Fusion of the Finite Element Method and the Radial Basis Function Neural Network
Introduction
In 1940, the Tacoma Narrows Bridge was one of the largest suspension bridges of its time, which cost about $6.4 million to construct. But just four months after opening, the bridge collapsed. And it wasn’t because of poor construction materials, but because the engineers did not account for the effect of wind forces. Back then, solving such problems was incredibly difficult, and the manual calculations could not handle such complexities. Sadly, the finite element method was not fully developed.
Fast forward to today, and we have the finite element method. But what exactly is this finite element method that could have saved $6.4 million in 1940?
The finite element method (FEM) is a numerical technique for solving complex partial differential equations using a process called discretization which means breaking down the problem into smaller parts, then we analyze. After this analysis, we reassemble the problems to find the approximate solution to the bigger problem. This technique can be used across different fields, like structural engineering for buildings and bridges, just in the case of the Tacoma Narrows Bridge. It can also be used in mechanical engineering for automobile designs and in other fields.
But this technique, the finite element method, has limitations. Limitations like high computational cost, memory usage, and sometimes poor accuracy in certain cases. And this is the area where our research comes in.
By hybridizing the finite element method with the Radial Basis Function Neural Networks (RBFNN), we can enhance the accuracy of solving partial differential equations (mathematical representation of real-world problem) and at the same time reduce the computational overhead we usually get when using FEM only.
But, how do we achieve this hybrid approach?
Method
– FEM –
Define a PDE in the form:
- ∇²u = 0 with initial condition
u(0,t) = 0, u(1,t) = 1
Applying FEM to the above equation will make us multiply it by a test function and integrate it by part to arrive at a “weak form”. Then, we discretize the domain. After discretization, we substitute the discretized domain into the weak form to arrive at a system of equations.
Solving the system of equations will result in the solution of the problem.
– RBFNN –
This is where things get interesting because this is the result we will loop into a NN to predict the next solution for us.
Before we do that, we should have an idea of what NN is.
On the left, we have the Biological Neuron & on the right, we have the Artificial Neuron.
To understand the ANN, we should have an idea of how the Biological Neuron works.
We all agree that our five senses gather information from our surroundings and load it into our brains as input, then the neurons train this data over time, and that is exactly how we could make predictions and decisions.
This is how we know “Apple” starts with “A”, “Boy” starts with “B”, and so on. This is exactly what we are trying to do with Artificial Neurons.
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To understand better, let’s look at this second illustration. We have three different layers here.
The first one is the input layer.
This is where the data will come in as input. In our case, it will be the Finite Element Method solution.
The hidden layer is where the training is going to take place. This is also the area where the neurons are.
Then, the final layer is the output layer, where the prediction is going to come out.
This prediction is also the key to the Hybrid approach solution.
Programming Environment
We employed the programming environment called Kaggle Computational Environment, and the programming language we used is called Python.
Some Python libraries we used include:
• FEniCS for solving FEM problems
• Matplotlib & Seaborn for the visualization tasks
• Numpy & Scipy for numerical tasks
• Scikit-learn, Torch, and TensorFlow for the NN tasks
Results
After applying this approad on some PDEs. The hybrid approach — FEM-RBFNNs eliminate complex meshing, enhance accuracy with data-driven learning, and capture sharp gradients better than FEM.
They adapt quickly to changes, reduce computation, and handle sparse or noisy data.
Incorporating observational data further boosts their reliability, making them a powerful and efficient alternative to traditional methods.
Conclusion
FEM is powerful when it comes to precision and structure. However, it might face challenges when facing problems with complex geometry, and computational time can often stretch endlessly.
On the other hand, RBFNNs are flexible, and their learning capabilities make the computational time small.
What we have done in this research is to combine the strengths of two different approaches and make it one to tackle the limitations of solving complex PDEs.
And, we did see evidence of that in problems we applied this approach.
Not only did the hybrid approach converge better, but it also gave us lower computation time compared to just FEM.
