Multiple shooting and neural ode in Rc models

New to Julia
1

Good morning everyone, I am working on using neural ode the optimization of RC models for house consumption.
In the following code I have defined a first state where P and T are exogenous variables. could you give me advice on how to improve the learning? an alternative could be to use physics informed neural networks (NeuralPDE.jl) but for these odes I have not succeeded yet.

using DifferentialEquations, Plots, Flux,Optim, DiffEqFlux, DataInterpolations,Random, ComponentArrays, Lux
using Optimization, OptimizationOptimisers, OptimizationOptimJL,OptimizationNLopt
rng = Random.default_rng()
using CSV
using DataFrames
using Plots
using Flux
using Statistics: mean
using DiffEqFlux, Optimization, OptimizationOptimJL,Plots
using ComponentArrays, Lux, DiffEqFlux, Optimization, OptimizationPolyalgorithms, DifferentialEquations, Plots
using DiffEqFlux: group_ranges
# Load the data
df = CSV.read("/Users/marco/OneDrive/Desktop/Tesi/Julia_files/measured_data_ex_00.csv", DataFrame)
df=repeat(df, outer=10)

registered_gas_flow = df[:, :2]
Registered_Temperature = df[:, 3]
sec_Temp = [mean(Registered_Temperature[i:i+59]) for i in 1:60:length(Registered_Temperature)-59]
sec_gas = [mean(registered_gas_flow[i:i+59]) for i in 1:60:length(registered_gas_flow)-59]
#create a 3600 time vector
function RC!(du,u,p,t)
    Rv, Ci,Rv2,Ci2 = p
    Text = ext_Temp(t)
    P= ext_power(t)
    P2= ext_power2(t)
    
    du[1] = 1/(Rv*Ci) .* (Text .- u[1]) .+ P/Ci
    end

u0= [20.0]


tspan= (0.0f0,3000.0f0)
datasize = 6000
tsteps= range(tspan[1], tspan[2], length = datasize)
p= [10000, 10, 10000, 10]

Ext_temperature = LinearInterpolation(sec_Temp,tsteps);
power = LinearInterpolation(sec_gas,tsteps);
power2= LinearInterpolation(sec_gas,tsteps);
function ext_Temp(tsteps)
return Ext_temperature(tsteps)
end
    
function ext_power(tsteps)
return power(tsteps)
end   

function ext_power2(tsteps)
return power2(tsteps)
end

prob= ODEProblem(RC!, u0, tspan, p)
sol= solve(prob, Tsit5(), saveat=tsteps)[1,:]

# Verify ODE solution
ode_data =Array(solve(prob, Tsit5(), saveat=tsteps))

plot(tsteps, ode_data[1,:], label = "data")


#resize ode data 
ode_data=ode_data[:,1:20:3000]
tsteps=tsteps[1:20:3000]

plot(tsteps, ode_data[1,:], label = "data")

ode_data_train=ode_data[:,1:75]
ode_data_test=ode_data[:,76:150]
tspan= (0.0f0, 75.0f0)
tsteps= range(tspan[1], tspan[2], length = 75)
datasize=75

plot(ode_data[1,1:75], label = "datatrain")
plot!(ode_data[1,76:150], label = "datatest")

anim = Plots.Animation()

# Define the Neural Network
nn = Lux.Chain(
                Lux.Dense(1, 84, tanh),
                Lux.Dense(84, 84, tanh),
                Lux.Dense(84, 44, tanh),
                Lux.Dense(44,1))
p_init, st = Lux.setup(rng, nn)

neuralode = NeuralODE(nn, tspan, Tsit5(), saveat = tsteps)
prob_node = ODEProblem((u,p,t)->nn(u,p,st)[1], u0, tspan, ComponentArray(p_init))

function plot_multiple_shoot(plt, preds, group_size)
	step = group_size-1
	ranges = group_ranges(datasize, group_size)

	for (i, rg) in enumerate(ranges)
		plot!(plt, tsteps[rg], preds[i][1,:], markershape=:circle)
	end
end

