Intro to InverseProblem

This example illustrates the construction of an "ensemble simulation" that can evaluate an ensemble of column models given an ensemble of free parameter sets. The example then builds an InverseProblem from observations, an ensemble simulation, and a set of free parameters, and illustrates its basic usage.

Install dependencies

First let's make sure we have all required packages installed.

using Pkg
pkg"add ParameterEstimocean, Oceananigans, Distributions, CairoMakie"

First we load few things

using ParameterEstimocean

using Oceananigans
using Oceananigans.Architectures: arch_array
using Oceananigans.Units
using Oceananigans.Models.HydrostaticFreeSurfaceModels: ColumnEnsembleSize
using Oceananigans.TurbulenceClosures: ConvectiveAdjustmentVerticalDiffusivity

using CairoMakie
using Distributions
using JLD2

We reuse some utilities from a previous example to build observations:

examples_path = joinpath(pathof(ParameterEstimocean), "..", "..", "examples")
include(joinpath(examples_path, "intro_to_observations.jl"))
data_path = generate_synthetic_observations()
observations = SyntheticObservations(data_path, field_names=:b, transformation=ZScore())

SyntheticObservations with fields (:b,)
├── times: [0 s, 4 hrs, 8 hrs, 12 hrs]
├── grid: 1×1×32 RectilinearGrid{Float64, Oceananigans.Grids.Flat, Oceananigans.Grids.Flat, Oceananigans.Grids.Bounded} on Oceananigans.Architectures.CPU with 0×0×1 halo
├── path: "convective_adjustment.jld2"
├── metadata: (:parameters, :grid, :coriolis, :closure)
└── transformation: Dict{Symbol, ParameterEstimocean.Transformations.Transformation{TimeIndices{UnitRange{Int64}}, Nothing, ZScore{Float64}}} with 1 entry

Building an "ensemble simulation"

Our next task is to construct a parameterized Oceananigans.Simulation that generates the "foward map" for an ensemble of free parameter sets. To generate an ensemble of column model model outputs efficiently, we construct one 3D Oceananigans.Simulation consisting of Nx by Ny independent column models.

The calibration problem then uses the ensemble simulation to find optimal parameters by minimizing the discrepency between the observations and the forward map.

"""
    extract_perfect_parameters(observations, Nensemble)

Extract parameters from a batch of "perfect" observations.
"""
function extract_perfect_parameters(observations, Nensemble)
    Nbatch = length(observations)
    Qᵘ, Qᵇ, N², f = [zeros(Nensemble, Nbatch) for i = 1:4]

    Nz = first(observations).grid.Nz
    Hz = first(observations).grid.Hz
    Lz = first(observations).grid.Lz
    Δt = first(observations).metadata.parameters.Δt

    for (j, obs) in enumerate(observations)
        Qᵘ[:, j] .= obs.metadata.parameters.Qᵘ
        Qᵇ[:, j] .= obs.metadata.parameters.Qᵇ
        N²[:, j] .= obs.metadata.parameters.N²
        f[:, j] .= obs.metadata.coriolis.f
    end

    file = jldopen(first(observations).path)
    closure = file["serialized/closure"]
    close(file)

    return Qᵘ, Qᵇ, N², f, Δt, Lz, Nz, Hz, closure
end

"""
    build_ensemble_simulation(observations, arch=CPU(); Nensemble=1)

Returns an `Oceananigans.Simulation` representing an `Nensemble × 1`
ensemble of column models designed to reproduce `observations`.
"""
function build_ensemble_simulation(observations, arch=CPU(); Nensemble=1)

    observations isa Vector || (observations = [observations]) # Singleton batch
    Nbatch = length(observations)

