LoadError when using interpolations as input for a neural ode

Hello,
I am trying to train a simple nn with an external input that depends on time. The input is interpolated using:

broadband = LinearInterpolation(time, U)

The network runs without any problem for the first iteration of the optimization process using ADAM. After calculating the loss, the program stops to work and I get the following error message:

ERROR: LoadError: ArgumentError: unable to check bounds for indices of type Intertions.WeightedAdjIndex{2,Float64}

Does the backpropagation algorithm fail when interpolation is involved?

Please find an excerpt of my code below:

# Define the neural network and ODE part
nn_model = FastChain(FastDense(4, 8, tanh), FastDense(8, 4))
p_model = initial_params(nn_model)

function dudt(u, p, t)
    nn_model(vcat(u[1], broadband(t), u[3], u[4]), p)
end

prob = ODEProblem(dudt, u0, [time_training[1], time_training[end]])

function predict_neuralode(p)
    _prob = remake(prob, p = p)
    sol = Array(solve(_prob, Tsit5(), saveat = t_step, abstol = 1e-8, reltol = 1e-6))
    sol = sol[3, :]
end

# Loss function defined as elementwise distance bewteen pred. actual data
function loss(p) # loss function
    sol = predict_neuralode(p)
    loss = sum(abs2, sol - Q_training)
    print("Loss: $loss")
    return loss
end

# start optimization
res0 = DiffEqFlux.sciml_train(loss, p_model, ADAM(0.01), maxiters = 100)

Can you share code that reproduces the error? I canā€™t see what would cause the error from what you posted.

Yes, below is a working example to reproduce the error. I had to replace my input which comes from a local file with some arbitray function that is interpolated, but the error is the same.

using DifferentialEquations
using Plots
using Flux
using DiffEqFlux
using Interpolations
using HDF5
import Statistics
import LinearAlgebra

# Define all required functions
# Define the double pendulum
function linear_target(du, u, p, t)
    Ļ•_1, Ļƒ_1, Ļ•_2, Ļƒ_2 = u
    Ī±, Ī², Ī³ = p
    du[1] = Ļƒ_1
    du[2] = Ī³ * broadband(t) - Ī± * Ļƒ_1 - Ļ•_1 + Ī² * (Ļ•_2 - Ļ•_1)
    du[3] = Ļƒ_2
    du[4] = -Ī± * Ļƒ_2 - Ļ•_2 - Ī² * (Ļ•_2 - Ļ•_1)
end


# Main
# Create some input data
t_step = 0.01125
time = 0:t_step:450
U = sin.(time)

# Interpolate signal in time for later usage in ODE
broadband = LinearInterpolation(time, U)

# Create target data for later optimization
p_ref = [1.1340, 0.5644, 3.8565]
tspan = (0.0, time[end])
u0 = [0.0, 0.0, 0.0, 0.0]
prob = ODEProblem(linear_target, u0, tspan, p_ref)
sol = solve(prob, AutoTsit5(Rosenbrock23()), p = p_ref, saveat = t_step)
Y_ref = sol[3, :] # Take only Ļƒ2 as target

# Define the neural network and ODE part
nn_model = FastChain(FastDense(4, 8, tanh), FastDense(8, 4))
p_model = initial_params(nn_model)

function dudt(u, p, t)
    nn_model(vcat(u[1], broadband(t), u[3], u[4]), p)
end

prob = ODEProblem(dudt, u0, [time[1], time[end]])

function predict_neuralode(p)
    _prob = remake(prob, p = p)
    sol = Array(solve(_prob, Tsit5(), saveat = t_step, abstol = 1e-8, reltol = 1e-6))
    sol = sol[3, :]
end

# Loss function defined as elementwise distance bewteen pred. actual data
function loss(p) # loss function
    sol = predict_neuralode(p)
    loss = sum(abs2, sol - Y_ref)
    print("Loss: $loss")
    return loss
end

# start optimization
res0 = DiffEqFlux.sciml_train(loss, p_model, ADAM(0.01), maxiters = 100)

Dear Community,
if anyone stumbles across this blog entry and may has an idea what the problem could be, please let me know.
Although this thread is old, the problem remains highly relevant to me!
Thank you very much in advance!

Okay, I can see how to get this working.