# Animate training, cannot make animation on CI server
# anim = Plots.Animation()
iter = 0
callback = function (p, l, preds; doplot = true)
  display(l)
  global iter
  iter += 1
  if doplot && iter%1 == 0
    # plot the original data
    plt = scatter(tsteps, ode_data_train[1,:], label = "Data")

    # plot the different predictions for individual shoot
    plot_multiple_shoot(plt, preds, group_size)

    frame(anim)
    display(plot(plt))
  end
  return false
end

# Define parameters for Multiple Shooting
group_size = 3
continuity_term = 100

function loss_function(data, pred)
	return sum(abs2, data - pred)
end

function loss_multiple_shooting(p)
    return multiple_shoot(p, ode_data_train, tsteps, prob_node, loss_function, Tsit5(),
                          group_size; continuity_term)
end

adtype = Optimization.AutoZygote()
optf = Optimization.OptimizationFunction((x,p) -> loss_multiple_shooting(x), adtype)
optprob = Optimization.OptimizationProblem(optf, ComponentArray(p_init))
res_ms = Optimization.solve(optprob, PolyOpt(), callback = callback)


gif(anim, "multiple_shooting.gif", fps=15)

optimized_params= res_ms.u

u1 = Float32[ode_data_train[1,end]]
tspan1=(75.0f0, 150.0f0)
tsteps1=range(tspan1[1], tspan1[2], length = datasize)

prob_neuralode2 = NeuralODE(nn, tspan1,Tsit5(), saveat = tsteps1)

function predict_neuralode2(p)
  Array(prob_neuralode2(u1, p, st)[1])
end
prediction=predict_neuralode2(optimized_params)
tstepsfinal=range(0.0f0, 150.0f0, length = 150)

plot(tstepsfinal, ode_data[1,:], label = "data")
plot!(tsteps1,ode_data_test[1,:], label = "data to check prediction")
scatter!(tsteps1, prediction[1,:], label = "prediction to check prediction")

plot_rc
plot_rc2

2

I cannot run this, the measured data is not available.

