⚡ electrical engineering

From RANS Prediction to Hydrofoil Optimization: A Benchmark of Neural Surrogates

This paper benchmarks four neural surrogate architectures (CNN-U-Net, PINN, FNO, and DeepONet) trained on 381 CFD simulations of hydrofoils, demonstrating that while DeepONet achieves the highest field and force accuracy and PINN offers the fastest inference, all models significantly accelerate hydrofoil optimization compared to OpenFOAM, though field accuracy alone does not guarantee optimal design outcomes.

Original authors: Aarav Dixit

Published 2026-10-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Aarav Dixit

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Designing a hydrofoil, the wing-like structure that lifts a boat out of the water to reduce drag, is a game of endless trial and error. To find the perfect shape, engineers must test thousands of variations, changing the curve, thickness, and angle of the foil. For each new shape, they must run a complex computer simulation that solves the physics of water flowing around the object. These simulations are incredibly accurate but also incredibly slow; running a single test can take hours on powerful computers. Because of this, exploring the full range of possible designs is often too expensive and time-consuming. In recent years, scientists have tried to speed this up by training artificial intelligence to act as a shortcut. Instead of solving the physics equations from scratch every time, these AI models learn to predict the flow of water and the resulting forces based on patterns they have seen before. The hope is that these "neural surrogates" can find the best designs in seconds rather than hours.

However, a new study published by independent researcher Aarav Dixit reveals a critical flaw in how we currently judge these AI shortcuts. For years, the standard way to decide which AI model was the best was to check how closely its predicted water pressure matched the slow, high-fidelity computer simulations. If the AI got the pressure map right, it was considered a success. But this paper argues that getting the pressure map right does not necessarily mean the AI will help an engineer find the best boat wing. The true test of a design tool is not just how well it mimics a map, but whether it leads to the same winning design as the slow, trusted method. To find out, the researcher set up a rigorous head-to-head competition between four different types of AI models, forcing them all to solve the same design problem and then checking their final answers with fresh, high-precision simulations.

The study focused on a specific set of hydrofoils based on standard shapes known as NACA profiles, which are defined by simple numbers describing their curve and thickness. The researcher generated a database of 381 detailed computer simulations of water flowing over these foils at different speeds and angles. From this data, four distinct families of neural networks were trained to predict the flow and the forces of lift and drag. The first type, a convolutional neural network, looks for local patterns in the flow like a human eye scanning a map. The second, a Fourier neural operator, analyzes the flow by breaking it down into global waves. The third, a deep operator network, separates the description of the shape from the description of the flow conditions. The fourth, a physics-informed neural network, is constrained by the fundamental laws of fluid motion, forcing it to respect the rules of physics even when guessing.

Once trained, these models were not just asked to predict a single case; they were handed the job of finding the best possible hydrofoil shape. Each model was given the same starting points and a strict limit of forty attempts to improve the design. The goal was to maximize the ratio of lift to drag, essentially making the boat go faster with less effort. Crucially, the optimization loop ran entirely on the AI predictions; the slow, high-fidelity computer was kept off during the search to see how the AI would behave when it was the only tool available. After the AI selected its best design, the researcher ran a fresh, high-precision simulation on that specific shape to see what the real physics said. This final check was compared against a baseline where the slow computer itself performed the optimization, providing a gold standard for what the best possible answer should be.

The results showed a surprising disconnect between accuracy and utility. In terms of raw prediction error, one model, the deep operator network, was the clear winner. It predicted the pressure, lift, and drag with the highest precision on unseen test cases, followed closely by the physics-informed network. The physics-informed model was also the fastest, offering a speedup of roughly 665 times compared to the traditional simulation. However, when these models were used to actually design a new foil, the picture changed. The deep operator network and the physics-informed network both found designs that performed exceptionally well, achieving lift-to-drag ratios between 36 and 37. In contrast, the model that looked for local patterns struggled significantly, finding designs that were far less efficient. The model that analyzed global waves was stable but consistently chose designs that were not as good as the others.

Perhaps the most important finding was that the performance of these AI models was not consistent. When the researcher ran the same optimization process multiple times with slightly different starting conditions, the final designs varied. The deep operator network and the physics-informed network produced different winners depending on the run, while the local-pattern model was unstable in both its predictions and its design choices. Furthermore, the study highlighted a hidden limitation in the verification process itself. Even when the AI found a great design, the final check using the high-precision simulation showed that the results were sensitive to the resolution of the computer grid. The drag force, in particular, changed by up to 26 percent when the simulation grid was made finer, suggesting that even the "truth" used to verify the AI has its own uncertainties.

This work demonstrates that the path from a good prediction to a good design is not straightforward. A model can be excellent at guessing the pressure at every point on a wing but still fail to guide an engineer to the optimal shape. The study concludes that the field of scientific machine learning needs to move beyond simple error metrics. To truly bridge the gap between artificial intelligence and engineering, these tools must be evaluated on the quality of the designs they produce and the robustness of those designs under different conditions. The best AI is not necessarily the one that draws the most accurate map, but the one that reliably leads to the best destination.

