Solving ODEs with NeuralPDE.jl

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

using NeuralPDE
using Lux
using OptimizationOptimisers
using OrdinaryDiffEq
using LinearAlgebra
using Random
using Plots
rng = Random.Xoshiro(0)

Random.Xoshiro(0xdb2fa90498613fdf, 0x48d73dc42d195740, 0x8c49bc52dc8a77ea, 0x1911b814c02405e8, 0x22a21880af5dc689)

Solve ODEs

The true function: \(u^{\prime} = cos(2 \pi t)\)

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

model (generic function with 1 method)

Prepare data

tspan = (0.0, 1.0)
u0 = 0.0
prob = ODEProblem(model, u0, tspan)

ODEProblem with uType Float64 and tType Float64. In-place: false
Non-trivial mass matrix: false
timespan: (0.0, 1.0)
u0: 0.0

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) |> f64

((layer_1 = (weight = [-0.04929668828845024; -0.3266667425632477; … ; -1.4946011304855347; -1.0391809940338135;;], bias = [-0.458548903465271, -0.8280583620071411, -0.38509929180145264, 0.32322537899017334, -0.32623517513275146]), layer_2 = (weight = [0.5656673908233643 -0.605137288570404 … 0.3129439055919647 0.22128699719905853], bias = [-0.11007555574178696])), (layer_1 = NamedTuple(), layer_2 = NamedTuple()))

Solve the ODE as in DifferentialEquations.jl, just change the solver algorithm to NeuralPDE.NNODE().

optimizer = OptimizationOptimisers.Adam(0.1)
alg = NeuralPDE.NNODE(chain, optimizer, init_params = ps)
sol = solve(prob, alg, maxiters=2000, saveat = 0.01)

┌ Warning: Mixed-Precision `matmul_cpu_fallback!` detected and Octavian.jl cannot be used for this set of inputs (C [Matrix{Float64}]: A [Base.ReshapedArray{Float32, 2, SubArray{Float32, 1, Vector{Float32}, Tuple{UnitRange{Int64}}, true}, Tuple{}}] x B [Matrix{Float64}]). Converting to common type to to attempt to use BLAS. This may be slow.
└ @ LuxLib.Impl ~/.julia/packages/LuxLib/3ewJc/src/impl/matmul.jl:194

retcode: Success
Interpolation: Trained neural network interpolation
t: 0.0:0.01:1.0
u: 101-element Vector{Float64}:
  0.0
  0.00959493022164767
  0.01908732527091774
  0.028460752516961535
  0.03769766390063074
  0.046779388904492084
  0.055686137820595706
  0.06439701685467354
  0.0728900566594053
  0.08114225591580405
  ⋮
 -0.07966677911884555
 -0.07121824118315172
 -0.062389224632537356
 -0.053190231869128524
 -0.043631724606658
 -0.03372410915291676
 -0.023477723480529164
 -0.012902825884776892
 -0.002009585045247463

Comparing to the regular solver

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

retcode: Success
Interpolation: 1st order linear
t: 101-element Vector{Float64}:
 0.0
 0.01
 0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0.09
 ⋮
 0.92
 0.93
 0.94
 0.95
 0.96
 0.97
 0.98
 0.99
 1.0
u: 101-element Vector{Float64}:
  0.0
  0.009993421557959134
  0.019947410479672478
  0.029822662260300302
  0.03958022071476466
  0.04918159908446656
  0.05858894195530296
  0.0677649690474175
  0.07667347363816583
  0.08527940825421805
  ⋮
 -0.07670552085637702
 -0.06779339219172714
 -0.05861083135542737
 -0.049195336960505105
 -0.039586192500697205
 -0.029824466350448206
 -0.019949141853139615
 -0.009995163933671121
 -1.7408033452440449e-6

plot(sol2, label = "Tsit5")
plot!(sol.t, sol.u, label = "NNODE")

Parameter estimation

using NeuralPDE, OrdinaryDiffEq, Lux, Random, OptimizationOptimJL, LineSearches, Plots
rng = Random.Xoshiro(0)

Random.Xoshiro(0xdb2fa90498613fdf, 0x48d73dc42d195740, 0x8c49bc52dc8a77ea, 0x1911b814c02405e8, 0x22a21880af5dc689)

NNODE only supports out-of-place functions

function lv(u, p, t)
    u₁, u₂ = u
    α, β, γ, δ = p
    du₁ = α * u₁ - β * u₁ * u₂
    du₂ = δ * u₁ * u₂ - γ * u₂
    [du₁, du₂]
end

lv (generic function with 1 method)

Generate data

tspan = (0.0, 5.0)
u0 = [5.0, 5.0]
true_p = [1.5, 1.0, 3.0, 1.0]
prob = ODEProblem(lv, u0, tspan, true_p)
sol_data = solve(prob, Tsit5(), saveat = 0.01)

t_ = sol_data.t
u_ = Array(sol_data)

