic modeling, machine learning, and algorithmic feature selection.

1. Mathematical and Combinatorial Foundations

Many subgrid selection problems are fundamentally combinatorial, involving the enumeration or optimization of discrete subsets of a structured grid under local constraints. In "Selections Without Adjacency on a Rectangular Grid" (Siehler, 2014), subgrid selection is formulated as counting the number of ways to choose $k$ squares from an $m\times n$ rectangular grid such that no two selected sites are immediately adjacent (either horizontally or vertically). The number of such selections, $T(m, n; k)$, is central to applications in resource placement, interference-avoiding sensor deployment, statistical design, and communication networks.

The paper provides an explicit formula for the $2\times n$ case:

$$T(2, n; k) = \sum_{r=1}^k 2^r \binom{k-1}{r-1} \binom{n-k+1}{r},$$

where enumerating over "runs" of consecutive selected columns and their possible top/bottom configurations captures the combinatorial complexity. For fixed $k$, $T(2, n; k)$ is a polynomial in $n$ of degree $k$, with an asymptotic form dominated by the term $(2^k / k!) n^k$.

Crucially, a unimodality theorem is proven: $T(2, n; k)$ attains its maximum at $k = \lceil n/2 \rceil$, reflecting that maximal diversity under adjacency constraints is achieved by selecting approximately half of the subgrid sites. Extensions to $3\times n$ grids and recurrence identities further generalize the mathematical underpinnings of subgrid selection in discrete settings.

2. Physical Parameterization and Subgrid Modeling in Simulations

In fields such as cosmology and fluid dynamics, subgrid selection defines how unresolved physics (e.g., supermassive black hole (SMBH) seeding and merging, turbulence, or eddy transport) are integrated into coarse-grained simulations.

For cosmological simulations with SMBHs (1006.2879), subgrid selection involves:

• Initial Seed Prescription: Choosing either a constant SMBH seed mass ($10^5\ M_\odot$) or one based on an empirical halo mass relation,
• Merger Criteria: Merging SMBHs either instantly when their host halos merge, or only when proximity and kinematic criteria are met,
• Frequency of Model Application: Applying subgrid operations every timestep versus at logarithmically spaced intervals in scale factor.

The choice among these subgrid strategies significantly alters emergent properties such as merger rates, cosmic mass density, and scaling relations (e.g., $M$-$\sigma$). Strikingly, with appropriately chosen subgrid rules, observed $z\sim0$ scaling relations between SMBH mass and halo properties can be replicated without explicit modeling of accretion or feedback, underscoring the profound impact of subgrid selection.

In turbulent flow simulations, subgrid-scale (SGS) selection is manifested in the construction and parameterization of SGS models, e.g., for Large Eddy Simulation (LES). A spectrum of approaches is available:

• Analytical SGS Closures (Cinlar model (Kara et al., 2015)): The subgrid stress tensor is decomposed into Leonard, cross, and Reynolds stresses, with the Reynolds stress analytically represented from a vortex-based stochastic field, parameterized by resolved strain rate and eddy statistics.
• Physically Consistent Model Selection (Silvis et al., 2016): Model frameworks are assessed and derived to preserve Navier–Stokes symmetries, wall scaling, realizability, and correct dissipation properties. Selection is subject to compatibility with constraints governing Galilean invariance, near-wall vanishing, and positive-definite Reynolds stress.
• Flow-Dependent Subgrid Characteristic Length (Trias et al., 2017): For anisotropic or unstructured grids, a new definition

$$\Delta = \sqrt{\frac{(\Delta_\text{tensor} G) : (\Delta_\text{tensor} G)}{G : G}}$$

is adopted, enabling SGS models (most notably eddy-viscosity closures) to reflect local mesh and flow features, mitigating anisotropy artifacts in subgrid selection.

3. Data-Driven and Machine Learning Approaches

Recent advances employ machine learning for subgrid selection either for statistical closure modeling or feature-space dimension reduction.

