The following defines the simple one-dimensional ode `dx_dt = -ωx`

, and a function `x(ω, t)`

to solve it. Fixing `t=T`

, I can readily differentiate `x(ω, t)`

with respect to `ω`

:

```
using DifferentialEquations
using SciMLSensitivity
using Zygote
const T = 19.7
dx_dt(x, ω, t) = -only(ω)*x
function x(ω, t)
problem = ODEProblem(
dx_dt,
[1.0,], # initial condition
(0, t), # time span
ω, # parameter (dimensions of 1/time)
)
solution = solve(
problem,
Tsit5(),
saveat=[t,], # only need final value
sensealg=InterpolatingAdjoint(; autojacvec = ZygoteVJP()),
)
return only(solution.u)
end
f(ω) = only(x(ω, T))
gradient(f, [1.0])
# ([-0.13590672273297327],)
```

However, certain invariance of my problem under rescaling means an equivalent solution is given as follows:

```
function x2(ω, t)
problem = ODEProblem(
dx_dt,
[1.0,], # initial condition
(0, t*only(ω)), # non-dimensionalized time span
[1.0,], # non-dimensionalized parameter
)
solution = solve(
problem,
Tsit5(),
saveat=[t*only(ω),], # only need final value
sensealg=InterpolatingAdjoint(; autojacvec = ZygoteVJP()),
)
return only(solution.u)
end
f2(ω) = only(x2(ω, T))
```

Indeed, we can see there is no difference between `f`

and f2`:

```
ω = [rand(),]
@assert f(ω) ≈ f2(ω)
```

And yet…

```
gradient(f2, [1.0])
# nothing
```

- Output of
`using Pkg; Pkg.status()`

```
(jl_Sd68uJ) pkg> status
Status `/private/var/folders/4n/gvbmlhdc8xj973001s6vdyw00000gq/T/jl_Sd68uJ/Project.toml`
[f6b3d34e] Bertalanffy v0.1.0 `~/Dropbox/Tumor/Bertalanffy`
[336ed68f] CSV v0.10.12
[b0b7db55] ComponentArrays v0.15.7
[a93c6f00] DataFrames v1.6.1
[0c46a032] DifferentialEquations v7.12.0
[d9f16b24] Functors v0.4.5
[b3c1a2ee] IterationControl v0.5.4
[3bd65402] Optimisers v0.3.1
[91a5bcdd] Plots v1.40.0
[1ed8b502] SciMLSensitivity v7.53.0
[860ef19b] StableRNGs v1.0.1
[2913bbd2] StatsBase v0.34.2
[bd369af6] Tables v1.11.1
[3a884ed6] UnPack v1.0.2
[e88e6eb3] Zygote v0.6.69
[9a3f8284] Random
[8dfed614] Test
```

```
julia> versioninfo()
Julia Version 1.10.0
Commit 3120989f39b (2023-12-25 18:01 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: macOS (x86_64-apple-darwin22.4.0)
CPU: 12 × Intel(R) Core(TM) i7-8850H CPU @ 2.60GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, skylake)
Threads: 17 on 12 virtual cores
Environment:
JULIA_LTS_PATH = /Applications/Julia-1.6.app/Contents/Resources/julia/bin/julia
JULIA_PATH = /Applications/Julia-1.10.app/Contents/Resources/julia/bin/julia
JULIA_EGLOT_PATH = /Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia
JULIA_NUM_THREADS = 12
JULIA_NIGHTLY_PATH = /Applications/Julia-1.10.app/Contents/Resources/julia/bin/julia
DYLD_FALLBACK_LIBRARY_PATH = /Users/anthony/.julia/artifacts/0233bb40b298b03aa3743cc339b4a5c6816ce583/lib:/Users/anthony/.julia/artifacts/1e901863cf8fbb1ee50a5d0976114a2371899331/lib:/Users/anthony/.julia/artifacts/dcc1b7719d5a106fba77bbc272d231e163d15fe5/lib:/Applications/Julia-1.10.app/Contents/Res
```

(truncated)