2×501 Matrix{Float64}:
 5.0  4.82567  4.65308  4.48283  4.31543  …  1.01959   1.03094   1.04248
 5.0  5.09656  5.18597  5.26791  5.34212     0.397663  0.389887  0.382307

Define a neural network

n = 15
chain = Chain(Dense(1, n, σ), Dense(n, n, σ), Dense(n, n, σ), Dense(n, 2))
ps, st = Lux.setup(rng, chain) |> f64

((layer_1 = (weight = [-0.04929668828845024; -0.3266667425632477; … ; -1.3531280755996704; -0.2917589843273163;;], bias = [0.28568029403686523, -0.4209803342819214, -0.24613642692565918, -0.9429000616073608, -0.3618292808532715, 0.077278733253479, 0.9969245195388794, 0.7939795255661011, 0.45440757274627686, -0.4830443859100342, -0.6861011981964111, -0.3221019506454468, -0.5597391128540039, -0.15051674842834473, 0.9440881013870239]), layer_2 = (weight = [-0.08606008440256119 -0.2168799191713333 … -0.3507671356201172 0.07374405115842819; 0.24009405076503754 -0.2372819483280182 … 0.34944412112236023 -0.21207459270954132; … ; 0.3976286053657532 0.28444960713386536 … -0.32817620038986206 0.396392285823822; -0.07926429808139801 0.35875916481018066 … -0.03593128174543381 -0.28511112928390503], bias = [-0.065037302672863, 0.18384626507759094, 0.17181798815727234, -0.17310386896133423, 0.06428726017475128, 0.09600061178207397, -0.08703552931547165, 0.06890828162431717, -0.16194558143615723, -0.14649711549282074, -0.14649459719657898, -0.04401325806975365, -0.015492657199501991, 0.1046019047498703, 0.15015578269958496]), layer_3 = (weight = [-0.2995997369289398 0.14921274781227112 … -0.011808237060904503 -0.3409591019153595; 0.4351722002029419 0.1286778748035431 … -0.20781198143959045 -0.030425485223531723; … ; -0.02206072397530079 0.14348538219928741 … -0.05763476341962814 -0.2672235071659088; 0.2975636124610901 -0.06781639903783798 … 0.4012162387371063 0.12123444676399231], bias = [0.04135546088218689, -0.2398381233215332, 0.1595604568719864, 0.08355490118265152, -0.06149742379784584, -0.06998120248317719, -0.008059235289692879, -0.10936713218688965, -0.18340998888015747, 0.06297893822193146, 0.04081515222787857, -0.04258332401514053, 0.11171907186508179, -0.21218737959861755, 0.07965957373380661]), layer_4 = (weight = [0.3909371793270111 -0.23473049700260162 … 0.07385867089033127 0.31727129220962524; -0.04396385699510574 0.1817844659090042 … -0.26729491353034973 0.24492913484573364], bias = [0.04966225475072861, -0.04299044609069824])), (layer_1 = NamedTuple(), layer_2 = NamedTuple(), layer_3 = NamedTuple(), layer_4 = NamedTuple()))

Loss function

additional_loss(phi, θ) = sum(abs2, phi(t_, θ) .- u_) / size(u_, 2)

additional_loss (generic function with 1 method)

NNODE solver

opt = LBFGS(linesearch = BackTracking())
alg = NNODE(chain, opt, ps; strategy = WeightedIntervalTraining([0.7, 0.2, 0.1], 500), param_estim = true, additional_loss)