3

I can’t upload the csv file

4

,Registered Gas Flow,Registered Temperature
0.0,1.7e-06,20.0
0.1,1.5e-06,20.0
0.2,2.2e-06,19.9
0.3,2.2e-06,19.8
0.4,2.6e-06,20.1
0.5,1.9e-06,20.3
0.6,2e-06,20.1
0.7,2.4e-06,20.1
0.8,1.4e-06,19.8
0.9,1.5e-06,19.6
1.0,2e-06,19.6
1.1,2.2e-06,19.8
1.2,1.5e-06,20.2
1.3,1.9e-06,19.9
1.4,1.5e-06,19.7
1.5,2e-06,20.1
1.6,2.3e-06,19.7
1.7,2.2e-06,20.1
1.8,2.6e-06,19.7
1.9,2.5e-06,20.1
2.0,2.1e-06,20.0
2.1,1.6e-06,20.0
2.2,2.2e-06,19.5
2.3,2e-06,20.0
2.4,2.3e-06,19.8
2.5,2.6e-06,19.9
2.6,2.4e-06,20.1
2.7,1.8e-06,20.1
2.8,1.5e-06,20.1
2.9,2.4e-06,20.0
3.0,1.9e-06,20.3
3.1,2.2e-06,19.9
3.2,2.4e-06,20.3
3.3,2.7e-06,20.2
3.4,1.9e-06,19.9
3.5,1.8e-06,20.5
3.6,2.7e-06,20.0
3.7,2.3e-06,19.7
3.8,2.4e-06,20.0
3.9,2.3e-06,20.2
4.0,1.6e-06,20.5
4.1,2.3e-06,20.4
4.2,2.6e-06,20.2
4.3,1.8e-06,20.4
4.4,2.1e-06,20.3
4.5,2.5e-06,20.1
4.6,2.1e-06,20.0
4.7,1.8e-06,20.5
4.8,1.9e-06,20.4
4.9,2e-06,20.8
5.0,1.7e-06,20.3
5.1,2.1e-06,20.3
5.2,2.7e-06,20.4
5.3,2.7e-06,20.1
5.4,1.6e-06,20.8
5.5,2.7e-06,20.6
5.6,2.2e-06,20.2
5.7,2.2e-06,20.3
5.8,1.9e-06,20.4
5.9,2.9e-06,20.8
6.0,2.2e-06,20.4
6.1,1.8e-06,20.1
6.2,1.7e-06,20.6
6.3,2.8e-06,20.3
6.4,1.6e-06,20.5
6.5,2.2e-06,20.5
6.6,2e-06,20.4
6.7,1.9e-06,20.5
6.8,2.8e-06,20.8
6.9,2.2e-06,20.3
7.0,2.2e-06,20.7
7.1,2.8e-06,20.9
7.2,1.6e-06,20.9
7.3,2.6e-06,20.6
7.4,2.6e-06,20.9
7.5,2.7e-06,20.7
7.6,1.7e-06,20.7
7.7,1.6e-06,20.9
7.8,2.5e-06,20.9
7.9,2.8e-06,21.4
8.0,2.4e-06,20.6
8.1,2.6e-06,21.2
8.2,2.6e-06,21.1
8.3,2.9e-06,20.7
8.4,2.8e-06,21.4
8.5,1.9e-06,21.3
8.6,2e-06,21.1
8.7,2.1e-06,21.4
8.8,2.3e-06,20.9
8.9,2.4e-06,21.1
9.0,2.7e-06,21.0
9.1,2.2e-06,21.5
9.2,2.7e-06,21.0
9.3,2e-06,21.6
9.4,2.2e-06,21.1
9.5,2.3e-06,21.3
9.6,2.2e-06,21.0
9.7,2e-06,21.4
9.8,1.9e-06,21.4
9.9,1.8e-06,21.0
10.0,2.4e-06,21.3
10.1,2.7e-06,21.3
10.2,1.7e-06,21.5
10.3,2.4e-06,21.4
10.4,2.3e-06,21.2
10.5,2.9e-06,21.4
10.6,2.6e-06,21.4
10.7,2.4e-06,21.5
10.8,2.9e-06,21.8
10.9,2.5e-06,21.7
11.0,2.8e-06,21.4
11.1,1.8e-06,21.6
11.2,1.9e-06,21.4
11.3,2.6e-06,21.4
11.4,1.8e-06,21.5
11.5,1.8e-06,22.0
11.6,2e-06,21.6
11.7,1.6e-06,21.4
11.8,2.2e-06,21.2
11.9,1.7e-06,21.8
12.0,2.6e-06,21.4
12.1,2.9e-06,21.4
12.2,1.9e-06,21.3
