MNIST Classification using Neural ODEs

To understand Neural ODEs, users should look up these lecture notes. We recommend users to directly use DiffEqFlux.jl, instead of implementing Neural ODEs from scratch.

Package Imports

using Lux, ComponentArrays, SciMLSensitivity, LuxCUDA, Optimisers, OrdinaryDiffEqTsit5,
      Random, Statistics, Zygote, OneHotArrays, InteractiveUtils, Printf
using MLDatasets: MNIST
using MLUtils: DataLoader, splitobs

CUDA.allowscalar(false)

Loading MNIST

function loadmnist(batchsize, train_split)
    # Load MNIST: Only 1500 for demonstration purposes
    N = 1500
    dataset = MNIST(; split=:train)
    imgs = dataset.features[:, :, 1:N]
    labels_raw = dataset.targets[1:N]

    # Process images into (H,W,C,BS) batches
    x_data = Float32.(reshape(imgs, size(imgs, 1), size(imgs, 2), 1, size(imgs, 3)))
    y_data = onehotbatch(labels_raw, 0:9)
    (x_train, y_train), (x_test, y_test) = splitobs((x_data, y_data); at=train_split)

    return (
        # Use DataLoader to automatically minibatch and shuffle the data
        DataLoader(collect.((x_train, y_train)); batchsize, shuffle=true),
        # Don't shuffle the test data
        DataLoader(collect.((x_test, y_test)); batchsize, shuffle=false)
    )
end

loadmnist (generic function with 1 method)

Define the Neural ODE Layer

First we will use the @compact macro to define the Neural ODE Layer.

function NeuralODECompact(
        model::Lux.AbstractLuxLayer; solver=Tsit5(), tspan=(0.0f0, 1.0f0), kwargs...)
    return @compact(; model, solver, tspan, kwargs...) do x, p
        dudt(u, p, t) = vec(model(reshape(u, size(x)), p))
        # Note the `p.model` here
        prob = ODEProblem(ODEFunction{false}(dudt), vec(x), tspan, p.model)
        @return solve(prob, solver; kwargs...)
    end
end

NeuralODECompact (generic function with 1 method)

We recommend using the compact macro for creating custom layers. The below implementation exists mostly for historical reasons when @compact was not part of the stable API. Also, it helps users understand how the layer interface of Lux works.

The NeuralODE is a ContainerLayer, which stores a model. The parameters and states of the NeuralODE are same as those of the underlying model.

struct NeuralODE{M <: Lux.AbstractLuxLayer, So, T, K} <: Lux.AbstractLuxWrapperLayer{:model}
    model::M
    solver::So
    tspan::T
    kwargs::K
end

function NeuralODE(
        model::Lux.AbstractLuxLayer; solver=Tsit5(), tspan=(0.0f0, 1.0f0), kwargs...)
    return NeuralODE(model, solver, tspan, kwargs)
end

Main.var"##225".NeuralODE

OrdinaryDiffEq.jl can deal with non-Vector Inputs! However, certain discrete sensitivities like ReverseDiffAdjoint can't handle non-Vector inputs. Hence, we need to convert the input and output of the ODE solver to a Vector.

function (n::NeuralODE)(x, ps, st)
    function dudt(u, p, t)
        u_, st = n.model(reshape(u, size(x)), p, st)
        return vec(u_)
    end
    prob = ODEProblem{false}(ODEFunction{false}(dudt), vec(x), n.tspan, ps)
    return solve(prob, n.solver; n.kwargs...), st
end

@views diffeqsol_to_array(l::Int, x::ODESolution) = reshape(last(x.u), (l, :))
@views diffeqsol_to_array(l::Int, x::AbstractMatrix) = reshape(x[:, end], (l, :))

diffeqsol_to_array (generic function with 2 methods)

Create and Initialize the Neural ODE Layer

function create_model(model_fn=NeuralODE; dev=gpu_device(), use_named_tuple::Bool=false,
        sensealg=InterpolatingAdjoint(; autojacvec=ZygoteVJP()))
    # Construct the Neural ODE Model
    model = Chain(FlattenLayer(),
        Dense(784 => 20, tanh),
        model_fn(
            Chain(Dense(20 => 10, tanh), Dense(10 => 10, tanh), Dense(10 => 20, tanh));
            save_everystep=false, reltol=1.0f-3,
            abstol=1.0f-3, save_start=false, sensealg),
        Base.Fix1(diffeqsol_to_array, 20),
        Dense(20 => 10))

    rng = Random.default_rng()
    Random.seed!(rng, 0)

    ps, st = Lux.setup(rng, model)
    ps = (use_named_tuple ? ps : ComponentArray(ps)) |> dev
    st = st |> dev

    return model, ps, st
end

create_model (generic function with 2 methods)

