Solve y'=y with neuralode

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1

Dear all, I am beginner in this topic and trying to solve simple y’=y (analytical solution: exp(t)) with y0=0 by modifying code on Neural Ordinary Differential Equations · DiffEqFlux.jl (sciml.ai), my code until now:
begin
# NeuralODE

u01 = Float32[2.; 0.]
n_samples = 16
tspan1 = (0.0f0, 1.0f0)

function trueODEfunc(du, u, p, t)
    true_A = [-0.1 2.0; -2.0 -0.1]
    du .= ((exp(t))'true_A)'
end
t = range(tspan1[1], tspan1[2], length=n_samples)
prob = DiffEqFlux.ODEProblem(trueODEfunc, u01, tspan1)
ode_data = Array(solve(prob, Tsit5(), saveat=t))

model = Chain(
    x -> exp(t),
    Dense(2, 50, tanh),
    Dense(50, 2),
)

n_ode = NeuralODE(model, tspan1, Tsit5(), saveat=t, reltol=1e-7, abstol=1e-9)
ps1 = Flux.params(n_ode)

loss_n_ode() = sum(abs2, ode_data .- n_ode(u01))

data = Iterators.repeated((), 25)  # epochs

Flux.train!(loss_n_ode, ps1, data, ADAM(0.1))
pred = n_ode(u01)

traces = [
    scatter(x=t, y=ode_data[1, :], name="u_1", mode="lines+markers"),
    scatter(x=t, y=ode_data[2, :], name="u_2", mode="lines+markers"),
    scatter(x=t, y=pred[1, :], name="u_1 pred", mode="lines+markers"),
    scatter(x=t, y=pred[2, :], name="u_2 pred", mode="lines+markers"),
]

plt = plot(traces)

end

of course it throws error but am unable to understand the setup, does anybody has idea how to start to structure the code correctly? thank you in advance

2

Are you trying to learn it just from data? x -> exp(t) won’t work because it would have to be a function of x, and it’s a somewhat ill-advised to assume the derivative is an exponential function (since it’s the solution that would be).

Your true ODE function isn’t y'=y here, so I an quite get what you’re trying to do.

3

thanks for your reply, the thing that i am trying to do is to solve simple ode y’=y for t =(0,1) with initial condition y0=1 by using PiNN and with Neural ODE:
begin
time_min = 0.0
time_max = 1.0
y0 = [1.0]
end;
function ode_problem(dy, y, params, t)
dy[1] = y[1]
end;
q_analytical(t) = exp(t)
numerical_solution = let
# Define the ODE problem
prob = ODEProblem(ode_problem, y0, (time_min, time_max))
# Solve the ODE problem
solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)
end;
this is my code prior to PiNN and Neural ODE, but I am quite confused how to put this to NN, should I simply generate data from exp() function and learn it? and if yes i am not sure how to implement it using available examples? thank you in advance

4

the other thing i tried is:

PiNN

begin
linear(u, p, t) = u
tspan = (0.0, 1.0)
u0 = 0.0
prob1 = ODEProblem(linear, u0, tspan)
end
begin
rng = Random.default_rng()
Random.seed!(rng, 0)
chain = Lux.Chain(Lux.Dense(1, 5, σ), Lux.Dense(5, 1))
ps, st = Lux.setup(rng, chain) |> Lux.f64
opt = OptimizationOptimisers.Adam(0.1)
alg = NeuralPDE.NNODE(chain, opt, init_params = ps)
end
sol = solve(prob1, alg, verbose = true, maxiters = 2000, saveat = 0.01)

which after couple of executions learns the function but not sure if there is better solution?

5

What are you trying to do with it? If you know the ODE, you don’t need a neural ODE.

Using a standard ODE solver will be much faster than a NN here.

1 Like
6

Hi yes agree with you but it is simply the task i need to do :slight_smile: even if it is kind of a useless, and to me it is unclear how to push it thru NN so i was asking for help - that’s all.

7

I’m not 100% sure what you’re even trying to do.