12.3,1.9e-06,21.9
12.4,2.1e-06,22.0
12.5,2.9e-06,22.0
12.6,2.6e-06,22.0
12.7,1.8e-06,21.6
12.8,2.6e-06,21.9
12.9,2.2e-06,22.0
13.0,2.2e-06,21.9
13.1,2.4e-06,22.0
13.2,1.9e-06,22.0
13.3,2.5e-06,22.0
13.4,2.9e-06,22.1
13.5,2.5e-06,22.1
13.6,3e-06,22.0
13.7,2.5e-06,22.6
13.8,1.8e-06,21.8
13.9,3.1e-06,22.1
14.0,2.3e-06,22.3
14.1,1.9e-06,22.3
14.2,2.5e-06,22.1
14.3,2.1e-06,22.6
14.4,2.4e-06,22.3
14.5,2.4e-06,22.4
14.6,3.1e-06,22.4
14.7,1.9e-06,22.4
14.8,2e-06,22.5
14.9,2.1e-06,22.5
15.0,2.6e-06,22.3
15.1,1.9e-06,23.0
15.2,2.6e-06,22.4
15.3,2.6e-06,22.5
15.4,2.8e-06,22.4
15.5,2e-06,22.6
15.6,2.4e-06,22.5
15.7,2.2e-06,22.3
15.8,2e-06,22.9
15.9,1.7e-06,22.6
16.0,3e-06,22.7
16.1,2.3e-06,22.7
16.2,1.9e-06,22.4
16.3,3.1e-06,22.7
16.4,2.3e-06,22.6
16.5,1.9e-06,22.9
16.6,3e-06,23.0
16.7,1.9e-06,23.3
16.8,2e-06,23.1
16.9,1.8e-06,23.2
17.0,2.2e-06,22.3
17.1,2.1e-06,22.8
17.2,1.9e-06,23.0
17.3,2.9e-06,22.9
17.4,2.4e-06,22.7
17.5,2.3e-06,23.2
17.6,2.4e-06,22.8
17.7,2.9e-06,22.9
17.8,3.1e-06,22.8
17.9,2.7e-06,23.4
18.0,2.1e-06,23.3
18.1,2.7e-06,23.3
18.2,3.1e-06,23.4
18.3,2.1e-06,23.3
18.4,3e-06,23.2
18.5,2.5e-06,23.9
18.6,2.5e-06,23.5
18.7,2.7e-06,23.9
18.8,2.9e-06,23.4
18.9,2.8e-06,23.3
19.0,2.5e-06,24.1
19.1,3.2e-06,23.7
19.2,2.7e-06,23.6
19.3,2.4e-06,23.7
19.4,3e-06,24.2
19.5,2.8e-06,24.2
19.6,2.8e-06,23.9
19.7,3.3e-06,24.2
19.8,3.2e-06,24.3
19.9,3e-06,23.8
20.0,2.2e-06,24.0
20.1,2.5e-06,24.1
20.2,2.6e-06,24.8
20.3,1.9e-06,24.0
20.4,2.4e-06,24.6
20.5,2.6e-06,24.3
20.6,3.1e-06,24.5
20.7,2.4e-06,24.3
20.8,2.9e-06,24.7
20.9,3.1e-06,24.4
21.0,2.8e-06,24.4
21.1,3e-06,24.9
21.2,2.8e-06,24.9
21.3,2.9e-06,25.0
21.4,1.8e-06,25.0
21.5,3.1e-06,24.8
21.6,2.6e-06,25.5
21.7,2.5e-06,25.2
21.8,1.9e-06,24.8
21.9,2.9e-06,25.0
22.0,2.5e-06,25.2
22.1,2.8e-06,24.7
22.2,2.8e-06,25.1
22.3,2.1e-06,25.0
22.4,3.1e-06,24.9
22.5,3.1e-06,25.2
22.6,3.1e-06,25.3
22.7,2.6e-06,25.7
22.8,2e-06,25.2
22.9,2.5e-06,25.5
23.0,3.3e-06,25.3
23.1,2.1e-06,25.8
23.2,2.5e-06,25.6
23.3,2.2e-06,25.2
23.4,2.6e-06,25.4
23.5,3.1e-06,25.5
23.6,2.4e-06,25.3
23.7,2.6e-06,26.4
23.8,3.1e-06,26.4
23.9,2.4e-06,25.9
24.0,2.1e-06,26.0
24.1,3.3e-06,25.9
24.2,2.1e-06,26.4
24.3,2.8e-06,26.3
24.4,2.8e-06,26.3
24.5,1.8e-06,26.0
