Solving ODEs with NeuralPDE.jl

Solving ODEs with NeuralPDE.jl#

From https://docs.sciml.ai/NeuralPDE/stable/tutorials/ode/

For example, to solve the ODE

\[ u^{\prime} = cos(2 \pi t) \]
using NeuralPDE
using Lux
using OptimizationOptimisers
using OrdinaryDiffEq
using LinearAlgebra
using Random
rng = Random.default_rng()
TaskLocalRNG()

The true function is a cosine wave in this example.

model(u, p, t) = cospi(2t)

tspan = (0.0f0, 1.0f0)
u0 = 0.0f0
prob = ODEProblem(model, u0, tspan)
ODEProblem with uType Float32 and tType Float32. In-place: false
timespan: (0.0f0, 1.0f0)
u0: 0.0f0

Construct a neural network to solve the problem.

chain = Lux.Chain(Lux.Dense(1, 5, σ), Lux.Dense(5, 1))
ps, st = Lux.setup(rng, chain)
((layer_1 = (weight = Float32[-0.6037462; -0.887221; … ; 0.21361578; 0.45948505;;], bias = Float32[0.0; 0.0; … ; 0.0; 0.0;;]), layer_2 = (weight = Float32[-0.51698124 0.916435 … 0.622723 0.2210002], bias = Float32[0.0;;])), (layer_1 = NamedTuple(), layer_2 = NamedTuple()))

We solve the ODE as before, just change the solver algorithm to NeuralPDE.NNODE().

optimizer = OptimizationOptimisers.Adam(0.1)
alg = NeuralPDE.NNODE(chain, optimizer, init_params = ps)
NNODE{Chain{@NamedTuple{layer_1::Dense{typeof(sigmoid_fast), Int64, Int64, typeof(glorot_uniform), typeof(zeros32), Static.True}, layer_2::Dense{typeof(identity), Int64, Int64, typeof(glorot_uniform), typeof(zeros32), Static.True}}, Nothing}, Adam, Nothing, Bool, Bool, Base.Pairs{Symbol, @NamedTuple{layer_1::@NamedTuple{weight::Matrix{Float32}, bias::Matrix{Float32}}, layer_2::@NamedTuple{weight::Matrix{Float32}, bias::Matrix{Float32}}}, Tuple{Symbol}, @NamedTuple{init_params::@NamedTuple{layer_1::@NamedTuple{weight::Matrix{Float32}, bias::Matrix{Float32}}, layer_2::@NamedTuple{weight::Matrix{Float32}, bias::Matrix{Float32}}}}}, Nothing, Nothing}(Chain{@NamedTuple{layer_1::Dense{typeof(sigmoid_fast), Int64, Int64, typeof(glorot_uniform), typeof(zeros32), Static.True}, layer_2::Dense{typeof(identity), Int64, Int64, typeof(glorot_uniform), typeof(zeros32), Static.True}}, Nothing}((layer_1 = Dense(1 => 5, sigmoid_fast), layer_2 = Dense(5 => 1)), nothing), Adam(0.1, (0.9, 0.999), 1.0e-8), nothing, false, true, nothing, false, nothing, Base.Pairs{Symbol, @NamedTuple{layer_1::@NamedTuple{weight::Matrix{Float32}, bias::Matrix{Float32}}, layer_2::@NamedTuple{weight::Matrix{Float32}, bias::Matrix{Float32}}}, Tuple{Symbol}, @NamedTuple{init_params::@NamedTuple{layer_1::@NamedTuple{weight::Matrix{Float32}, bias::Matrix{Float32}}, layer_2::@NamedTuple{weight::Matrix{Float32}, bias::Matrix{Float32}}}}}(:init_params => (layer_1 = (weight = [-0.6037462; -0.887221; … ; 0.21361578; 0.45948505;;], bias = [0.0; 0.0; … ; 0.0; 0.0;;]), layer_2 = (weight = [-0.51698124 0.916435 … 0.622723 0.2210002], bias = [0.0;;]))))
sol = solve(prob, alg, verbose=true, maxiters=2000, saveat = 0.01)
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retcode: Success
Interpolation: Trained neural network interpolation
t: 0.0:0.01:1.0
u: 101-element Vector{Float64}:
  0.0
  0.00944402213521234
  0.018817099250277716
  0.028100628704810757
  0.0372748440818813
  0.04631881431473818
  0.05521045415817861
  0.06392654761776762
  0.07244278599223654
  0.08073382219436322
  0.0887733429860173
  0.09653416068613183
  0.10398832577841613
  ⋮
 -0.10296169296570452
 -0.09577268539225432
 -0.08815974116293056
 -0.08013165631047155
 -0.07169722175587259
 -0.06286520663555657
 -0.053644343975717865
 -0.04404331845383631
 -0.03407075600697663
 -0.02373521506586277
 -0.013045179212622202
 -0.002009051078320445

Comparing to the regular solver

sol2 = solve(prob, Tsit5(), saveat=sol.t)

using LinearAlgebra
norm(sol.u .- sol2.u, Inf)
0.004960657473632363