Define Utility Functions

const logitcrossentropy = CrossEntropyLoss(; logits=Val(true))

function accuracy(model, ps, st, dataloader)
    total_correct, total = 0, 0
    st = Lux.testmode(st)
    for (x, y) in dataloader
        target_class = onecold(y)
        predicted_class = onecold(first(model(x, ps, st)))
        total_correct += sum(target_class .== predicted_class)
        total += length(target_class)
    end
    return total_correct / total
end

accuracy (generic function with 1 method)

Training

function train(model_function; cpu::Bool=false, kwargs...)
    dev = cpu ? cpu_device() : gpu_device()
    model, ps, st = create_model(model_function; dev, kwargs...)

    # Training
    train_dataloader, test_dataloader = loadmnist(128, 0.9) |> dev

    tstate = Training.TrainState(model, ps, st, Adam(0.001f0))

    ### Lets train the model
    nepochs = 9
    for epoch in 1:nepochs
        stime = time()
        for (x, y) in train_dataloader
            _, _, _, tstate = Training.single_train_step!(
                AutoZygote(), logitcrossentropy, (x, y), tstate)
        end
        ttime = time() - stime

        tr_acc = accuracy(model, tstate.parameters, tstate.states, train_dataloader) * 100
        te_acc = accuracy(model, tstate.parameters, tstate.states, test_dataloader) * 100
        @printf "[%d/%d]\tTime %.4fs\tTraining Accuracy: %.5f%%\tTest \
                 Accuracy: %.5f%%\n" epoch nepochs ttime tr_acc te_acc
    end
end

train(NeuralODECompact)

[1/9]	Time 129.1843s	Training Accuracy: 37.48148%	Test Accuracy: 40.00000%
[2/9]	Time 0.5219s	Training Accuracy: 58.22222%	Test Accuracy: 57.33333%
[3/9]	Time 0.6924s	Training Accuracy: 67.85185%	Test Accuracy: 70.66667%
[4/9]	Time 0.4994s	Training Accuracy: 74.29630%	Test Accuracy: 74.66667%
[5/9]	Time 0.5288s	Training Accuracy: 76.29630%	Test Accuracy: 76.00000%
[6/9]	Time 0.5490s	Training Accuracy: 78.74074%	Test Accuracy: 80.00000%
[7/9]	Time 0.4892s	Training Accuracy: 82.22222%	Test Accuracy: 81.33333%
[8/9]	Time 0.4916s	Training Accuracy: 83.62963%	Test Accuracy: 83.33333%
[9/9]	Time 0.4918s	Training Accuracy: 85.18519%	Test Accuracy: 82.66667%

train(NeuralODE)

[1/9]	Time 36.2287s	Training Accuracy: 37.48148%	Test Accuracy: 40.00000%
[2/9]	Time 0.5229s	Training Accuracy: 57.18519%	Test Accuracy: 57.33333%
[3/9]	Time 0.7165s	Training Accuracy: 68.37037%	Test Accuracy: 68.00000%
[4/9]	Time 0.5129s	Training Accuracy: 73.77778%	Test Accuracy: 75.33333%
[5/9]	Time 0.5013s	Training Accuracy: 76.14815%	Test Accuracy: 77.33333%
[6/9]	Time 0.7354s	Training Accuracy: 79.48148%	Test Accuracy: 80.66667%
[7/9]	Time 0.5105s	Training Accuracy: 81.25926%	Test Accuracy: 80.66667%
[8/9]	Time 0.5098s	Training Accuracy: 83.40741%	Test Accuracy: 82.66667%
[9/9]	Time 0.7640s	Training Accuracy: 84.81481%	Test Accuracy: 82.00000%

We can also change the sensealg and train the model! GaussAdjoint allows you to use any arbitrary parameter structure and not just a flat vector (ComponentArray).

