A Machine Learning approach to Finite Element Methods, using U-Net inspired architectures.
This project has been developed as part of my master thesis for in Imperial College London with the supervision of Prof. A. Bharath and Dr. Masouros.
What's missing:
Some of the trained models were too heavy to be updated on GitHub. This means that in order to run some of the plotting functions you will have to run the Main and train the models yourself. While the databases to perform ShapeToShape and CoordToShape reconstruction are generated on the fly, the ones containing the Von Misses stresses have been produced using Marc Mentat software (a FEA software), and are not given here. If you are interested in reproducing this work using the same data, please don't hesitate to contact me.
How to run the code
Run the main to train the model and display some of the key parameters to judge the performance of the model. The main is divided into commented sections that perform different tasks (as described bellow). Comment/Uncomment the appropriate section to perform the task needed.
In order to print sample reconstructions there are a number of functions available:
- Plot_Sample.py will plot a sample reconstruction using the model you have just trained, which is saved automatically in the folder "Saved_Models" with the name: modellino.model
- Plot_Image_Report.py will plot a comparison of different models for a certain task
- Plot_Image_Report_ForceToStress.py will plot the recontructions from the three different architectures developed here.
Finite Element Methods (FEM) are currently amongst the most exploited techniques to approximate the solutions of physical problems. While they realise the validating step for automated systems that promise to revolutionise the functional design sector such as generative designs and personalised implants, they are extremely tedious and generally require extensive human intervention. Recent advancements in artificial intelligence and the wider availability of computational power have motivated a shift towards predictive approaches using machine learning tools. However, the irregular and variable structures of the input data and the complexity of underlying physical principles make this task remarkably complex. Previous approaches have failed to create robust models capable of dealing with inputs as fed to FEM, which are characterised by their variable size and contain connectivity information. To this end, we propose two approaches: a U-Net based neural network suitable for variable size input stress reconstruction, and a DGCNN based network able to process adjacency matrices and non-constant input sizes as well as infer physical properties of geometrical shapes. We introduce the Multi-Poly dataset composed of polygons with 4 to 8 vertices, and demonstrate the enhanced performance of U-Net models in geometrical shape reconstruction (2.3e-5 MSE and 1.8e-4 ID) and variable-morphology input handling. In the stress reconstruction task our model outperforms previous ones, achieving over ten times higher accuracy with respect to ID score (0.076 ID and 17.7 AMSE). A modified version of the Multi-Poly dataset is then used to demonstrate that our DGCNN derived model can correctly classify (98.4% accuracy) convex and concave polygons. U-Net based models proved to be a viable alternative for stress estimation on shapes with variable input size; DGCNN networks similarly demonstrated this ability to process/ competency in processing data of variable size while in addition embedding an adjacency matrix. In light of these results, both may prove to be promising FEM surrogate candidates.
This works adapts the U-Net architecture to the tasks of shape reconstruction and stress distribution inference. The general structure of the model consists of a traditional contracting network supplemented with a series of up-sampling layers. The large number of feature channels in the up-sampling part ensures that context information and physical characteristics are propagated to higher resolution layers.
A total of five architectures have been implemented and explored in this study. Three of them (Minimal, Shallow and Full U-Nets) include both an encoding and a decoding part; the other two (Half U-Net for shape reconstruction and Half U-Net for stress estimation) comprise only the decoder. Below we report the details of the two best performers of each type.
Shallow U-Net architecture:
The Shallow U-Net takes a 64x64 matrix input and returns either the reconstructed shape of the polygon, or the distribution of stresses. Between other modifications, this network is fundamentally different from the original U-Net in the application of padding and strides, such that the input size is maintained to the output.
Its contracting path (left of figure) consists of 3 convolutional blocks each with two 3x3 padded 2D convolutions followed by a rectified linear unit (ReLU) and of a 2x2 max pooling step with stride 1 for down-sampling. Similarly, the bottleneck is made of a 3x3 padded convolution with a ReLU. In the expansive path (right side in left of Figure 8.a), every block consists of an up-sampling deconvolution of the feature map with filter size 2x2 followed by two 3x3 2D convolutions. After each up-sampling operation, there is the concatenation with the cropped feature map from the corresponding convolutional layer. The final layer consists of a 1x1 convolution to reduce the number of feature maps of the output to 1. No fully connected layer is used in the network.
Half U-Net architecture:
The other architecture, here named Half U-Net, is designed to start from a low dimensional input and hence consists of the expansive part only. However, as this network is constrained in its input size, it presents similar limitations to previous solutions and is reported here only as an alternative approach to former studies. It consists of 6 deconvolutional blocks, each with an up-sampling deconvolution followed by two 3x3 2D convolutions. The first 5 up-sampling operations have filter size 2x2, while the last one is applied with a 1x2 filter. After the second upsampling, a convolution with strides (2,1) is applied in order to obtain the desired output size. The final layer consists of a 1x1 convolution.
