From 4f8d54949719f1ad13f371e1672099fc1ad63a92 Mon Sep 17 00:00:00 2001 From: DrWatson Date: Wed, 10 Feb 2021 21:42:07 +0100 Subject: [PATCH 1/5] Clean up --- DelayLotkaVolterra/Manifest.toml | 1653 ----------------- DelayLotkaVolterra/Project.toml | 19 - DelayLotkaVolterra/VolterraExp.jl | 259 --- DelayLotkaVolterra/eval_loop.jl | 63 - DelayLotkaVolterra/knowledge_enhanced_NN.jld2 | Bin 164118 -> 0 bytes DelayLotkaVolterra/loop_runs.jld2 | Bin 2280268 -> 0 bytes DelayLotkaVolterra/run_loop.jl | 191 -- DelayLotkaVolterra/sindy_extrapolation.pdf | Bin 58365 -> 0 bytes DelayLotkaVolterra/sindy_loop_training.pdf | Bin 540753 -> 0 bytes 9 files changed, 2185 deletions(-) delete mode 100644 DelayLotkaVolterra/Manifest.toml delete mode 100644 DelayLotkaVolterra/Project.toml delete mode 100644 DelayLotkaVolterra/VolterraExp.jl delete mode 100644 DelayLotkaVolterra/eval_loop.jl delete mode 100644 DelayLotkaVolterra/knowledge_enhanced_NN.jld2 delete mode 100644 DelayLotkaVolterra/loop_runs.jld2 delete mode 100644 DelayLotkaVolterra/run_loop.jl delete mode 100644 DelayLotkaVolterra/sindy_extrapolation.pdf delete mode 100644 DelayLotkaVolterra/sindy_loop_training.pdf diff --git a/DelayLotkaVolterra/Manifest.toml b/DelayLotkaVolterra/Manifest.toml deleted file mode 100644 index 1f3927a..0000000 --- a/DelayLotkaVolterra/Manifest.toml +++ /dev/null @@ -1,1653 +0,0 @@ -# This file is machine-generated - 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-[deps] -DataDrivenDiffEq = "2445eb08-9709-466a-b3fc-47e12bd697a2" -DiffEqFlux = "aae7a2af-3d4f-5e19-a356-7da93b79d9d0" -DiffEqSensitivity = "41bf760c-e81c-5289-8e54-58b1f1f8abe2" -DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa" -Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c" -JLD2 = "033835bb-8acc-5ee8-8aae-3f567f8a3819" -Juno = "e5e0dc1b-0480-54bc-9374-aad01c23163d" -Lasso = "b4fcebef-c861-5a0f-a7e2-ba9dc32b180a" -ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78" -Optim = "429524aa-4258-5aef-a3af-852621145aeb" -OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" -Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" -Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c" diff --git a/DelayLotkaVolterra/VolterraExp.jl b/DelayLotkaVolterra/VolterraExp.jl deleted file mode 100644 index b5883a7..0000000 --- a/DelayLotkaVolterra/VolterraExp.jl +++ /dev/null @@ -1,259 +0,0 @@ -cd(@__DIR__) -using Pkg; Pkg.activate("."); Pkg.instantiate() - -# Single experiment, move to ensemble further on -# Some good parameter values are stored as comments right now -# because this is really good practice - -using OrdinaryDiffEq -using ModelingToolkit -using DataDrivenDiffEq -using LinearAlgebra, DiffEqSensitivity, Optim -using DiffEqFlux, Flux -using Plots -gr() - -function lotka(du, u, p, t) - α, β, γ, δ = p - du[1] = α*u[1] - β*u[2]*u[1] - du[2] = γ*u[1]*u[2] - δ*u[2] -end - -# Define the experimental parameter -tspan = (0.0f0,3.0f0) -u0 = Float32[0.44249296,4.6280594] -p_ = Float32[1.3, 0.9, 0.8, 1.8] -prob = ODEProblem(lotka, u0,tspan, p_) -solution = solve(prob, Vern7(), abstol=1e-12, reltol=1e-12, saveat = 0.1) - -scatter(solution, alpha = 0.25) -plot!(solution, alpha = 0.5) - -# Ideal data -X = Array(solution) -# Add noise to the data -println("Generate noisy data") -Xₙ = X + Float32(1e-3)*randn(eltype(X), size(X)) - -# Define the neueral network which learns L(x, y, y(t-τ)) -# Actually, we do not care about overfitting right now, since we want to -# extract the derivative information without numerical differentiation. -L = FastChain(FastDense(2, 32, tanh),FastDense(32, 32, tanh), FastDense(32, 2)) -p = initial_params(L) - -function dudt_(u, p,t) - x, y = u - z = L(u,p) - [p_[1]*x + z[1], - -p_[4]*y + z[2]] -end - -prob_nn = ODEProblem(dudt_,u0, tspan, p) -sol_nn = solve(prob_nn, Tsit5(), u0 = u0, p = p, saveat = solution.t) - -plot(solution) -plot!(sol_nn) - -function predict(θ) - Array(solve(prob_nn, Vern7(), u0 = u0, p=θ, saveat = solution.t, - abstol=1e-6, reltol=1e-6, - sensealg = InterpolatingAdjoint(autojacvec=ReverseDiffVJP()))) -end - -# No regularisation right now -function loss(θ) - pred = predict(θ) - sum(abs2, Xₙ .- pred), pred -end - -# Test -loss(p) - -const losses = [] - -callback(θ,l,pred) = begin - push!(losses, l) - if length(losses)%50==0 - println("Current loss after $(length(losses)) iterations: $(losses[end])") - end - false -end - -# First train with ADAM for better convergence -res1 = DiffEqFlux.sciml_train(loss, p, ADAM(0.01), cb=callback, maxiters = 200) - -# Train with BFGS -res2 = DiffEqFlux.sciml_train(loss, res1.minimizer, BFGS(initial_stepnorm=0.01), cb=callback, maxiters = 10000) - -println("Final training loss after $(length(losses)) iterations: $(losses[end])") - -# Plot the losses -plot(losses, yaxis = :log, xaxis = :log, xlabel = "Iterations", ylabel = "Loss") - -# Plot the data and the approximation -NNsolution = predict(res2.minimizer) -# Trained on noisy data vs real solution -plot(solution.t, NNsolution') -plot!(solution.t, X') - -# Ideal derivatives -DX = Array(solution(solution.t, Val{1})) - -prob_nn2 = ODEProblem(dudt_,u0, tspan, res2.minimizer) -_sol = solve(prob_nn2, Tsit5()) -DX_ = Array(_sol(solution.t, Val{1})) - -# The learned derivatives -plot(DX') -plot!(DX_') - -# Ideal data -L̄ = [-p_[2]*(X[1,:].*X[2,:])';p_[3]*(X[1,:].*X[2,:])'] -# Neural network guess -L̂ = L(Xₙ,res2.minimizer) - -scatter(L̄') -plot!(L̂') - -# Plot the error -scatter(abs.(L̄-L̂)', yaxis = :log) - -## Sparse Identification - -# Create a Basis -@variables u[1:2] -# Lots of polynomials -polys = Any[1] - -for i ∈ 1:5 - push!(polys, u[1]^i) - push!(polys, u[2]^i) - for j ∈ i:5 - if i != j - push!(polys, (u[1]^i)*(u[2]^j)) - push!(polys, u[2]^i*u[1]^i) - end - end -end - -# And some other stuff -h = [cos.(u)...; sin.(u)...; polys...] -basis = Basis(h, u) - -# Create an optimizer for the SINDy problem -opt = SR3() -# Create the thresholds which should be used in the search process -λ = exp10.(-7:0.1:3) -# Target function to choose the results from; x = L0 of coefficients and L2-Error of the model -g(x) = x[1] < 1 ? Inf : norm(x, 2) -# Test on original data and without further knowledge -println("SINDy on full ideal, unavailable data") -Ψ = SINDy(Xₙ[:, :], DX[:, :], basis, λ, opt, g = g, maxiter = 10000) # Fail -println(Ψ) -print_equations(Ψ) - -# Test on ideal derivative data for unknown function ( not available ) -println("SINDy on partial ideal, unavailable data") -Ψ = SINDy(Xₙ[:, 1:end], L̄[:, 1:end], basis, λ,opt, g = g, maxiter = 10000) # Succeed -println(Ψ) -print_equations(Ψ) - -# Test on uode derivative data -println("SINDy on learned, partial, available data") -Ψ = SINDy(Xₙ[:, 2:end], L̂[:, 2:end], basis, λ, opt, g = g, maxiter = 10000, normalize = true, denoise = true) # Succeed -println(Ψ) -print_equations(Ψ) - -# Extract the parameter -p̂ = parameters(Ψ) -println("First parameter guess : $(p̂)") - -# The parameters are a bit off, but the equations are recovered -# Start another SINDy run to get closer to the ground truth -# Create function -unknown_sys = ODESystem(Ψ) -unknown_eq = ODEFunction(unknown_sys) - -# Just the equations -b = Basis((u, p, t)->unknown_eq(u, [1.; 1.], t), u) - -# Retune for better parameters -> we could also use DiffEqFlux or other parameter estimation tools here. -Ψf = SINDy(Xₙ[:, 2:end], L̂[:, 2:end], b, STRRidge(0.01), maxiter = 100, convergence_error = 1e-18) # Succeed -println(Ψf) -p̂ = parameters(Ψf) -println("Second parameter guess : $(p̂)") - -# Create function -recovered_sys = ODESystem(Ψf) -recovered_eq = ODEFunction(recovered_sys) - -# Build a ODE for the estimated system -function dudt(du, u, p, t) - # Add SINDy Term - α, δ, β, γ = p - z = recovered_eq(u, [β; γ], t) - du[1] = α*u[1] + z[1] - du[2] = -δ*u[2] + z[2] -end - -# Create the approximated problem and solution -ps = [p_[[1,4]]; p̂] -approximate_prob = ODEProblem(dudt, u0, tspan, ps) -approximate_solution = solve(approximate_prob, Tsit5(), saveat = 0.01) - -# Plot -plot(solution) -plot!