NeuralPDE.NNODE{Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_4::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, Optim.LBFGS{Nothing, LineSearches.InitialStatic{Float64}, LineSearches.BackTracking{Float64, Int64}, Returns{Nothing}}, @NamedTuple{layer_1::@NamedTuple{weight::Matrix{Float64}, bias::Vector{Float64}}, layer_2::@NamedTuple{weight::Matrix{Float64}, bias::Vector{Float64}}, layer_3::@NamedTuple{weight::Matrix{Float64}, bias::Vector{Float64}}, layer_4::@NamedTuple{weight::Matrix{Float64}, bias::Vector{Float64}}}, Bool, NeuralPDE.WeightedIntervalTraining{Float64}, Bool, typeof(Main.var"##225".additional_loss), Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}(Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_4::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}((layer_1 = Dense(1 => 15, σ), layer_2 = Dense(15 => 15, σ), layer_3 = Dense(15 => 15, σ), layer_4 = Dense(15 => 2)), nothing), Optim.LBFGS{Nothing, LineSearches.InitialStatic{Float64}, LineSearches.BackTracking{Float64, Int64}, Returns{Nothing}}(10, LineSearches.InitialStatic{Float64}
  alpha: Float64 1.0
  scaled: Bool false
, LineSearches.BackTracking{Float64, Int64}
  c_1: Float64 0.0001
  ρ_hi: Float64 0.5
  ρ_lo: Float64 0.1
  iterations: Int64 1000
  order: Int64 3
  maxstep: Float64 Inf
  cache: Nothing nothing
, nothing, Returns{Nothing}(nothing), Optim.Flat(), true), (layer_1 = (weight = [-0.04929668828845024; -0.3266667425632477; … ; -1.3531280755996704; -0.2917589843273163;;], bias = [0.28568029403686523, -0.4209803342819214, -0.24613642692565918, -0.9429000616073608, -0.3618292808532715, 0.077278733253479, 0.9969245195388794, 0.7939795255661011, 0.45440757274627686, -0.4830443859100342, -0.6861011981964111, -0.3221019506454468, -0.5597391128540039, -0.15051674842834473, 0.9440881013870239]), layer_2 = (weight = [-0.08606008440256119 -0.2168799191713333 … -0.3507671356201172 0.07374405115842819; 0.24009405076503754 -0.2372819483280182 … 0.34944412112236023 -0.21207459270954132; … ; 0.3976286053657532 0.28444960713386536 … -0.32817620038986206 0.396392285823822; -0.07926429808139801 0.35875916481018066 … -0.03593128174543381 -0.28511112928390503], bias = [-0.065037302672863, 0.18384626507759094, 0.17181798815727234, -0.17310386896133423, 0.06428726017475128, 0.09600061178207397, -0.08703552931547165, 0.06890828162431717, -0.16194558143615723, -0.14649711549282074, -0.14649459719657898, -0.04401325806975365, -0.015492657199501991, 0.1046019047498703, 0.15015578269958496]), layer_3 = (weight = [-0.2995997369289398 0.14921274781227112 … -0.011808237060904503 -0.3409591019153595; 0.4351722002029419 0.1286778748035431 … -0.20781198143959045 -0.030425485223531723; … ; -0.02206072397530079 0.14348538219928741 … -0.05763476341962814 -0.2672235071659088; 0.2975636124610901 -0.06781639903783798 … 0.4012162387371063 0.12123444676399231], bias = [0.04135546088218689, -0.2398381233215332, 0.1595604568719864, 0.08355490118265152, -0.06149742379784584, -0.06998120248317719, -0.008059235289692879, -0.10936713218688965, -0.18340998888015747, 0.06297893822193146, 0.04081515222787857, -0.04258332401514053, 0.11171907186508179, -0.21218737959861755, 0.07965957373380661]), layer_4 = (weight = [0.3909371793270111 -0.23473049700260162 … 0.07385867089033127 0.31727129220962524; -0.04396385699510574 0.1817844659090042 … -0.26729491353034973 0.24492913484573364], bias = [0.04966225475072861, -0.04299044609069824])), false, true, NeuralPDE.WeightedIntervalTraining{Float64}([0.7, 0.2, 0.1], 500), true, Main.var"##225".additional_loss, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}())

Solve the problem Use verbose=true to see the fitting process

sol = solve(prob, alg, verbose = false, abstol = 1e-8, maxiters = 5000, saveat = t_)

retcode: Success
Interpolation: Trained neural network interpolation
t: 501-element Vector{Float64}:
 0.0
 0.01
 0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0.09
 ⋮
 4.92
 4.93
 4.94
 4.95
 4.96
 4.97
 4.98
 4.99
 5.0
u: 501-element Vector{Vector{Float64}}:
 [5.0, 5.0]
 [4.825219433412094, 5.096168844830861]
 [4.652540064572287, 5.185057682551678]
 [4.482318996752367, 5.266456345722116]
 [4.314965228356471, 5.340186557623263]
 [4.150904890444599, 5.406105763965478]
 [3.9905535903720475, 5.46411050953679]
 [3.8342950530751816, 5.514140329714225]
 [3.68246566750306, 5.556181807688908]
 [3.5353445948175324, 5.590271868408185]
 ⋮
 [0.9501419393420445, 0.453755580223798]
 [0.9584773587584268, 0.44504763407907877]
 [0.9668286501471819, 0.4365451623089873]
 [0.9751918701835995, 0.42824087366862784]
 [0.9835631730078926, 0.4201276511532237]
 [0.9919388102156157, 0.4121985503144687]
 [1.0003151307194824, 0.4044467974495518]
 [1.0086885804904049, 0.39686578767248903]
 [1.017055702183347, 0.38944908287801816]

See the fitted parameters

println(sol.k.u.p)

[1.498960037473685, 0.9999148676519805, 2.9935328565732076, 0.9982063915725597]

Visualize the fit

plot(sol, labels = ["u1_pinn" "u2_pinn"])
plot!(sol_data, labels = ["u1_data" "u2_data"])

---

This notebook was generated using Literate.jl.