Parametric Generalized Hamiltonian Neural Networks #207
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| using HDF5 | ||
| using GeometricMachineLearning | ||
| using GeometricMachineLearning: QPT, QPT2, Activation, ParametricLoss, SymbolicNeuralNetwork, SymbolicPullback | ||
| using CairoMakie | ||
| using NNlib: relu | ||
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| # PARAMETERS | ||
| omega = 1.0 # natural frequency of the harmonic Oscillator | ||
| Omega = 3.5 # frequency of the external sinusoidal forcing | ||
| F = .9 # amplitude of the external sinusoidal forcing | ||
| ni_dim = 10 # number of initial conditions per dimension (so ni_dim^2 total) | ||
| T = 2π * 20 | ||
| nt = 1000 # number of time steps | ||
| dt = T/nt # time step | ||
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| # Generating the initial condition array | ||
| IC = vec( [(q=q0, p=p0) for q0 in range(-1, 1, ni_dim), p0 in range(-1, 1, ni_dim)] ) | ||
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| # Generating the solution array | ||
| ni = ni_dim^2 | ||
| q = zeros(Float64, ni, nt+1) | ||
| p = zeros(Float64, ni, nt+1) | ||
| t = collect(dt * range(0, nt, step=1)) | ||
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| """ | ||
| Turn a vector of numbers into a vector of `NamedTuple`s to be used by `ParametricDataLoader`. | ||
| """ | ||
| function turn_parameters_into_correct_format(t::AbstractVector, IC::AbstractVector{<:NamedTuple}) | ||
| vec_of_params = NamedTuple[] | ||
| for time_step ∈ t | ||
| time_step == t[end] || push!(vec_of_params, (t = time_step, )) | ||
| end | ||
| vcat((vec_of_params for _ in axes(IC, 1))...) | ||
| end | ||
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| for i in 1:nt+1 | ||
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| for j=1:ni | ||
| q[j,i] = ( IC[j].p - Omega*F/(omega^2-Omega^2) )/ omega *sin(omega*t[i]) + IC[j].q*cos(omega*t[i]) + F/(omega^2-Omega^2)*sin(Omega*t[i]) | ||
| p[j,i] = -omega^2*IC[j].q*sin(omega*t[i]) + ( IC[j].p - Omega*F/(omega^2-Omega^2) )*cos(omega*t[i]) + Omega*F/(omega^2-Omega^2)*cos(Omega*t[i]) | ||
| # q[j,i] = ( IC[j].p - Omega*F/(omega^2-Omega^2) )/ omega *exp(-omega*t[i]) - IC[j].q*exp(-omega*t[i]) + F/(omega^2-Omega^2)*exp(-Omega*t[i]) | ||
| # p[j,i] = -omega^2*IC[j].q*exp(-omega*t[i]) + ( IC[j].p + Omega*F/(omega^2-Omega^2) )*exp(-omega*t[i]) - Omega*F/(omega^2-Omega^2)*exp(-Omega*t[i]) | ||
| end | ||
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| end | ||
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| @doc raw""" | ||
| Turn a `NamedTuple` of ``(q,p)`` data into two tensors of the correct format. | ||
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| This is the tricky part as the structure of the input array(s) needs to conform with the structure of the parameters. | ||
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| Here the data are rearranged in an array of size ``(n, 2, t_f - 1)`` where ``[t_0, t_1, \ldots, t_f]`` is the vector storing the time steps. | ||
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| If we deal with different initial conditions as well, we still put everything into the third (parameter) axis. | ||
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| # Example | ||
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| ```jldoctest | ||
| using GeometricMachineLearning | ||
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| q = [1. 2. 3.; 4. 5. 6.] | ||
| p = [1.5 2.5 3.5; 4.5 5.5 6.5] | ||
| qp = (q = q, p = p) | ||
| turn_q_p_data_into_correct_format(qp) | ||
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| # output | ||
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| (q = [1.0 2.0; 4.0 5.0;;; 2.0 3.0; 5.0 6.0], p = [1.5 2.5; 4.5 5.5;;; 2.5 3.5; 5.5 6.5]) | ||
| ``` | ||
| """ | ||
| function turn_q_p_data_into_correct_format(qp::QPT2{T, 2}) where {T} | ||
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| number_of_time_steps = size(qp.q, 2) - 1 # not counting t₀ | ||
| number_of_initial_conditions = size(qp.q, 1) | ||
| q_array = zeros(T, 1, 2, number_of_time_steps * number_of_initial_conditions) | ||
| p_array = zeros(T, 1, 2, number_of_time_steps * number_of_initial_conditions) | ||
| for initial_condition_index ∈ 0:(number_of_initial_conditions - 1) | ||
| for time_index ∈ 1:number_of_time_steps | ||
| q_array[:, 1, initial_condition_index * number_of_time_steps + time_index] .= qp.q[initial_condition_index + 1, time_index] | ||
| q_array[:, 2, initial_condition_index * number_of_time_steps + time_index] .= qp.q[initial_condition_index + 1, time_index + 1] | ||
