How to fit a FitzHugh-Nagumo model with DiffEqFlux.jl?

PikaPei · April 7, 2022, 6:59pm

I try to apply the DiffEqFlux to the FitzHugh-Nagumo model, an excitable neuron model.

Basically, I only replace the differential equations in this tutorial (Neural Ordinary Differential Equations with sciml_train) with the FitzHugh-Nagumo model and use Rosenbrock23() solver.

However, I can’t get a good result (see the last figure).
I don’t know where it goes wrong or where I can improve it.
I will appreciate any suggestion or reference. Thank you!

Here is my code:

using DiffEqFlux, DifferentialEquations, Plots


function fitzhugh_nagumo!(du, u, p, t)
    v, w = u
    a, b, c = p
    du[1] = v * (a - v) * (v - 1) - w
    du[2] = b * v - c * w
end


# ground truth data
u0 = [0.5, 0.0]
p_true = [-0.1, 0.01, 0.02]
tspan = (0.0, 200.0)
datasize = 100
tsteps = range(tspan[1], tspan[2], length=datasize)

prob_trueode = ODEProblem(fitzhugh_nagumo!, u0, tspan, p_true)
ode_data = Array(solve(prob_trueode, Rosenbrock23(), saveat=tsteps))

dudt2 = FastChain(FastDense(2, 32, tanh),
                  FastDense(32, 2))

prob_neuralode = NeuralODE(dudt2, tspan, Rosenbrock23(), saveat=tsteps)


function loss_neuralode(p)
    pred = Array(prob_neuralode(u0, p))
    loss = sum(abs2, ode_data .- pred)
    return loss, pred
end


callback(p, l, pred; doplot=true) = begin
    display(l)

    # plot current prediction against data
    plt = plot(tsteps, ode_data', label = "data")
    scatter!(plt, tsteps, pred', label = "prediction")
    if doplot
        display(plot(plt))
    end

    return false
end

result_neuralode = DiffEqFlux.sciml_train(loss_neuralode, prob_neuralode.p,
                                          cb=callback, maxiters=200)

ChrisRackauckas · April 7, 2022, 7:37pm

Did you try multiple shooting?

PikaPei · April 9, 2022, 1:47pm

1:
Thank you, Chris! Multiple shooting works! (although most trials fell into local minima, I’m still trying to improve it.)
ezgif.com-resize

But even in successful trials, the training process still raised an error:
I’m not sure what is wrong, so I put all the error messages here.