(approximate_solution) - -## Simulation - -# Look at long term prediction -t_long = (0.0, 50.0) -approximate_prob = ODEProblem(dudt, u0, t_long, ps) -approximate_solution_long = solve(approximate_prob, Tsit5(), saveat = 0.1) # Using higher tolerances here results in exit of julia -plot(approximate_solution_long) - -true_prob = ODEProblem(lotka, u0, t_long, p_) -true_solution_long = solve(true_prob, Tsit5(), saveat = approximate_solution_long.t) -plot!(true_solution_long) - - - -using JLD2 -@save "knowledge_enhanced_NN.jld2" solution recovered_sys approximate_solution_long NNsolution L true_solution_long Xₙ L̄ L̂ -@load "knowledge_enhanced_NN.jld2" solution recovered_sys approximate_solution_long NNsolution L true_solution_long Xₙ L̄ L̂ - -c1 = 3 # RGBA(174/255,192/255,201/255,1) # Maroon -c2 = :orange # RGBA(132/255,159/255,173/255,1) # Red -c3 = :blue # RGBA(255/255,90/255,0,1) # Orange -c4 = :purple # RGBA(153/255,50/255,204/255,1) # Purple - -p1 = plot(0.1:0.1:tspan[end],abs.(Array(solution)[:,2:end] .- NNsolution[:,2:end])' .+ eps(Float32), - lw = 3, yaxis = :log, title = "Timeseries of UODE Error", - color = [3 :orange], xlabel = "t", - label = ["x(t)" "y(t)"], - titlefont = "Helvetica", legendfont = "Helvetica", - legend = :topright) - -# Plot L₂ -p2 = plot(X[1,:], X[2,:], L̂[2,:], lw = 3, - title = "Neural Network Fit of U2(t)", color = c1, - label = "Neural Network", xaxis = "x", yaxis="y", - titlefont = "Helvetica", legendfont = "Helvetica", - legend = :bottomright) -plot!(X[1,:], X[2,:], L̄[2,:], lw = 3, label = "True Missing Term", color=c2) - -p3 = scatter(solution, color = [c1 c2], label = ["x data" "y data"], - title = "Extrapolated Fit From Short Training Data", - titlefont = "Helvetica", legendfont = "Helvetica", - markersize = 5) - -plot!(p3,true_solution_long, color = [c1 c2], linestyle = :dot, lw=5, label = ["True x(t)" "True y(t)"]) -plot!(p3,approximate_solution_long, color = [c3 c4], lw=1, label = ["Estimated x(t)" "Estimated y(t)"]) -plot!(p3,[2.99,3.01],[0.0,maximum(hcat(Array(solution),Array(approximate_solution_long)))],lw=2,color=:black) -annotate!([(1.5,9,text("Training \nData", 10, :center, :top, :black, "Helvetica"))]) -l = @layout [grid(1,2) - grid(1,1)] -plot(p1,p2,p3,layout = l) - -savefig("sindy_extrapolation.pdf") diff --git a/DelayLotkaVolterra/eval_loop.jl b/DelayLotkaVolterra/eval_loop.jl deleted file mode 100644 index 99ae9f4..0000000 --- a/DelayLotkaVolterra/eval_loop.jl +++ /dev/null @@ -1,63 +0,0 @@ -cd(@__DIR__) -using Pkg; Pkg.activate("."); Pkg.instantiate() - -# Single experiment, move to ensemble further on -# Some good parameter values are stored as comments right now -# because this is really good practice - -using JLD2 -using LinearAlgebra -using Statistics -using Plots -gr() - - - -@load "loop_runs.jld2" loop_losses loop_error loop_sparsity failures - -# Compute some Statistics over the training loops -l_ends = [l[end] for l in loop_losses] -l_length = [length(l) for l in loop_losses] -mean(l_ends) -std(l_ends) -minimum(l_length) -maximum(l_length) -# Failures in % -failures / length(loop_losses) * 100 - - -p1 = plot(xlabel = "Iterations", ylabel = "Loss") - -for l in loop_losses - plot!(l, alpha = 0.35, yaxis = :log, legend = nothing) -end - -display(p1) - -savefig("sindy_training.pdf") - -f1 = Array{Float32}(undef, 2, length(loop_error)) -f2 = similar(f1) -f3 = similar(f1) -i = 1 - -for (e_, s_) in zip(loop_error, loop_sparsity) - global f1, f2, i - f1[:, i] = [e_[1]; s_[1]] - f2[:, i] = [e_[2]; s_[2]] - f3[:, i] = [sum(e_); loop_losses[i][end]] - i += 1 -end - -p2 = scatter(f1[1, :], f2[1, :], - xlabel = "L2 Error of Equation 1", ylabel = "L2 Error of Equation 2", - alpha = 0.3, legend = nothing, xlim = (0, 0.5), ylim = (0, 0.5)) - -p3 = scatter(f3[1, : ], f3[2, :], - ylabel = "Final Training Loss", xlabel = "Summed L2 Error of Model", - legend = nothing, alpha = 0.3, yaxis = :log) - - -p = plot(p1, p3) - -savefig("sindy_loop_training.pdf") diff --git 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zb3+sTGmt7MBPedn{xLEB*QWpa0}J`tzp(!wPmWAy=O@blx4TsSXP1(}p`l?*f(m#>JKhd&?h|q}8z1sgoqN@KSqy9-&hIvP<42+B-%OQkD`gsQiuZmc< fQsg?k{^!8{G+y+d?;)(TfLH%#P5)lHpzGfNl1XP( diff --git a/DelayLotkaVolterra/run_loop.jl b/DelayLotkaVolterra/run_loop.jl deleted file mode 100644 index b351d02..0000000 --- a/DelayLotkaVolterra/run_loop.jl +++ /dev/null @@ -1,191 +0,0 @@ -cd(@__DIR__) -using Pkg; Pkg.activate("."); Pkg.instantiate() - -# Single experiment, move to ensemble further on -# Some good parameter values are stored as comments right now -# because this is really good practice - -using OrdinaryDiffEq -using ModelingToolkit -using DataDrivenDiffEq -using LinearAlgebra, DiffEqSensitivity, Optim -using DiffEqFlux, Flux -using JLD2 - -## Simulation - -println("Generate data") - -function lotka(du, u, p, t) - α, β, γ, δ = p - du[1] = α * u[1] - β * u[2] * u[1] - du[2] = γ * u[1] * u[2] - δ * u[2] -end - -# Define the experimental parameter -tspan = (0.0f0, 3.0f0) -u0 = Float32[0.44249296,4.6280594] -p_ = Float32[1.3, 0.9, 0.8, 1.8] -prob = ODEProblem(lotka, u0, tspan, p_) -solution = solve(prob, Vern7(), abstol=1e-12, reltol=1e-12, saveat=0.1) - -# Ideal data -X = Array(solution) - -## Generate Basis for Sparse regression - - -# Create a Basis -@variables u[1:2] -# Lots of polynomials -polys = Operation[1] - -for i ∈ 1:5 - push!(polys, u[1]^i) - push!(polys, u[2]^i) - for j ∈ i:5 - if i != j - push!(polys, (u[1]^i) * (u[2]^j)) - push!(polys, u[2]^i * u[1]^i) - end - end -end - -# And some other stuff -h = [cos.(u)...; sin.(u)...; polys...] -basis = Basis(h, u) - - -## LOOP - -loop_losses = [] -loop_sparsity = [] -loop_error = [] -failures = 0 - -for i in 1:75 - - global loop_losses, loop_sparsity, loop_error, basis, X, failures - - println("Starting loop $i") - Xₙ = X + Float32(1e-3) * randn(eltype(X), size(X)) - -# Define the neueral network which learns L(x, y, y(t-τ)) -# Actually, we do not care about overfitting right now, since we want to -# extract the derivative information without numerical differentiation. - L = FastChain(FastDense(2, 32, tanh), FastDense(32, 32, tanh), FastDense(32, 2)) - p = initial_params(L) - - function dudt_(u, p, t) - x, y = u - z = L(u, p) - [p_[1] * x + z[1], - -p_[4] * y + z[2]] - end - - prob_nn = ODEProblem(dudt_, u0, tspan, p) - sol_nn = solve(prob_nn, Tsit5(), u0=u0, p=p, saveat=solution.t) - - function predict(θ) - Array(solve(prob_nn, Vern7(), u0 = u0, p = θ, saveat=solution.t, - abstol=1e-6, reltol=1e-6, - sensealg=InterpolatingAdjoint(autojacvec=ReverseDiffVJP()))) - end - -# No regularisation right now - function loss(θ) - pred = predict(θ) - sum(abs2, Xₙ .