Automatic Differentiation Backends

In BoundaryValueDiffEq.jl, automatic differentiation backend should only be wrapped in BVPJacobianAlgorithm(diffmode, bc_diffmode, nonbc_diffmode). BVPJacobianAlgorithm(diffmode, bc_diffmode, nonbc_diffmode) supports user-specified mixed automatic differentiation backends in differrent part of a boundary value problem, and AD choice should depended on the type of boundary value problem:

• BVProblem: Differentiation mode for boundary condition part and non boundary condition part should be specified, for example, BVPJacobianAlgorithm(; bc_diffmode, nonbc_diffmode), default to BVPJacobianAlgorithm(; bc_diffmode = AutoForwardDiff(), nonbc_diffmode = AutoSparse(AutoForwardDiff())).
• TwoPointBVProblem: Differentiation mode for overall solving should be specified, for example, BVPJacobianAlgorithm(; diffmode), default to BVPJacobianAlgorithm(; diffmode = AutoSparse(AutoForwardDiff())).

In BoundaryValueDiffEq.jl, we require AD to obtain the Jacobian of the loss function which contains the collocation equation and boundary condition equations. For TwoPointBVProblem, the Jacobian of the loss function is a sparse banded matrix with known sparsity pattern, but for general multi-points BVProblem, the Jacobian of the loss function is an almost banded matrix, which has the first several rows as the boundary conditions and the rest as a sparse banded matrix with known sparsity pattern but without the first several rows. In this case, we can specify mixed AD backend in BVPJacobianAlgorithm to make the most of the different sparsity pattern to accelerate BVP solving process.

Summary of Finite Differencing Backends

• AutoFiniteDiff: Finite differencing using FiniteDiff.jl, not optimal but always applicable.

Summary of Forward Mode AD Backends

• AutoForwardDiff: The best choice for dense problems.
• AutoPolyesterForwardDiff: Might be faster than AutoForwardDiff for large problems. Requires PolyesterForwardDiff.jl to be installed and loaded.
• AutoEnzyme(; mode = Enzyme.Forward): Source transformation forward mode AD.

Summary of Reverse Mode AD Backends

• AutoEnzyme(; mode = Enzyme.Reverse): Source transformation reverse mode AD.
• AutoMooncake(; config = nothing): Source transformation reverse mode AD.