train(NeuralODE; sensealg=GaussAdjoint(; autojacvec=ZygoteVJP()), use_named_tuple=true)

[1/9]	Time 42.8908s	Training Accuracy: 37.48148%	Test Accuracy: 40.00000%
[2/9]	Time 0.4939s	Training Accuracy: 58.44444%	Test Accuracy: 58.00000%
[3/9]	Time 0.5104s	Training Accuracy: 66.96296%	Test Accuracy: 68.00000%
[4/9]	Time 0.6748s	Training Accuracy: 72.44444%	Test Accuracy: 73.33333%
[5/9]	Time 0.4703s	Training Accuracy: 76.37037%	Test Accuracy: 76.00000%
[6/9]	Time 0.4817s	Training Accuracy: 78.81481%	Test Accuracy: 79.33333%
[7/9]	Time 0.6777s	Training Accuracy: 80.51852%	Test Accuracy: 81.33333%
[8/9]	Time 0.5007s	Training Accuracy: 82.74074%	Test Accuracy: 83.33333%
[9/9]	Time 0.5049s	Training Accuracy: 85.25926%	Test Accuracy: 82.66667%

But remember some AD backends like ReverseDiff is not GPU compatible. For a model this size, you will notice that training time is significantly lower for training on CPU than on GPU.

train(NeuralODE; sensealg=InterpolatingAdjoint(; autojacvec=ReverseDiffVJP()), cpu=true)

[1/9]	Time 99.1523s	Training Accuracy: 37.48148%	Test Accuracy: 40.00000%
[2/9]	Time 3.9228s	Training Accuracy: 58.74074%	Test Accuracy: 56.66667%
[3/9]	Time 4.6773s	Training Accuracy: 69.92593%	Test Accuracy: 71.33333%
[4/9]	Time 3.3143s	Training Accuracy: 72.81481%	Test Accuracy: 74.00000%
[5/9]	Time 4.7579s	Training Accuracy: 76.37037%	Test Accuracy: 78.66667%
[6/9]	Time 10.1299s	Training Accuracy: 79.03704%	Test Accuracy: 80.66667%
[7/9]	Time 7.9662s	Training Accuracy: 81.62963%	Test Accuracy: 80.66667%
[8/9]	Time 8.7852s	Training Accuracy: 83.33333%	Test Accuracy: 80.00000%
[9/9]	Time 8.3199s	Training Accuracy: 85.40741%	Test Accuracy: 82.00000%

For completeness, let's also test out discrete sensitivities!

train(NeuralODE; sensealg=ReverseDiffAdjoint(), cpu=true)

[1/9]	Time 42.9006s	Training Accuracy: 37.48148%	Test Accuracy: 40.00000%
[2/9]	Time 17.7529s	Training Accuracy: 58.66667%	Test Accuracy: 57.33333%
[3/9]	Time 20.4639s	Training Accuracy: 69.70370%	Test Accuracy: 71.33333%
[4/9]	Time 18.0215s	Training Accuracy: 72.74074%	Test Accuracy: 74.00000%
[5/9]	Time 16.3387s	Training Accuracy: 76.14815%	Test Accuracy: 78.66667%
[6/9]	Time 18.2039s	Training Accuracy: 79.03704%	Test Accuracy: 80.66667%
[7/9]	Time 20.5332s	Training Accuracy: 81.55556%	Test Accuracy: 80.66667%
[8/9]	Time 19.2303s	Training Accuracy: 83.40741%	Test Accuracy: 80.00000%
[9/9]	Time 18.3113s	Training Accuracy: 85.25926%	Test Accuracy: 81.33333%

Alternate Implementation using Stateful Layer

Starting v0.5.5, Lux provides a StatefulLuxLayer which can be used to avoid the Boxing of st. Using the @compact API avoids this problem entirely.

struct StatefulNeuralODE{M <: Lux.AbstractLuxLayer, So, T, K} <:
       Lux.AbstractLuxWrapperLayer{:model}
    model::M
    solver::So
    tspan::T
    kwargs::K
end

function StatefulNeuralODE(
        model::Lux.AbstractLuxLayer; solver=Tsit5(), tspan=(0.0f0, 1.0f0), kwargs...)
    return StatefulNeuralODE(model, solver, tspan, kwargs)
end

function (n::StatefulNeuralODE)(x, ps, st)
    st_model = StatefulLuxLayer{true}(n.model, ps, st)
    dudt(u, p, t) = st_model(u, p)
    prob = ODEProblem{false}(ODEFunction{false}(dudt), x, n.tspan, ps)
    return solve(prob, n.solver; n.kwargs...), st_model.st
end

