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2_ebm.jl
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### A Pluto.jl notebook ###
# v0.19.40
using Markdown
using InteractiveUtils
# ╔═╡ 1fbb5414-3483-11ef-2f91-affe69c5f577
begin
using Plots, CSV, DataFrames, NaNStatistics, DifferentialEquations, ModelingToolkit
import PlotlyJS
plotlyjs()
end
# ╔═╡ d584e21d-3c8e-4fd9-9818-33cadb782ac9
md"""# Earth Energy Balance Model
**Main Topics:**
- The [SciML](https://sciml.ai/) Organization
- DataFrames
- Data Visualization
- Differential Equations
- [Flux.jl](https://fluxml.ai/Flux.jl/stable/) and Neural ODE
"""
# ╔═╡ c5765cae-28fd-439e-b5ce-844b5637d2e4
begin
CO2_historical_path = download("https://www.epa.gov/system/files/other-files/2022-07/ghg-concentrations_fig-1.csv")
offset = findfirst(startswith("Year"), readlines(CO2_historical_path))
CO2_historical_data = CSV.read(
CO2_historical_path, DataFrame;
header=offset,
skipto=offset+2,
)
end
# ╔═╡ 503a198e-6021-43f2-b30e-9b10372c86d1
@view CO2_historical_data[end-250:end, :] # avoid copying data
# ╔═╡ d6ecc837-be55-4db7-bac7-8446e7a3cfa2
begin
CO2_ts = CO2_historical_data[CO2_historical_data.Year .>= 1850, :]
values = replace(Matrix(CO2_ts[:,2:end]), missing=>NaN)
CO2_ts.CO2 .= nanmean(values, dims=2)
select!(CO2_ts, :Year, :CO2) # drop other columns in-place
first(CO2_ts, 5), last(CO2_ts, 5)
end
# ╔═╡ 96e1e5c6-f4b2-48f1-8a85-c031c915e1d5
md"More information on how to use dataframes in Julia, in comparison with Pandas in Python: [https://dataframes.juliadata.org/stable/man/comparisons/](https://dataframes.juliadata.org/stable/man/comparisons/)."
# ╔═╡ 52891ead-46a3-4ceb-8be9-e6539fb5aa54
md"#### Fit a cubic curve to the CO2 time series"
# ╔═╡ 8264a6cf-803c-4f37-b516-b82be1000a38
begin
CO2_PreIndust = first(CO2_ts).CO2
time_features(ts) = let x = ts .- 1850; [x x.^2 x.^3] end
CO2(t) = time_features(t) * CO2_coefs .+ CO2_PreIndust
CO2_coefs = let
X = time_features(CO2_ts[:, "Year"])
y = CO2_ts[:, "CO2"] .- CO2_PreIndust
p = X \ y # equivalent to pinv(X) * y, gives the least squares approximation
end
CO2(2017:2027)
end
# ╔═╡ aa1a215a-3a32-48d9-a7f6-8af49a703354
CO2_coefs
# ╔═╡ f539ddd6-041f-4b84-91c0-aca74131c960
begin
years = 1850:2030
plot(CO2_ts[:, "Year"] , CO2_ts[:, "CO2"],
label="Global atmospheric CO₂ concentration")
plot!(years, CO2(years), label="Fitted curve", legend=:bottomright)
title!("CO₂ observations and fit")
end
# ╔═╡ e39a69fe-0c82-482d-9a69-a5b028610c62
md"""
**Incoming Solar Insolation:**
The incoming solar insolation is proportional to the cross-sectional area ``S=\pi R^2`` of the Earth.
**Outgoing thermal radiation:**
The outgoing thermal radiation is primarily a function of the global temperature, involving processes like blackbody radiation, water vapor feedback, etc. Since it's very complicated, we use a linear model to approximate it.
**Greenhouse effect:**
Empirically, the greenhouse effect is known to be a logarithmic function of carbon dioxide (CO₂) concentrations in the atmosphere.