First of all, unsurprisingly, it seems like the error is as a result of trying to differentiate LinearInterpolation.

julia> gradient(broadband, 0.5)
ERROR: ArgumentError: unable to check bounds for indices of type Interpolations.WeightedAdjIndex{2,Float64}

Some part of that package apparently doesnā€™t play well with autodiff.

The implication of this is that you canā€™t compute d(du/dt)/dt.

Now, fortunately, you donā€™t need to! It just so happens that you only need d(du/dt)/dt to backpropagate through an ODE wrt the initial time point tspan[1], which in your case is fixed. (Not a fact thatā€™s terribly obvious, I donā€™t think. I only know this because Iā€™ve worked on problems of this exact type before, and had to write the gradient calculations from scratch.)

As a result, it should suffice to simply dummy the gradient wrt t, and the rest of the gradients should be calculated just fine:

import Zygote

function dudt(u, p, t)
    b = Zygote.ignore(()->broadband(t))
    nn_model(vcat(u[1], b, u[3], u[4]), p)
end

This is a bit of a hack, of course, but it seems to work.

This aside, neural differential equations of this type (with time-dependent input) are known as neural controlled differential equations. Thereā€™s been a line of work on this you might find interesting:
Neural Controlled Differential Equations for Irregular Time Series, NeurIPS 2020
Neural Rough Differential Equations for Long Time Series, ICML 2021
PyTorch library: torchcde.
(+ one more paper coming soon)

And more tangentially (kind of just for fun wrt this discussion), using a SDE+CDE as a generator-discriminator pair in a GAN:
Neural SDEs as Infinite-Dimensional GANs, ICML 2021.

I hope that helps.

3 Likes

Caveat emptor:

I do note that there is also Zygote.dropgrad. Iā€™m not completely certain of the difference between dropgrad and ignore, but I found that dropgrad gave some other obscure error. Possibly dropgrad is the appropriate tool here, but the documentation is pretty sparse. I havenā€™t tried training the above model more than a few steps to see if it converges.


As an alternative, you can try implementing the interpolation yourself. (Such that the autodiff works with it.) Linear interpolation is pretty straightforward, after all.

Also have a think about using the tstops argument. If youā€™re using linear interpolation then your vector field has kinks (derivative discontinuities) wrt time, and the solver will have a slightly easier time of it if it doesnā€™t have to discover these for itelf.

3 Likes

Yeah no worries, feel free to bump it quicker than that. Sometimes threads just get lost in the text wave. Patrickā€™s answer is a good one. Iā€™d also mention that GitHub - PumasAI/DataInterpolations.jl: A library of data interpolation and smoothing functions has a bunch of derivative overloads, which can be important for interpolations since straight differentiation of many interpolants is not necessarily numerically stable (Iā€™m looking at you Lagrange interpolation). But in your case, it would be best to just ignore the derivative of that term.

I usually tend to do it via wrapping it in a quick @nograd function, but I think they all lead to the same pond. Maybe @dhairyagandhi96 could enlighten us with how they differ.

Interpolations.jl recently got Zygote integration in Compatibility with ChainRules, Drop Julia 1.0 support by rick2047 Ā· Pull Request #414 Ā· JuliaMath/Interpolations.jl Ā· GitHub, that may be helpful to try out here

But in this case, is the fact that it needs to exist just an issue of missing DCE?

Dear @patrick-kidger, @ChrisRackauckas and @dhairyagandhi96,

thank you so much for your replies! For now I just tried to implement the ā€œhackā€ suggested by @patrick-kidger and an initial loss is returned, but then Julia throws a new error stating that

" LoadError: ArgumentError: tuple must be non-empty in expression starting atā€¦" and the location in my newly adapted file (simply replaced dudt). The line of code that seems to stop working is

res0 = DiffEqFlux.sciml_train(loss, p_model, ADAM(0.01), maxiters = 100)

Which confuses me quite a bit. @patrick-kidger you mentioned you had the code running for a few iterations so I assume you didnā€™t encounter this problem?

Update again. Thatā€™s a hiccup from yesterdayā€™s ChainRules update stuff (oopsā€¦)

Yeah suspected soā€¦ I used Pkg.update() but it still doesnā€™t workā€¦ Should I wait and try again in a few days?

]st? It should be good now.

Hmmm - I restarted Julia and tried it agian, now im throwing compiling errors for the DiffEqFlux package right of the start.

