Orbits/julia_code/BellmanResidualMinimization.jl

### A Pluto.jl notebook ###
# v0.17.0

using Markdown
using InteractiveUtils

# ╔═╡ 0829eb90-1d74-46b6-80ec-482a2b71c6fe
using PlutoUI, Flux,LinearAlgebra

# ╔═╡ 66f0e667-d722-4e1e-807b-84a39cbc41b1
md"""
# Bellman Residual Minimization


"""

# ╔═╡ 9fa41b7c-1923-4c1e-bfc6-20ce4a1a2ede
md"""
Number of Constellations: $(const N_constellations = 3)

Number of Debris Trackers: $(const N_debris = 1)

Number of Overall States: $(const N_states = N_constellations + N_debris)
"""

# ╔═╡ 90446134-4e45-471c-857d-4e165e51937a
begin
	md"""
	Parameterization
	"""
	abstract type PhysicalParameters end
	
	#setup physical model
	struct BasicModel <: PhysicalParameters
	    #rate at which debris hits satellites
	    debris_collision_rate::Real
	    #rate at which satellites of different constellations collide
	    satellite_collision_rates::Matrix{Float64}
	    #rate at which debris exits orbits
	    decay_rate::Real
	    #rate at which satellites
	    autocatalysis_rate::Real
	    #ratio at which a collision between satellites produced debris
	    satellite_collision_debris_ratio::Real
	    #Ratio at which launches produce debris
	    launch_debris_ratio::Real
	end

		#Getting loss parameters together.
	loss_param = 2e-3;
	loss_weights = loss_param*(ones(N_constellations,N_constellations) - LinearAlgebra.I);
	
	#orbital decay rate
	decay_param = 0.01;
	
	#debris generation parameters
	autocatalysis_param = 0.001;
	satellite_loss_debris_rate = 5.0;
	launch_debris_rate = 0.05;
	
	#Todo, wrap physical model as a struct with the parameters
	bm = BasicModel(
	    loss_param
	    ,loss_weights
	    ,decay_param
	    ,autocatalysis_param
	    ,satellite_loss_debris_rate
	    ,launch_debris_rate
	)
end

# ╔═╡ 5b45b29e-f0f4-41e9-91e7-d444687feb4e
#implement survival function
function survival(
		stocks::Array
		,debris::Array
		,physical_model::BasicModel
	)	
    return exp.(
		-physical_model.satellite_collision_rates * stocks 
		.- (physical_model.debris_collision_rate*debris)
	)
end

# ╔═╡ 152f3a3c-a565-41bb-8e59-6ab0d2315ffb
#debris evolution 
function H(
		stocks::Array
		,debris::Array
		,launches::Array
		, physical_model::BasicModel
	)
    #get changes in debris from natural dynamics
    natural_debris_dynamics = (1-physical_model.decay_rate+physical_model.autocatalysis_rate) * debris 
    
    #get changes in debris from satellite loss
    satellite_loss_debris = physical_model.satellite_collision_debris_ratio * (1 .- survival(stocks,debris,physical_model))'*stocks 
    
    #get changes in debris from launches
    launch_debris = physical_model.launch_debris_ratio*sum(launches)
    
    #return total debris level
    return natural_debris_dynamics .+ satellite_loss_debris .+ launch_debris
end

# ╔═╡ 25ac9438-2b1d-4f6b-9ff1-1695e1d52b51
#stock update rules
function G(
		stocks::Array
		,debris::Array
		,launches::Array
		, physical_model::BasicModel
)
    return LinearAlgebra.diagm(survival(stocks,debris,physical_model) .- physical_model.decay_rate)*stocks + launches
end

# ╔═╡ 29ff1777-d276-4e8f-8582-4ca191f2e2ff
md"""
## Setup Neural Networks
"""

# ╔═╡ f7aabe43-9a2c-4fe0-8099-c29cdf66566c
function value_function_generator(number_params=10)
	return Flux.Chain(
	    Flux.Parallel(vcat
		        #parallel joins together stocks and debris
		        ,Flux.Chain(
					Flux.Dense(N_constellations, N_states*2,Flux.relu)
		            #,Flux.Dense(N_states*2, N_states*2,Flux.σ)
		        )
		        ,Flux.Chain(
					Flux.Dense(N_debris, N_states,Flux.relu)
		            #,Flux.Dense(N_states, N_states)
				)
			)
		    #Apply some transformations
			,Flux.Dense(N_states*3,number_params,Flux.σ)
			,Flux.Dense(number_params,1,Flux.σ)
		)

