PaperlayerView exact revision on arXiv

Source-linked AI summary

Enhancing generalization in endwall film cooling prediction: Incorporating the superposition principle into transformer-based neural operators

Qineng Wang, Liming Song, Tianyuan Liu, Zhendong Guo

arXiv:2609.31633v1cs.LGphysics.comp-phphysics.flu-dyn

TL;DR

Endwall film-cooling prediction must handle complex flows and layouts with hole counts outside the training range. The paper proposes SDNO, combining SDF encoding and neural operators through trainable superposition. It reports a 0.0048 K error for 10-hole samples, a 79% reduction relative to the fully supervised baseline.

  • Problem

    Endwall film-cooling prediction is challenging in complex flows, and data-driven models struggle to generalize to hole counts outside their training data.

    Source: §1 · p. 2

  • Method

    SDNO uses SDF layout encoding, a Transformer-based operator to predict sub-layout fields, and a trainable FNO to combine them by superposition.

    Source: §1 · p. 4, §3.2 · p. 9, §4.1 · p. 13

  • Results

    0.0048 K average absolute temperature error for 10-hole validation samples, a 79% reduction relative to the fully supervised baseline.

    Source: §6 · p. 27

  • Takeaways & Limitations

    The study concludes that superposition-based prediction can extrapolate to unseen Pak-B endwall layouts containing up to 20 holes.

    Source: §6 · p. 27

    Training still relies on high-fidelity CFD data; future work could add multi-fidelity data, physical constraints, and operating-condition and hole-geometry inputs.

    Source: §6 · p. 28

Abstract

from arXiv · show

In this study, a physics-enhanced neural operator framework is proposed to enhance the generalization prediction ability of the cooling layout of a turbine endwall with variable number of film holes. Specifically, inspired by the film cooling superposition principle, we propose a film cooling prediction model, namely superposition-based deep neural operator (SDNO), that divides the endwall temperature field prediction into two stages. In the first stage, the cooling layout of a turbine endwall is divided into several sub-parts with randomly assigned film holes, and a Transformer-based neural operator network, namely Calculate Net, is designed to predict the temperature field of each sub-part. Then, in the second stage, another neural operator network, i.e., Super Net, is trained to combine the temperature fields predicted by Calculate Net for each sub-part and obtain the superposed temperature field of the full cooling layout. Additionally, instead of directly taking the film cooling contours as pixel plots, a signed distance function (SDF) which is sensitive to the variable locations of cooling holes, is designed to encode the location information of cooling holes. Furthermore, the proposed endwall film cooling prediction model is trained with the samples that changing the number of film holes from 1-5 with variable locations. Then, the trained prediction shows excellent generalization prediction ability, which can accurately predict the film effectiveness of the cooling layout with 10-20 film cooling holes that are unseen in the training samples. The proposed SDNO also improves prediction accuracy relative to the fully supervised baseline. With the above, the effectiveness of our proposed prediction model has been well demonstrated.

1. Introduction

Endwall film-cooling prediction is difficult in complex flows, and data-driven models struggle to generalize to unseen hole counts without impractically large datasets. This study combines the film-cooling superposition principle with neural operators to improve prediction accuracy and extrapolation.

1. Introduction

  • Complex endwall vortices and pressure gradients make film-cooling prediction nonlinear, while CFD requires substantial time and computational resources.Surrogate models can be faster, but traditional ones predict a scalar cooling effect rather than local temperature distributions.Source: §1 · p. 2
  • Data-driven models can lose accuracy on unseen hole counts, while building general-purpose models requires datasets that may be impractical for engineering.The introduction contrasts this with physical models that adapt to varying hole numbers.Source: §1 · pp. 3–4
  • SDNO combines the film-cooling superposition principle with neural operators in a decomposition-computing-superposition strategy to improve prediction accuracy and extrapolation.The strategy represents the physical principle as a neural operator with trainable parameters.Source: §1 · p. 4
  • The proposed contributions include SDF encoding of small geometric features, integration of superposition into the operator architecture, and a tailored training strategy.Earlier deep-learning studies explored film-cooling distributions using approaches including CNNs, FNO, and UNet.Source: §1 · pp. 3–4

2. Problem Setup

The study uses a high-fidelity Pak-B endwall simulation and parameterized hole layouts to prepare a dataset for assessing generalization to layouts with unseen hole counts.