ERROR: LoadError: Failed to precompile DiffEqFlux [aae7a2af-3d4f-5e19-a356-7da93b79d9d0] to C:\Users\grego\.julia\compiled\v1.5\DiffEqFlux\BdO4p_6n1vG.ji

I feel like that at this point it maybe simpler to just reinstall Julia cleanā€¦

What is it saying in that?

I wonder if the issue is v1.6. Additions of some compiler tools (Cassette) might be involved in something here.

Hereā€™s the full error message (with my file path removed :slight_smile: )

LoadError: Failed to precompile DiffEqFlux [aae7a2af-3d4f-5e19-a356-7da93b79d9d0] to C:\Users\grego.julia\compiled\v1.5\DiffEqFlux\BdO4p_6n1vG.ji.

in expression starting at C:\Users\grego\ [ā€¦] \min_working_example.jl:3

error(::String) at error.jl:33

compilecache(::Base.PkgId, ::String) at loading.jl:1305

_require(::Base.PkgId) at loading.jl:1030

require(::Base.PkgId) at loading.jl:928

require(::Module, ::Symbol) at loading.jl:923

include_string(::Function, ::Module, ::String, ::String) at loading.jl:1088

Try updating again. I found I made a version bounding issue that would cause a problem in v1.5 (we really need a new LTS :sweat_smile:).

Iā€™m afraid it didnā€™t workt out, but at least the error message changed to some extend/got longerā€¦

[ Info: Precompiling DiffEqFlux [aae7a2af-3d4f-5e19-a356-7da93b79d9d0]  
ERROR: LoadError: LoadError: UndefVarError: HasBranchingCtx not defined
Stacktrace:
 [1] getproperty(::Module, ::Symbol) at .\Base.jl:26
 [2] top-level scope at C:\Users\grego\.julia\packages\DiffEqFlux\W9Cwj\src\fast_layers.jl:195
 [3] include(::Function, ::Module, ::String) at .\Base.jl:380
 [4] include at .\Base.jl:368 [inlined]
 [5] include(::String) at C:\Users\grego\.julia\packages\DiffEqFlux\W9Cwj\src\DiffEqFlux.jl:1
 [6] top-level scope at C:\Users\grego\.julia\packages\DiffEqFlux\W9Cwj\src\DiffEqFlux.jl:84
 [7] include(::Function, ::Module, ::String) at .\Base.jl:380
 [8] include(::Module, ::String) at .\Base.jl:368
 [9] top-level scope at none:2
 [10] eval at .\boot.jl:331 [inlined]
 [11] eval(::Expr) at .\client.jl:467
 [12] top-level scope at .\none:3
in expression starting at C:\Users\grego\.julia\packages\DiffEqFlux\W9Cwj\src\fast_layers.jl:195
in expression starting at C:\Users\grego\.julia\packages\DiffEqFlux\W9Cwj\src\DiffEqFlux.jl:84
ERROR: LoadError: Failed to precompile DiffEqFlux [aae7a2af-3d4f-5e19-a356-7da93b79d9d0] to C:\Users\grego\.julia\compiled\v1.5\DiffEqFlux\BdO4p_6n1vG.ji.
Stacktrace:
 [1] error(::String) at .\error.jl:33
 [2] compilecache(::Base.PkgId, ::String) at .\loading.jl:1305
 [3] _require(::Base.PkgId) at .\loading.jl:1030
 [4] require(::Base.PkgId) at .\loading.jl:928
 [5] require(::Module, ::Symbol) at .\loading.jl:923
 [6] include_string(::Function, ::Module, ::String, ::String) at .\loading.jl:1088
in expression starting at C:\Users\grego\ [...] \min_working_example.jl:3

Could you please post ]st -m? Thatā€™s going to be a lot more helpful than going error-by-error. My guess is you donā€™t have DiffEqSensitivity v6.52.0, but that would be weird since thereā€™s no clear bounds that would cause that?

Yes sure! Please find the console output below:

Status `C:\Users\grego\.julia\environments\v1.5\Manifest.toml`
  [c3fe647b] AbstractAlgebra v0.17.1
  [621f4979] AbstractFFTs v1.0.1
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  [764a87c0] BoundaryValueDiffEq v2.7.1
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  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
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  [9abbd945] Profile
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  [10745b16] Statistics
  [4607b0f0] SuiteSparse
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  [cf7118a7] UUIDs
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