end

# ╔═╡ d816b252-bdca-44ba-ac5c-cb21163a1e9a
function policy_function_generator(number_params=10)
	return Flux.Chain(
	    Flux.Parallel(vcat
		        #parallel joins together stocks and debris
		        ,Flux.Chain(
					Flux.Dense(N_constellations, N_states*2,Flux.relu)
		            #,Flux.Dense(N_states*2, N_states*2,Flux.σ)
		        )
		        ,Flux.Chain(
					Flux.Dense(N_debris, N_states,Flux.relu)
		            #,Flux.Dense(N_states, N_states)
				)
			)
		    #Apply some transformations
			,Flux.Dense(N_states*3,number_params,Flux.σ)
			,Flux.Dense(number_params,N_constellations,Flux.σ)
		)

end

# ╔═╡ 95bfc9d8-8427-41d6-9f0f-f155296eef91
#not needed
begin
	#=
	Test a return in tuples. Just to see what can happen.
	=#
	passthrough(x::Array) = x
	Tuple(a::Array,b::Array) = (a,b)
	Flux.Parallel(Tuple,
		passthrough,passthrough
	)(([1,2],[3,4]));
end

# ╔═╡ fb6aacff-c42d-4ec1-88cb-5ce1b2e8874f
begin
	value = value_function_generator();
	policy = policy_function_generator();
end

# ╔═╡ 206ac4cc-5102-4381-ad8a-777b02dc4d5a
begin
	abstract type EconomicParameters end
	struct EconModel1 <: EconomicParameters
		β::Real
		payoff_array::Array{Real}
		policy_costs::Array{Real}
	end
end

# ╔═╡ 65e0b1fa-d5e1-4ff6-8736-c9d6b5f40150
em = EconModel1(0.95, [1 0 0 ], [5 0 0 ])

# ╔═╡ 1cbaa2e5-55e4-46f9-82d0-04b481470094
function payoff1(
	s::Vector
	,d::Vector
	,a::Vector
	,em::EconModel1
)
	return em.payoff_array*s - em.policy_costs*a
end

# ╔═╡ 41271ab4-1ec7-431f-9efb-0f7c3da2d8b4
function Ξ(
	s::Vector
	,d::Vector
	, physical_model::PhysicalParameters
	,econ_params::EconomicParameters
	,payoff_fn::Function
)
	a = policy((s,d))
	s′ = G(s,d,a,physical_model)
	d′ = H(s,d,a,physical_model)

	bellman_residuals = value((s,d)) - payoff_fn(s,d,a,econ_params) - econ_params.β*value((s′,d′))
	maximization_condition = - payoff_fn(s,d,a,econ_params) - econ_params.β*value((s′,d′))

	return sum([bellman_residuals.^2 maximization_condition])
end

# ╔═╡ a20959be-65e4-4b69-9521-503bc59f0854
begin
	N=12
	data = [(rand(1:500, N_constellations),rand(1:500, N_debris)) for n=1:N]
end

# ╔═╡ dff642d9-ec5a-4fed-a059-6c07760a3a58
#loss function
loss(s,d) = Ξ(s,d,bm,em,payoff1)

# ╔═╡ 20c777b5-4295-4478-8f53-b18cd409c8ae
s1 = ones(N_constellations)

# ╔═╡ 1d65707c-6333-4252-ace6-bad47146ba06
d1 = ones(N_debris)

# ╔═╡ 43b99708-0052-4b78-886c-92ac2b532f29
Ξ(s1,d1,bm,em,payoff1)

# ╔═╡ 39433c1a-c3ac-45b0-b1bf-ff2d42ca9cbb


# ╔═╡ 6bf8d29a-7990-4e91-86e6-d9894ed3db27
#optimizer
ADAM = Flux.Optimise.ADAM()

# ╔═╡ 74f5fde3-0593-46fc-a688-f1db7ab28c64
for epoch in 1:200
	#train the policy funciton
	Flux.Optimise.train!(loss, params(policy), data, ADAM)

	#Sweep through the value functions:w
	
	#Train the value functions
	Flux.Optimise.train!(loss, params(value), data, ADAM)
end

# ╔═╡ 02f3fe78-e7a7-453f-9ddf-acddf08d8676
begin
	accum = 0.0
	for d in data
		accum += loss(d...)
	end
	accum/N
end

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uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"

