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Geometry-Aware Neural Preconditioner

Registry ID: geometry_preconditioner
Categories: acceleration, geometry
Architecture: learned geometry-conditioned preconditioner embedded in an iterative numerical solver.

Method architecture

flowchart LR
    X["Discrete system<br/>A x = b"] --> G["Geometry and mesh<br/>feature encoder"]
    G --> P["Learned neural<br/>preconditioner"]
    X --> I["Iterative numerical solver"]
    P --> I
    I --> R{"Residual and<br/>convergence test"}
    R -- continue --> I
    R -- converged --> Y["Numerical solution"]

    classDef input fill:#ffffff,stroke:#167d9a,stroke-width:1.5px,color:#17324d;
    classDef core fill:#e9f8fb,stroke:#167d9a,stroke-width:1.5px,color:#17324d;
    classDef decision fill:#fff4e8,stroke:#c97932,stroke-width:1.5px,color:#5a3518;
    classDef output fill:#eef8f0,stroke:#34855c,stroke-width:1.5px,color:#173f2b;
    class X input;
    class G,P,I core;
    class R decision;
    class Y output;

The numerical residual and stopping criterion remain explicit. The learned component accelerates the solver but does not remove the solver's convergence check.

flowchart LR
    D["Matrices, meshes and<br/>solver histories"] --> AD["Canonical acceleration adapter"]
    AD --> S["CFDSample + geometry<br/>and linear-system metadata"]
    S --> C["Dataset-conditioned configuration<br/>dimension · mesh · feature width"]
    B["Matched tolerance and<br/>wall-clock benchmark plan"] --> C
    C --> H["ModelHub.load('geometry_preconditioner')"]
    H --> M["Neural preconditioner"]
    M --> E["Approved iterative-solver adapter"]
    E --> P["Converged solution and<br/>iteration history"]
    P --> V["Residual, iteration-count and<br/>wall-clock evaluation"]
    V --> F["Figures + experiment manifest"]

    classDef data fill:#ffffff,stroke:#506784,stroke-width:1.3px,color:#17263a;
    classDef config fill:#e8f6fb,stroke:#1580a0,stroke-width:1.3px,color:#17324d;
    classDef execute fill:#eef8f0,stroke:#34855c,stroke-width:1.3px,color:#173f2b;
    classDef verify fill:#fff4e8,stroke:#c97932,stroke-width:1.3px,color:#5a3518;
    class D,AD,S,B data;
    class C,H,M config;
    class E,P execute;
    class V,F verify;

Fair acceleration evidence

Compare methods at matched residual tolerance and report setup cost, iteration count, convergence failures, and wall-clock time across held-out geometries and meshes.

Value

The discrete residual and convergence criterion remain available, making this class attractive for trustworthy CFD acceleration.

Required evidence

Cross-mesh, cross-geometry, solver-setting, Reynolds-number, iteration-count, and wall-clock tests.

Reference

Lee et al., geometry-aware hybrid iterative solvers, 2025.