24.6,1.8e-06,26.3
24.7,3e-06,25.8
24.8,3e-06,26.5
24.9,2.3e-06,26.0
25.0,2.5e-06,26.6
25.1,2.9e-06,26.5
25.2,2.9e-06,26.5
25.3,2.2e-06,26.6
25.4,3.2e-06,26.8
25.5,2.9e-06,26.6
25.6,3.1e-06,26.4
25.7,2e-06,26.6
25.8,1.9e-06,26.6
25.9,3e-06,26.5
26.0,2.8e-06,26.6
26.1,2.1e-06,26.5
26.2,2.2e-06,26.7
26.3,2e-06,27.2
26.4,2.5e-06,27.4
26.5,3.4e-06,27.0
26.6,3.3e-06,26.8
26.7,2.5e-06,27.2
26.8,2.3e-06,27.0
26.9,2.9e-06,26.9
27.0,2.3e-06,27.0
27.1,3e-06,26.9
27.2,3.1e-06,27.8
27.3,2.5e-06,27.2
27.4,2.3e-06,27.4
27.5,2.2e-06,27.4
27.6,1.9e-06,27.6
27.7,2.2e-06,27.7
27.8,3.4e-06,27.2
27.9,2.2e-06,27.2
28.0,3.1e-06,27.2
28.1,2.1e-06,28.0
28.2,2.5e-06,27.3
28.3,2.8e-06,27.7
28.4,2.2e-06,28.0
28.5,3.2e-06,27.4
28.6,2.1e-06,28.1
28.7,3.5e-06,28.1
28.8,2.3e-06,27.6
28.9,2.2e-06,28.0
29.0,2e-06,28.0
29.1,2.2e-06,27.8
29.2,2.9e-06,27.9
29.3,3.1e-06,28.6
29.4,3.2e-06,28.4
29.5,3e-06,28.2
29.6,3.1e-06,28.5
29.7,3e-06,28.4
29.8,2.9e-06,28.4
29.9,2e-06,28.2
30.0,2.8e-06,28.0
30.1,2.3e-06,28.7
30.2,2.4e-06,28.8
30.3,3e-06,28.4
30.4,2.1e-06,28.2
30.5,3.2e-06,29.0
30.6,2.7e-06,28.5
30.7,2.8e-06,28.5
30.8,2.9e-06,29.1
30.9,2.4e-06,29.0
31.0,3e-06,28.9
31.1,2.1e-06,29.0
31.2,2.1e-06,28.6
31.3,2.7e-06,28.8
31.4,3.4e-06,29.5
31.5,3.2e-06,28.7
31.6,2.8e-06,29.0
31.7,3.1e-06,28.8
31.8,2.4e-06,29.4
31.9,2.1e-06,29.1
32.0,2.8e-06,29.2
32.1,2.4e-06,29.3
32.2,2.8e-06,29.1
32.3,2.8e-06,29.4
32.4,2.6e-06,29.7
32.5,2.4e-06,29.8
32.6,2.5e-06,29.6
32.7,2.6e-06,30.3
32.8,2.6e-06,29.4
32.9,2.7e-06,29.8
33.0,2.5e-06,29.5
33.1,2.3e-06,29.3
33.2,2.1e-06,29.9
33.3,2.6e-06,30.0
33.4,2.4e-06,30.5
33.5,3e-06,29.7
33.6,3e-06,29.8
33.7,2.8e-06,30.1
33.8,3.1e-06,30.0
33.9,3.1e-06,29.8
34.0,3.5e-06,30.2
34.1,2.7e-06,30.6
34.2,3.5e-06,30.4
34.3,2.5e-06,30.5
34.4,2.7e-06,30.3
34.5,2.8e-06,30.2
34.6,3e-06,30.2
34.7,2.6e-06,31.0
34.8,3e-06,31.0
34.9,3.6e-06,31.3
35.0,2.2e-06,30.7
35.1,3.1e-06,30.9
35.2,2.5e-06,30.8
35.3,2.8e-06,30.7
35.4,2.8e-06,30.9
35.5,3.3e-06,30.8
35.6,2e-06,31.1
35.7,2.4e-06,31.6
35.8,3e-06,31.2
35.9,2.9e-06,31.5
36.0,3.5e-06,31.5
36.1,2.4e-06,31.8
36.2,3.6e-06,31.5
36.3,2.4e-06,32.0
36.4,3.4e-06,31.7
36.5,3.1e-06,31.8
36.6,2.8e-06,32.3
36.7,3.3e-06,31.7
36.8,2e-06,31.9