• Deep Learning-Based Parameterizations (Pawar et al., 2019, Kang et al., 2022): Feedforward networks, convolutional architectures, and hybrid CNN-ANN methods are trained to learn mappings from resolved-scale quantities (velocities, gradients, resolved stresses) to SGS stress tensors or dissipation rates, often outperforming traditional models (e.g., dynamic Smagorinsky) in accuracy and computational efficiency. Notable is the demonstration that including richer input features (neighboring stencil values or resolved stress tensors) enhances alignment with "true" SGS stress, and carefully normalized networks robustly generalize across Reynolds numbers and spatial resolutions.
• Non-local Parameterization of Atmospheric Subgrid Processes (Wang et al., 2022): Subgrid predictions in climate models benefit from incorporating non-local (spatially adjacent column) inputs, as many subgrid atmospheric processes (e.g., organized convection) span multiple grid boxes. Neural networks trained on 3×3 supercolumns outperform single-column models—particularly for momentum transport—by exploiting horizontal gradients and divergence structure. Layer-wise relevance propagation reveals that non-local wind information is critical even when vertical velocity is provided explicitly.
• Stochastic and Generative Models (Perezhogin et al., 2023): Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) learn the conditional distribution of subgrid forcing $S$ given resolved state $\overline{q}$, capturing spatially correlated stochasticity (not mere pointwise white noise). This approach significantly improves the statistical fidelity and stability of parameterized climate simulations at very coarse resolutions, compared to deterministic or local-noise closures.

4. Algorithmic and Optimization Methods for Subgrid Selection

In computational frameworks, subgrid selection is also addressed through dynamic algorithmic schemes that prioritize computational resources or enforce constraints:

• Gradient-Guided Subgrid Feature Selection in BoostCNN (Fang et al., 30 Jul 2025): Within boosting ensembles of convolutional neural networks, subgrid selection involves dynamically selecting informative spatial subgrids (subsets of rows and columns) by computing per-pixel importance via the gradient of the loss with respect to each input. This subgrid is used as the input for subsequent weak learners, focusing learning on regions contributing most to residual error. Boosting weights are embedded into network training via a least squares loss (main effect: $g^*(x_i) = \beta \cdot w(x_i, z_i)$), and importance sampling accelerates convergence by concentrating updates. This methodology yields faster convergence, improved predictive accuracy, and memory/computation savings, as weaker learners are not required to process the full image.
• Optimal Clipping for Structural SGS Models (Prakash et al., 2022): To control numerical instability from excessive backscatter in structural SGS models, the paper introduces "optimal clipping," a constrained minimization that minimally adjusts predicted SGS tensors to enforce non-positive local energy transfer. This maintains high correlation with the resolved SGS structure, improves stability, and obviates the blunt over-damping of standard clipping approaches.

5. Subgrid Modeling in Specialized Physical Contexts

In some disciplines, subgrid selection is coextensive with the choice of physical closure at scales unresolvable in simulations but essential for macroscopic dynamics.

• Neutrino Quantum Kinetics in Astrophysics (Nagakura et al., 2023, Johns, 26 Jan 2024): Fast neutrino flavor oscillations in core-collapse supernovae and neutron-star merger simulations occur on subgrid scales. The BGK subgrid model replaces the high-frequency oscillatory terms with a relaxation toward an asymptotic state, parameterized by a relaxation time $\tau_a$ and an asymptotic (mixed) distribution $f^a$:

$p^\mu \partial_\mu f = -\frac{1}{\tau_a}(f - f^a).$

Moment-based formulations integrate this term into standard energy and flux equations, preserving compatibility with classical transport schemes. The more rigorous "miscidynamic" approach defines a self-consistent local mixing equilibrium sufficing both local and global constraints, allowing for systematic nonadiabatic corrections analogous to hydrodynamic viscosity.

• Subgrid Iceberg-Sea Ice Coupling (Mehlmann et al., 27 Jul 2025): In climate models, the stationary fast ice fraction—determined by anchoring from subgrid icebergs—cannot be directly resolved. A hybrid method models icebergs as Lagrangian particles that exert localized drag on the Eulerian sea-ice continuum via a Stokeslet-based Green’s function kernel integrated using finite elements. This approach preserves stability, exhibits bounded energy, and reproduces realistic fast-ice formation, polynya openings, and spatial sea-ice features absent in standard models.

6. Implications, Limitations, and Future Directions

Subgrid selection approaches directly influence the physical fidelity, stability, computational efficiency, and interpretability of simulations and learning architectures. The domain-specific choice of selection method (analytical, empirical, algorithmic, stochastic, or data-driven) must balance:

• Physical/theoretical consistency (symmetries, scaling, realizability),
• Numerical efficiency and stability (especially in high-dimensional or multiscale regimes),
• Capacity to match empirical observations or high-resolution benchmarks,
• Robustness to changing resolution, parameter regimes, and input statistics.

Key open challenges include reconciliation of conflicting model constraints (as in SGS turbulence models), systematic validation against empirical data across scales (e.g., comparing simulated and observed GMCs (Li et al., 2020)), and extending subgrid selection strategies (such as non-local or multiscale-learning frameworks (Otness et al., 2023)) to emerging domains and methods. Continued integration of machine learning with physical and combinatorial models promises to advance both the theoretical understanding and practical performance of subgrid selection across disciplines.