- pred), pred - end - -# Test - loss(p) - - losses = [] - - callback(θ, l, pred) = begin - push!(losses, l) - # if length(losses) % 50 == 0 - # println("Current loss after $(length(losses)) iterations: $(losses[end])") - # end - false - end - - # First train with ADAM for better convergence - res1 = DiffEqFlux.sciml_train(loss, p, ADAM(0.01), cb=callback, maxiters=200) - println("Training loss after $(length(losses)) iterations: $(losses[end])") - - if losses[end] > 1e5 || any(isnan.(res1.minimizer)) || any(abs.(res1.minimizer) .> 1e5) - failures += 1 - continue - end - - # Train with BFGS - res2 = try - DiffEqFlux.sciml_train(loss, res1.minimizer, BFGS(initial_stepnorm=0.01), cb=callback, maxiters=10000) - catch - @warn "Optimization in loop $i failed" - failures += 1 - continue - end - - println("Final training loss after $(length(losses)) iterations: $(losses[end])") - push!(loop_losses, losses) - -# Plot the data and the approximation - NNsolution = predict(res2.minimizer) - -# Ideal derivatives - DX = Array(solution(solution.t, Val{1})) - - prob_nn2 = ODEProblem(dudt_, u0, tspan, res2.minimizer) - _sol = solve(prob_nn2, Tsit5()) - DX_ = Array(_sol(solution.t, Val{1})) - -# Ideal data - L̄ = [-p_[2] * (X[1,:] .* X[2,:])';p_[3] * (X[1,:] .* X[2,:])'] -# Neural network guess - L̂ = L(Xₙ, res2.minimizer) - -## Sparse Identification - -# Create an optimizer for the SINDy problem - opt = SR3() -# Create the thresholds which should be used in the search process - λ = exp10.(-7:0.1:3) -# Target function to choose the results from; x = L0 of coefficients and L2-Error of the model - g(x) = x[1] < 1 ? Inf : norm(x, 2) - -# Test on uode derivative data - println("Starting sparse regression") - Ψ = SINDy(Xₙ[:, 2:end], L̂[:, 2:end], basis, λ, opt, g = g, maxiter = 10000, normalize = true, denoise = true) # Succeed - - -# Extract the parameter - p̂ = parameters(Ψ) - -# The parameters are a bit off, but the equations are recovered -# Start another SINDy run to get closer to the ground truth -# Create function - unknown_sys = ODESystem(Ψ) - unknown_eq = ODEFunction(unknown_sys) - -# Just the equations - b = Basis((u, p, t) -> unknown_eq(u, [1.; 1.], t), u) - -# Retune for better parameters -> we could also use DiffEqFlux or other parameter estimation tools here. - Ψf = SINDy(Xₙ[:, 2:end], L̂[:, 2:end], b, STRRidge(0.01), maxiter=100, convergence_error=1e-18) # Succeed - - push!(loop_error, get_error(Ψf)) - push!(loop_sparsity, get_sparsity(Ψf)) - - println("Statistics for run $i : \n Error $(loop_error[end])\n Sparsity $(loop_sparsity[end])") - - if i%25 == 0 - println("Saving...") - 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LotkaVolterra/Manifest.toml | 1775 +++++++++++++++++ LotkaVolterra/Scenario_1_full_plot.pdf | Bin 0 -> 30130 bytes .../Scenario_1_full_plot_adapted.pdf | Bin 0 -> 54094 bytes LotkaVolterra/Scenario_1_recovery_0.01.jld2 | Bin 0 -> 192907 bytes LotkaVolterra/Scenario_1_recovery_0.05.jld2 | Bin 0 -> 192427 bytes LotkaVolterra/scenario_1.jl | 244 +++ LotkaVolterra/scenario_1_loop.jl | 200 ++ 7 files changed, 2219 insertions(+) create mode 100644 LotkaVolterra/Manifest.toml create mode 100644 LotkaVolterra/Scenario_1_full_plot.pdf create mode 100644 LotkaVolterra/Scenario_1_full_plot_adapted.pdf create mode 100644 LotkaVolterra/Scenario_1_recovery_0.01.jld2 create mode 100644 LotkaVolterra/Scenario_1_recovery_0.05.jld2 create mode 100644 LotkaVolterra/scenario_1.jl create mode 100644 LotkaVolterra/scenario_1_loop.jl diff --git a/LotkaVolterra/Manifest.toml b/LotkaVolterra/Manifest.toml new file mode 100644 index 0000000..c835beb --- /dev/null +++ b/LotkaVolterra/Manifest.toml @@ -0,0 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Network prediction + du[1] = p_true[1]*u[1] + û[1] + du[2] = -p_true[4]*u[2] + û[2] +end + +# Closure with the known parameter +nn_dynamics!(du,u,p,t) = ude_dynamics!(du,u,p,t,p_) +# Define the problem +prob_nn = ODEProblem(nn_dynamics!,Xₙ[:, 1], tspan, p) + +## Function to train the network +# Define a predictor +function predict(θ, X = Xₙ[:,1], T = t) + Array(solve(prob_nn, Vern7(), u0 = X, p=θ, + tspan = (T[1], T[end]), saveat = T, + abstol=1e-6, reltol=1e-6, + sensealg = ForwardDiffSensitivity() + )) +end + +# Simple L2 loss +function loss(θ) + X̂ = predict(θ) + sum(abs2, Xₙ .- X̂) +end + +# Container to track the losses +losses = Float32[] + +# Callback to show the loss during training +callback(θ,l) = begin + push!(losses, l) + if length(losses)%50==0 + println("Current loss after $(length(losses)) iterations: $(losses[end])") + end + false +end + +## Training + +# First train with ADAM for better convergence -> move the parameters into a +# favourable starting positing for BFGS +res1 = DiffEqFlux.sciml_train(loss, p, ADAM(0.1f0), cb=callback, maxiters = 200) +println("Training loss after $(length(losses)) iterations: $(losses[end])") +# Train with BFGS +res2 = DiffEqFlux.sciml_train(loss, res1.minimizer, BFGS(initial_stepnorm=0.01f0), cb=callback, maxiters = 10000) +println("Final training loss after $(length(losses)) iterations: $(losses[end])") + +# Plot the losses +pl_losses = plot(1:200, losses[1:200], yaxis = :log10, xaxis = :log10, xlabel = "Iterations", ylabel = "Loss", label = "ADAM", color = :blue) +plot!(201:length(losses), losses[201:end], yaxis = :log10, xaxis = :log10, xlabel = "Iterations", ylabel = "Loss", label = "BFGS", color = :red) +# Rename the best candidate +p_trained = res2.minimizer + +## Analysis of the trained network +# Plot the data and the approximation +X̂ = predict(p_trained, Xₙ[:,1], t[1]:0.05f0:t[end]) +# Trained on noisy data vs real solution +pl_trajectory = scatter(t[1]:0.05f0:t[end], transpose(X̂), ylabel = "t", xlabel ="x(t), y(t)", color = :red, label = ["UDE Approximation" nothing]) +plot!(t, transpose(Xₙ), color = :black, label = ["Measurements" nothing]) + +# Ideal unknown interactions of the predictor +Ȳ = [-p_[2]*(X̂[1,:].*X̂[2,:])';p_[3]*(X̂[1,:].*X̂[2,:])'] +# Neural network guess +Ŷ = U(X̂,p_trained) + +pl_reconstruction = scatter(t[1]:0.05f0:t[end], transpose(Ŷ), ylabel = "t", xlabel ="I1(t), I2(t)", color = :red, label = ["UDE Approximation" nothing]) +plot!(t[1]:0.05f0:t[end], transpose(Ȳ), color = :black, label = ["True Interaction" nothing]) + +# Plot the error +pl_reconstruction_error = scatter(t[1]:0.05f0:t[end], norm.(eachcol(Ȳ-Ŷ)), yaxis = :log, xlabel = "t", ylabel = "L2-Error", color = :black) + +## Symbolic regression via sparse regression ( SINDy based ) + +# Create a Basis +@variables u[1:2] +# Generate the basis functions, multivariate polynomials up to deg 5 +# and sine +b = [polynomial_basis(u, 5); sin.(u)] +basis = Basis(b, u) + +# Create an optimizer for the SINDy problem +opt = SR3(Float32(1e-2), Float32(0.1)) +# Create the thresholds which should be used in the search process +λ = exp10.