[[Libdl]]
uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb"

[[LinearAlgebra]]
deps = ["Libdl"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[LogExpFunctions]]
deps = ["ChainRulesCore", "DocStringExtensions", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"]
git-tree-sha1 = "6193c3815f13ba1b78a51ce391db8be016ae9214"
uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
version = "0.3.4"

[[Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "3d3e902b31198a27340d0bf00d6ac452866021cf"
uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
version = "0.5.9"

[[Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Media]]
deps = ["MacroTools", "Test"]
git-tree-sha1 = "75a54abd10709c01f1b86b84ec225d26e840ed58"
uuid = "e89f7d12-3494-54d1-8411-f7d8b9ae1f27"
version = "0.5.0"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.2"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"

[[NNlib]]
deps = ["Adapt", "ChainRulesCore", "Compat", "LinearAlgebra", "Pkg", "Requires", "Statistics"]
git-tree-sha1 = "5203a4532ad28c44f82c76634ad621d7c90abcbd"
uuid = "872c559c-99b0-510c-b3b7-b6c96a88d5cd"
version = "0.7.29"

[[NNlibCUDA]]
deps = ["CUDA", "LinearAlgebra", "NNlib", "Random", "Statistics"]
git-tree-sha1 = "04490d5e7570c038b1cb0f5c3627597181cc15a9"
uuid = "a00861dc-f156-4864-bf3c-e6376f28a68d"
version = "0.1.9"

[[NaNMath]]
git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.5"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"

[[OpenLibm_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "05823500-19ac-5b8b-9628-191a04bc5112"

[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
version = "0.5.5+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "ae4bbcadb2906ccc085cf52ac286dc1377dceccc"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.1.2"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlutoUI]]
deps = ["AbstractPlutoDingetjes", "Base64", "Dates", "Hyperscript", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"]
git-tree-sha1 = "57312c7ecad39566319ccf5aa717a20788eb8c1f"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.18"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.2"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[Profile]]
deps = ["Printf"]
uuid = "9abbd945-dff8-562f-b5e8-e1ebf5ef1b79"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[Random123]]
deps = ["Libdl", "Random", "RandomNumbers"]
git-tree-sha1 = "0e8b146557ad1c6deb1367655e052276690e71a3"
uuid = "74087812-796a-5b5d-8853-05524746bad3"
version = "1.4.2"

[[RandomNumbers]]
deps = ["Random", "Requires"]
git-tree-sha1 = "043da614cc7e95c703498a491e2c21f58a2b8111"
uuid = "e6cf234a-135c-5ec9-84dd-332b85af5143"
version = "1.5.3"

[[RealDot]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "9f0a1b71baaf7650f4fa8a1d168c7fb6ee41f0c9"
uuid = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
version = "0.1.0"

[[Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.1.3"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "f0bccf98e16759818ffc5d97ac3ebf87eb950150"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "1.8.1"

[[Static]]
deps = ["IfElse"]
git-tree-sha1 = "e7bc80dc93f50857a5d1e3c8121495852f407e6a"
uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
version = "0.4.0"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.13"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "eb35dcc66558b2dda84079b9a1be17557d32091a"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.12"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[TimerOutputs]]
deps = ["ExprTools", "Printf"]
git-tree-sha1 = "7cb456f358e8f9d102a8b25e8dfedf58fa5689bc"
uuid = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
version = "0.5.13"

[[TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "216b95ea110b5972db65aa90f88d8d89dcb8851c"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.6"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[ZipFile]]
deps = ["Libdl", "Printf", "Zlib_jll"]
git-tree-sha1 = "3593e69e469d2111389a9bd06bac1f3d730ac6de"
uuid = "a5390f91-8eb1-5f08-bee0-b1d1ffed6cea"
version = "0.9.4"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zygote]]
deps = ["AbstractFFTs", "ChainRules", "ChainRulesCore", "DiffRules", "Distributed", "FillArrays", "ForwardDiff", "IRTools", "InteractiveUtils", "LinearAlgebra", "MacroTools", "NaNMath", "Random", "Requires", "SpecialFunctions", "Statistics", "ZygoteRules"]
git-tree-sha1 = "0fc9959bcabc4668c403810b4e851f6b8962eac9"
uuid = "e88e6eb3-aa80-5325-afca-941959d7151f"
version = "0.6.29"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
"""

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