1.6615035429439715  # loss from callback function
1.6608717651905747
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
┌ Warning: First function call produced NaNs. Exiting.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/initdt.jl:203
┌ Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq ~/.julia/packages/OrdinaryDiffEq/iOlmD/src/solve.jl:510
┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome.
└ @ SciMLBase ~/.julia/packages/SciMLBase/GW7GW/src/integrator_interface.jl:325
ERROR: MethodError: Cannot `convert` an object of type Nothing to an object of type Float32
Closest candidates are:
  convert(::Type{T}, ::Static.StaticFloat64{N}) where {N, T<:AbstractFloat} at ~/.julia/packages/Static/pkxBE/src/float.jl:26
  convert(::Type{T}, ::LLVM.GenericValue, ::LLVM.LLVMType) where T<:AbstractFloat at ~/.julia/packages/LLVM/tVv0H/src/execution.jl:39
  convert(::Type{T}, ::LLVM.ConstantFP) where T<:AbstractFloat at ~/.julia/packages/LLVM/tVv0H/src/core/value/constant.jl:103
  ...
Stacktrace:
  [1] fill!(dest::Vector{Float32}, x::Nothing)
    @ Base ./array.jl:351
  [2] copyto!
    @ ./broadcast.jl:921 [inlined]
  [3] materialize!
    @ ./broadcast.jl:871 [inlined]
  [4] materialize!(dest::Vector{Float32}, bc::Base.Broadcast.Broadcasted{Base.Broadcast.DefaultArrayStyle{0}, Nothing, typeof(identity), Tuple{Base.RefValue{Nothing}}})
    @ Base.Broadcast ./broadcast.jl:868
  [5] (::GalacticOptim.var"#268#278"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}})(::Vector{Float32}, ::Vector{Float32})
    @ GalacticOptim ~/.julia/packages/GalacticOptim/fow0r/src/function/zygote.jl:8
  [6] (::GalacticOptim.var"#144#152"{OptimizationProblem{false, OptimizationFunction{false, GalacticOptim.AutoZygote, OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, GalacticOptim.var"#268#278"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#271#281"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#276#286", Nothing, Nothing, Nothing}, Vector{Float32}, SciMLBase.NullParameters, Nothing, Nothing, Nothing, Nothing, Base.Pairs{Symbol, var"#17#19", Tuple{Symbol}, NamedTuple{(:cb,), Tuple{var"#17#19"}}}}, GalacticOptim.var"#143#151"{OptimizationProblem{false, OptimizationFunction{false, GalacticOptim.AutoZygote, OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, GalacticOptim.var"#268#278"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#271#281"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#276#286", Nothing, Nothing, Nothing}, Vector{Float32}, SciMLBase.NullParameters, Nothing, Nothing, Nothing, Nothing, Base.Pairs{Symbol, var"#17#19", Tuple{Symbol}, NamedTuple{(:cb,), Tuple{var"#17#19"}}}}, OptimizationFunction{false, GalacticOptim.AutoZygote, OptimizationFunction{false, GalacticOptim.AutoZygote, OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, GalacticOptim.var"#268#278"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#271#281"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#276#286", Nothing, Nothing, Nothing}, GalacticOptim.var"#268#278"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#271#281"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#276#286", Nothing, Nothing, Nothing}}, OptimizationFunction{false, GalacticOptim.AutoZygote, OptimizationFunction{false, GalacticOptim.AutoZygote, OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, GalacticOptim.var"#268#278"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#271#281"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#276#286", Nothing, Nothing, Nothing}, GalacticOptim.var"#268#278"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#271#281"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#276#286", Nothing, Nothing, Nothing}})(G::Vector{Float32}, θ::Vector{Float32})
    @ GalacticOptim ~/.julia/packages/GalacticOptim/fow0r/src/solve/optim.jl:93
  [7] value_gradient!!(obj::TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}, x::Vector{Float32})
    @ NLSolversBase ~/.julia/packages/NLSolversBase/cfJrN/src/interface.jl:82
  [8] value_gradient!(obj::TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}, x::Vector{Float32})
    @ NLSolversBase ~/.julia/packages/NLSolversBase/cfJrN/src/interface.jl:69
  [9] value_gradient!(obj::Optim.ManifoldObjective{TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}}, x::Vector{Float32})
    @ Optim ~/.julia/packages/Optim/wFOeG/src/Manifolds.jl:50
 [10] (::LineSearches.var"#ϕdϕ#6"{Optim.ManifoldObjective{TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}}, Vector{Float32}, Vector{Float32}, Vector{Float32}})(α::Float32)
    @ LineSearches ~/.julia/packages/LineSearches/Ki4c5/src/LineSearches.jl:84
