# Knowledge-Constrained Shape Optimization with a Mixture-of-Experts Neural Operator for High-Confidence Design

arXiv: https://arxiv.org/abs/2607.09763 · HTML: https://mlsanity.com/paper/2607.09763/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in MoE theory and gating functions, one of 12 directions in Mixture-of-experts.
- Builds on 22 of its 87 references that are in the map.
- First posted on arXiv 7 Jul 2026.

## Placed in

- Mixture-of-experts / [MoE theory and gating functions](https://mlsanity.com/moe/moe-theory-and-gating-functions/)

## Builds on (most cited first, 22 of 22)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Retrieval-Augmented Generation for Large Language Models: A Survey](https://mlsanity.com/paper/2312.10997/) (2023)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Point Transformer V3: Simpler, Faster, Stronger](https://mlsanity.com/paper/2312.10035/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Hunyuan3D 2.5: Towards High-Fidelity 3D Assets Generation with Ultimate Details](https://mlsanity.com/paper/2506.16504/) (2025)
- [DrivAerNet++: A Large-Scale Multimodal Car Dataset with Computational Fluid Dynamics Simulations and Deep Learning Benchmarks](https://mlsanity.com/paper/2406.09624/) (2024)
- [Machine Learning in Aerodynamic Shape Optimization](https://mlsanity.com/paper/2202.07141/) (2022)
- [AI Agents in Engineering Design: A Multi-Agent Framework for Aesthetic and Aerodynamic Car Design](https://mlsanity.com/paper/2503.23315/) (2025)
- [DrivAerStar: An Industrial-Grade CFD Dataset for Vehicle Aerodynamic Optimization](https://mlsanity.com/paper/2510.16857/) (2025)
- [Optimization and Generation in Aerodynamics Inverse Design](https://mlsanity.com/paper/2602.03582/) (2026)
- [Geometry-Aware Neural Optimizer for Shape Optimization and Inversion](https://mlsanity.com/paper/2605.04474/) (2026)
- [Asynchronous Parallel Reinforcement Learning for Optimizing Propulsive Performance in Fin Ray Control](https://mlsanity.com/paper/2401.11349/) (2024)
- [A Closed-Loop Multi-Agent Framework for Aerodynamics-Aware Automotive Styling Design](https://mlsanity.com/paper/2508.03370/) (2025)
- [RETO: A Rotary-Enhanced Transformer Operator for High-Fidelity Prediction of Automotive Aerodynamics](https://mlsanity.com/paper/2605.00062/) (2026)
- [A Multi-AI-agent Framework Enabling End-to-end Finite Element Analysis for Solid Mechanics Problems](https://mlsanity.com/paper/2606.00138/) (2026)
- [A Geometry-Aware Triplane Field Network for Vehicle Aerodynamic Prediction](https://mlsanity.com/paper/2606.07724/) (2026)

## Abstract (arXiv)

Engineering shape optimization faces challenges in both expert-dependent problem setup and surrogate-model reliability. In practical aerodynamic design, optimization settings such as editable regions, deformation ranges, and design-preservation constraints are typically specified manually by experienced engineers, while surrogate-based optimization may become unreliable for heterogeneous geometry databases and out-of-distribution designs. To address these challenges, we propose a knowledge-constrained shape-optimization framework that translates knowledge-based constraints and user intent into quantifiable parameters of DFFD-based deformation operators, enabling engineering-aware and controllable constrained optimization. We further develop a Mixture-of-Experts Neural Operator (MoE-NO) to improve drag prediction and trend consistency over heterogeneous aerodynamic datasets. Based on the MoE-NO encoder and Mahalanobis distance, an uncertainty-estimation strategy is introduced to detect out-of-distribution geometries and selectively trigger physics-solver feedback for local sample enrichment. Experiments on in-house MPV, SUV, and Sedan datasets show that MoE-NO achieves a test-set MAPE of 1.16% and a trend-prediction accuracy of 94.34%, outperforming the best baseline results of 1.52% and 90.34%, respectively. Vehicle shape-optimization experiments further yield CFD-validated drag coefficient reductions of approximately 4% to 10%.