An ad hoc dataset, which will be referred to here as Multi-Poly dataset, was generated to train the networks. It consists of 12,000 planar shapes, equally divided between polygons with 4 to 8 corners. Vertices are generated starting from a circle of radius 1. Each vertex falls at a random distance between 0.8-1 from the centre. The angles between each vertex, initially set to 360/n, are perturbated following a normal distribution with variance (90/n)^2, where n is the number of corners for a given polygon. The coordinates are then scaled based on the canvas size (or image size). Each polygon of the Multi-Poly dataset is described by 3 objects as follows:
- The cartesian coordinates of the polygon in the form nx2;
- A 64x64 matrix with ones in correspondence to corners and zeros otherwise;
- A 64x64 matrix with ones inside the polygon and zeros otherwise.
In addition, we tested our models on the dataset created by Benkirane and Laskowski, which is composed of 10,000 pentagons with their respective Von Misses stresses. This contains information relative to corner coordinates, boundary conditions and forces applied to them, and the corresponding Von Misses stresses in the form of a 64x64 matrix (more information about the dataset can be found here).
Several experiments were run to test model performance on different tasks. In the first instance, Minimal, Shallow and Full U-Net were trained on the Shape-to-Shape and Stress-to-Stress reconstruction to check the models’ understanding of the underlying physics. When trained to estimate the shape and stresses of the polygon, we used the Multi-Poly and Benkirane-Laskowski’s datasets respectively. In both tasks, models were trained on the whole datasets with a training/validation split of 75/25 using random weights initializations, Adam optimiser (learning rate = 0.001, beta = 0.9, beta2 = 0.999), and batch size of 50. The networks were trained over 10 epochs. Since we are using an encoder-decoder model and working with images, mean absolute error (MAE) and mean squared error (MSE) metrics are good fits for the loss function. The former produced the best results and was chosen for our experiment:
where y_i is the target output for a pixel, x_i is the predicted output and n is the number of pixels. Secondly, the models were trained to reconstruct polygons starting from corner coordinates. This task tests the ability of the networks to generate a high dimensional output from a low dimensionality input, therefore requiring the decoder to exploit the geometrical relationship between the two. The networks have been trained both on a fraction of the Multi-Poly dataset containing purely pentagons (2,400 data) and on the entire variety of shapes in the Multi-Poly dataset. The same training parameters as before were found to be optimal and were therefore adopted. Lastly, we tested the networks on the stress estimation task using a modified version of the Shallow U-Net with 4 input channels. The network was fed with corner coordinates, boundary conditions, and the forces applied. Forces were considered to be planar, and all data was organised on 4 64x64 matrices as follows: a binary matrix with ones in correspondence of the corners; a scalar field indicating location and magnitude of forces in the horizontal direction (non-zero only at corners); a scalar field indicating location and magnitude of forces in the vertical direction; a binary matrix containing the boundary conditions information, indicating what corners are fixed (fixation was applied in all directions). The normalisation function implemented accounted for the nature of the physics of the problem. Forces and Von Misses stresses were normalised by:
were F_x and F_y are the forces applied to each vertex, so that G is the maximum vector force applied to the polygon. In contrast to previous approaches, we do not assume to know a priori the output stress and do not use it for normalization, which results in significantly greater employability of the model in real analysis. Again, the same training parameters were applied for this task.
To verify the networks’ understanding of underlying physics we are interested in testing their ability to compress the shape or stress distribution information into a bottleneck, then decode it back to the original shape.
Reconstruction of polygon shapes using different U-Net based models. New sample polygons unseen by the network were generated using the same method proposed for the Multi-Poly dataset. From the right: the target image (Target) and the reconstructions by the Minimal U-Net, by the Full U-Net, and the Shallow U-Net:
Reconstruction of the stress distribution on polygons using different U-Net based models. Stress is in MPa:
All models took less than 0.1 seconds to reconstruct a single 64x64 image on a recent GPU. During these tasks, different metrics were recorded to assess the performances of the models: training loss, testing loss, mean squared error, mean percentage error and image difference, with the latter obtained by calculating the sum of the difference between predicted output and ground truth at the end divisible by the test dataset size.
Next, we looked at the ability of the networks to reconstruct polygon shapes from the coordinates input in order to verify their understanding of geometrical relationships. Whilst this is a trivial task for humans, or can be hardcoded easily, a model must correctly infer geometrical principles. Half U-Net was used on pentagons only due to its fixed input size. The networks were trained as describe above over the Multi-Poly dataset.
Reconstruction of different polygons using the proposed networks. The input is a 64x64 matrix with ones in correspondence of the corners and zeros otherwise. The ground truth image (Target), is a 64x64 matrix with ones inside the polygon and zeros otherwise:
Lastly, we employed the networks in the stress estimation task, involving complex physical input-output relationships. The networks were trained as described above over the Benkirane and Laskowski dataset.
Estimate of stress distributions using the proposed networks. The input comprises coordinates, forces and boundary conditions in the form of 64x64 matrices. Output is Von Misses stresses in 64x64 matrix form:
For better comparison with previous methods, the average maximum stress error (AMSE) was computed, obtained by calculating the sum of the difference between predicted and ground truth maximum stresses, at the end divisible by the test dataset size.
Compares stress estimation results obtained with Shallow U-Net (left) and the best results obtained by M&L (right) on the dataset of highest complexity. Stress reconstruction by M&L are available online .