Train the new Stateful Neural ODE

train(StatefulNeuralODE)

[1/9]	Time 39.5600s	Training Accuracy: 37.48148%	Test Accuracy: 40.00000%
[2/9]	Time 0.6338s	Training Accuracy: 58.22222%	Test Accuracy: 55.33333%
[3/9]	Time 0.5006s	Training Accuracy: 68.29630%	Test Accuracy: 68.66667%
[4/9]	Time 0.4926s	Training Accuracy: 73.11111%	Test Accuracy: 76.00000%
[5/9]	Time 0.4840s	Training Accuracy: 75.92593%	Test Accuracy: 76.66667%
[6/9]	Time 0.7907s	Training Accuracy: 78.96296%	Test Accuracy: 80.66667%
[7/9]	Time 0.4676s	Training Accuracy: 80.81481%	Test Accuracy: 81.33333%
[8/9]	Time 0.5421s	Training Accuracy: 83.25926%	Test Accuracy: 82.66667%
[9/9]	Time 0.4722s	Training Accuracy: 84.59259%	Test Accuracy: 82.00000%

We might not see a significant difference in the training time, but let us investigate the type stabilities of the layers.

Type Stability

model, ps, st = create_model(NeuralODE)

model_stateful, ps_stateful, st_stateful = create_model(StatefulNeuralODE)

x = gpu_device()(ones(Float32, 28, 28, 1, 3));

NeuralODE is not type stable due to the boxing of st

@code_warntype model(x, ps, st)

MethodInstance for (::Lux.Chain{@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Main.var"##225".NeuralODE{Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, Tuple{Float32, Float32}, Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing})(::CUDA.CuArray{Float32, 4, CUDA.DeviceMemory}, ::ComponentArrays.ComponentVector{Float32, CUDA.CuArray{Float32, 1, CUDA.DeviceMemory}, Tuple{ComponentArrays.Axis{(layer_1 = 1:0, layer_2 = ViewAxis(1:15700, Axis(weight = ViewAxis(1:15680, ShapedAxis((20, 784))), bias = 15681:15700)), layer_3 = ViewAxis(15701:16240, Axis(layer_1 = ViewAxis(1:210, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)), layer_2 = ViewAxis(211:320, Axis(weight = ViewAxis(1:100, ShapedAxis((10, 10))), bias = 101:110)), layer_3 = ViewAxis(321:540, Axis(weight = ViewAxis(1:200, ShapedAxis((20, 10))), bias = 201:220)))), layer_4 = 16241:16240, layer_5 = ViewAxis(16241:16450, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)))}}}, ::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}, layer_4::@NamedTuple{}, layer_5::@NamedTuple{}})
  from (c::Lux.Chain)(x, ps, st::NamedTuple) @ Lux /var/lib/buildkite-agent/builds/gpuci-12/julialang/lux-dot-jl/src/layers/containers.jl:480
Arguments
  c::Lux.Chain{@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Main.var"##225".NeuralODE{Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, Tuple{Float32, Float32}, Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}
  x::CUDA.CuArray{Float32, 4, CUDA.DeviceMemory}
  ps::ComponentArrays.ComponentVector{Float32, CUDA.CuArray{Float32, 1, CUDA.DeviceMemory}, Tuple{ComponentArrays.Axis{(layer_1 = 1:0, layer_2 = ViewAxis(1:15700, Axis(weight = ViewAxis(1:15680, ShapedAxis((20, 784))), bias = 15681:15700)), layer_3 = ViewAxis(15701:16240, Axis(layer_1 = ViewAxis(1:210, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)), layer_2 = ViewAxis(211:320, Axis(weight = ViewAxis(1:100, ShapedAxis((10, 10))), bias = 101:110)), layer_3 = ViewAxis(321:540, Axis(weight = ViewAxis(1:200, ShapedAxis((20, 10))), bias = 201:220)))), layer_4 = 16241:16240, layer_5 = ViewAxis(16241:16450, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)))}}}
  st::Core.Const((layer_1 = NamedTuple(), layer_2 = NamedTuple(), layer_3 = (layer_1 = NamedTuple(), layer_2 = NamedTuple(), layer_3 = NamedTuple()), layer_4 = NamedTuple(), layer_5 = NamedTuple()))
Body::TUPLE{CUDA.CUARRAY{FLOAT32, 2, CUDA.DEVICEMEMORY}, NAMEDTUPLE{(:LAYER_1, :LAYER_2, :LAYER_3, :LAYER_4, :LAYER_5), <:TUPLE{@NAMEDTUPLE{}, @NAMEDTUPLE{}, ANY, @NAMEDTUPLE{}, @NAMEDTUPLE{}}}}
1 ─ %1 = Base.getproperty(c, :layers)::@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Main.var"##225".NeuralODE{Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, Tuple{Float32, Float32}, Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}
│   %2 = Lux.applychain(%1, x, ps, st)::TUPLE{CUDA.CUARRAY{FLOAT32, 2, CUDA.DEVICEMEMORY}, NAMEDTUPLE{(:LAYER_1, :LAYER_2, :LAYER_3, :LAYER_4, :LAYER_5), <:TUPLE{@NAMEDTUPLE{}, @NAMEDTUPLE{}, ANY, @NAMEDTUPLE{}, @NAMEDTUPLE{}}}}
└──      return %2