    Qᵘ, Qᵇ, N², f, Δt, Lz, Nz, Hz, closure = extract_perfect_parameters(observations, Nensemble)

    column_ensemble_size = ColumnEnsembleSize(Nz=Nz, ensemble=(Nensemble, Nbatch), Hz=Hz)
    ensemble_grid = RectilinearGrid(arch, size = column_ensemble_size, topology = (Flat, Flat, Bounded), z = (-Lz, 0))

    coriolis_ensemble = arch_array(arch, [FPlane(f=f[i, j]) for i = 1:Nensemble, j=1:Nbatch])
    closure_ensemble = arch_array(arch, [deepcopy(closure) for i = 1:Nensemble, j=1:Nbatch])

    Qᵘ, Qᵇ, N² = Tuple(arch_array(arch, p) for p in (Qᵘ, Qᵇ, N²))

    u_bcs = FieldBoundaryConditions(top = FluxBoundaryCondition(Qᵘ))
    b_bcs = FieldBoundaryConditions(top = FluxBoundaryCondition(Qᵇ), bottom = GradientBoundaryCondition(N²))

    tracers = first(observations).metadata.parameters.tracers

    ensemble_model = HydrostaticFreeSurfaceModel(grid = ensemble_grid,
                                                 tracers = tracers,
                                                 buoyancy = BuoyancyTracer(),
                                                 boundary_conditions = (; u=u_bcs, b=b_bcs),
                                                 coriolis = coriolis_ensemble,
                                                 closure = closure_ensemble)

    ensemble_simulation = Simulation(ensemble_model; Δt=Δt, stop_time=first(observations).times[end])

    return ensemble_simulation, closure
end

Main.build_ensemble_simulation

The following illustrations uses a simple ensemble simulation with two ensemble members:

ensemble_simulation, closure★ = build_ensemble_simulation(observations; Nensemble=3)

(Simulation of HydrostaticFreeSurfaceModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 10 seconds
├── Elapsed wall time: 0 seconds
├── Stop time: 12 hours
├── Stop iteration : Inf
├── Wall time limit: Inf
├── Callbacks: OrderedDict with 4 entries:
│   ├── stop_time_exceeded => Callback of stop_time_exceeded on IterationInterval(1)
│   ├── stop_iteration_exceeded => Callback of stop_iteration_exceeded on IterationInterval(1)
│   ├── wall_time_limit_exceeded => Callback of wall_time_limit_exceeded on IterationInterval(1)
│   └── nan_checker => Callback of NaNChecker for u on IterationInterval(100)
├── Output writers: OrderedDict with no entries
└── Diagnostics: OrderedDict with no entries, ConvectiveAdjustmentVerticalDiffusivity{Oceananigans.TurbulenceClosures.VerticallyImplicitTimeDiscretization}(background_κz=0.0001 convective_κz=1.0 background_νz=1.0e-5 convective_νz=0.9))

Free parameters

We construct some prior distributions for our free parameters. We found that it often helps to constrain the prior distributions so that neither very high nor very low values for diffusivities can be drawn out of the distribution.

priors = (convective_κz = lognormal(mean=0.3, std=0.05),
          background_κz = lognormal(mean=2.5e-4, std=0.25e-4))

free_parameters = FreeParameters(priors)

FreeParameters with 2 parameters
├── names: (:convective_κz, :background_κz)
└── priors: Dict{Symbol, Any}
    ├── convective_κz => LogNormal{Float64}(μ=-1.2176722914199933, σ=0.16552635496534787)
    └── background_κz => LogNormal{Float64}(μ=-8.299024805528612, σ=0.0997513451195927)

We also take the opportunity to collect a named tuple of the optimal parameters

θ★ = (convective_κz = closure★.convective_κz,
      background_κz = closure★.background_κz)

(convective_κz = 1.0, background_κz = 0.0001)

Visualizing the priors

We visualize our prior distributions by plotting a huge number of samples:

using ParameterEstimocean.Parameters: unconstrained_prior, transform_to_constrained