Technical Summary: From RANS Prediction to Hydrofoil Optimization

Problem Statement
In hydrofoil design, shape optimization is an iterative process wrapped around a flow solver. Each modification to camber, thickness, or incidence typically requires a new Reynolds-averaged Navier–Stokes (RANS) solution, making even modest searches computationally expensive. While neural surrogates offer a way to bypass the solver by learning the map from geometry and operating conditions to flow fields and forces, a critical gap remains in the literature: standard benchmarks evaluate models based on field prediction errors (e.g., pressure or velocity fields) or isolated force accuracy. These metrics do not necessarily predict whether a surrogate will make the same design decisions as the high-fidelity solver when embedded in an optimization loop. The central question addressed is not merely which network minimizes field error, but which architecture preserves the specific quantities that determine a useful design and remains robust across different optimization seeds.

Methodology
The study conducts a controlled benchmark of four representative neural surrogate families on a dataset of 381 OpenFOAM RANS simulations of hydrofoils in water. The dataset includes various NACA profiles (0006–0021, 2412–2415, 4412–4421) across multiple angles of attack and Reynolds numbers.

Model Architectures:
- CNN-U-Net: A convolutional network emphasizing local multiscale structure.
- Fourier Neural Operator (FNO): A spectral method mixing information globally in Fourier space.
- Deep Operator Network (DeepONet): Separates parameter and coordinate representations using branch and trunk networks.
- Physics-Informed Neural Network (PINN): Constrained by differential equations (RANS continuity and momentum) alongside supervised loss.
Experimental Protocol:
- Data Partitioning: The dataset is split into disjoint training, validation, and final-test sets based on complete hydrofoil families to ensure no geometric overlap between sets.
- Training: All models are trained with three different random seeds on the same data split.
- Optimization Loop: Each trained surrogate drives a four-parameter hydrodynamic shape optimization (HSO) loop using the Nelder–Mead algorithm. The objective is to maximize the lift-to-drag ratio ( $L/D$ ) at a fixed Reynolds number. The optimizer queries the surrogate exclusively; OpenFOAM is not used inside the loop.
- Verification: The "winning" designs from each surrogate's optimization are validated by generating new meshes and running independent, high-fidelity OpenFOAM simulations. A direct-CFD baseline, using the same optimization protocol but with OpenFOAM as the solver, serves as the reference.

Key Contributions

End-to-End Benchmarking: Unlike prior studies focusing on field or force errors, this work evaluates surrogates through the entire design loop, from initial geometry to the final optimized shape.
Seed Robustness Analysis: The study explicitly tests the stability of optimization outcomes across different random seeds, revealing that high field accuracy does not guarantee consistent design decisions.
Simulator Verification vs. Grid Independence: The paper highlights the distinction between verifying a surrogate's selected design against a fresh CFD solve and achieving grid-independent validation, noting that drag predictions remain sensitive to mesh resolution even after optimization.

Results

Prediction Accuracy: On the untouched test families, DeepONet achieved the lowest mean errors for pressure ( $C_p$ ), lift ( $C_L$ ), and drag ( $C_D$ ). The PINN was a close second in accuracy and demonstrated the fastest inference speed.
Computational Speed: All models provided significant speedups over OpenFOAM, ranging from 384× to 665× faster inference.
Optimization Outcomes:
- DeepONet and PINN produced the highest optimized $L/D$ ratios (37.47 and 36.85, respectively) in the fresh OpenFOAM endpoint checks.
- CNN-U-Net showed instability, with high sensitivity to random seeds and lower performance in both force prediction and optimization.
- FNO was more repeatable across seeds but selected designs with lower performance compared to DeepONet and PINN.
- The direct-CFD baseline achieved an $L/D$ of 29.89, indicating that while surrogates can find superior designs, the results vary significantly by architecture and seed.
Cavitation Screening: DeepONet also outperformed other models in predicting pressure-based cavitation inception risk (F1 score of 0.900).

Significance and Claims
The paper claims that standard benchmarks based on field and force errors are insufficient for evaluating neural flow surrogates in engineering contexts. While DeepONet excels in static prediction accuracy and the compact PINN dominates in speed, the study reveals a "critical blind spot": isolated test accuracy does not reliably predict downstream design decisions or seed robustness.

The author concludes that to truly bridge machine learning and scientific engineering, surrogates must be evaluated on their operational consequences (i.e., the quality of the designs they select) and the numerical uncertainty of the verifying simulator. The work demonstrates that field accuracy, force accuracy, search quality, and simulator verification are related but distinct properties that cannot be used interchangeably. Future work is identified as extending this paradigm to three-dimensional flows and active cavitation-constrained optimization.

Technical Summary: From RANS Prediction to Hydrofoil Optimization

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