(-7:0.1:5) +# Target function to choose the results from; x = L0 of coefficients and L2-Error of the model +g(x) = x[1] < 1 ? Inf : norm(x, 2) + +# Test on ideal derivative data for unknown function ( not available ) +println("SINDy on partial ideal, unavailable data") +Ψ = SINDy(X̂, Ȳ, basis, λ, opt, g = g, maxiter = 10000) # Succeed +println(Ψ) +print_equations(Ψ) + +# Test on uode derivative data +println("SINDy on learned, partial, available data") +Ψ = SINDy(X̂, Ŷ, basis, λ, opt, g = g, maxiter = 50000, normalize = true, denoise = true, convergence_error = Float32(1e-10)) # Succeed +println(Ψ) +print_equations(Ψ) + +# Extract the parameter +p̂ = parameters(Ψ) +println("First parameter guess : $(p̂)") + +# Just the equations +b = Basis((u, p, t)->Ψ(u, [1.; 1.], t), u) + +# Retune for better parameters -> we could also use DiffEqFlux or other parameter estimation tools here. +Ψf = SINDy(X̂, Ŷ, b, STRRidge(0.01f0), maxiter = 100, convergence_error = Float32(1e-18)) # Succeed +println(Ψf) +p̂ = parameters(Ψf) +println("Second parameter guess : $(p̂)") + +# Define the recovered, hyrid model +function recovered_dynamics!(du,u, p, t, p_true) + û = Ψf(u, p) # Network prediction + du[1] = p_true[1]*u[1] + û[1] + du[2] = -p_true[4]*u[2] + û[2] +end + +# Closure with the known parameter +estimated_dynamics!(du,u,p,t) = recovered_dynamics!(du,u,p,t,p_) + +estimation_prob = ODEProblem(estimated_dynamics!, u0, tspan, p̂) +estimate = solve(estimation_prob, Tsit5(), saveat = t) + +# Plot +plot(solution) +plot!(estimate) + +## Simulation + +# Look at long term prediction +t_long = (0.0f0, 50.0f0) +estimation_prob = ODEProblem(estimated_dynamics!, u0, t_long, p̂) +estimate_long = solve(estimation_prob, Tsit5(), saveat = 0.1) # Using higher tolerances here results in exit of julia +plot(approximate_solution_long) + +true_prob = ODEProblem(lotka!, u0, t_long, p_) +true_solution_long = solve(true_prob, Tsit5(), saveat = approximate_solution_long.t) +plot!(true_solution_long) + +## Save the results +save("$(svname)recovery_$(noise_magnitude).jld2", + "solution", solution, "X", Xₙ, "t" , t, "neural_network" , U, "initial_parameters", p, "trained_parameters" , p_trained, # Training + "losses", losses, "result", Ψf, "recovered_parameters", p̂, # Recovery + "long_solution", true_solution_long, "long_estimate", estimate_long) # Estimation + + +## Post Processing and Plots + +c1 = 3 # RGBA(174/255,192/255,201/255,1) # Maroon +c2 = :orange # RGBA(132/255,159/255,173/255,1) # Red +c3 = :blue # RGBA(255/255,90/255,0,1) # Orange +c4 = :purple # RGBA(153/255,50/255,204/255,1) # Purple + +p1 = plot(t,abs.(Array(solution) .- estimate)' .+ eps(Float32), + lw = 3, yaxis = :log, title = "Timeseries of UODE Error", + color = [3 :orange], xlabel = "t", + label = ["x(t)" "y(t)"], + titlefont = "Helvetica", legendfont = "Helvetica", + legend = :topright) + +# Plot L₂ +p2 = plot3d(X̂[1,:], X̂[2,:], Ŷ[2,:], lw = 3, + title = "Neural Network Fit of U2(t)", color = c1, + label = "Neural Network", xaxis = "x", yaxis="y", + titlefont = "Helvetica", legendfont = "Helvetica", + legend = :bottomright) +plot!(X̂[1,:], X̂[2,:], Ȳ[2,:], lw = 3, label = "True Missing Term", color=c2) + +p3 = scatter(solution, color = [c1 c2], label = ["x data" "y data"], + title = "Extrapolated Fit From Short Training Data", + titlefont = "Helvetica", legendfont = "Helvetica", + markersize = 5) + +plot!(p3,true_solution_long, color = [c1 c2], linestyle = :dot, lw=5, label = ["True x(t)" "True y(t)"]) +plot!(p3,estimate_long, color = [c3 c4], lw=1, label = ["Estimated x(t)" "Estimated y(t)"]) +plot!(p3,[2.99,3.01],[0.0,10.0],lw=1,color=:black, label = nothing) +annotate!([(1.5,13,text("Training \nData", 10, :center, :top, :black, "Helvetica"))]) +l = @layout [grid(1,2) + grid(1,1)] +plot(p1,p2,p3,layout = l) + +savefig("$(svname)full_plot.pdf") diff --git a/LotkaVolterra/scenario_1_loop.jl b/LotkaVolterra/scenario_1_loop.jl new file mode 100644 index 0000000..3f6c7b3 --- /dev/null +++ b/LotkaVolterra/scenario_1_loop.jl @@ -0,0 +1,200 @@ +## Environment and packages +cd(@__DIR__) +using Pkg; Pkg.activate("."); Pkg.instantiate() + +using OrdinaryDiffEq +using ModelingToolkit +using DataDrivenDiffEq +using LinearAlgebra, DiffEqSensitivity, Optim +using DiffEqFlux, Flux +using Plots +gr() +using JLD2, FileIO +using Statistics +# Set a random seed for reproduceable behaviour +using Random +Random.seed!(1234) + + +# Create a function to adapt the noise magnitude +function noisy_magnitude(iteration) + iteration <= 40 && return Float32(1e-3) + iteration <= 80 && return Float32(5e-3) + iteration <= 120 && return Float32(1e-2) + iteration <= 160 && return Float32(2.5e-2) + return Float32(5e-2) +end + +## Data generation +function lotka!(du, u, p, t) + α, β, γ, δ = p + du[1] = α*u[1] - β*u[2]*u[1] + du[2] = γ*u[1]*u[2] - δ*u[2] +end + +# Define the experimental parameter +tspan = (0.0f0,3.0f0) +u0 = Float32[0.44249296,4.6280594] +p_ = Float32[1.3, 0.9, 0.8, 1.8] +prob = ODEProblem(lotka!, u0,tspan, p_) +solution = solve(prob, Vern7(), abstol=1e-12, reltol=1e-12, saveat = 0.1) + +# Ideal data +X = Array(solution) +t = solution.t + +# Add noise in terms of the mean +x̄ = mean(X, dims = 2)*5e-2 + +## Define the network +# Gaussian RBF as activation +rbf(x) = exp.(-(x.^2)) + +function recover_dynamics(X, t, svname, grpname) + + @info "Started run $(grpname)." + + # Define the network 2->5->5->5->2 + U = FastChain( + FastDense(2, 5, rbf), + FastDense(5, 5, rbf), + FastDense(5, 5, rbf), + FastDense(5, 2), + ) + # Get the initial parameters + p = initial_params(U) + + # Define the hybrid model + function ude_dynamics!(du, u, p, t, p_true) + û = U(u, p) # Network prediction + du[1] = p_true[1] * u[1] + û[1] + du[2] = -p_true[4] * u[2] + û[2] + end + + # Closure with the known parameter + nn_dynamics!(du, u, p, t) = ude_dynamics!(du, u, p, t, p_) + # Define the problem + prob_nn = ODEProblem(nn_dynamics!, X[:, 1], (t[1], t[end]), p) + + ## Function to train the network + # Define a predictor + function predict(θ, X = X[:, 1], T = t) + Array( + solve( + prob_nn, + Vern7(), + u0 = X, + p = θ, + tspan = (T[1], T[end]), + saveat = T, + abstol = 1e-6, + reltol = 1e-6, + sensealg = ForwardDiffSensitivity(), + ), + ) + end + + # Simple L2 loss + function loss(θ) + X̂ = predict(θ) + sum(abs2, X .- X̂) + Float32(1e-4)*sum(abs2, θ)/length(θ) + end + + # Container to track the losses + losses = Float32[] + + # Callback to show the loss during training + callback(θ, l) = begin + push!(losses, l) + false + end + + ## Training + + # First train with ADAM for better convergence -> move the parameters into a + # favourable starting positing for BFGS + res1 = DiffEqFlux.sciml_train( + loss, + p, + ADAM(0.1f0), + cb = callback, + maxiters = 200, + ) + println("Training loss after $(length(losses)) iterations: $(losses[end])") + # Train with BFGS + res2 = DiffEqFlux.sciml_train( + loss, + res1.minimizer, + BFGS(initial_stepnorm = 0.01f0), + cb = callback, + maxiters = 10000, + ) + println( + "Final training loss after $(length(losses)) iterations: $(losses[end])", + ) + + p_trained = res2.minimizer + + X̂ = predict(p_trained) + Ŷ = U(X̂, p_trained) + + # Create a Basis + @variables u[1:2] + # Generate the basis functions, multivariate polynomials up to deg 5 + # and sine + b = [polynomial_basis(u, 5); sin.(u)] + basis = Basis(b, u) + + + # Create an optimizer for the SINDy problem + opt = SR3(Float32(1e-2), Float32(0.1)) + # Create the thresholds which should be used in the search process + λ = exp10.