test: test lazy initialization in remake for supported problem types

test/downstream/modelingtoolkit_remake.jl

@@ -337,11 +337,45 @@ end
     @test sccprob4.p !== sccprob4.probs[2].p
 end
 
+# TODO: Rewrite this test when MTK build initialization for everything
 @testset "Lazy initialization" begin
-    @variables x(t) [guess = 1.0] y(t) [guess = 1.0]
-    @parameters p=missing [guess = 1.0]
-    @mtkbuild sys = ODESystem([D(x) ~ x, x + y ~ p], t)
-    prob = ODEProblem(sys, [x => 1.0, y => 1.0], (0.0, 1.0))
-    prob2 = remake(prob; u0 = [x => 2.0])
-    @test prob2.ps[p] ≈ 3.0
+    @variables _x(..) [guess = 1.0] y(t) [guess = 1.0]
+    @parameters p=1.0 [guess = 1.0]
+    @brownian a
+    x = _x(t)
+
+    initprob = NonlinearProblem(nothing) do args...
+        return 0.0
+    end
+    iprobmap = (_...) -> [1.0, 1.0]
+    iprobpmap = function (orig, sol)
+        ps = parameter_values(orig)
+        setp(orig, p)(ps, 3.0)
+        return ps
+    end
+    initdata = SciMLBase.OverrideInitData(initprob, nothing, iprobmap, iprobpmap)
+    @test SciMLBase.is_trivial_initialization(initdata)
+
+    @testset "$Problem" for (SystemT, rhss, Problem, Func) in [
+        (ODESystem, 0.0, ODEProblem, ODEFunction),
+        (System, a, SDEProblem, SDEFunction),
+        (ODESystem, _x(t - 0.1), DDEProblem, DDEFunction),
+        (System, _x(t - 0.1) + a, SDDEProblem, SDDEFunction),
+        (NonlinearSystem, y + 2, NonlinearProblem, NonlinearFunction),
+        (NonlinearSystem, y + 2, NonlinearLeastSquaresProblem, NonlinearFunction)
+    ]
+        is_nlsolve = SystemT == NonlinearSystem
+        D = is_nlsolve ? (v) -> v^3 : Differential(t)
+        sys_args = is_nlsolve ? () : (t,)
+        prob_args = is_nlsolve ? () : ((0.0, 1.0),)
+
+        @mtkbuild sys = SystemT([D(x) ~ x + rhss, x + y ~ p], sys_args...)
+        prob = Problem(sys, [x => 1.0, y => 1.0], prob_args...)
+        func_args = isdefined(prob.f, :g) ? (prob.f.g,) : ()
+        func = Func{true, SciMLBase.FullSpecialize}(
+            prob.f.f, func_args...; initialization_data = initdata, sys = prob.f.sys)
+        prob2 = remake(prob; f = func)
+        @test SciMLBase.is_trivial_initialization(prob2)
+        @test prob2.ps[p] ≈ 3.0
+    end
 end