36.9,2.2e-06,32.2
37.0,3.1e-06,31.8
37.1,3.2e-06,32.6
37.2,2.7e-06,31.9
37.3,2.9e-06,32.2
37.4,3.2e-06,32.1
37.5,3.6e-06,32.3
37.6,3.1e-06,32.0
37.7,2.4e-06,32.6
37.8,3.3e-06,32.8
37.9,2.2e-06,32.4
38.0,2e-06,32.9
38.1,2.8e-06,32.6
38.2,2.1e-06,32.3
38.3,2.2e-06,32.6
38.4,3.5e-06,32.6
38.5,3.3e-06,32.7
38.6,3.3e-06,33.3
38.7,2.7e-06,32.6
38.8,2e-06,32.9
38.9,3e-06,33.1
39.0,2.8e-06,33.0
39.1,2.1e-06,32.8
39.2,2.9e-06,33.1
39.3,2.6e-06,33.2
39.4,2.3e-06,32.9
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90.9,1.8e-06,46.6
91.0,1.7e-06,45.7
91.1,1.7e-06,46.4
91.2,2.3e-06,45.0
91.3,1.5e-06,46.3
91.4,2.1e-06,46.7
91.5,1.4e-06,46.7
91.6,1.9e-06,45.9
91.7,1.8e-06,46.5
91.8,1.5e-06,45.0
91.9,2.3e-06,46.5
92.0,2.2e-06,46.3
92.1,1.4e-06,45.8
92.2,2.3e-06,45.9
92.3,2e-06,45.5
92.4,1.7e-06,46.1
92.5,1.8e-06,46.1
92.6,1.7e-06,46.1
92.7,1.9e-06,45.9
92.8,2.2e-06,46.1
92.9,2.3e-06,45.6
93.0,1.6e-06,46.0
93.1,1.9e-06,45.7
93.2,1.8e-06,45.9
93.3,1.4e-06,46.7
93.4,1.4e-06,46.1
93.5,1.6e-06,45.8
93.6,2e-06,46.2
93.7,2.3e-06,45.7
93.8,1.8e-06,46.3
93.9,2.3e-06,45.4
94.0,1.7e-06,46.1
94.1,1.5e-06,46.0
94.2,1.7e-06,45.3
94.3,2.4e-06,46.6
94.4,2e-06,45.1
94.5,1.7e-06,45.2
94.6,1.9e-06,46.0
94.7,2.1e-06,46.3
94.8,2.3e-06,44.9
94.9,1.8e-06,45.6
95.0,1.4e-06,45.5
95.1,1.8e-06,45.6
95.2,2e-06,46.2
95.3,2.2e-06,45.8
95.4,2.3e-06,46.1
95.5,1.4e-06,45.2
95.6,1.4e-06,45.6
95.7,2e-06,44.9
95.8,2.2e-06,45.1
95.9,2e-06,45.7
96.0,2e-06,45.4
96.1,2.4e-06,45.3
96.2,1.3e-06,44.9
96.3,2.3e-06,45.2
96.4,1.5e-06,45.1
96.5,2.4e-06,45.5
96.6,1.9e-06,45.7
96.7,1.6e-06,45.5
96.8,2.1e-06,45.4
96.9,1.9e-06,45.3
97.0,2.2e-06,45.5
97.1,2.2e-06,45.5
97.2,2e-06,44.9
97.3,2.1e-06,44.9
97.4,2e-06,45.8
97.5,1.6e-06,44.7
97.6,1.9e-06,45.2
97.7,2e-06,45.1
97.8,1.6e-06,44.4
97.9,2.2e-06,44.8
98.0,1.7e-06,45.1
98.1,2.3e-06,45.7
98.2,1.5e-06,44.8
98.3,2.3e-06,44.8
98.4,1.4e-06,44.2
98.5,1.7e-06,44.2
98.6,2.1e-06,44.2
98.7,2.3e-06,44.5
98.8,1.6e-06,45.3
98.9,1.8e-06,45.1
99.0,1.8e-06,45.7
99.1,2.1e-06,44.0
99.2,2.2e-06,44.7
99.3,1.9e-06,45.0
99.4,1.5e-06,44.5
99.5,2.2e-06,44.8
99.6,1.9e-06,44.7
99.7,1.4e-06,45.0
99.8,1.5e-06,44.7
99.9,2.2e-06,44.8
100.0,1.4e-06,45.0
100.1,1.4e-06,44.6