2.1. Numerical simulation and validation

  • A coupled heat-transfer simulation supplies Pak-B endwall temperature fields for varying film-cooling layouts, with main flow entering above and coolant injected from the plenum.The process includes both gas-film coverage and heat transfer.Source: §2.1 · p. 5, §2 · p. 5

2.2. Film-cooling holes layout generation

  • The layout parameterization generates hole positions, maps them to endwall coordinates, and sets injection angles while keeping hole diameter at 4 mm and inclination at 30 degrees.The minimum relative distance between holes is constrained to exceed 0.15 to avoid interference during geometric modeling.Source: §2.2 · p. 7

2.3. Data preparation

  • The dataset separates lower-hole-count training layouts from higher-count validation layouts to test extrapolation beyond the training range.The study defines extrapolation generalization as adapting to scenarios beyond the training dataset.Source: §2.3 · pp. 7–8
  • The dataset contains 2,730 samples: 2,000 training samples for 1, 2, 3, and 5 holes, 700 validation samples across counts, and 30 fine-tuning test samples for higher counts.There are 500 training samples per trained hole count and 110 generated samples for each of the 10-, 15-, and 20-hole configurations.Source: §2.3 · p. 8

3. Fully Supervised training and Empirical Superposition Principle

The section evaluates fully supervised prediction and classical film-cooling superposition for endwall cases, including extrapolation beyond training conditions. It also describes SDF preprocessing and a Transformer-based temperature-field predictor.

3.1. Data preprocessing

  • SDF preprocessing records each pixel’s distance to the nearest hole edge, addressing jagged boundaries and information voids from binary encoding of small holes.Binary encoding marks inside and outside hole locations as 0 and 1, respectively.Source: §3.1 · p. 9

3.2. Establishment and Training of Fully Supervised Network

  • The Transformer-based neural operator predicts endwall temperature from an SDF channel and two spatial-coordinate channels on a 128 × 128 grid.Its architecture uses three Galerkin-based simplified linear-attention layers, residual connections, and multilayer perceptrons.Source: §3.2 · pp. 9–10
  • Outside the training range, fully supervised prediction errors rise 20 to 100 times, and errors also increase as hole counts grow at the same sample size.Within the training set, the model shows high-precision prediction.Source: §3.2 · p. 10, Figure 5 · p. 12

3.3. Film cooling superposition Principle

  • Classical superposition combines individual-hole temperature fields, but is limited in complex flow and heat transfer; on the Pak-B endwall, Seller predicts lower temperatures than CFD.The traditional formula attributes all cooling to gas-film coverage, underestimating the superposed temperature.Source: §3.3 · pp. 11–12, Figure 6 · p. 12

4. Deep Operator Architecture Integrating Superposition Principle

The SDNO combines a generalized film-cooling superposition principle with calculation and superposition networks. Its training strategy decomposes layouts into subsamples, predicts their temperature fields, and combines them into a full-layout prediction.

4.1. Generalization and Concretization of the Film cooling Superposition Principle

  • The generalized superposition operation combines fields over continuous subfields and is designed to retain commutative and associative properties.The formulation removes assumptions of explicit direct computation and point-local computation.Source: §4.1 · p. 13
  • SDNO pairs a Transformer-based calculation network with an FNO superposition network to predict and combine temperature fields.The superposition network takes two normalized physical fields and reconstructs one temperature field.Source: §4.1 · p. 13, §4.2 · p. 14

4.2. The training strategy of superposition neural network

  • In superposition mode, layouts are decomposed into subsamples, calculated separately, then combined through pairwise superposition to produce one final temperature field.The number of subsamples is N = 2^tsuper; Figure 8 illustrates decompositions of a 10-hole sample.Source: §4.2 · pp. 14–15, Figure 8 · p. 16
  • Both networks are updated during superposition training, and the authors report improved prediction accuracy and generalization; multiple prediction paths also reduce required sample counts.The same sample can contribute through different decompositions and superposition strategies.Source: §4.2 · pp. 15, 17
  • Training concurrently uses normal and superposition strategies for 700 steps, then freezes the calculation network for 100 fine-tuning iterations at learning rate 0.00001.Fine-tuning uses tsuper = 1, 2, 3.Source: §4.2 · pp. 15, 17

5. Results and discussion

Superposition improves out-of-distribution endwall predictions, and fine-tuning sharply reduces spatial error across 10-, 15-, and 20-hole layouts. The results also identify uniform hole allocation at inference and a slight in-distribution accuracy trade-off.

