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tfNeuralODE-Spirals

This vignette runs through a small example of how to use tfNeuralODE to learn differential equations. We start by importing all of the correct packages, and instantiating a few constants.

# initial checks for python
library(reticulate)
if(!py_available()){
  install_python()
}

#checking for tensorflow and keras installation
#checking for tensorflow installation
if(!py_module_available("tensorflow")){
  tensorflow::install_tensorflow()
}
library(tensorflow)
library(tfNeuralODE)
library(keras)
library(deSolve)

# constants
data_size = 1000
batch_time = 20 # this seems to works the best ...
niters = 200
batch_size = 16

Now we create the neural network that will define our system. We also instantiate a few more constants that define our system.

# ODE Model with time input

OdeModel(keras$Model) %py_class% {
  initialize <- function() {
    super$initialize()
    self$block_1 <- layer_dense(units = 50, activation = 'tanh')
    self$block_2 <- layer_dense(units = 2, activation = 'linear')
  }

  call <- function(inputs) {
    x<- inputs ^ 3
    x <- self$block_1(x)
    self$block_2(x)
  }
}
model<- OdeModel()
# more constants, time vectors
tsteps <- seq(0, 25, by = 25/data_size)
true_y0 = c(2., 0.)
true_A = rbind(c(-0.1, 2.0), c(-2.0, -0.1))

Now we solve the ODE and plot our results.

# solving a spiral ode
trueODEfunc<- function(du, u, p, t){
  true_A = rbind(c(-0.1, 2.0), c(-2.0, -0.1))
  du <- (u^3) %*% true_A
  return(list(du))
}

# solved ode output
prob_trueode <- lsode(func = trueODEfunc, y = true_y0, times = tsteps)

Let’s start looking at how we’re going to train our model. We instantiate an optimizer and create a batching function.

#optimizer
optimizer = tf$keras$optimizers$legacy$Adam(learning_rate = 1e-3)

# batching function
get_batch<- function(prob_trueode, tsteps){
  starts = sample(seq(1, data_size - batch_time), size = batch_size, replace = FALSE)
  batch_y0 <- as.matrix(prob_trueode[starts,])
  batch_yN <- as.matrix(prob_trueode[starts + batch_time,])
  batch_y0 <- tf$cast((batch_y0), dtype = tf$float32)
  batch_yN <- tf$cast((batch_yN), dtype = tf$float32)
  return(list(batch_y0, batch_yN))
}

Now we can train our neural ODE, using naive backpropagation.

# Training Neural ODE
for(i in 1:niters){
  #print(paste("Iteration", i, "out of", niters, "iterations."))
  inp = get_batch(prob_trueode[,2:3], tsteps)
  pred = forward(model, inputs = inp[[1]], tsteps = tsteps[1:batch_time])
  with(tf$GradientTape() %as% tape, {
    tape$watch(pred)
    loss = tf$reduce_mean(tf$abs(pred - inp[[2]]))
  })
  #print(paste("loss:", as.numeric(loss)))
  dLoss = tape$gradient(loss, pred)
  list_w = backward(model, tsteps[1:batch_time], pred, output_gradients = dLoss)
  optimizer$apply_gradients(zip_lists(list_w[[3]], model$trainable_variables))

  # graphing the Neural ODE
  if(i %% 200 == 0){
    pred_y = forward(model = model, inputs = tf$cast(t(as.matrix(true_y0)), dtype = tf$float32),
                     tsteps = tsteps, return_states = TRUE)
    pred_y_c<- k_concatenate(pred_y[[2]], 1)
    p_m<- as.matrix(pred_y_c)
    plot(p_m, main = paste("iteration", i), type = "l", col = "red")
    lines(prob_trueode[,2], prob_trueode[,3], col = "blue")
  }
}
plot of the final result
plot of the final result

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.