In academia, people at times encounter research happening in various different frontiers. Specialists of similar fields, working at their own respective frontiers, are able to communicate with surprising ease and efficiency. Naturally, knowing and understanding complicated words and their scientific or mathematical meaning is important. But it is commonplace, as a newcomer in a research domain, to be lost in the jargon. I begin the documentation to give the motivation and background in a simple and friendly manner.
Let us first look into the example of 2D linear elasticity problem in the paper. A neural network is trained which has inputs
Boundary geometry: Chosen from a set of polygons of 6-8 edges
Boundary condition: Each edge can have only one of the following three types of boundary conditions.
Material parameters: Poisson’s ratio and density in a given range
Point in the domain
The output of the NN is the target area upper bound A. The training is carried out in a supervised fashion the key steps to be followed are given below.
Create a list of problems by choosing the boundary geometry, boundary condition, Poisson’ ratio etc and parameterise them. So, each problem can be assigned a set of numbers.
Numerically solve each problem for a Low Density Uniform Mesh (LDUM) to get a Low Accuracy Solution (LAS) and a High Density Uniform Mesh (HDUM) to get a High Accuracy Solution (HAS).
Let P_i be the parameters of the i-th problem. x_j be the center of the j-th element in the LDUM of the i-th problem and E_ij be the error approximate for the j-th element in the LDUM of the i-th problem. To find E_ij the HAS and LAS are used. The NN is trained to minimise the loss
mean square (NN(P_i, x_j) - 1/E_ij).
Now for an unseen problem we can get the target area upper bound A. This is used by the Triangle package to form the non-uniform mesh.