We avoid the problem entirely by using StatefulNeuralODE

@code_warntype model_stateful(x, ps_stateful, st_stateful)

MethodInstance for (::Lux.Chain{@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Main.var"##225".StatefulNeuralODE{Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, Tuple{Float32, Float32}, Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing})(::CUDA.CuArray{Float32, 4, CUDA.DeviceMemory}, ::ComponentArrays.ComponentVector{Float32, CUDA.CuArray{Float32, 1, CUDA.DeviceMemory}, Tuple{ComponentArrays.Axis{(layer_1 = 1:0, layer_2 = ViewAxis(1:15700, Axis(weight = ViewAxis(1:15680, ShapedAxis((20, 784))), bias = 15681:15700)), layer_3 = ViewAxis(15701:16240, Axis(layer_1 = ViewAxis(1:210, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)), layer_2 = ViewAxis(211:320, Axis(weight = ViewAxis(1:100, ShapedAxis((10, 10))), bias = 101:110)), layer_3 = ViewAxis(321:540, Axis(weight = ViewAxis(1:200, ShapedAxis((20, 10))), bias = 201:220)))), layer_4 = 16241:16240, layer_5 = ViewAxis(16241:16450, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)))}}}, ::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}, layer_4::@NamedTuple{}, layer_5::@NamedTuple{}})
  from (c::Lux.Chain)(x, ps, st::NamedTuple) @ Lux /var/lib/buildkite-agent/builds/gpuci-12/julialang/lux-dot-jl/src/layers/containers.jl:480
Arguments
  c::Lux.Chain{@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Main.var"##225".StatefulNeuralODE{Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, Tuple{Float32, Float32}, Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}
  x::CUDA.CuArray{Float32, 4, CUDA.DeviceMemory}
  ps::ComponentArrays.ComponentVector{Float32, CUDA.CuArray{Float32, 1, CUDA.DeviceMemory}, Tuple{ComponentArrays.Axis{(layer_1 = 1:0, layer_2 = ViewAxis(1:15700, Axis(weight = ViewAxis(1:15680, ShapedAxis((20, 784))), bias = 15681:15700)), layer_3 = ViewAxis(15701:16240, Axis(layer_1 = ViewAxis(1:210, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)), layer_2 = ViewAxis(211:320, Axis(weight = ViewAxis(1:100, ShapedAxis((10, 10))), bias = 101:110)), layer_3 = ViewAxis(321:540, Axis(weight = ViewAxis(1:200, ShapedAxis((20, 10))), bias = 201:220)))), layer_4 = 16241:16240, layer_5 = ViewAxis(16241:16450, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)))}}}
  st::Core.Const((layer_1 = NamedTuple(), layer_2 = NamedTuple(), layer_3 = (layer_1 = NamedTuple(), layer_2 = NamedTuple(), layer_3 = NamedTuple()), layer_4 = NamedTuple(), layer_5 = NamedTuple()))
Body::Tuple{CUDA.CuArray{Float32, 2, CUDA.DeviceMemory}, @NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}, layer_4::@NamedTuple{}, layer_5::@NamedTuple{}}}
1 ─ %1 = Base.getproperty(c, :layers)::@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Main.var"##225".StatefulNeuralODE{Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, Tuple{Float32, Float32}, Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}
│   %2 = Lux.applychain(%1, x, ps, st)::Tuple{CUDA.CuArray{Float32, 2, CUDA.DeviceMemory}, @NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}, layer_4::@NamedTuple{}, layer_5::@NamedTuple{}}}
└──      return %2

Note, that we still recommend using this layer internally and not exposing this as the default API to the users.

