Port Symbolics extension to the Symbolics v7 APIs

[ChrisRackauckas-Claude]

Important

This PR should be ignored until reviewed by @ChrisRackauckas.

Investigation of #69 revealed the Symbolics extension has been entirely non-functional on Symbolics v7 — which is the only version the compat allows — because it still uses the v6 extension APIs:

1. Symbolics.array_term does not exist in any Symbolics v7 release. Symbolically calling an array-output (N_out >= 1) interpolation throws UndefVarError: array_term not defined.
2. Scalar-output terms had Unknown shape. SymbolicUtils.term(interp, t...; type = Real) leaves shape = Unknown(-1), which expand_derivatives rejects with Differentiation with array expressions is not yet supported. SymbolicUtils v4 needs promote_shape (or an explicit shape) for non-builtin callables.
3. The derivative registrations were dead code. The Symbolics.derivative(op, args, ::Val{I}) overload API was removed in Symbolics v7 in favor of @register_derivative/derivative_rule (the same migration DataInterpolations.jl did in its Symbolics ext behind a pkgversion(Symbolics) >= v"7" check). Defining methods on the old function does nothing, so derivatives of interpolation terms were left as unexpanded Differential operators.

Why CI never caught this: test/runtests.jl has an Extensions group containing test_symbolics_ext.jl, but test/test_groups.toml only declares Core and QA, so the extension tests never ran on CI. (Locally, on current main with Symbolics v7.26, test_symbolics_ext.jl errors immediately at interp(x, y).)

Changes

• Build terms with SymbolicUtils.term with explicit type and shape, and define SymbolicUtils.promote_shape alongside promote_symtype for both NDInterpolation and DifferentiatedNDInterpolation.
• Register derivatives with @register_derivative; partial derivatives are still represented by DifferentiatedNDInterpolation so arbitrary mixed orders compose.
• Add the Extensions group to test_groups.toml so these tests actually run on CI.
• Extend tests: scalar-output (N_out == 0) interpolations (the Differentiating through DataInterpolationsND functions throws error #69 use case — previously untested), full expand_derivatives chain rule through both arguments, second/mixed derivatives, and numeric evaluation of every generated expression. Also fixed an ex isa BasicSymbolic{Vector{Real}} assertion that is invalid under SymbolicUtils v4 (the type parameter is now the variant, not the symtype).

All APIs used (@register_derivative, Symbolics.SymbolicT, SymbolicUtils.ShapeVecT) exist since Symbolics v7.0.0, so no compat change is needed.

What this does and does not fix for #69

With this PR, directly embedding NDInterpolation objects in MTK equations works end-to-end: the #69 MWE rewritten with direct embedding compiles via mtkcompile (9-equation fully determined system) and solves successfully with Rodas5P (verified locally).

The original MWE uses MTK callable parameters ((itp::NDInterpolation)(..)), and that form still fails: in the symbolic expression the operation is the symbolic stand-in for the parameter (symtype FnType{Tuple, Real, NDInterpolation}), not an NDInterpolation instance, so no instance-dispatched derivative rule can ever fire and expand_derivatives leaves Differential(p(t))(itp(p(t), T(t))) behind, which validate_operator then rejects. That gap is upstream (Symbolics needs a hook to dispatch derivative rules on the FnType's callable type) and affects DataInterpolations.jl 1-D the same way whenever a callable parameter is called with an unknown rather than t (its ODEProblem construction fails at the initialization system even though mtkcompile happens to pass). I am filing that separately upstream.

Local test output

GROUP=Extensions Pkg.test():
Test Summary:       | Pass  Total   Time
Symbolics Extension |   23     23  44.3s

GROUP=Core Pkg.test():
Interpolations |  206    206
Derivatives    |  369    369
DataInterpolations |   28     28
Interface      |   21     21

Runic.jl run on the changed files.

🤖 Generated with Claude Code

[ChrisRackauckas-Claude]

CI status note: all 9 Extensions jobs (the tests validating this fix) pass on ubuntu/macos/windows × julia lts/1/pre. The remaining red is unrelated to this PR:

• Downgrade: pre-existing on main (PrecompileTools lower bound), fixed by Raise PrecompileTools compat lower bound to fix Downgrade CI #73 where the same job passes.
• QA (macos/ubuntu): pre-existing Aqua failures on main, fixed by Fix Aqua QA failures: unbound type parameters in validate_cache and missing Pkg compat #70.
• Core (julia pre, ubuntu): tests passed (Testing DataInterpolationsND tests passed in the log); the job then failed on a runner cache-upload authentication error — needs a maintainer re-run, nothing to fix in the branch.

🤖 Generated with Claude Code