(-7:0.1:5) + # Target function to choose the results from; x = L0 of coefficients and L2-Error of the model + g(x) = x[1] < 1 ? Inf : norm(x, 2) + + # Test on uode derivative data + Ψ = SINDy(X̂, Ŷ, basis, λ, opt, g = g, maxiter = 50000, normalize = true, denoise = true, convergence_error = Float32(1e-10)) # Succeed + + # Extract the parameter + p̂ = parameters(Ψ) + + # Just the equations + b = Basis((u, p, t) -> Ψ(u, ones(Float32, length(p̂)), t), u) + + # Retune for better parameters -> we could also use DiffEqFlux or other parameter estimation tools here. + Ψf = SINDy( + X̂, + Ŷ, + b, + STRRidge(0.01), + maxiter = 100, + convergence_error = 1e-18, + ) # Succeed + + p̂ = parameters(Ψf) + + @info "Found equations : $(Ψf.equations.eqs)" + @info "Parameter estimation : $(p̂)" + + ## Save the results + + jldopen("$(svname)recovery_loop.jld2", "a+") do file + mygroup = JLD2.Group(file, grpname) + mygroup["X"] = X + mygroup["t"] = t + mygroup["initial_parameters"] = p + mygroup["trained_parameters"] = p_trained + mygroup["losses"] = losses + mygroup["result"] = Ψf + end + + @info "Finished run $(grpname)." + + return +end + +for i in 1:200 + Xₙ = X .+ (noisy_magnitude(i)*x̄) .* randn(eltype(X), size(X)...) + recover_dynamics(Xₙ, t, "Scenario_1_broadnoise_", "$i") +end From 0e3ca32af91a9a2529be98a17646000f8a7e9df9 Mon Sep 17 00:00:00 2001 From: DrWatson Date: Wed, 10 Feb 2021 21:43:33 +0100 Subject: [PATCH 3/5] Add Scenario 2 --- LotkaVolterra/Scenario_2_full_plot.pdf | Bin 0 -> 29435 bytes .../Scenario_2_full_plot_adapted.pdf | Bin 0 -> 52858 bytes LotkaVolterra/Scenario_2_recovery_0.01.jld2 | Bin 0 -> 196867 bytes LotkaVolterra/scenario_2.jl | 276 ++++++++++++++++++ 4 files changed, 276 insertions(+) create mode 100644 LotkaVolterra/Scenario_2_full_plot.pdf create mode 100644 LotkaVolterra/Scenario_2_full_plot_adapted.pdf create mode 100644 LotkaVolterra/Scenario_2_recovery_0.01.jld2 create mode 100644 LotkaVolterra/scenario_2.jl diff --git a/LotkaVolterra/Scenario_2_full_plot.pdf b/LotkaVolterra/Scenario_2_full_plot.pdf new file mode 100644 index 0000000000000000000000000000000000000000..558862c4ee50e86c7ee49b3e44bed8ea83b4e1a3 GIT binary patch literal 29435 zcmZU(b6{Of_XgTFjn&3hW7|$++ewoX+eTyCwynlVV>Y%gfec$i7I2t%^HH9iHyKtNAmt#1ay%?(2*WM`=7U~NZ0E-I-=@o&@A z+UmQWgCPO=cTPrnMh1FDdJYB#W;O;Ekbs<>wSgnR@E-*N76MSu1O|4dPKI_AFuc4l zhE@jeJv08#2uiLth6Hp%dJcLP*2bU?^o$Md37A0w3NUoCjt&;4R-iW1-}e7?@ZSE9 ziT`x)A5oBv){a&V1oZzi+N{|Le1Yp}n=E9RTDsH#Y&Dh&AZR-#^R*Z13g~fQF$Hc5x6@ba>CfdyYgE8Cm|y z2ni6-)6#RW(=%}}63_wk{-^b>^)B~cL59Bq4Da2&Ka6=d@t2-gK@FLr}Z zgdLWDmODmQHc%midiHG1>{>aCAOs54k_aWS@>`5O&V`hX$S(Mk<=g zz4>Ht@kLwua-Lt<>%4IqMdCcd@<3k&j;S*4Hz1 z-yhvnJ^7~C9uLupJlpHM2&d-VkI)WW_DN;dn1~eFjcc%4ucKfsy;-->mM?5kp+vnN zo;OgA76%JFrFQxRAJmp;s68zJ16;hfn0iu7RZ&+i@?I!o)t3zTT;(3eCumh(Ql>=| zT@T_0>L}wo2C1bz?Os{Vj-uhivKP~n)S0Hg{IeLJ?1}4IjDs|7Zjq)3n4Q`H#`vbY z54Mro+E6Bi>dp(8dRiBUk7}{g_1VwVwbY?|c8t=q8#eGSO-tyT9B^a!?JY{Ox!%uV z2@=M=@l_fHxnFZWQ**tU?5iT%uA>wS1C-P4Fvkwy498S`4OY0LV)mwvM;f+eRXn5~a9dQc zUly-q=>A6AFBL&d#9T)s2{#pO=Mx_$C_h}M%DIGqOfDL+V7#AzqWOEv#kHqgeCzlU zEF_BkIp4NWk+AU@X-eM+R-1tleHJIrMAEpG+duP!^$QzX90tI|cd*2zFL-rNY~C*R zu@`N3!sozE0(RA9IKVVxHPtPuJ5UM1Um;Sv`a4-xrwq<^2@wr5ZKTC$i4yJ1p3R}V zuU$1ftvE~sqkW}4Mwa8S#ah>#PyG=chp67yBukc31sQmnG_^MmHT{yQ+Gn>Rmc#SV zRnGY|oDz}|LKf-%lGq=D89fm@{xkZaD}qmv%iK1zpX_DXlBVM>sOOss&EjLHEtzi= zjZLcdi)O8-S1yAqz1qEa^ryq>qo?KEg=O5qg6-nBSTR{rQLyqxM-s<) zj8}!I3YRg_CBH{7{F$Rj4L{nk^=i`-+yU4I^gW`G72@SF26jzuGOnK+V|EsK9RJx1 z&tmRuc(Pc$hzotAE)v_leIi@r^1t#;AE%*u&1siBtHh*4HHjY-<+YYLx(;tvwpB(>8G_Bi1@Aw1et&<1HjzqFJT~MHw8R zJicUQy&)bM?p<)=eVgJg?(c_3-)J(4HX-Bkj>e|A9yJQdr3VR>5)|ho*ZY+~tkomz z9I8qsH7E2#I?Rzvs;RjY3XyXPRN4)-z;7qbjegOGCl+X~JDMh2pATz@s~Zq9eK*PZ z%Ese*EY;Z`a@mVe=FP43V(d!X&15WbGn^lX~*_dTl6!Ch}4icR-C3c1BBDaqV zc+)RyiVB}5klBXYDklzkxfZ4ZM#S|yVjZp)FFAA(dvTvcy5p;0wd(DgSG^`+cVIGX zySr5m<}lzQcD?t{8q+^=of(++30@w=jIT{j`(s zu?bL?hiF8m*7LKZACPxg z*=mp>Erqe$@y`Nm`ya>f@!3alwCJj1dBbE<=F^NXR@ku=MM^Sz9nwyKJW1)!N1?u` zIFd*+5-3++x9AQ(>`2TlMjCTKv!LZsQ44kDx=dhrteKKpKn>H`8vaQ!<_$z$sdQ@d z4UjRFu<=4tN9(ias2h)2z_+@2YAXu^92bd^va*LZZ@OUc3<^=zR~n<`ki@JXlNhPQ zU&Pj`rb3Pp51&n79t5w-Z8j4Pkjz~mA*@ME`(D-s*Rp4hlrXCYxxcL0$r}h`DM4rP z_7b*gdNjBc{<0BKtiTt3yGOr*(OzAQW!Q*T9hefla109wx$+1sQ#dxW-5zDD(et*P zKx{va&-z~Q4DmuLqINZuw^={i=A&@DIxLx<I!1S0JgN#*{9EAyOs~Aj~2v)Vs{u`8IzJGZP*~LV!NE;xkO{`9y zqvx6Y2huG~rXrC;5SLC@xpFGfW#&Rf3w30X9pyia)!G7(J!YoK6qE|x7mz;gI@B+f z8+6x4RB4nR2qPV**3b{K(a&qKmpxXirO32IbG94cB0DRVnHF3_c2`OgzHs75DLk~P z0xpxXQZnB4ityUYS_A{e4W94C)2%ZWlIu7S+HD)Q0_60`EfEV*ll&Fow7pIEro#dR z{uBs*$`74)wX{v85y`tU@68s3oOZ@QFTjmzJvnp%viU7HX%|u z2Lu^v4e?CDq88Oi^-rdE=XVQks^1MFKJqeGib$=bj%OBIj9Yv(gtqkW#)>4cwZtyx zni`!nL|lDH3^Vib@(e%1P^0e9;69su!dS3@-m@aC(HY>U#EmSm2R;j~P$;6mQQyi1 zw4EyOglyps{EqZZ$Oo*WF(r)GnP3w8+N^zafP>qJM)4<_jHEX>z z>nL+w4OQa3OXyr!;N|f4C)I&#Tmer@&MIVcdeA3&CtPXI$ECh}{WjQ&rV86j^MT;; zBs;i?CHTnISdtpP7_D;M5MPCw>`(mEO1|-Ur!H(kFZ1VdB5RD9goY-EEwaIqYaDRF ztkbam`vD=TM5QA3yoR^s%b<5nou6j{0+EFV&9E#uc>+O*`dm zTrqy9AxWh4-Jk|&F+(m9es$-=dB4Jotwgog93rU0`CILnw-Nsf0u7(7D{okDLX-8! 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z4t%ET(KE|+J?4W7UAHzJ)AiWmNAGpcEnSbC>e1Q>y7zdqn8$R~b&EA#*Q4Ii_vpgm zy7^bUo4-)7n?NcYCAEDOO##9F5iWsFAi-mmKY1 zYwfmvTAMD-Xl-ND4gXIYo0FNIYfsJ0NO;AukJ`7iy@tdkMfZaJV6g+bw&d{2NY2ep zN`EpV**-lpJ8jN(M`~b8s!wW0sy&tDtfcIuDarQa?40Z!_vWx?C#7a2PySsFznzX8 z{5H`3w@uH?$w|)HQF^atC+Fm*+t(jiIBMQel9oPsPVbxlyWYRfKiu`-$kQJa literal 0 HcmV?d00001 diff --git a/LotkaVolterra/scenario_2.jl b/LotkaVolterra/scenario_2.jl new file mode 100644 index 0000000..5ada3d0 --- /dev/null +++ b/LotkaVolterra/scenario_2.jl @@ -0,0 +1,276 @@ +## Environment and packages +cd(@__DIR__) +using Pkg; Pkg.activate("."); Pkg.instantiate() + +using OrdinaryDiffEq +using ModelingToolkit +using DataDrivenDiffEq +using LinearAlgebra, DiffEqSensitivity, Optim +using DiffEqFlux, Flux +using Plots +gr() +using JLD2, FileIO +using Statistics +# Set a random seed for reproduceable behaviour +using Random + +Random.seed!() + +# Create a name for saving ( basically a prefix ) +svname = "Scenario_2_" + +## Data generation +function lotka!(du, u, p, t) + α, β, γ, δ = p + du[1] = α*u[1] - β*u[2]*u[1] + du[2] = γ*u[1]*u[2] - δ*u[2] +end + +# Define the experimental parameter +tspan = (0.0f0,6.0f0) +u0 = Float32[0.44249296,4.6280594] +p_ = Float32[1.3, 0.9, 0.8, 1.8] +prob = ODEProblem(lotka!, u0,tspan, p_) +solution = solve(prob, Vern7(), abstol=1e-6, reltol=1e-6, saveat = 0.1) + +scatter(solution, alpha = 0.25) +plot!(solution, alpha = 0.5) + +# Ideal data +X = Array(solution) +t = solution.t +# Add noise in terms of the mean +x̄ = mean(X, dims = 2) +noise_magnitude = Float32(1e-2) +Xₙ = X .+ (noise_magnitude*x̄) .* randn(eltype(X), size(X)) + +# Subsample the data in y +# We assume we have only 5 measurements in y, evenly distributed +ty = collect(t[1]:Float32(6/5):t[end]) +# Create datasets for the different measurements +round(Int64, mean(diff(ty))/mean(diff(t))) +XS = zeros(eltype(X), length(ty)-1, floor(Int64, mean(diff(ty))/mean(diff(t)))+1) # All x data +TS = zeros(eltype(t), length(ty)-1, floor(Int64, mean(diff(ty))/mean(diff(t)))+1) # Time data +YS = zeros(eltype(X), length(ty)-1, 2) # Just two measurements in y + +for i in 1:length(ty)-1 + idxs = ty[i].<= t .<= ty[i+1] + XS[i, :] = Xₙ[1, idxs] + TS[i, :] = t[idxs] + YS[i, :] = [Xₙ[2, t .== ty[i]]'; Xₙ[2, t .== ty[i+1]]] +end + +XS + +scatter!(t, transpose(Xₙ)) +plot(t, transpose(X .- Xₙ)) +## Define the network +# Gaussian RBF as activation +rbf(x) = exp.