# Multi-Fidelity deep learning for predicting the nonlinear buckling behaviour of concrete thin-shells

## Overview

This repository contains scripts and data supporting the following research article:
Pollet, M., Shepherd, P., & Hawkins, W. (2026). Multi-Fidelity deep learning for predicting the nonlinear buckling behaviour of concrete thin-shells.
Engineering Applications of Artificial Intelligence, 174, 114490. https://doi.org/10.1016/j.engappai.2026.114490

It contains the trained Multilayer Perceptrons used to obtain the results presented in the article, as well as Python scripts that were used to train these models and assess their performance.
Running these scripts necessitates the ConcreteShellFEA dataset (available at: https://doi.org/10.15125/BATH-01519) to be downloaded.
Additionally the path to this dataset needs to be specified in each script, under the DATASET_ROOT variable.
Further details about this data can be found in the related research article.

## Article abstract

This research introduces a novel approach to rapidly estimate the nonlinear buckling behaviour of concrete thin-shells, using Multi-Fidelity deep learning.
These statistical models, which can make fast predictions, could potentially be used to improve design space exploration during the structural shape optimisation phase.
Indeed, the use of nonlinear Finite Element analysis for estimating the buckling factor is unpractical in such settings because of the high computational cost.
This research considers the use of Multi-Fidelity models to mitigate the computational cost required to constitute a sufficiently large dataset for training deep learning models.
Two datasets - a low-fidelity dataset and a high-fidelity dataset – that contain concrete thin-shells with various shapes and material properties were therefore generated.
The buckling factor of the 20,000 thin-shells in the low-fidelity dataset were obtained through linear eigenvalue Finite Element analysis, which has a low computational cost.
Additionally, the buckling factors of the 5,000 thin-shells in the high-fidelity dataset were obtained using computationally expensive nonlinear Finite Element analyses.
These datasets were used to train Multi-Fidelity Multilayer Perceptrons in two different manners.
The first approach relies on two sequentially connected models, each trained with data of different fidelity.
The second approach is based on Transfer Learning and relies on the pre-training and re-training of a single model with data of different fidelity.
The two Multi-Fidelity approaches are found to be more accurate when the high-fidelity training set size is below 750.
For higher training set sizes, the results of the three approaches are nearly equivalent, even though the baseline is slightly more accurate.
The Multi-Fidelity are therefore most beneficial in situations where the computational resources are limited.
When the full datasets are used, the three approaches are able to predict accurately the nonlinear buckling factors, as their Mean Absolute Errors are consistently below 0.65%.
Regarding speed, the slowest Multilayer Perceptron is found to be more than 97,000 times quicker than the average time required for a nonlinear FE analysis.
These results demonstrate that surrogate models are a viable method for the fast estimation of the nonlinear buckling behaviour of concrete thin-shells.

## Repository structure

```
MultiFidelityDeepLearningForPredictingTheNonlinearBucklingBehaviourOfConcreteThinShells/
│── models/                                        # Trained models
│   ├── full_dataset/                              # Models trained using the full dataset
│   │   ├── hf_MLP_best_model.pth                  # Baseline MLP used in paper
│   │   ├── lf_MLP_best_model.pth                  # Low-fidelity MLP used in paper
│   │   ├── mf_sequential_best_model.pth           # Multi-Fidelity (sequential) MLP used in paper
│   │   ├── mf_transfer_best_model.pth             # Multi-Fidelity (transfer learning) MLP used in paper
│   ├── varying_dataset_size/                      # Models trained with varying amounts of data
│── results/raw_predictions/                       # Predictions on testing set
│   │   ├── hf_MLP_best_preds.pth                  # Predictions made by Baseline MLP
│   │   ├── lf_MLP_best_preds.pth                  # Predictions made by Low-fidelity MLP
│   │   ├── mf_sequential_best_preds.pth           # Predictions made by Multi-Fidelity (sequential) MLP
│   │   ├── mf_transfer_best_preds.pth             # Predictions made by Multi-Fidelity (transfer learning) MLP
│── scripts/                                       # Scripts for model training and evaluation
│   ├── datasets.py                                # Definition of dataset classes
│   ├── metrics.py                                 # Definition of metric classes
│   ├── models.py                                  # Definition of model classes
│   ├── tools.py                                   # Definition of various tool classes and functions
│   ├── train_hf_MLP.py                            # Nonlinear buckling prediction using Baseline MLP
│   ├── train_lf_MLP.py                            # Linear buckling prediction using Low-fidelity MLP
│   ├── train_mf_sequential_MLP.py                 # Nonlinear buckling prediction using Multi-Fidelity (sequential) MLP
│   ├── train_mf_transfer_MLP.py                   # Nonlinear buckling prediction using Multi-Fidelity (transfer learning) MLP
│   ├── hf_MLP_varying_dataset_size.py             # Evaluation of Baseline MLPs trained with varying amounts of data
│   ├── lf_MLP_varying_dataset_size.py             # Evaluation of Low-fidelity MLPs trained with varying amounts of data
│   ├── mf_sequential_MLP_varying_dataset_size.py  # Evaluation of Multi-Fidelity (sequential) MLPs trained with varying amounts of data
│   ├── mf_transfer_MLP_varying_dataset_size.py    # Evaluation of Multi-Fidelity (transfer learning) MLPs trained with varying amounts of data
│── requirements.txt                               # List of dependencies
│── README.md                                      # Repository documentation
```

## License

The contents of this repository is licensed under Creative Commons: Attribution 4.0 (CC BY), which allows for the derivation of new resources from the data, and for the redistribution of the data and their derivations, both openly and commercially, as long as the creators of this repository are credited for the original creation.

## Crediting/citing the creators of this dataset

The creators of this repository can be acknowledged by citing the related research article:

Pollet, M., Shepherd, P., & Hawkins, W. (2026). Multi-Fidelity deep learning for predicting the nonlinear buckling behaviour of concrete thin-shells.
Engineering Applications of Artificial Intelligence, 174, 114490. https://doi.org/10.1016/j.engappai.2026.114490

or using BibTeX:

```
@article{Pollet2026,
   author = {Maxime Pollet and Paul Shepherd and Will Hawkins},
   doi = {10.1016/j.engappai.2026.114490},
   issn = {0952-1976},
   journal = {Engineering Applications of Artificial Intelligence},
   month = {6},
   pages = {114490},
   publisher = {Pergamon},
   title = {Multi-Fidelity deep learning for predicting the nonlinear buckling behaviour of concrete thin-shells},
   volume = {174},
   url = {https://linkinghub.elsevier.com/retrieve/pii/S0952197626007712},
   year = {2026}
}
```