# Learning Nonlinear Responses in PET Bottle Buckling with a Hybrid DeepONet-Transolver Framework

arXiv: https://arxiv.org/abs/2509.13520 · HTML: https://mlsanity.com/paper/2509.13520/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 35 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [PointCNN: Convolution On X-Transformed Points](https://mlsanity.com/paper/1801.07791/) (2018)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Rethinking Network Design and Local Geometry in Point Cloud: A Simple Residual MLP Framework](https://mlsanity.com/paper/2202.07123/) (2022)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Deep Generative Models in Engineering Design: A Review](https://mlsanity.com/paper/2110.10863/) (2021)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [Latent Neural Operator for Solving Forward and Inverse PDE Problems](https://mlsanity.com/paper/2406.03923/) (2024)
- [DrivAerNet++: A Large-Scale Multimodal Car Dataset with Computational Fluid Dynamics Simulations and Deep Learning Benchmarks](https://mlsanity.com/paper/2406.09624/) (2024)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [DrivAerML: High-Fidelity Computational Fluid Dynamics Dataset for Road-Car External Aerodynamics](https://mlsanity.com/paper/2408.11969/) (2024)
- [DoMINO: A Decomposable Multi-scale Iterative Neural Operator for Modeling Large Scale Engineering Simulations](https://mlsanity.com/paper/2501.13350/) (2025)
- [X-MeshGraphNet: Scalable Multi-Scale Graph Neural Networks for Physics Simulation](https://mlsanity.com/paper/2411.17164/) (2024)
- [Physics-Informed Geometry-Aware Neural Operator](https://mlsanity.com/paper/2408.01600/) (2024)
- [Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solving](https://mlsanity.com/paper/2408.02198/) (2024)
- [WindsorML: High-Fidelity Computational Fluid Dynamics Dataset For Automotive Aerodynamics](https://mlsanity.com/paper/2407.19320/) (2024)
- [Fusion-DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic and Supersonic Flows](https://mlsanity.com/paper/2501.01934/) (2025)
- [Learning solution operators of PDEs defined on varying domains via MIONet](https://mlsanity.com/paper/2402.15097/) (2024)

## Abstract (arXiv)

Neural surrogates and operator networks for solving partial differential equation (PDE) problems have attracted significant research interest in recent years. However, most existing approaches are limited in their ability to generalize solutions across varying non-parametric geometric domains. In this work, we address this challenge in the context of Polyethylene Terephthalate (PET) bottle buckling analysis, a representative packaging design problem conventionally solved using computationally expensive finite element analysis (FEA). We introduce a hybrid DeepONet-Transolver framework that simultaneously predicts nodal displacement fields and the time evolution of reaction forces during top load compression. Our methodology is evaluated on two families of bottle geometries parameterized by two and four design variables. Training data is generated using nonlinear FEA simulations in Abaqus for 254 unique designs per family. The proposed framework achieves mean relative L2 errors of 2.5-13% for displacement fields and approximately 2.4% for time-dependent reaction forces for the four-parameter bottle family. Point-wise error analyses further show absolute displacement errors on the order of 10-4-10-3, with the largest discrepancies confined to localized geometric regions. Importantly, the model accurately captures key physical phenomena, such as buckling behavior, across diverse bottle geometries. These results highlight the potential of our framework as a scalable and computationally efficient surrogate, particularly for multi-task predictions in computational mechanics and applications requiring rapid design evaluation.