Julia Differential Algebraic Equation as a Boundary Value Problem

Question

Does Julia support boundary value differential algebraic equations? I have an implicit ODE with a variable mass matrix that is sometimes singular, so I have to use the DAEProblem. My problem is two coupled second order ODEs for x1(t) and x2(t) that I have transformed into four first order equations by setting x1'(t) = y1(t) and x2'(t)=y2(t). I have values for x1 and x2 at the start and end of my domain, but don't have values for y1 or y2 anywhere, so I have a need for both a DAE and a BVP.

This github post suggests that this is possible, but I'm afraid I don't understand the machinery well enough to understand how to couple DAEProblem with BVProblem.

I've had success writing multiple shooting code following numerical recipes to solve the problem, but it's fairly clunky. Ultimately, I would like to pair this with DiffEqFlux (I have quite a few measurements of x1 and x2 along the domain and don't know the exact form of the differential equation), but I suspect that would be much simpler if there was a more direct approach to linking BVProblem with DAEProblem.

Answer 1

Just go directly to DiffEqFlux, since parameter estimation encompasses BVPs. Write the boundary conditions as part of the loss function on a DAEProblem (i.e. that the starting values should equal x and the final values should equal y), and optimize the initial conditions at the same time as any parameters. Optimizing only the intial conditions and not any parameters is equivalent to a single shooting BVP solver in this form, and this allows simultaneous parameter estimation. Or use the multiple shooting layer functions to do multiple shooting. Or use a BVProblem with a mass matrix.

For any more help, you'll need to share code for what you tried and didn't work, since it's not anything more difficult than that so it's hard to give more generic help than just "use constructor x".