 [11] (::LineSearches.HagerZhang{Float64, Base.RefValue{Bool}})(ϕ::Function, ϕdϕ::LineSearches.var"#ϕdϕ#6"{Optim.ManifoldObjective{TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}}, Vector{Float32}, Vector{Float32}, Vector{Float32}}, c::Float32, phi_0::Float32, dphi_0::Float32)
    @ LineSearches ~/.julia/packages/LineSearches/Ki4c5/src/hagerzhang.jl:139
 [12] HagerZhang
    @ ~/.julia/packages/LineSearches/Ki4c5/src/hagerzhang.jl:101 [inlined]
 [13] perform_linesearch!(state::Optim.BFGSState{Vector{Float32}, Matrix{Float32}, Float32, Vector{Float32}}, method::BFGS{LineSearches.InitialStatic{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Nothing, Float64, Flat}, d::Optim.ManifoldObjective{TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}})
    @ Optim ~/.julia/packages/Optim/wFOeG/src/utilities/perform_linesearch.jl:59
 [14] update_state!(d::TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}, state::Optim.BFGSState{Vector{Float32}, Matrix{Float32}, Float32, Vector{Float32}}, method::BFGS{LineSearches.InitialStatic{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Nothing, Float64, Flat})
    @ Optim ~/.julia/packages/Optim/wFOeG/src/multivariate/solvers/first_order/bfgs.jl:139
 [15] optimize(d::TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}, initial_x::Vector{Float32}, method::BFGS{LineSearches.InitialStatic{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Nothing, Float64, Flat}, options::Optim.Options{Float64, GalacticOptim.var"#_cb#150"{var"#17#19", BFGS{LineSearches.InitialStatic{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Nothing, Float64, Flat}, Base.Iterators.Cycle{Tuple{GalacticOptim.NullData}}}}, state::Optim.BFGSState{Vector{Float32}, Matrix{Float32}, Float32, Vector{Float32}})
    @ Optim ~/.julia/packages/Optim/wFOeG/src/multivariate/optimize/optimize.jl:54
 [16] optimize(d::TwiceDifferentiable{Float32, Vector{Float32}, Matrix{Float32}, Vector{Float32}}, initial_x::Vector{Float32}, method::BFGS{LineSearches.InitialStatic{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Nothing, Float64, Flat}, options::Optim.Options{Float64, GalacticOptim.var"#_cb#150"{var"#17#19", BFGS{LineSearches.InitialStatic{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Nothing, Float64, Flat}, Base.Iterators.Cycle{Tuple{GalacticOptim.NullData}}}})
    @ Optim ~/.julia/packages/Optim/wFOeG/src/multivariate/optimize/optimize.jl:36
 [17] ___solve(prob::OptimizationProblem{false, OptimizationFunction{false, GalacticOptim.AutoZygote, OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, GalacticOptim.var"#268#278"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#271#281"{GalacticOptim.var"#267#277"{OptimizationFunction{true, GalacticOptim.AutoZygote, DiffEqFlux.var"#84#89"{typeof(loss_multiple_shooting)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}, Nothing}}, GalacticOptim.var"#276#286", Nothing, Nothing, Nothing}, Vector{Float32}, SciMLBase.NullParameters, Nothing, Nothing, Nothing, Nothing, Base.Pairs{Symbol, var"#17#19", Tuple{Symbol}, NamedTuple{(:cb,), Tuple{var"#17#19"}}}}, opt::BFGS{LineSearches.InitialStatic{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Nothing, Float64, Flat}, data::Base.Iterators.Cycle{Tuple{GalacticOptim.NullData}}; cb::Function, maxiters::Nothing, maxtime::Nothing, abstol::Nothing, reltol::Nothing, progress::Bool, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ GalacticOptim ~/.julia/packages/GalacticOptim/fow0r/src/solve/optim.jl:129
 [18] #__solve#141
    @ ~/.julia/packages/GalacticOptim/fow0r/src/solve/optim.jl:49 [inlined]
 [19] #solve#480
    @ ~/.julia/packages/SciMLBase/GW7GW/src/solve.jl:3 [inlined]
 [20] sciml_train(::typeof(loss_multiple_shooting), ::Vector{Float32}, ::Nothing, ::Nothing; lower_bounds::Nothing, upper_bounds::Nothing, maxiters::Nothing, kwargs::Base.Pairs{Symbol, var"#17#19", Tuple{Symbol}, NamedTuple{(:cb,), Tuple{var"#17#19"}}})
    @ DiffEqFlux ~/.julia/packages/DiffEqFlux/UPHk9/src/train.jl:111
 [21] top-level scope
    @ ~/project/odenet/test4.jl:115

2:
I wonder if the NeuralODE can be trained with different external inputs?
For example:

function fitzhugh_nagumo!(du, u, p, t)
    v, w = u
    a, b, c, input = p
    du[1] = v * (a - v) * (v - 1) - w + input
    du[2] = b * v - c * w
end

p1 = [-0.1, 0.01, 0.02, 0.0]
p2 = [-0.1, 0.01, 0.02, 0.1]

Then I use p1 to generate ode_data1 and p2 to generate ode_data2 (with the same u0 and tspan).
Is it possible to apply these two data to train the network?

PikaPei · April 10, 2022, 10:14am

Sorry for not checking these pages first. The problem and solution are really well-documented.

Just for the record, I put links here.