| p_array[:, 1, initial_condition_index * number_of_time_steps + time_index] .= qp.p[initial_condition_index + 1, time_index] | ||
| p_array[:, 2, initial_condition_index * number_of_time_steps + time_index] .= qp.p[initial_condition_index + 1, time_index + 1] | ||
| end | ||
| end | ||
| (q = q_array, p = p_array) | ||
| end | ||
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| # SAVING TO FILE | ||
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| # h5 = h5open(path, "w") | ||
| # write(h5, "q", q) | ||
| # write(h5, "p", p) | ||
| # write(h5, "t", t) | ||
| # | ||
| # attrs(h5)["ni"] = ni | ||
| # attrs(h5)["nt"] = nt | ||
| # attrs(h5)["dt"] = dt | ||
| # | ||
| # close(h5) | ||
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| """ | ||
| This takes time as a single additional parameter (third axis). | ||
| """ | ||
| function load_time_dependent_harmonic_oscillator_with_parametric_data_loader(qp::QPT{T}, t::AbstractVector{T}, IC::AbstractVector) where {T} | ||
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| qp_reformatted = turn_q_p_data_into_correct_format(qp) | ||
| t_reformatted = turn_parameters_into_correct_format(t, IC) | ||
| ParametricDataLoader(qp_reformatted, t_reformatted) | ||
| end | ||
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| # This sets up the data loader | ||
| dl = load_time_dependent_harmonic_oscillator_with_parametric_data_loader((q = q, p = p), t, IC) | ||
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| # This sets up the neural network | ||
| width::Int = 1 | ||
| nhidden::Int = 1 | ||
| n_integrators::Int = 2 | ||
| # sigmoid_linear_unit(x::T) where {T<:Number} = x / (T(1) + exp(-x)) | ||
| arch1 = ForcedGeneralizedHamiltonianArchitecture(2; activation = tanh, width = width, nhidden = nhidden, n_integrators = n_integrators, parameters = turn_parameters_into_correct_format(t, IC)[1], forcing_type = :P) | ||
| arch2 = ForcedGeneralizedHamiltonianArchitecture(2; activation = tanh, width = width, nhidden = nhidden, n_integrators = n_integrators, parameters = turn_parameters_into_correct_format(t, IC)[1], forcing_type = :Q) | ||
| arch3 = ForcedGeneralizedHamiltonianArchitecture(2; activation = tanh, width = 2width, nhidden = nhidden, n_integrators = n_integrators, parameters = turn_parameters_into_correct_format(t, IC)[1], forcing_type = :QP) | ||
| nn1 = NeuralNetwork(arch1) | ||
| nn2 = NeuralNetwork(arch2) | ||
| nn3 = NeuralNetwork(arch3) | ||
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| # This is where training starts | ||
| batch_size = 128 | ||
| n_epochs = 200 | ||
| batch = Batch(batch_size) | ||
| o1 = Optimizer(AdamOptimizer(), nn1) | ||
| o2 = Optimizer(AdamOptimizer(), nn2) | ||
| o3 = Optimizer(AdamOptimizer(), nn3) | ||
| loss = ParametricLoss() | ||
| _pb = SymbolicPullback(nn1, loss, turn_parameters_into_correct_format(t, IC)[1]); | ||
| _pb = SymbolicPullback(nn2, loss, turn_parameters_into_correct_format(t, IC)[1]); | ||
| _pb = SymbolicPullback(nn3, loss, turn_parameters_into_correct_format(t, IC)[1]); | ||
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| function train_network() | ||
| o1(nn1, dl, batch, n_epochs, loss, _pb) | ||
| o2(nn2, dl, batch, n_epochs, loss, _pb) | ||
| o3(nn3, dl, batch, n_epochs, loss, _pb) | ||
| end | ||
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| loss_array = train_network() | ||
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| trajectory_number = 20 | ||
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| # Testing the network | ||
| initial_conditions = (q = q[trajectory_number, 1], p = p[trajectory_number, 1]) | ||
| n_steps = nt | ||
| trajectory = (q = zeros(1, n_steps), p = zeros(1, n_steps)) | ||
| trajectory.q[:, 1] .= initial_conditions.q | ||
| trajectory.p[:, 1] .= initial_conditions.p | ||
| # note that we have to supply the parameters as a named tuple as well here: | ||
| for t_step ∈ 0:(n_steps-2) | ||
| qp_temporary = nn3.model((q = [trajectory.q[1, t_step+1]], p = [trajectory.p[1, t_step+1]]), (t = t[t_step+1],), nn3.params) | ||
| trajectory.q[:, t_step+2] .= qp_temporary.q | ||
| trajectory.p[:, t_step+2] .= qp_temporary.p | ||
| end | ||
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| fig = Figure() | ||
| ax = Axis(fig[1,1]) | ||
| lines!(ax, trajectory.q[1,:]; label="nn") | ||
| lines!(ax, q[trajectory_number,:]; label="analytic") | ||
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This functionality should most likely be in
GeometricProblemsorGMLDataSets.