Tsinghua University · 2026

Reliable and efficient steady CFD from surrogate predictions through Newton-Krylov correction

Mingcheng Lei1, Weishao Tang1, Yufei Zhang1,2,*, Haixin Chen1

1 School of Aerospace Engineering, Tsinghua University 2 State Key Laboratory of Advanced Space Propulsion, Tsinghua University * Corresponding author

Open the Interactive Project Demo Upload or modify an airfoil geometry and run the solver-coupled workflow.
Comparison of standalone surrogate prediction, conventional cold-start CFD, and solver-coupled Surrogate-Newton CFD

Surrogate-Newton CFD uses a learned full-field prediction to enter the convergence region of the target solver, then applies Newton-Krylov updates until the native residual satisfies the solver tolerance.

Abstract

Neural surrogates can evaluate flow fields rapidly, but their reliability degrades under out-of-distribution geometries. We introduce a solver-coupled surrogate-Newton framework that treats a surrogate prediction as a high-quality initial state for Newton-Krylov correction. The learned model supplies global flow structure; the numerical solver supplies residual-based verification and terminal accuracy.

On an optimization-derived transonic airfoil benchmark, the framework reduces the median residual L2 ratio by more than seven orders of magnitude while lowering field and aerodynamic errors. In supercritical-airfoil optimization it improves online reliability with a 15.5-fold generation-level speedup over CFD, and the same principle extends to three-dimensional flying-wing configurations.

Method

Prediction supplies structure; the solver supplies acceptance.

Geometry and flow conditions are first mapped to a complete steady-flow state. That prediction is then injected into the target CFD discretization and corrected until its native nonlinear residual satisfies the prescribed tolerance.

  1. 01PredictInfer the full flow and turbulence state.
  2. 02EvaluateMeasure the native discrete residual.
  3. 03CorrectApply Jacobian-free Newton–Krylov updates.
  4. 04AcceptStop at the target solver tolerance.

Surrogate initializationÛ = S(G, θ)

Local correctionJ(Uk)ΔU = −R(Uk)

Numerical acceptance‖R(UNK)‖ ≤ ε

Optimization-derived OOD benchmark

The evaluation set is built from intermediate airfoils visited by actual transonic optimization trajectories. Its geometric shift is measured explicitly in the 26-dimensional CST representation.

Laminar training, supercritical training, and optimization benchmark airfoil populations
(a) Geometry populations. Optimization generates shapes that differ systematically from the training families.
Nearest-neighbour distance distributions for training, validation, and benchmark geometries
(b) Nearest-neighbour distance. Most benchmark geometries lie beyond the training distribution's high-percentile region.

Solver consistency under geometric shift

Newton-Krylov correction reduces both the native residual and physical errors. The individual panels below separate aggregate benchmark statistics from representative pressure-field recovery.

Residual, field, lift, and drag error distributions before and after correction
Result 1 Residual and aerodynamic errors decrease together. Terminal and staged correction shift the distributions of residual ratio, field MSE, lift error, and drag error toward the converged CFD reference.
Pressure fields and absolute errors along a geometry sequence from RAE2822 to an out-of-distribution ellipse
Result 2 A representative geometry sequence into the OOD regime. From the reference RAE2822 airfoil toward a strongly shifted ellipse geometry, Newton-Krylov correction recovers the converged pressure structure and sharply reduces shock-related error.

Reliable online aerodynamic optimization

A fixed Newton-Krylov budget transfers the offline recovery gains to iterative design, where each generation evaluates candidate geometries outside the original training distribution.

Optimization trajectories for RAE2822 and OAT15A with surrogate-only and surrogate-Newton evaluation
(a) Optimization histories. Correction brings the online estimate closer to converged CFD verification for both design tasks.

Extension to three-dimensional wings

We further evaluate the same coupling principle on flying-wing configurations with planforms deliberately shifted beyond the training distribution. Full-wing surface fields show the recovered three-dimensional structure, while spanwise sections expose the corresponding pressure-field correction in detail.

Training, validation, and out-of-distribution flying-wing planform examples and distance distributions
(a) Controlled planform shift. Five OOD groups move systematically beyond the training and validation planform support.
Full flying-wing surface pressure comparison and errors for CFD, surrogate prediction, and Newton-Krylov correction
(c) Three-dimensional surface recovery. Full-wing surface pressure maps show that NK15 restores the converged spatial structure and strongly suppresses the surrogate error.
Three flying-wing sectional pressure fields comparing CFD, surrogate prediction, Newton-Krylov correction, and their errors
(d) Spanwise pressure-field recovery. At three wing sections, NK15 reduces the surrogate pressure-field error and recovers the converged CFD structure.

Code and interactive system

The public repository contains the static project page, demo application, scheduling layer, tests, and deployment documentation. The compute-backed interactive interface runs independently on the project server.

Open the interactive demo →

BibTeX

@article{lei2026surrogate_newton,
  title   = {Reliable and efficient steady CFD from surrogate predictions
             through Newton--Krylov correction},
  author  = {Lei, Mingcheng and Tang, Weishao and Zhang, Yufei and Chen, Haixin},
  journal = {arXiv preprint arXiv:2608.04400},
  year    = {2026},
  url     = {https://arxiv.org/abs/2608.04400}
}