Finally checking the compact model

model_compact, ps_compact, st_compact = create_model(NeuralODECompact)

@code_warntype model_compact(x, ps_compact, st_compact)

MethodInstance for (::Lux.Chain{@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.CompactLuxLayer{:₋₋₋no_special_dispatch₋₋₋, Main.var"##225".var"#2#3", Nothing, @NamedTuple{model::Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}}, Lux.CompactMacroImpl.ValueStorage{@NamedTuple{}, @NamedTuple{solver::Returns{OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}}, tspan::Returns{Tuple{Float32, Float32}}}}, Tuple{Tuple{Symbol}, Tuple{Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing})(::CUDA.CuArray{Float32, 4, CUDA.DeviceMemory}, ::ComponentArrays.ComponentVector{Float32, CUDA.CuArray{Float32, 1, CUDA.DeviceMemory}, Tuple{ComponentArrays.Axis{(layer_1 = 1:0, layer_2 = ViewAxis(1:15700, Axis(weight = ViewAxis(1:15680, ShapedAxis((20, 784))), bias = 15681:15700)), layer_3 = ViewAxis(15701:16240, Axis(model = ViewAxis(1:540, Axis(layer_1 = ViewAxis(1:210, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)), layer_2 = ViewAxis(211:320, Axis(weight = ViewAxis(1:100, ShapedAxis((10, 10))), bias = 101:110)), layer_3 = ViewAxis(321:540, Axis(weight = ViewAxis(1:200, ShapedAxis((20, 10))), bias = 201:220)))),)), layer_4 = 16241:16240, layer_5 = ViewAxis(16241:16450, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)))}}}, ::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{model::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}, solver::OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, tspan::Tuple{Float32, Float32}, ₋₋₋kwargs₋₋₋::Lux.CompactMacroImpl.KwargsStorage{@NamedTuple{kwargs::Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}}}, layer_4::@NamedTuple{}, layer_5::@NamedTuple{}})
  from (c::Lux.Chain)(x, ps, st::NamedTuple) @ Lux /var/lib/buildkite-agent/builds/gpuci-12/julialang/lux-dot-jl/src/layers/containers.jl:480
Arguments
  c::Lux.Chain{@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.CompactLuxLayer{:₋₋₋no_special_dispatch₋₋₋, Main.var"##225".var"#2#3", Nothing, @NamedTuple{model::Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}}, Lux.CompactMacroImpl.ValueStorage{@NamedTuple{}, @NamedTuple{solver::Returns{OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}}, tspan::Returns{Tuple{Float32, Float32}}}}, Tuple{Tuple{Symbol}, Tuple{Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}
  x::CUDA.CuArray{Float32, 4, CUDA.DeviceMemory}
  ps::ComponentArrays.ComponentVector{Float32, CUDA.CuArray{Float32, 1, CUDA.DeviceMemory}, Tuple{ComponentArrays.Axis{(layer_1 = 1:0, layer_2 = ViewAxis(1:15700, Axis(weight = ViewAxis(1:15680, ShapedAxis((20, 784))), bias = 15681:15700)), layer_3 = ViewAxis(15701:16240, Axis(model = ViewAxis(1:540, Axis(layer_1 = ViewAxis(1:210, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)), layer_2 = ViewAxis(211:320, Axis(weight = ViewAxis(1:100, ShapedAxis((10, 10))), bias = 101:110)), layer_3 = ViewAxis(321:540, Axis(weight = ViewAxis(1:200, ShapedAxis((20, 10))), bias = 201:220)))),)), layer_4 = 16241:16240, layer_5 = ViewAxis(16241:16450, Axis(weight = ViewAxis(1:200, ShapedAxis((10, 20))), bias = 201:210)))}}}
  st::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{model::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}, solver::OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, tspan::Tuple{Float32, Float32}, ₋₋₋kwargs₋₋₋::Lux.CompactMacroImpl.KwargsStorage{@NamedTuple{kwargs::Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}}}, layer_4::@NamedTuple{}, layer_5::@NamedTuple{}}
Body::Tuple{CUDA.CuArray{Float32, 2, CUDA.DeviceMemory}, @NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{model::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}, solver::OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, tspan::Tuple{Float32, Float32}, ₋₋₋kwargs₋₋₋::Lux.CompactMacroImpl.KwargsStorage{@NamedTuple{kwargs::Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}}}, layer_4::@NamedTuple{}, layer_5::@NamedTuple{}}}
1 ─ %1 = Base.getproperty(c, :layers)::@NamedTuple{layer_1::Lux.FlattenLayer{Nothing}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.CompactLuxLayer{:₋₋₋no_special_dispatch₋₋₋, Main.var"##225".var"#2#3", Nothing, @NamedTuple{model::Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(tanh), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}}, Lux.CompactMacroImpl.ValueStorage{@NamedTuple{}, @NamedTuple{solver::Returns{OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}}, tspan::Returns{Tuple{Float32, Float32}}}}, Tuple{Tuple{Symbol}, Tuple{Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}}}, layer_4::Lux.WrappedFunction{Base.Fix1{typeof(Main.var"##225".diffeqsol_to_array), Int64}}, layer_5::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}
│   %2 = Lux.applychain(%1, x, ps, st)::Tuple{CUDA.CuArray{Float32, 2, CUDA.DeviceMemory}, @NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{model::@NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}, solver::OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, tspan::Tuple{Float32, Float32}, ₋₋₋kwargs₋₋₋::Lux.CompactMacroImpl.KwargsStorage{@NamedTuple{kwargs::Base.Pairs{Symbol, Any, NTuple{5, Symbol}, @NamedTuple{save_everystep::Bool, reltol::Float32, abstol::Float32, save_start::Bool, sensealg::SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensitivity.ZygoteVJP}}}}}}, layer_4::@NamedTuple{}, layer_5::@NamedTuple{}}}
└──      return %2