5.1. Joint Training Results of the Computational Network and the Superposition Network

  • Joint training uses random hole decompositions to enrich sample combinations and improve the computational network, but its OOD accuracy alone remains below engineering requirements.The reported accuracy improvement excludes single-hole samples.Source: §5.1 · p. 18, Figure 9 · p. 18
  • Uniformly distributing holes among inference sub-samples minimizes OOD prediction errors, so the study adopts this fixed allocation strategy.The authors contrast fixed, uniform inference allocation with random training allocation used to increase diversity.Source: Figure 10 · p. 19, §5.1 · p. 19
  • SDNO reduces 10-hole prediction error by 79% versus fully supervised training on the same dataset.With one superposition invocation, samples within twice the training-set hole count achieve errors below 0.0050K; the 10-hole example reaches 0.0048K when training uses at most five holes.Source: §5.1 · p. 19, Figure 11 · p. 20
  • SDNO captures upstream low-temperature regions and downstream cold films, including overlapping low-temperature regions formed by several holes near the suction surface.The authors report accurate capture despite the coupled flow associated with overlapping hole influence.Source: §5.1 · p. 21

5.2. Fine-Tuning Training Results of the Superposition Network

  • Fine-tuning slightly decreases in-distribution accuracy while notably improving OOD accuracy; 20-hole fine-tuning samples also improve 10-hole predictions.The reported comparisons use a superposition count of two.Source: §5.2 · pp. 21, 23
  • With three superposition levels, fine-tuned MRVE is 1.084%, 1.383%, and 2.756% versus baseline 41.884%, 37.297%, and 33.697% for 10-, 15-, and 20-hole validation layouts.Each OOD configuration therefore has a relative MRVE reduction exceeding 90%.Source: Table 5 · p. 24, §5.2 · p. 24
  • Fine-tuning yields maximum temperature-field prediction errors below 0.5K for each illustrated 10-, 15-, and 20-hole sample.The final fine-tuning step removes the magnitude bias in the axial effectiveness curve after the superposition model estimates its trend with slight deviations.Source: §5.2 · p. 24, Figure 16 · p. 25, Figure 17 · p. 26

6. Conclusions

The superposition model predicts endwall temperature fields for unseen layouts with up to 20 holes, outperforming a fully supervised baseline. It reduces sample requirements, but training still relies on high-fidelity CFD data.

6. Conclusions

  • 0.0048 K average absolute temperature error on 10-hole validation samples, a 79% reduction relative to the fully supervised baseline.The superposition model’s MRVE values were 1.084%, 1.383%, and 2.756% for 10-, 15-, and 20-hole samples, versus baseline values of 41.884%, 37.297%, and 33.697%, respectively.Source: §6 · p. 27
  • The model integrates film-cooling superposition with a deep neural operator; SDF preprocessing represents hole geometries, a Transformer predicts sub-layout fields, and a local FNO combines them.It was pre-trained on layouts with 1, 2, 3, and 5 holes and evaluated on layouts with up to 20 holes.Source: §6 · p. 27
  • Fine-tuning with a small additional dataset further improves extrapolation, and the reported OOD configurations achieve more than 90% relative MRVE reduction.The approach reduces sample requirements, though training still relies on high-fidelity CFD data.Source: §6 · pp. 27–28
  • Future work could combine multi-fidelity data and additional physical constraints, and extend inputs to operating conditions, hole diameter, and injection angle.These extensions would support assessment of more varied turbomachinery cooling layouts.Source: §6 · p. 28
Loading 2609.31633v1…