(-(x.^2)) + +# Define the network 2->5->5->5->2 +U = FastChain( + FastDense(2,5,rbf), FastDense(5,5, rbf), FastDense(5,5, rbf), FastDense(5,2) +) +# Get the initial parameters, first is linear decay +p = [rand(Float32); initial_params(U)] + +# Define the hybrid model +function ude_dynamics!(du,u, p, t, p_true) + û = U(u, p[2:end]) # Network prediction + du[1] = p_true[1]*u[1] + û[1] + # We assume a linear decay rate for the predator + du[2] = -p[1]*u[2] + û[2] +end + +# Closure with the known parameter +nn_dynamics!(du,u,p,t) = ude_dynamics!(du,u,p,t,p_) +# Define the problem +prob_nn = ODEProblem(nn_dynamics!,Xₙ[:, 1], tspan, p) + +## Function to train the network +# Define a predictor +function predict(θ, X = Xₙ[:,1], T = t) + Array(solve(prob_nn, Vern7(), u0 = X, p=θ, + tspan = (T[1], T[end]), saveat = T, + abstol=1e-6, reltol=1e-6, + sensealg = ForwardDiffSensitivity() + )) +end + +# Multiple shooting like loss +function loss(θ) + # Start with a regularization on the network + l = convert(eltype(θ), 1e-3)*sum(abs2, θ[2:end]) ./ length(θ[2:end]) + for i in 1:size(XS,1) + X̂ = predict(θ, [XS[i,1], YS[i,1]], TS[i, :]) + # Full prediction in x + l += sum(abs2, XS[i,:] .- X̂[1,:]) + # Add the boundary condition in y + l += abs(YS[i, 2] .- X̂[2, end]) + end + + return l +end + +# Container to track the losses +losses = Float32[] + +# Callback to show the loss during training +callback(θ,l) = begin + push!(losses, l) + if length(losses)%50==0 + println("Current loss after $(length(losses)) iterations: $(losses[end])") + end + false +end + +## Training + +# First train with ADAM for better convergence -> move the parameters into a +# favourable starting positing for BFGS +res1 = DiffEqFlux.sciml_train(loss, p, ADAM(0.1f0), cb=callback, maxiters = 200) +println("Training loss after $(length(losses)) iterations: $(losses[end])") +# Train with BFGS +res2 = DiffEqFlux.sciml_train(loss, res1.minimizer, BFGS(initial_stepnorm=0.01f0), cb=callback, maxiters = 10000) +println("Final training loss after $(length(losses)) iterations: $(losses[end])") + +# Plot the losses +pl_losses = plot(1:200, losses[1:200], yaxis = :log10, xaxis = :log10, xlabel = "Iterations", ylabel = "Loss", label = "ADAM", color = :blue) +plot!(201:length(losses), losses[201:end], yaxis = :log10, xaxis = :log10, xlabel = "Iterations", ylabel = "Loss", label = "BFGS", color = :red) +# Rename the best candidate +p_trained = res2.minimizer + +## Analysis of the trained network +# Plot the data and the approximation +X̂ = predict(p_trained) +# Trained on noisy data vs real solution +pl_trajectory = scatter(t, transpose(X̂), ylabel = "t", xlabel ="x(t), y(t)", color = :red, label = ["UDE Approximation" nothing]) +plot!(t, transpose(Xₙ), color = :black, label = ["Measurements" nothing]) + +# Ideal unknown interactions of the predictor +Ȳ = [-p_[2]*(X̂[1,:].*X̂[2,:])';p_[3]*(X̂[1,:].*X̂[2,:])'] +# Neural network guess +Ŷ = U(X̂,p_trained[2:end]) + +pl_reconstruction = scatter(t, transpose(Ŷ), ylabel = "t", xlabel ="I1(t), I2(t)", color = :red, label = ["UDE Approximation" nothing]) +plot!(t, transpose(Ȳ), color = :black, label = ["True Interaction" nothing]) + +# Plot the error +pl_reconstruction_error = scatter(t, norm.(eachcol(Ȳ-Ŷ)), yaxis = :log, xlabel = "t", ylabel = "L2-Error", color = :black) + +## Symbolic regression via sparse regression ( SINDy based ) + +# Create a Basis +@variables u[1:2] + +# Generate the basis functions, multivariate polynomials up to deg 5 +# and sine +b = [polynomial_basis(u, 5); sin.(u)] +basis = Basis(b, u) + +# Create an optimizer for the SINDy problem +opt = SR3(Float32(1e-2), Float32(1e-2)) +# Create the thresholds which should be used in the search process +λ = Float32.(exp10.(-7:0.1:3)) +# Target function to choose the results from; x = L0 of coefficients and L2-Error of the model +g(x) = x[1] < 1 ? Inf : norm(x, 2) + +# Test on ideal derivative data for unknown function ( not available ) +println("SINDy on partial ideal, unavailable data") +Ψ = SINDy(X̂, Ȳ, basis, λ, opt, g = g, maxiter = 10000) # Succeed +println(Ψ) +print_equations(Ψ) + +# Test on uode derivative data +println("SINDy on learned, partial, available data") +Ψ = SINDy(X̂, Ŷ, basis, λ, opt, g = g, maxiter = 10000, normalize = true, denoise = true) # Succeed +println(Ψ) +print_equations(Ψ) + +# Extract the parameter +p̂ = parameters(Ψ) +println("First parameter guess : $(p̂)") + +# Just the equations +b = Basis((u, p, t)->Ψ(u, [1.; 1.], t), u) + +# Retune for better parameters -> we could also use DiffEqFlux or other parameter estimation tools here. +Ψf = SINDy(X̂, Ŷ, b, STRRidge(0.01), maxiter = 100, convergence_error = 1e-18) # Succeed +println(Ψf) +p̂ = parameters(Ψf) +println("Second parameter guess : $(p̂)") + +# Define the recovered, hyrid model +function recovered_dynamics!(du,u, p, t, p_true) + û = Ψf(u, p) # Network prediction + du[1] = p_true[1]*u[1] + û[1] + du[2] = -p_trained[1]*u[2] + û[2] +end + +# Closure with the known parameter +estimated_dynamics!(du,u,p,t) = recovered_dynamics!(du,u,p,t,p_) + +estimation_prob = ODEProblem(estimated_dynamics!, u0, tspan, p̂) +estimate = solve(estimation_prob, Tsit5(), saveat = t) + +# Plot +plot(solution) +plot!(estimate) + +## Simulation + +# Look at long term prediction +t_long = (0.0f0, 50.0f0) +estimation_prob = ODEProblem(estimated_dynamics!, u0, t_long, p̂) +estimate_long = solve(estimation_prob, Tsit5(), saveat = 0.1) # Using higher tolerances here results in exit of julia +plot(approximate_solution_long) + +true_prob = ODEProblem(lotka!, u0, t_long, p_) +true_solution_long = solve(true_prob, Tsit5(), saveat = approximate_solution_long.t) +plot!(true_solution_long) + +## Save the results +save("$(svname)recovery_$(noise_magnitude).jld2", + "solution", solution, "X", Xₙ, "t" , t, "neural_network" , U, "initial_parameters", p, "trained_parameters" , p_trained, # Training + "losses", losses, "result", Ψf, "recovered_parameters", p̂, # Recovery + "long_solution", true_solution_long, "long_estimate", estimate_long) # Estimation + + +## Post Processing and Plots + +c1 = 3 # RGBA(174/255,192/255,201/255,1) # Maroon +c2 = :orange # RGBA(132/255,159/255,173/255,1) # Red +c3 = :blue # RGBA(255/255,90/255,0,1) # Orange +c4 = :purple # RGBA(153/255,50/255,204/255,1) # Purple + +p1 = plot(t,abs.(Array(solution) .- estimate)' .+ eps(Float32), + lw = 3, yaxis = :log, title = "Timeseries of UODE Error", + color = [3 :orange], xlabel = "t", + label = ["x(t)" "y(t)"], + titlefont = "Helvetica", legendfont = "Helvetica", + legend = :topright) + +# Plot L₂ +p2 = plot(X[1,:], X[2,:], Ŷ[2,:], lw = 3, + title = "Neural Network Fit of U2(t)", color = c1, + label = "Neural Network", xaxis = "x", yaxis="y", + titlefont = "Helvetica", legendfont = "Helvetica", + legend = :bottomright) +plot!(X[1,:], X[2,:], Ȳ[2,:], lw = 3, label = "True Missing Term", color=c2) + +p3 = scatter(t, Xₙ[1,:], color = [c1], label = "x data", + title = "Extrapolated Fit From Incomplete Training Data", + titlefont = "Helvetica", legendfont = "Helvetica", + markersize = 5) + +scatter!(ty, [YS[:, 1]; YS[end, end]], color = c2, label = "y data", markersize = 5) +plot!(p3,true_solution_long, color = [c1 c2], linestyle = :dot, lw=5, label = ["True x(t)" "True y(t)"]) +plot!