A computational study of a reacting hydrogen jet in cross-flow found that model choice should depend on the quantity being predicted. An MLP, or multilayer perceptron, was used to map injector spacing to scalar outputs such as wall heat transfer and hydrogen throughput. Parametric SINDy was used for time-dependent, field-level statistics such as Reynolds stress. The authors’ conclusion was that direct surrogates suit scalar response maps, while reduced dynamical models suit temporal evolution and field reconstruction.
The dataset contained CFD cases at five nondimensional injector spacings, µ=2, 4, 6, 8 and 10, for a reacting hydrogen jet-in-cross-flow configuration. The scalar analysis covered mean unburnt-hydrogen throughput, mean area-averaged wall heat transfer and mean bulk temperature concentration; the field analysis reconstructed Reynolds-stress profiles.
Scalar predictions had a clear trade-off
The simulated scalar responses were non-monotonic. At µ=8, normalized unburnt-hydrogen throughput was 0.542 and wall heat transfer was 0.392 MW/m². At µ=10, bulk temperature concentration was 1.077. No single tested spacing was lowest for every measure.
To build the scalar surrogate, the MLP used five hidden layers with widths 16–32–32–24–12. It was trained for 200 epochs with Adam, a normalized-output mean-squared-error loss and a 10−2 learning rate; 20 models meeting an acceptance threshold below 0.05 were retained.
The comparison baseline was PCHIP, a shape-preserving interpolation method. In leave-one-sample-out testing, one spacing was omitted at a time. Omitting an endpoint forced one-sided extrapolation, which produced negative heat-transfer predictions and errors above 100%. Aggregate error was about 34% for interpolation-only cases and 65.4% across the full comparison.
The MLP put the heat-transfer minimum near µ≈8.1 and the unburnt-hydrogen minimum near µ≈7.8; PCHIP put both at µ=8. The models therefore pointed to similar intermediate spacing, although the exact minima depended on the method. The results describe the modeled CFD relationship and do not establish that spacing caused the observed differences.
Accuracy also varied sharply by output. Aggregate leave-one-sample-out MLP relative error was 27.2% for heat transfer and 21.7% for unburnt-hydrogen throughput, compared with 1.46% for bulk temperature concentration. Every MLP temperature-concentration error was below 4%.
Fields exposed a different trade-off
For the field problem, five POD modes were retained to compress the flow description before SINDy modeled its reduced dynamics. The configuration used normalized spacing, a 21-point third-order Savitzky–Golay derivative, and cubic-polynomial and harmonic terms through K=8, with sequential thresholded ridge regression.
Hyperparameters were selected by grid search. Candidate models were screened for derivative accuracy, correlation, amplitude and stability, integrated over training spacings, and retained only when their reduced-coordinate rollouts were stable. Reported reconstructions were reintegrated with ode45 at tolerance 10−8.
That reduced temporal model increased error overall rather than lowering it. Across all spacings and locations, SINDy’s spacing-mean range-normalized RMSE was 6.7% higher than POD-only reconstruction. At x/D=0, POD’s aggregate nRMSE was 0.088, while SINDy’s was 19.3% higher, mainly because of overprediction at µ=4 and 8.
Results still depended on spacing. SINDy improved on the spacing mean only at µ=10, where its error was 0.139, a 7.1% decrease from POD; the largest degradation was at µ=8, with a 16.7% increase. The five-mode POD basis captured 92.42% of POD energy but did not fully recover downstream Reynolds-stress minima and maxima at x/D=2.
SINDy was also used to predict field-level profiles at unsimulated µ=3, 5, 7 and 9. The predictions varied along the flow: µ=3 had local minima at x/D=0 and 1; µ=5 was the minimum at x/D=2 and consistently had higher peaks; µ=7 and 9 had similar trends and magnitudes. These are model predictions rather than additional CFD cases, and independent CFD validation at those values was not reported.
The lesson is to match model to task
The evidence remains narrow: one reacting hydrogen jet-in-cross-flow configuration and five simulated spacings. The MLP ensemble spread represents variation across accepted initializations, not a calibrated predictive interval. How well either workflow would perform outside this configuration and spacing range remains open.
The authors’ reading is therefore pragmatic. MLPs fit fast scalar parameter-to-response maps, while POD–SINDy preserves information needed for temporal evolution and field reconstruction, even when its aggregate Reynolds-stress error is higher.
The document is arXiv:2608.26064v1, an arXiv preprint dated 26 Aug 2026. The authors acknowledge support from the National Energy Technology Laboratory’s University Turbine System Research Program under grant DE-FE0032080 and technical input and feedback from Solar Turbines Incorporated.
Paper data and sources
Original title: Neural-Network and Reduced-order Modeling Workflows for AI-Driven CFD: Fast Response Surfaces, Reduced Dynamics and Jet in Cross-flow Examples
Authors: Kaku E. Eduku, Pavel P. Popov, Gustaaf Jacobs
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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