Support immutable output types in out-of-place scalar→vector gradient

`finite_difference_gradient` (out-of-place, cached) for scalar `x` previously
delegated to `finite_difference_gradient!` which uses `@. df = result / epsilon`.
This in-place broadcast fails when the output buffer contains immutable array
types (e.g. `ArrayPartition{SVector}` from `SecondOrderODEProblem` in
OrdinaryDiffEq.jl) because `setindex!` is not defined for `SVector`.

Extract the scalar→vector case into `_scalar_gradient_oop` which computes the
finite difference result purely out-of-place using `@. (a - b) / h` (which
allocates a new array via `copy` rather than mutating via `copyto!`).

Fixes SciML/OrdinaryDiffEq.jl#3444

Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Author: ChrisRackauckas

src/gradients.jl

@@ -256,12 +256,46 @@ function finite_difference_gradient(
         dir = true) where {T1, T2, T3, T4, fdtype, returntype, inplace}
     if typeof(x) <: AbstractArray
         df = zero(returntype) .* x
+        finite_difference_gradient!(
+            df, f, x, cache, relstep = relstep, absstep = absstep, dir = dir)
+        df
     else
-        df = zero(cache.c1)
+        # Scalar x: compute out-of-place to support immutable output types
+        # (e.g. ArrayPartition{SVector} from SecondOrderODEProblem).
+        _scalar_gradient_oop(f, x, cache, fdtype, returntype, inplace;
+            relstep = relstep, absstep = absstep, dir = dir)
+    end
+end
+
+# Out-of-place scalar→vector gradient that never mutates the result,
+# so it works even when f returns immutable arrays (SVector, etc.).
+function _scalar_gradient_oop(
+        f, x::Number, cache, fdtype, returntype, inplace;
+        relstep, absstep, dir)
+    fx, c1, c2 = cache.fx, cache.c1, cache.c2
+
+    if fdtype == Val(:forward)
+        epsilon = compute_epsilon(Val(:forward), x, relstep, absstep, dir)
+        _c1 = inplace == Val(true) ? (f(c1, x + epsilon); c1) : f(x + epsilon)
+        if typeof(fx) != Nothing
+            @. (_c1 - fx) / epsilon
+        else
+            _c2 = inplace == Val(true) ? (f(c2, x); c2) : f(x)
+            @. (_c1 - _c2) / epsilon
+        end
+    elseif fdtype == Val(:central)
+        epsilon = compute_epsilon(Val(:central), x, relstep, absstep, dir)
+        _c1 = inplace == Val(true) ? (f(c1, x + epsilon); c1) : f(x + epsilon)
+        _c2 = inplace == Val(true) ? (f(c2, x - epsilon); c2) : f(x - epsilon)
+        @. (_c1 - _c2) / (2 * epsilon)
+    elseif fdtype == Val(:complex) && returntype <: Real
+        epsilon_complex = eps(real(eltype(x)))
+        _c1 = inplace == Val(true) ?
+              (f(c1, x + im * epsilon_complex); c1) : f(x + im * epsilon_complex)
+        @. imag(_c1) / epsilon_complex
+    else
+        fdtype_error(returntype)
     end
-    finite_difference_gradient!(
-        df, f, x, cache, relstep = relstep, absstep = absstep, dir = dir)
-    df
 end
 
 # vector of derivatives of a vector->scalar map by each component of a vector x

test/finitedifftests.jl

@@ -290,6 +290,40 @@ complex_cache = FiniteDiff.GradientCache(df, x, Val{:complex})
     @test err_func(FiniteDiff.finite_difference_gradient!(df, f, x, complex_cache), df_ref) < 1e-15
 end
 
+@time @testset "Gradient of f:scalar->vector with immutable output" begin
+    # Regression test: finite_difference_gradient with scalar x must work
+    # when f returns immutable arrays, since the out-of-place API should
+    # never need to mutate the result. This failed previously because the
+    # cached version called finite_difference_gradient! which used @. df = ...
+    # to write into an immutable buffer.
+    #
+    # We use a wrapper around a regular Vector that blocks setindex! to
+    # simulate immutable array types (like StaticArrays.SVector or
+    # ArrayPartition containing SVectors).
+    struct ReadOnlyVec{T} <: AbstractVector{T}
+        data::Vector{T}
+    end
+    Base.size(v::ReadOnlyVec) = size(v.data)
+    Base.getindex(v::ReadOnlyVec, i::Int) = v.data[i]
+    Base.setindex!(::ReadOnlyVec, _, ::Int) = error("ReadOnlyVec does not support setindex!")
+    Base.similar(v::ReadOnlyVec) = ReadOnlyVec(zeros(eltype(v), length(v)))
+    Base.zero(v::ReadOnlyVec) = ReadOnlyVec(zeros(eltype(v), length(v)))
+    # Out-of-place broadcast returns a plain Vector (like SVector .+ SVector returns SVector)
+    Base.BroadcastStyle(::Type{<:ReadOnlyVec}) = Broadcast.DefaultArrayStyle{1}()
+
+    g(t) = ReadOnlyVec([sin(t), cos(t)])
+    t0 = 1.0
+    g_ref = [cos(t0), -sin(t0)]
+
+    # Out-of-place cached version (the bug path)
+    df_template = similar(g(t0))
+    for fdtype in (Val(:forward), Val(:central))
+        cache = FiniteDiff.GradientCache(df_template, t0, fdtype, Float64, Val(false))
+        result = FiniteDiff.finite_difference_gradient(g, t0, cache)
+        @test err_func(collect(result), g_ref) < 1e-4
+    end
+end
+
 f(df, x) = (df[1] = sin(x); df[2] = cos(x); df)
 x = (2π * rand()) * (1 + im)
 fx = fill(zero(typeof(x)), 2)