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Using automatic differentiation

This tutorial shows how to compute the power spectrum $P(k; \theta)$ and its (logarithmic) derivatives

\[\frac{\partial \lg P}{\partial \lg \theta_i}\]

using automatic differentiation with ForwardDiff.jl. This technique can also differentiate any other quantity.

1. Wrap the evaluation

First, we must decide which parameters $\theta$ the power spectrum $P(k; \theta)$ should be considered a function of. To do so, let us write a small wrapper function that calculates the power spectrum as a function of the parameters $(T_{\gamma 0}, \Omega_{c0}, \Omega_{b0}, N_\textrm{eff}, h, Y_p)$, following the Getting started tutorial:

using SymBoltz
M = ΛCDM(K = nothing)
pars = [M.γ.T₀, M.c.Ω₀, M.b.Ω₀, M.ν.Neff, M.g.h, M.b.YHe, M.h.m_eV, M.I.ln_As1e10, M.I.ns]
prob0 = CosmologyProblem(M, Dict(pars .=> NaN))

probgen = parameter_updater(prob0, pars)
P(k, θ) = spectrum_matter(probgen(θ), k; verbose = true, ptopts = (reltol = 1e-3,))
P (generic function with 1 method)

It is now easy to evaluate the power spectrum:

using Unitful, UnitfulAstro
θ = [2.7, 0.27, 0.05, 3.0, 0.7, 0.25, 0.02, 3.0, 0.95]
ks = 10 .^ range(-3, 0, length=100) / u"Mpc"
Ps = P(ks, θ)
100-element Vector{Unitful.Quantity{Float64, 𝐋^3, Unitful.FreeUnits{(Mpc^3,), 𝐋^3, nothing}}}:
 15091.286437529841 Mpc^3
 16046.453127293555 Mpc^3
 17053.396362358268 Mpc^3
 18113.462913616688 Mpc^3
 19228.35306259395 Mpc^3
 20398.740506580954 Mpc^3
 21625.697204340187 Mpc^3
 22909.558262625527 Mpc^3
 24250.614451400896 Mpc^3
 25648.743451015856 Mpc^3
     ⋮
   234.84497055604393 Mpc^3
   200.77838421189972 Mpc^3
   171.15808908742756 Mpc^3
   145.987773603961 Mpc^3
   124.34227589034302 Mpc^3
   105.58332904125727 Mpc^3
    89.81639254132381 Mpc^3
    76.33366869239902 Mpc^3
    64.76071526314873 Mpc^3

This can be plotted with

using Plots
plot(log10.(ks/u"1/Mpc"), log10.(Ps/u"Mpc^3"); xlabel = "lg(k/Mpc⁻¹)", ylabel = "lg(P/Mpc³)", label = nothing)
Example block output

2. Calculate the derivatives

To get $\partial \lg P / \partial \lg \theta$, we can simply pass the wrapper function P(k, θ) through ForwardDiff.jacobian:

using ForwardDiff
lgP(lgθ) = log10.(P(ks, 10 .^ lgθ) / u"Mpc^3") # in log-space
dlgP_dlgθs = ForwardDiff.jacobian(lgP, log10.(θ))
100×9 Matrix{Float64}:
 -0.148129  -1.23995  -0.229643  -0.00985249  …  -0.0045606   3.0  -3.71642
 -0.16216   -1.23398  -0.228542  -0.0108759      -0.0045308   3.0  -3.65014
 -0.177471  -1.22759  -0.227354  -0.0119988      -0.00452774  3.0  -3.58385
 -0.194118  -1.2206   -0.226059  -0.0132194      -0.00455568  3.0  -3.51756
 -0.212186  -1.21307  -0.224661  -0.0145463      -0.00461834  3.0  -3.45128
 -0.231784  -1.20507  -0.223178  -0.0159908   …  -0.00472004  3.0  -3.38499
 -0.252907  -1.19627  -0.221541  -0.0175545      -0.00486063  3.0  -3.3187
 -0.275637  -1.18659  -0.219748  -0.0192396      -0.00503862  3.0  -3.25242
 -0.300228  -1.17647  -0.217887  -0.0210669      -0.00525512  3.0  -3.18613
 -0.326558  -1.16547  -0.215847  -0.0230376      -0.00550646  3.0  -3.11984
  ⋮                                           ⋱                    
 -5.9802     1.88732  -0.368798  -0.383752       -0.0253149   3.0   2.31565
 -6.02133    1.90445  -0.360193  -0.383383       -0.0253166   3.0   2.38194
 -6.0466     1.92119  -0.366931  -0.385121       -0.0253174   3.0   2.44823
 -6.08018    1.93611  -0.367469  -0.386857       -0.0253198   3.0   2.51451
 -6.11169    1.94946  -0.365861  -0.387967    …  -0.0253204   3.0   2.5808
 -6.14017    1.9635   -0.36696   -0.389302       -0.0253215   3.0   2.64709
 -6.1655     1.97766  -0.367343  -0.389805       -0.0253213   3.0   2.71337
 -6.18939    1.99097  -0.368437  -0.390854       -0.0253216   3.0   2.77966
 -6.21662    2.00424  -0.36816   -0.39222        -0.0253226   3.0   2.84595

The matrix element dlgP_dlgθs[i, j] now contains $\partial \lg P(k_i) / \partial \lg \theta_j$. We can plot them all at once:

plot(
    log10.(ks/u"1/Mpc"), dlgP_dlgθs;
    xlabel = "lg(k/Mpc⁻¹)", ylabel = "∂ lg(P) / ∂ lg(θᵢ)",
    labels = "θᵢ=" .* ["Tγ0" "Ωc0" "Ωb0" "Neff" "h" "YHe" "mh" "ln(10¹⁰As)" "ns"]
)
Example block output

Get values and derivatives together

The above example showed how to calculate the power spectrum values and their derivatives through two separate calls. If you need both, it is faster to calculate them simultaneously with the package DiffResults.jl:

using DiffResults

# Following DiffResults documentation:
Pres = DiffResults.JacobianResult(ks/u"1/Mpc", θ) # allocate buffer for values+derivatives for a function with θ-sized input and ks-sized output
Pres = ForwardDiff.jacobian!(Pres, lgP, log10.(θ)) # evaluate values+derivatives of lgP(log10.(θ)) and store the results in Pres
lgPs = DiffResults.value(Pres) # extract values
dlgP_dlgθs = DiffResults.jacobian(Pres) # extract derivatives

p1 = plot(
   log10.(ks/u"1/Mpc"), lgPs;
   ylabel = "lg(P/Mpc³)", label = nothing
)
p2 = plot(
   log10.(ks/u"1/Mpc"), dlgP_dlgθs;
   xlabel = "lg(k/Mpc⁻¹)", ylabel = "∂ lg(P) / ∂ lg(θᵢ)",
   labels = "θᵢ=" .* ["Tγ0" "Ωc0" "Ωb0" "Neff" "h" "YHe" "mh" "ln(10¹⁰As)" "ns"]
)
plot(p1, p2, layout=(2, 1), size = (600, 600))
Example block output

General approach

The technique shown here can be used to calculate the derivative of any SymBoltz.jl output quantity:

  1. Write a wrapper function output(input) that calculates the desired output quantities from the desired input quantities.
  2. Use ForwardDiff.derivative(output, input) (scalar-to-scalar), ForwardDiff.gradient(output, input) (vector-to-scalar) or ForwardDiff.jacobian(output, input) (vector-to-vector) to evaluate the derivative of output at the values input. Or use the similar functions in DiffResults to calculate the value and derivatives simultaneously.