# Fusion-DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic and Supersonic Flows

arXiv: https://arxiv.org/abs/2501.01934 · HTML: https://mlsanity.com/paper/2501.01934/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 47 references that are in the map.
- First posted on arXiv 3 Jan 2025.

## Builds on (most cited first, 16 of 16)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024)
- [RiemannONets: Interpretable Neural Operators for Riemann Problems](https://mlsanity.com/paper/2401.08886/) (2024)
- [Derivative-enhanced Deep Operator Network](https://mlsanity.com/paper/2402.19242/) (2024)
- [Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO)](https://mlsanity.com/paper/2307.09072/) (2023)
- [Enhanced data efficiency using deep neural networks and Gaussian processes for aerodynamic design optimization](https://mlsanity.com/paper/2008.06731/) (2020)
- [MMGP: a Mesh Morphing Gaussian Process-based machine learning method for regression of physical problems under non-parameterized geometrical variability](https://mlsanity.com/paper/2305.12871/) (2023)
- [Diffeomorphism Neural Operator for various domains and parameters of partial differential equations](https://mlsanity.com/paper/2402.12475/) (2024)
- [Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2403.08220/) (2024)

## Built on by (most cited first, 13 of 13)

- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [ArGEnT: Arbitrary Geometry-encoded Transformer for Operator Learning](https://mlsanity.com/paper/2602.11626/) (2026)
- [Shock-Aware Physics-Guided Fusion-DeepONet Operator for Rarefied Micro-Nozzle Flows](https://mlsanity.com/paper/2510.17887/) (2025)
- [Convolutional-neural-operator-based transfer learning for solving PDEs](https://mlsanity.com/paper/2512.17969/) (2025)
- [Crack Path Prediction with Operator Learning using Discrete Particle System data Generation](https://mlsanity.com/paper/2506.01976/) (2025)
- [Learning Nonlinear Responses in PET Bottle Buckling with a Hybrid DeepONet-Transolver Framework](https://mlsanity.com/paper/2509.13520/) (2025)
- [Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries](https://mlsanity.com/paper/2605.00760/) (2026)
- [UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning](https://mlsanity.com/paper/2605.12700/) (2026)
- [Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions](https://mlsanity.com/paper/2605.19823/) (2026)
- [No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study](https://mlsanity.com/paper/2607.23667/) (2026)
- [HI-MeshGraphNets: Efficient and Accurate Mesh-based Physics Learning with Hierarchical Multi-scale Graph Neural Networks](https://mlsanity.com/paper/2608.13827/) (2026)
- [APPSolver: Adaptive Patch Partitioning for Point-Wise Ship Flow Prediction on Unstructured Meshes](https://mlsanity.com/paper/2608.29355/) (2026)
- [CANTO: CAD-Native Transformer Operators for AI-Aided Engineering](https://mlsanity.com/paper/2609.36806/) (2026)

## Abstract (arXiv)

Shape optimization is essential in aerospace vehicle design, including reentry systems, and propulsion system components, as it directly influences aerodynamic efficiency, structural integrity, and overall mission success. Rapid and accurate prediction of external and internal flows accelerates design iterations. To this end, we develop a new variant of DeepONet, called Fusion-DeepONet as a fast surrogate model for geometry-dependent hypersonic and supersonic flow fields. We evaluated Fusion-DeepONet in learning two external hypersonic flows and a supersonic shape-dependent internal flow problem. First, we compare the performance of Fusion-DeepONet with state-of-the-art neural operators to learn inviscid hypersonic flow around semi-elliptic blunt bodies for two grid types: uniform Cartesian and irregular grids. Fusion-DeepONet provides comparable accuracy to parameter-conditioned U-Net on uniform grids while outperforming MeshGraphNet and Vanilla-DeepONet on irregular grids. Fusion-DeepONet requires significantly fewer trainable parameters than U-Net, MeshGraphNet, and FNO. For the second hypersonic problem, we set up Fusion-DeepONet to map from geometry and free stream Mach number to the temperature field around a reentry capsule traveling at hypersonic speed. This fast surrogate is then improved to predict the spatial derivative of the temperature, resulting in an accurate prediction of heat flux at the surfaces of the capsule. To enhance the accuracy of spatial derivative prediction, we introduce a derivative-enhanced loss term with the least computation overhead. For the third problem, we show that Fusion-DeepONet outperforms MeshGraphNet in learning geometry-dependent supersonic flow in a converging-diverging nozzle configuration. For all the problems, we used high-fidelity simulations with a high-order entropy-stable DGSEM solver to generate training datasets with limited samples.