A>ODE Function

using DifferentialEquations
function f(du,u,p,t)
  du[1] = dx = p[1]*u[1] - u[1]*u[2]
  du[2] = dy = -3*u[2] + u[1]*u[2]
end

u0 = [1.0;1.0]
tspan = (0.0,10.0)
p = [1.5]
prob = ODEProblem(f,u0,tspan,p)

B>Data generation [X,Y] using 'Tsit5()' algorithm

sol = solve(prob,Tsit5())
t = collect(range(0,stop=10,length=400))

using RecursiveArrayTools 
randomized = VectorOfArray([(sol(t[i])) for i in 1:length(t)])
data = convert(Array,randomized)

C>Creating [x], [y]

x=Vector{Float64}(undef,400)
y=Vector{Float64}(undef,400)

for i in range(1,stop=400)
  x[i]=data[1,i]
end

for i in range(1,stop=400)
  y[i]=data[2,i]
end

D>Creating object(x_hat and y_hat) to approximate RBF

using BasisFunctionExpansions

rbf = UniformRBFE(t,10, normalize=true)
bfa_x = BasisFunctionApproximation(x,t,rbf,1)
bfa_y = BasisFunctionApproximation(y,t,rbf,1)

x_hat  = bfa_x(t)
y_hat  = bfa_y(t)

E>Graph of ODE solution vs RBF Approximation(Number of parameter =10).

using Plots

scatter(t,x, lab="x (ODE slo)")
scatter!(t,x_hat, lab="x_hat (RBF apporx.)")
scatter!(t,y,lab="y (ODE slo)")
scatter!(t,y_hat, lab="y_hat (RBF apporx.)")

F> (ODE solution-RBF approximation) at each point for number of parameter 2 to 100

nv=2:100

lcurve_x = map(nv) do n
  rbf = UniformRBFE(t, n, normalize = true)
  bfa = BasisFunctionApproximation(x,t,rbf,1)
  (x-bfa(t))
end

lcurve_y = map(nv) do n
  rbf = UniformRBFE(t, n, normalize = true)
  bfa = BasisFunctionApproximation(y,t,rbf,1)
  (y-bfa(t))
end

G>L1 error on x for number of parameter 2 to 100

i=1
j=1
x_err = Array{Float64}(undef, 99)
for i in range(1,stop=99)
  d=0
  for j in range(1,stop=400)
    if lcurve_x[i][j]>0
      d=lcurve_x[i][j]+d
    else
      d=-lcurve_x[i][j]+d
    end
  x_err[i]=d                                   
  end
end

H>L1 error on y for number of parameter 2 to 100

y_err = Array{Float64}(undef, 99)
for i in range(1,stop=99)
  d=0
  for j in range(1,stop=400)
    if lcurve_y[i][j]>0
      d=lcurve_y[i][j]+d
    else
      d=-lcurve_y[i][j]+d
    end
  y_err[i]=d                                   
  end
end 

K>L1error vs no. of parameter used in RBF

scatter(nv,x_err,lab="x_err")
scatter!(nv,y_err,lab="y_err")

L>Number of parameter at which L1 error is least

function f(d)
  i=1
  j=2
  h=999
  for i in range(1,stop=99)
    if d[i]<=h
      h=d[i]      
      j=i
    end
  end
  return j
end 

M>BEST solution

min_x=f(x_err)
min_y=f(y_err)

rbf_x = UniformRBFE(t,min_x, normalize=true)
rbf_y = UniformRBFE(t,min_y, normalize=true)
bfa_x = BasisFunctionApproximation(x,t,rbf_x,1)
bfa_y = BasisFunctionApproximation(y,t,rbf_y,1)

x_hat_new  = bfa_x(t)
y_hat_new  = bfa_y(t)

N>Graph of ODE solution vs RBF Approximation(Number of parameter min_x,min_y).

scatter(t,x, lab="x (ODE slo)")
scatter!(t,x_hat_new, lab="x_hat (RBF apporx.)")
scatter!(t,y,lab="y (ODE slo)")
scatter!(t,y_hat_new, lab="y_hat (RBF apporx.)")

O>L2 error

i=1
j=1
x_err = Array{Float64}(undef, 99)
for i in range(1,stop=99)
  d=0
  for j in range(1,stop=400)
    d=(lcurve_x[i][j])^2+d
  x_err[i]=(d^0.5)                                   
  end
end
y_err = Array{Float64}(undef, 99)
for i in range(1,stop=99)
  d=0
  for j in range(1,stop=400)
    d=(lcurve_y[i][j])^2+d
  y_err[i]=(d^0.5)                                   
  end
end