Frontmatter

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using DifferentialEquations, Plots, PlutoUI
48.4 s

Initial Condition
md"
Initial Condition
"
192 μs
1.0
@bind u0 Slider(0:.1:2; default=1, show_value=true)
617 ms

Parameter
md"
Parameter
"
198 μs
1.0
@bind p Slider(-2:.1:2; default=1, show_value=true)
1.2 ms
f (generic function with 1 method)
f(u,p,t) = 10*sin(t*30) + p*u
543 μs
1.0
p
11.2 μs
begin
tspan = (0.0,1.0)
prob = ODEProblem(f,u0,tspan,p)
sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)
nothing
end
1.8 s
begin
begin
plot(sol,linewidth=5,title="Solution to u'=f(u,t)",
xaxis="Time (t)",yaxis="u(t)",label="solution", ylims=(0,3))
end
end
9.5 s

A Simple scalar Neural Network
md"
A Simple scalar Neural Network
"
299 μs
simple (generic function with 1 method)
begin
W1 = rand(n)
W2 = rand(1,n)
simple(x) = (W2 *tanh.(x*W1))[1]
end
35.1 ms
3.0831909010016787
simple(1)
217 ms
using Flux
26.5 sn
10
n = 10
15.1 μsNNODE
Chain(
  Main.var"#1#2"{DataType}(Float32),
  Dense(1 => 10, tanh),                 # 20 parameters
  Dense(10 => 1),                       # 11 parameters
  first,
)                   # Total: 4 arrays, 31 parameters, 404 bytes.
NNODE = Chain(x -> [Float32(x)], # Take in a scalar and transform it into an array
Dense(1 => n,tanh),
Dense(n => 1),
first) # Take first value, i.e. return a scalar
141 ms
NNODE.layers
---
10×1 Matrix{Float32}:
  0.3943369
  0.018199405
 -0.0168786
 -0.38904595
  0.44548574
 -0.2726122
  0.5850382
 -0.6337153
 -0.4097277
  0.34402916
NNODE[2].weight
16.8 μs
1×10 Matrix{Float32}:
 -0.0337934  0.635426  0.498114  0.141992  …  0.253938  -0.705633  -0.137575
NNODE[3].weight
18.2 μs
Flux.params(NNODE).params
13.7 ms

Let's solve u'=sin(2πt), u(0)=1
md"
Let's solve u'=sin(2πt), u(0)=1
"
248 μs
g (generic function with 1 method)
g(t) = t*NNODE(t) + 1f0
476 μs
loss (generic function with 1 method)
begin
using Statistics # for the loss
ϵ = sqrt(eps(Float32))
loss() = mean(abs2(((g(t+ϵ)-g(t))/ϵ) - sin(2π*t)) for t in 0:1f-2:1f0)
end
1.5 ms
begin
opt = Flux.Descent(0.01)
data = Iterators.repeated((), 5000)
iter = 0
cb = function () #callback function to observe training
global iter += 1
if iter % 500 == 0
display(loss())
end
end
display(loss())
Flux.train!(loss, Flux.params(NNODE), data, opt; cb=cb)
end
100% 97.2 s
begin
t = 0:0.001:1.0
plot(t,g.(t),label="NN")
plot!(t,1.0 +1/(2π) .- cos.(2π.*t)/2π, label = "True Solution")
end
653 ms