**Reference:**
- [https://computationalthinking.mit.edu/Fall24/climate\_science/our\_first\_climate_model/](https://computationalthinking.mit.edu/Fall24/climate_science/our_first_climate_model/)
- [https://florianboergel.github.io/climateoftheocean/2020-11-11-energy-model.html](https://florianboergel.github.io/climateoftheocean/2020-11-11-energy-model.html)"""
# ╔═╡ c5d362b6-e0ed-4742-9ec6-1d1a9d03ac58
begin
@parameters t α a S B C T_0
@variables T(t) R(t) G(T, t)
absorbed_solar_radiation = (1 - α) * S / 4
outgoing_thermal_radiation = G
greenhouse_effect = a * log(R)
D = Differential(t)
eqs = [
C * D(T) ~
absorbed_solar_radiation -
outgoing_thermal_radiation +
greenhouse_effect,
G ~ absorbed_solar_radiation + B * (T_0 - T),
R ~ only(CO2(t) / CO2(1850)), # takes the value of a single-element array
]
end
# ╔═╡ 70db5d4d-e346-476a-8bba-50070260abe5
@mtkbuild sys = ODESystem(eqs, t)
# ╔═╡ 4084d556-5673-444f-bb9f-e77a502799d5
begin
T0 = 14.0
ini = [T => T0] # initial condition
ps = [ # parameters
a => 5., # CO2 forcing coefficient [W/m^2]
α => 0.3, # albedo
C => 51., # atmosphere and upper-ocean heat capacity
S => 1368., # solar insolation [W/m^2]
B => -1.3, # climate feedback parameter [W/m^2/°C]
T_0 => T0, # preindustrial time (assume thermal equilibrium)
]
tspan = (1850, 2024)
prob = ODEProblem(sys, ini, tspan, ps)
end
# ╔═╡ 6b608521-626c-4366-b1ce-5520d2c71529
solution = solve(prob)
# ╔═╡ f314a58a-3d15-4035-bb75-68897c714071
solution(2020:2030)
# ╔═╡ 29f5d3a1-b805-4c90-b602-0d2de83d0152
begin
temps = vec(solution(1880:2030))
plot(1880:2030, temps, lw=2, legend=:bottomright,
label="Predicted Temperature from model")
title!("Global Average Temperature")
xlabel!("year")
ylabel!("Temp °C")
end
# ╔═╡ b29d66d4-ef28-4de8-acc1-7d4fe607742b
begin
T_url = "https://data.giss.nasa.gov/gistemp/graphs/graph_data/Global_Mean_Estimates_based_on_Land_and_Ocean_Data/graph.txt"
s = read(download(T_url), String)
io = replace(s, r" +" => " ") |> IOBuffer
T_df = CSV.read(io, DataFrame, header=false, skipto=6);
T_df = rename(T_df[:,1:2], :Column1=>:year, :Column2=>:temp)
T_df.temp .+= 14.15
T_df
end
# ╔═╡ 7f4af72c-9ca1-4b65-8557-9d59880ba261
plot!(T_df[:, :year], T_df[:, :temp], color=:black, label="NASA Observations")
# ╔═╡ 9295aed5-1fcc-455f-83fd-8e8996dae6c6
sum(abs2, vec(solution(T_df.year)) - T_df.temp) / size(T_df, 1) # MSE
# ╔═╡ a2bd2e6c-4908-472a-9cd7-b7bf22c313e2
md"""The reason why the predicted temperature is first higher than the observation and becomes lower later is probably due to the coarse estimation of the climate feedbacks (linearly parametrized by ``B`` ).
Therefore, let's use a neural ODE to better approximate the real observations. We will use a Flux neural network to approximate the outgoing radiation term, replacing the linear climate feedback."""
# ╔═╡ 68b04370-7d3e-4b6a-a07a-e11b90ce440e
begin
@variables G_NN(T, t)
new_sys = substitute(sys, [ps; G => G_NN; R => sys.eqs[3].rhs])
end
# ╔═╡ 9a93642b-7fca-4346-914b-e9a1ab6c2bc9
new_dT = simplify(new_sys.eqs[1].rhs, expand=true) # formula of dT/dt
# ╔═╡ 981f864e-da3f-4f27-996b-3172ae0fe76f
begin
using Flux, SciMLSensitivity, Random
Random.seed!(32)
init = Flux.kaiming_normal
nn = Chain(Dense(3,16,selu,init=init),
Dense(16,2,init=init)) |> f64 # float32 by default
params, reconstruct = Flux.destructure(nn)
"The outgoing thermal radiation term `G_NN(T, t)` estimated by the NN."
