# ANN for Boundary Value Problem

I have a question regarding solving Boundary Value Problems (BVP) using ANNs.

My understanding is that this is currently a challenging task. Most scientific literature on the subject is interested in solving cases with simple boundary conditions which can be approximated analytically, before solving for the domain using the PDE.

I am interested in a hybride approach between Data driven training (providing example of the solution) and Physics based training (i.e. using the PDE and boundary condition as training data).

The idea would be to use an FEM software to obtain the solution of a specific BVP. I could then use that data to train an ANN for that specific BVP. The only inputs to my ANN would be based on the BVP, so mostly space coordinates and time.

My question is, how likely is it that the ANN will be able to solve a very similar BVP (e.g. change a boundary condition by 1%) once it has undergone additional training using only the PDE and the new boundary condition ?

I feel like if the change in the output fuction (say the pressure field) from the given change in boundary condition (say 1%) is small enough, starting with the ANN trained using the FEM results should be able to find a solution to the new problem within just a few iterations of backpropagation.

I essence, I guess what I am asking is how easy is it to train an ANN by changing the error function slightly. Will the network be able to "overwrite" the learning it has received and adjust its weight to minimize the new error function.

For those of you interested in the end goal of all this, the idea would be to be able to do sensitivity analysis more efficiently. One would just run a few computationally intensive FEM and then interpolate between them using the approach described, by using small increments in boundary conditions.

Thank you all for your suggestions and comments.