I am hitting a weird situation that I can not figure out why – perhaps it’s a bug within NeuralPDE or maybe just need to tweak my settings.
Given the Lorenzs system in this example (i.e., eqs) I am able to do forward simulation using this code with the parameter values specified
strategy = NeuralPDE.QuadratureTraining(; abstol = 1e-6, reltol = 1e-6, batch = 200)
discretization = NeuralPDE.PhysicsInformedNN([chain1, chain2, chain3], strategy, param_estim=false)
@named pdesystem = PDESystem(eqs, bcs, domains, [t], [x(t), y(t), z(t)],
[σ_, ρ, β], defaults = Dict([σ_ => 10.0, ρ => 9.0, β => 8/3]))
prob = discretize(pdesystem, discretization)
opt = BFGS(linesearch = BackTracking())
res = Optimization.solve(prob, opt; maxiters = 2000)
The neural network solution matches the numerical solution (see below) using traditional solvers in OrdinaryDiffEq. (Side note: the retcode from solve is actually a ‘Failure’)
Now, if I change the parameter to ρ => 10.0, it seems to fail all of a sudden? i.e., running
@named pdesystem = PDESystem(eqs, bcs, domains, [t], [x(t), y(t), z(t)],
[σ_, ρ, β], defaults = Dict([σ_ => 10.0, ρ => 10.0, β => 8/3]))
produces the wrong solution (but what’s weird is that now the retcode is a 'Success`).
so the question is whydoes changing this parameter mess up the results? For values of ρ between 1 to 9, it works fine but as soon as you change to ρ >= 10, it messes up.
Even more weirdness is that I can use the PINN solver to estimate these parameters (like in the example linked above). So if I start with all parameters Dict([σ_ => 1.0, ρ => 1.0, β => 1.0])) and set param_estim = true, it can recover the parameters!!!
(Aside: @ChrisRackauckas I know you said not to use the PINN/PDEsystem API for ODE systems but I was just testing and learning the API to get a deeper understanding!)
Happy to provide full reproducible code