function nn_term(nn::Chain, T, t)
x = (t - 1850) / 200
c = nn([x, x^2, x^3])
return c[1] + c[2]T
end
function dTdt(dT, T, ps, yr)
nn = reconstruct(ps)
g = nn_term(nn, T[1], yr) # estimation of longwave radiation
dT[1] = substitute(new_dT, Dict(G_NN => g, t => yr))
end
nn_prob = ODEProblem(dTdt, [T0], tspan, params)
end
# ╔═╡ af156b54-629a-48e7-8065-e426b17e8224
begin
true_temp = T_df[!, :temp]
mask = T_df.year[1] .<= range(tspan...) .<= T_df.year[end]
predict_temp() = vec(solve(nn_prob, Tsit5(), p=params, saveat=1))[mask]
loss() = sum(abs2, predict_temp() .- true_temp) / length(true_temp)
losses = []
history = []
callback = function ()
push!(losses, loss())
push!(history, predict_temp())
end
# training phase 1 (initial error is huge)
opt = Adam(0.5)
data = Iterators.repeated((), 100) # () is the input to the loss function
Flux.train!(loss, Flux.params(params), data, opt, cb=callback)
foreach(println, losses)
# training phase 2 (for faster convergence)
opt = Adam(0.01)
data = Iterators.repeated((), 200) # () is the input to the loss function
empty!(losses)
Flux.train!(loss, Flux.params(params), data, opt, cb=callback)
end
# ╔═╡ 512c0d2e-25ed-44fe-a0f9-f09af0d1be49
plot(losses, title="training loss", xlabel="epoch", legend=false)
# ╔═╡ d16cea4f-8139-423a-af49-cdadac7e116f
md"Average error: $(round(sqrt(loss()), digits=2))°C"
# ╔═╡ 840b3aae-3ea6-4726-bfea-569ddb69e103
@gif for p in history[2:2:end] # plot each frame in this for-loop to make animation
plot(T_df[:, :year], p, ylim=(13.6,15.4), label="Predicted Temperature", legend=:bottomright)
plot!(T_df[:, :year], T_df[:, :temp], color=:black, label="NASA Observations")
title!("Globle Average Temperature")
end
# ╔═╡ ea8427d8-37f5-4e69-ab1f-e53f4031610d
begin # forecast into the future
ts = (1880, 2050)
pred = Vector(solve(nn_prob, Tsit5(), p=params, tspan=ts, saveat=1))
plot(range(ts...), pred, label="Predicted Temperature", legend=:bottomright)
plot!(T_df[:, :year], T_df[:, :temp], color=:black, label="NASA Observations")
end
# ╔═╡ 6731f051-55c0-4a52-9995-05e975d5fb36
md"""Our result shows that if GHG emissions keep the current tendency of growth, it's likely that we will approach **2°C** global warming above the pre-industrial level.
The following figure shows global warming predictions in the [IPCC 6th Assessment Report](https://www.ipcc.ch/report/ar6/syr/) ([Figure SPM.4](https://www.ipcc.ch/report/ar6/syr/figures/figure-spm-4)) for reference. We can see that even with the *intermediate* emission scenario, a 2°C of global warming is very likely to happen by 2050. The prediction of our model is in accord with this result.
"""
# ╔═╡ 587767ef-3a3a-4e29-9427-7d8fcb806abd
md"""## Further Reading
- [The Fast Track to Julia](https://cheatsheet.juliadocs.org/)
- [Introduction to Computational Thinking at MIT](https://computationalthinking.mit.edu/)
- [A Degree of Concern: Why Global Temperatures Matter](https://climate.nasa.gov/news/2865/a-degree-of-concern-why-global-temperatures-matter/)
- [What is the Paris climate agreement and why does 1.5C matter?](https://www.bbc.co.uk/news/science-environment-35073297)
- [Why did the IPCC choose 2° C as the goal for limiting global warming?](https://climate.mit.edu/ask-mit/why-did-ipcc-choose-2deg-c-goal-limiting-global-warming)
"""
# ╔═╡ e39b3544-b348-4d93-ad5d-02a774576f0e
md"## Take-home Exercise"
# ╔═╡ 461898ec-3f89-4795-a238-f217ca072607
md"""
!!! danger "Task"
Given the observed temperatures `T(t)` from 1880 to 2023, calculate the outgoing thermal radiation `G(T(t), t)` for both the original ODE and the neural ODE. Plot the results together in a single figure.
"""
# ╔═╡ 856ee0d5-b1db-4843-942c-937758feb656
let
md"(Show this cell for a sample solution)"
# trained_nn = reconstruct(params)
# olr(temp) = substitute(sys.eqs[2].rhs, [T => temp, ps...])
# G_orig = olr.(T_df.temp)
# olr1(temp, year) = nn_term(trained_nn, temp, year)
# G_node = olr1.(T_df.temp, T_df.year)
# plot(T_df.year, G_orig, label="OLR in original ODE", ylabel="W/m²")
# plot!(T_df.year, G_node, label="OLR in neural ODE")
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b"
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c"
ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78"
NaNStatistics = "b946abbf-3ea7-4610-9019-9858bfdeaf2d"
PlotlyJS = "f0f68f2c-4968-5e81-91da-67840de0976a"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
SciMLSensitivity = "1ed8b502-d754-442c-8d5d-10ac956f44a1"
[compat]
CSV = "~0.10.14"
DataFrames = "~1.6.1"
DifferentialEquations = "~7.13.0"
Flux = "~0.14.16"
ModelingToolkit = "~9.19.0"
NaNStatistics = "~0.6.36"
PlotlyJS = "~0.18.13"
Plots = "~1.40.4"
SciMLSensitivity = "~7.61.1"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
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