Appendix

using InteractiveUtils
InteractiveUtils.versioninfo()

if @isdefined(MLDataDevices)
    if @isdefined(CUDA) && MLDataDevices.functional(CUDADevice)
        println()
        CUDA.versioninfo()
    end

    if @isdefined(AMDGPU) && MLDataDevices.functional(AMDGPUDevice)
        println()
        AMDGPU.versioninfo()
    end
end

Julia Version 1.10.6
Commit 67dffc4a8ae (2024-10-28 12:23 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 48 × AMD EPYC 7402 24-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 48 default, 0 interactive, 24 GC (on 2 virtual cores)
Environment:
  JULIA_CPU_THREADS = 2
  JULIA_DEPOT_PATH = /root/.cache/julia-buildkite-plugin/depots/01872db4-8c79-43af-ab7d-12abac4f24f6
  LD_LIBRARY_PATH = /usr/local/nvidia/lib:/usr/local/nvidia/lib64
  JULIA_PKG_SERVER = 
  JULIA_NUM_THREADS = 48
  JULIA_CUDA_HARD_MEMORY_LIMIT = 100%
  JULIA_PKG_PRECOMPILE_AUTO = 0
  JULIA_DEBUG = Literate

CUDA runtime 12.6, artifact installation
CUDA driver 12.6
NVIDIA driver 560.35.3

CUDA libraries: 
- CUBLAS: 12.6.3
- CURAND: 10.3.7
- CUFFT: 11.3.0
- CUSOLVER: 11.7.1
- CUSPARSE: 12.5.4
- CUPTI: 2024.3.2 (API 24.0.0)
- NVML: 12.0.0+560.35.3

Julia packages: 
- CUDA: 5.5.2
- CUDA_Driver_jll: 0.10.3+0
- CUDA_Runtime_jll: 0.15.4+0

Toolchain:
- Julia: 1.10.6
- LLVM: 15.0.7

Environment:
- JULIA_CUDA_HARD_MEMORY_LIMIT: 100%

1 device:
  0: NVIDIA A100-PCIE-40GB MIG 1g.5gb (sm_80, 3.920 GiB / 4.750 GiB available)

---

This page was generated using Literate.jl.