Nsamples = 50000000

sample(prior) = [transform_to_constrained(prior, X) for X in rand(unconstrained_prior(prior), Nsamples)]

convective_κz_samples = sample(priors.convective_κz)
background_κz_samples = sample(priors.background_κz)

fig = Figure()
ax_top = Axis(fig[1, 1], xlabel = "convective κᶻ [m² s⁻¹]", ylabel = "Density")
density!(ax_top, convective_κz_samples)
xlims!(ax_top, 0, 10)

ax_bottom = Axis(fig[2, 1], xlabel = "background κᶻ [m² s⁻¹]", ylabel = "Density")
density!(ax_bottom, background_κz_samples)

save("prior_visualization.svg", fig)

The InverseProblem

We can construct the inverse problem $y = G(θ) + η$. Here, $y$ are the observations and $G$ is the ensemble_model.

calibration = InverseProblem(observations, ensemble_simulation, free_parameters)

InverseProblem{ConcatenatedOutputMap} with free parameters (:convective_κz, :background_κz)
├── observations: SyntheticObservations of (:b,) on 1×1×32 RectilinearGrid{Float64, Oceananigans.Grids.Flat, Oceananigans.Grids.Flat, Oceananigans.Grids.Bounded} on Oceananigans.Architectures.CPU with 0×0×1 halo
├── simulation: Simulation on 3×1×32 RectilinearGrid{Float64, Oceananigans.Grids.Flat, Oceananigans.Grids.Flat, Oceananigans.Grids.Bounded} on Oceananigans.Architectures.CPU with 0×0×1 halo with Δt=10.0
├── free_parameters: (:convective_κz, :background_κz)
└── output map: ConcatenatedOutputMap

Using InverseProblem to compute forward_map

As a sanity check we apply the forward_map on the calibration after we initialize all ensemble members with the true parameter values. We then confirm that the output of the forward_map matches the observations to machine precision.

θ¹ = (convective_κz = 0.8 * θ★.convective_κz,
      background_κz = 9.0 * θ★.background_κz)

θ² = (convective_κz = 2.0 * θ★.convective_κz,
      background_κz = 0.1 * θ★.background_κz)

θ_ensemble = [θ★, θ¹, θ²]

G = forward_map(calibration, θ_ensemble)
y = observation_map(calibration)

96×1 Matrix{Float64}:
 -1.827652672852313
 -1.6951720731798834
 -1.562691473507566
 -1.430210873835191
 -1.2977302741632166
 -1.1652496744909402
 -1.0327690748188671
 -0.900288475146917
 -0.7678078754746136
 -0.635327275802567
  ⋮
  0.3899979925638174
  0.38938525034955224
  0.38868351035616716
  0.38789276851434884
  0.38701302133046905
  0.3860442658889935
  0.3849864998546511
  0.38383972147427636
  0.3826039295782583

The forward_map output G is a two-dimensional matrix whose first dimension is the size of the state space and whose second dimension is the ensemble_size. Here, we ensure that first ensemble member of the mapped output, which was run with the "true" parameters, is identical to the mapped observations:

G[:, 1] ≈ y

true

Visualizing forward model output

Next we visualize the discrepency between solutions generated by true and non-optimal parameter sets θ¹ and θ². Time-series data from the ensemble run is collected by calibration.time_series_collector:

time_series_collector = calibration.time_series_collector
times = time_series_collector.times

# Extract last save point and plot each solution component
Nt = length(times)

b = time_series_collector.field_time_serieses.b[Nt]
t = times[Nt]
z = znodes(b)

# The ensemble varies along the first, or `x`-dimension:
b★ = interior(b)[1, 1, :]
b¹ = interior(b)[2, 1, :]
b² = interior(b)[3, 1, :]

fig = Figure()
ax = Axis(fig[1, 1],
          xlabel = "Buoyancy [m s⁻²]",
          ylabel = "Depth [m]")

b★_label = "true b at t = " * prettytime(t)
b¹_label = "b with $θ¹"
b²_label = "b with $θ²"

lines!(ax, b★, z; label=b★_label, linewidth=2)
lines!(ax, b¹, z; label=b¹_label, linewidth=2)
lines!(ax, b², z; label=b²_label, linewidth=2)

axislegend(ax, position=:lt)

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