(p3,estimate_long, color = [c3 c4], lw=2, label = ["Estimated x(t)" "Estimated y(t)"]) +plot!(p3,[5.99,6.01],[0.0,maximum(hcat(Array(solution),Array(estimate_long)))],lw=1,color=:black, label = nothing) +annotate!([(3,12,text("Training \nData", 10, :center, :top, :black, "Helvetica"))]) +l = @layout [grid(1,2) + grid(1,1)] +plot(p1,p2,p3,layout = l) + +savefig("$(svname)full_plot.pdf") From 558e4f101179a173a0d03f552441320abf7149ab Mon Sep 17 00:00:00 2001 From: DrWatson Date: Wed, 10 Feb 2021 21:48:50 +0100 Subject: [PATCH 4/5] Add Fisher KPP --- 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z#m-q&2cC!4I`ystETV%ip1n3aTRYY9(YR$=cK6FMEIq$8!MSH9$ zKtwOfX3T;ce_>02pJzj;Ch9zen(ji)vGC*$3@QQt9>%?|==NcAJh#1=K>lbrLQI-9 zW)q_%*ajuAxxyE)(wd>Q!HeQ*euu*;e42b^2HL^q(}z@`t>ne< zcH?Bj{d^O#C0WAv^Xp8d?*%x1{{EikV5{=_1(an=!TZQ}atCOw5{kg%|Dm;K+z+)92$VY)z;X!=7 zWECFF`&z5;wtS|>>_6wGZw2f7TbPb>qBS2|( -2.0f0 * ones(Nx), 1=> ones(Nx-1), -1 => ones(Nx-1)) ./ dx^2) +#Periodic BC +lap[1,end] = 1.0f0/dx^2 +lap[end,1] = 1.0f0/dx^2 + +function rc_ode(rho, p, t) + #finite difference + D * lap * rho + reaction.(rho) +end + +probpde = ODEProblem(rc_ode, rho_ic(Delta), (0.0f0, T), saveat=dt) +solution = solve(probpde, Tsit5()); + +# Ideal data +X = Array(solution) +t = solution.t + +# Add noise in terms of the mean +x̄ = mean(X, dims = 2) +noise_magnitude = Float32(1e-2) +Xₙ = X .+ (noise_magnitude*x̄) .* randn(eltype(X), size(X)) + +pl_solution_1 = plot(solution, color = :black, ylabel = "rho(x,t)", label = ["True Data" [nothing for i in 1:Nx-1]...]) +scatter!(t, transpose(Xₙ), color = :red,label = ["Noisy Data" [nothing for i in 1:Nx-1]...], legend =:bottomright) +# Create dataset plot +pl_contour = contour(t,x,Xₙ, xlabel = "t", ylabel = "x", fill = (true, :thermal)) +pl_initial_data = plot(pl_solution_1, pl_contour, layout = (1,2)) +savefig(pl_initial_data, "$(svname)full_data_$(noise_magnitude).pdf") + +## Neural Network setup +## Define the network +# Gaussian RBF as activation +rbf(x) = exp.(-(x.^2)) + +# Create the network to recover the reaction +rx_nn = FastChain( + FastDense(1, 5, rbf), + FastDense(5, 5, rbf), + FastDense(5, 5, rbf), + FastDense(5, 1) +) + +#initialize D0 close to D/dx^2 +D0 = Float32[6.5] +p1s = initial_params(rx_nn) +p2s = zeros(Float32, 4) +p = [p1s;p2s;D0] + +function nn_ode(u,p,t) + # This is the dircetional derivative + u_cnn_1 = [p[end-4] * u[end] + p[end-3] * u[1] + p[end-2] * u[2]] + u_cnn = [p[end-4] * u[i-1] + p[end-3] * u[i] + p[end-2] * u[i+1] for i in 2:size(u, 1)-1] + u_cnn_end = [p[end-4] * u[end-1] + p[end-3] * u[end] + p[end-2] * u[1]] + + [rx_nn([ui], p[1:length(p1s)])[1] for ui in u] + p[end] * vcat(u_cnn_1, u_cnn, u_cnn_end) +end + +prob_nn = ODEProblem(nn_ode, Xₙ[:, 1], (t[1], t[end]), p) + + +## Necessary functions + +function predict_pde(θ, X = Xₙ[:, 1], T = t) + # No ReverseDiff if using Flux + Array(solve(prob_nn,Tsit5(), + u0 = X, p = θ, tspan = (T[1], T[end]), saveat=T, + sensealg=ForwardDiffSensitivity() + )) +end + +#match data and force the weights of the CNN to add up to zero +function objective_pde(p) + pred = predict_pde(p) + sum(abs2, pred .- Xₙ) + abs(sum(p[end-4 : end-2])) +end + +# Container to track the loss +losses = Float32[] + +# Callback to see the training +callback(θ,l) = begin + push!(losses, l) + if length(losses)%1==0 + println("Current loss after $(length(losses)) iterations: $(losses[end])") + end + false +end + +## Train the model +# First train with ADAM +res1 = DiffEqFlux.sciml_train(objective_pde, p, ADAM(0.1f0), cb=callback, maxiters = 200) +# Train with BFGS +res2 = DiffEqFlux.sciml_train(objective_pde, res1.minimizer, BFGS(initial_stepnorm=0.1f0), cb=callback, maxiters = 10000) + +p_trained = res2.minimizer + +## Data evaluation +tsample = t[1]:1.0*mean(diff(t)):t[end] +X̂ = predict_pde(p_trained, Xₙ[:,1], tsample) + +# Create estimates +R̂ = similar(X̂) + +for i in 1:size(X̂, 1), j in 1:size(X̂, 2) + R̂[i,j] = rx_nn(X̂[i,j], p_trained[1:length(p1s)])[1] +end +plot(tsample, R̂', legend = nothing) + +## Symbolic regression via sparse regression / SINDy using DataDrivenDiffEq + +# Reshape the reactions +@variables u[1:1] +b = Basis(monomial_basis(u, 10), u) +# We assume (via the modeling) a common mapping of the states +# via the reaction. In other words a global function. Hence, we can +# use all measurements as one variable +Xs = vcat(eachrow(X̂[1:end, :])...) +Rs = vcat(eachrow(R̂[1:end, :])...) +opt = STRRidge()#SR3(Float32(1e-2), Float32(1e-2)) +λs = Float32.(exp10.(-3:0.01:2)) +g(x) = x[1] < 1 ? Inf : norm(x,2) +Ψ = SINDy(Xs, Rs,b, λs, opt, denoise = true, normalize = true, g = g, maxiter = 20000) +println(Ψ) +print_equations(Ψ) + + +## Save the results + +save("$(svname)recovery_$(noise_magnitude).jld2", + "solution", solution, "X", Xₙ, "t" , t, "neural_network" , rx_nn, "initial_parameters", p, "trained_parameters" , p_trained, # Training + "losses", losses, "result", Ψ, # Recovery + ) # Estimation From 6804d187791fc82ecda3a0371246161adeec5d0f Mon Sep 17 00:00:00 2001 From: DrWatson Date: Wed, 10 Feb 2021 21:49:17 +0100 Subject: [PATCH 5/5] Add Hudson Bay example --- LotkaVolterra/Hudson_Bay_recovery.jld2 | Bin 0 -> 85422 bytes LotkaVolterra/hudson_bay.jl | 246 +++++++++++++++++++++++++ LotkaVolterra/hudson_bay_data.dat | 21 +++ 3 files changed, 267 insertions(+) create mode 100644 LotkaVolterra/Hudson_Bay_recovery.jld2 create mode 100644 LotkaVolterra/hudson_bay.jl create mode 100644 LotkaVolterra/hudson_bay_data.dat diff --git a/LotkaVolterra/Hudson_Bay_recovery.jld2 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packages +cd(@__DIR__) +using Pkg; Pkg.activate("."); Pkg.instantiate() + +using OrdinaryDiffEq +using ModelingToolkit +using DataDrivenDiffEq +using LinearAlgebra, DiffEqSensitivity, Optim +using DiffEqFlux, Flux +using Plots +gr() +using JLD2, FileIO +using Statistics +using DelimitedFiles +# Set a random seed for reproduceable behaviour +using Random +Random.seed!(5443) + +## Data Preprocessing +# The data has been taken from https://jmahaffy.sdsu.edu/courses/f00/math122/labs/labj/q3v1.htm +# Originally published in +hudson_bay_data = readdlm("hudson_bay_data.dat", '\t', Float32, '\n') +# Measurements of prey and predator +Xₙ = Matrix(transpose(hudson_bay_data[:, 2:3])) +plot(t, transpose(Xₙ)) +# Normalize the data; since the data domain is strictly positive +# we just need to divide by the maximum +xscale = maximum(Xₙ, dims =2) +Xₙ .= 1f0 ./ xscale .* Xₙ +# Time from 0 -> n +t = hudson_bay_data[:, 1] .- hudson_bay_data[1, 1] +tspan = (t[1], t[end]) + +# Plot the data +scatter(t, transpose(Xₙ), xlabel = "t [a]", ylabel = "x(t), y(t)") +plot!(t, transpose(Xₙ), xlabel = "t", ylabel = "x(t), y(t)") + +## Direct Identification via SINDy + Collocation +# Create a collocation +dx̂,x̂ = collocate_data(Xₙ,t, GaussianKernel()) +# Look at the collocation +plot(t, dx̂') +# Perform sindy + +# Create a Basis +@variables u[1:2] + +# Generate the basis functions, multivariate polynomials up to deg 5 +# and sine +b = [polynomial_basis(u, 5); sin.(u)] +basis = Basis(b, u) +# Create an optimizer for the SINDy problem +opt = SR3(Float32(1e-2), Float32(1e-2)) +# Create the thresholds which should be used in the search process +λ = Float32.(exp10.