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Boundary value problem (BVP) solvers for scientific machine learning (SciML)

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Opening an issue so that I remember to write it once #155 is finished. The combination of options for BVP Jacobians are quite a lot so even I forget which...

I am trying to implement the "[shooting method](https://en.wikipedia.org/wiki/Shooting_method)" to solve a simple differential equation (\ddot{x} = -x*(1-x^2) ) as a boundary value problem. I also need to optimize the initial...

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**Describe the example** A clear and concise description of what the example is. **Minimal Reproducible Example 👇** Without MRE, we would only be able to help you to a limited...

## Checklist - [ ] Appropriate tests were added - [ ] Any code changes were done in a way that does not break public API - [ ] All...

A common problem in optimal control is when you want to achieve some objective state but do not have a specific time by which that objective must be attained. One...

### Main Idea Place Nodes at points the BC is evaluate to minimize ODE Sensitivity Cost ### How it is done? - Create a wrapper around the ODESolution and capture...

Using the parameter estimation functionality: http://docs.juliadiffeq.org/latest/analysis/parameter_estimation.html#Parameter-Estimation-1 we can solve BVPs with more equations than solutions by changing `problem_new_parameters` http://docs.juliadiffeq.org/latest/analysis/parameter_estimation.html#The-Problem-Generator-Function-1 to change the initial condition (and maybe some parameters) as part...

At the moment `BoundaryValueDiffEq` doesn't seem to support derivative boundary conditions(neumann, robin) at the boundaries in ODEs of BVP form. Requesting maintainers for same !

With #89 finally merged, let's push BoundaryValueDiffEq.jl to the next level!🎉🎉🎉🎉 Here are some issues we need to address in the near future: - [x] The code style is a...

While trying ```MIRK``` solvers and ```Shooting``` solvers with example problem in README, the two kinds of solvers gave two different solutions: ```julia using BoundaryValueDiffEq, OrdinaryDiffEq function simplependulum!(du, u, p, t)...