(-7:0.1:3)) +# Target function to choose the results from; x = L0 of coefficients and L2-Error of the model +g(x) = x[1] < 1 ? Inf : norm(x, 2) +# Test on derivative data +Ψ = SINDy(x̂, dx̂, basis, λ, opt, g = g, maxiter = 50000, normalize = true, denoise = true) # Succeed +println(Ψ) +print_equations(Ψ) # Fails +b2 = Basis((u,p,t)->Ψ(u,ones(length(parameters(Ψ))),t),u, linear_independent = true) +Ψ = SINDy(x̂, dx̂, b2, λ, opt, g = g, maxiter = 50000, normalize = true, denoise = true) # Succeed +println(Ψ) +print_equations(Ψ) # Fails +parameters(Ψ) +## UDE Approach +# Subsample the data in y -> initial fitting strategy (batching) +# We assume we have only 5 measurements in y, evenly distributed +ty = collect(t[1]:Float32(t[end]/5):t[end]) +# Create datasets for the different measurements +t +XS = zeros(Float32, length(ty)-1, floor(Int64, mean(diff(ty))/mean(diff(t)))+1) # All x data +TS = zeros(Float32, length(ty)-1, floor(Int64, mean(diff(ty))/mean(diff(t)))+1) # Time data +YS = zeros(Float32, length(ty)-1, 2) # Just two measurements in y + +for i in 1:length(ty)-1 + idxs = ty[i].<= t .<= ty[i+1] + XS[i, :] = Xₙ[1, idxs] + TS[i, :] = t[idxs] + YS[i, :] = [Xₙ[2, t .== ty[i]]'; Xₙ[2, t .== ty[i+1]]] +end + +## Define the network +# Gaussian RBF as activation +rbf(x) = exp.(-(x.^2)) + +# Define the network 2->5->5->5->2 +U = FastChain( + FastDense(2,5,rbf), FastDense(5,5, rbf), FastDense(5,5, tanh), FastDense(5,2) +) + +# Get the initial parameters, first two is linear birth / decay of prey and predator +p = [rand(Float32,2); initial_params(U)] + +# Define the hybrid model +function ude_dynamics!(du,u, p, t) + û = U(u, p[3:end]) # Network prediction + # We assume a linear birth rate for the prey + du[1] = p[1]u[1] + û[1] + # We assume a linear decay rate for the predator + du[2] = -p[2]*u[2] + û[2] +end + +# Define the problem +prob_nn = ODEProblem(ude_dynamics!,Xₙ[:, 1], tspan, p) + +## Function to train the network +# Define a predictor +function predict(θ, X = Xₙ[:,1], T = t) + Array(solve(prob_nn, Vern7(), u0 = X, p=θ, + tspan = (T[1], T[end]), saveat = T, + abstol=1e-6, reltol=1e-6, + sensealg = ForwardDiffSensitivity() + )) +end + +# Multiple shooting like loss +function shooting_loss(θ) + # Start with a regularization on the network + l = convert(eltype(θ), 1e-3)*sum(abs2, θ[3:end]) ./ length(θ[3:end]) + for i in 1:size(XS,1) + X̂ = predict(θ, [XS[i,1], YS[i,1]], TS[i, :]) + # Full prediction in x + l += sum(abs2, XS[i,:] .- X̂[1,:]) + # Add the boundary condition in y + l += abs2(YS[i, 2] .- X̂[2, end]) + end + + return l +end + +function loss(θ) + X̂ = predict(θ) + sum(abs2, Xₙ - X̂) + convert(eltype(θ), 1e-3)*sum(abs2, θ[3:end]) ./ length(θ[3:end]) +end + +# Container to track the losses +losses = Float32[] + +# Callback to show the loss during training +callback(θ,l) = begin + push!(losses, l) + if length(losses)%5==0 + println("Current loss after $(length(losses)) iterations: $(losses[end])") + end + false +end + +## Training -> First shooting / batching to get a rough estimate + +# First train with ADAM for better convergence -> move the parameters into a +# favourable starting positing for BFGS +res1 = DiffEqFlux.sciml_train(shooting_loss, p, ADAM(0.1f0), cb=callback, maxiters = 100) +println("Training loss after $(length(losses)) iterations: $(losses[end])") +# Train with BFGS to achieve partial fit of the data +res2 = DiffEqFlux.sciml_train(shooting_loss, res1.minimizer, BFGS(initial_stepnorm=0.01f0), cb=callback, maxiters = 200) +println("Training loss after $(length(losses)) iterations: $(losses[end])") +# Full L2-Loss for full prediction +res3 = DiffEqFlux.sciml_train(loss, res2.minimizer, BFGS(initial_stepnorm=0.01f0), cb=callback, maxiters = 10000) +println("Final training loss after $(length(losses)) iterations: $(losses[end])") + +# Rename the best candidate +p_trained = res3.minimizer + +## Analysis of the trained network +# Interpolate the solution +tsample = t[1]:0.5:t[end] +X̂ = predict(p_trained, Xₙ[:,1], tsample) +# Trained on noisy data vs real solution +plot(t, transpose(Xₙ), color = :black, label = ["Measurements" nothing]) +plot!(tsample, transpose(X̂), color = :red, label = ["Interpolation" nothing]) + +# Neural network guess +Ŷ = U(X̂,p_trained[3:end]) + +scatter(tsample, transpose(Ŷ), xlabel = "t", ylabel ="I1(t), I2(t)", color = :red, label = ["UDE Approximation" nothing]) + +## Symbolic regression via sparse regression ( SINDy based ) + +# Create a Basis +@variables u[1:2] + +# Generate the basis functions, multivariate polynomials up to deg 5 +# and sine +b = [polynomial_basis(u, 5); sin.(u)] +basis = Basis(b, u) + +# Create an optimizer for the SINDy problem +opt = SR3(Float32(1e-2), Float32(1e-2)) +# Create the thresholds which should be used in the search process +λ = Float32.(exp10.(-7:0.1:3)) +# Target function to choose the results from; x = L0 of coefficients and L2-Error of the model +g(x) = x[1] < 1 ? Inf : norm(x, 2) + +# Test on uode derivative data +println("SINDy on learned, partial, available data") +Ψ = SINDy(X̂, Ŷ, basis, λ, opt, g = g, maxiter = 50000, normalize = true, denoise = true) +println(Ψ) +print_equations(Ψ) + +# Extract the parameter +p̂ = parameters(Ψ) +println("First parameter guess : $(p̂)") + +# Just the equations -> we reiterate on sindy here +# searching all linear independent components again +b = Basis((u, p, t)->Ψ(u, ones(length(p̂)), t), u, linear_independent = true) +println(b) +# Retune for better parameters -> we could also use DiffEqFlux or other parameter estimation tools here. +opt = SR3(Float32(1e-2), Float32(1e-2)) +Ψf = SINDy(X̂, Ŷ, b, opt, maxiter = 10000, normalize = true, convergence_error = eps()) # Succeed +println(Ψf) +print_equations(Ψf) +p̂ = parameters(Ψf) +println("Second parameter guess : $(p̂)") + +# Define the recovered, hyrid model with the rescaled dynamics +function recovered_dynamics!(du,u, p, t) + û = Ψf(u, p[3:4]) # Network prediction + du[1] = p[1]*u[1] + û[1] + du[2] = -p[2]*u[2] + û[2] +end + +p_model = [p_trained[1:2];p̂] +estimation_prob = ODEProblem(recovered_dynamics!, Xₙ[:, 1], tspan, p_model) +estimate = solve(estimation_prob, Tsit5(), saveat = 0.1) + +# Plot +plot(t, transpose(Xₙ)) +plot!(estimate) + +## Simulation + +# Look at long term prediction +t_long = (0.0f0, 50.0f0) +estimation_prob = ODEProblem(recovered_dynamics!, Xₙ[:, 1], t_long, p_model) +estimate_long = solve(estimation_prob, Tsit5(), saveat = 0.25) +plot(estimate_long) + +## Save the results +save("Hudson_Bay_recovery.jld2", + "X", Xₙ, "t" , t, "neural_network" , U, "initial_parameters", p, "trained_parameters" , p_trained, # Training + "losses", losses, "result", Ψf, "recovered_parameters", p̂, # Recovery + "model", recovered_dynamics!, "model_parameter", p_model, + "long_estimate", estimate_long) # Estimation diff --git a/LotkaVolterra/hudson_bay_data.dat b/LotkaVolterra/hudson_bay_data.dat new file mode 100644 index 0000000..f1fe86b --- /dev/null +++ b/LotkaVolterra/hudson_bay_data.dat @@ -0,0 +1,21 @@ +1900 30 4 +1901 47.2 6.1 +1902 70.2 9.8 +1903 77.4 35.2 +1904 36.3 59.4 +1905 20.6 41.7 +1906 18.1 19 +1907 21.4 13 +1908 22 8.3 +1909 25.4 9.1 +1910 27.1 7.4 +1911 40.3 8 +1912 57 12.3 +1913 76.6 19.5 +1914 52.3 45.7 +1915 19.5 51.1 +1916 11.2 29.7 +1917 7.6 15.8 +1918 14.6 9.7 +1919 16.2 10.1 +1920 24.7 8.6

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