---
title: "GPU-Accelerated Ordinary Differential Equations (ODE) in R with diffeqr"
author: "Chris Rackauckas"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{GPU-Accelerated Ordinary Differential Equations (ODE) in R with diffeqr}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

In many cases one is interested in solving the same ODE many times over many
different initial conditions and parameters. In diffeqr parlance this is called
an ensemble solve. diffeqr inherits the parallelism tools of the
[SciML ecosystem](https://sciml.ai/) that are used for things like
[automated equation discovery and acceleration](https://arxiv.org/abs/2001.04385).
Here we will demonstrate using these parallel tools to accelerate the solving
of an ensemble.

First, let's define the JIT-accelerated Lorenz equation like before:

```R
de <- diffeqr::diffeq_setup()
lorenz <- function (u,p,t){
  du1 = p[1]*(u[2]-u[1])
  du2 = u[1]*(p[2]-u[3]) - u[2]
  du3 = u[1]*u[2] - p[3]*u[3]
  c(du1,du2,du3)
}
u0 <- c(1.0,1.0,1.0)
tspan <- c(0.0,100.0)
p <- c(10.0,28.0,8/3)
prob <- de$ODEProblem(lorenz,u0,tspan,p)
fastprob <- diffeqr::jitoptimize_ode(de,prob)
```

Now we use the `EnsembleProblem` as defined on the
[ensemble parallelism page of the documentation](https://diffeq.sciml.ai/stable/features/ensemble/):
Let's build an ensemble by utilizing uniform random numbers to randomize the
initial conditions and parameters:

```R
prob_func <- function (prob,i,rep){
  de$remake(prob,u0=runif(3)*u0,p=runif(3)*p)
}
ensembleprob = de$EnsembleProblem(fastprob, prob_func = prob_func, safetycopy=FALSE)
```

Now we solve the ensemble in serial:

```R
sol = de$solve(ensembleprob,de$Tsit5(),de$EnsembleSerial(),trajectories=10000,saveat=0.01)
```

To add GPUs to the mix, we need to bring in [DiffEqGPU](https://github.com/SciML/DiffEqGPU.jl).
The `diffeqr::diffeqgpu_setup()` helper function will install CUDA for you and
bring all of the bindings into the returned object:

```R
degpu <- diffeqr::diffeqgpu_setup("CUDA")
```

#### Note: `diffeqr::diffeqgpu_setup` can take awhile to run the first time as it installs the drivers!

Now we simply use `EnsembleGPUKernel(degpu$CUDABackend())` with a
GPU-specialized ODE solver `GPUTsit5()` to solve 10,000 ODEs on the GPU in 
parallel:

```R
sol <- de$solve(ensembleprob,degpu$GPUTsit5(),degpu$EnsembleGPUKernel(degpu$CUDABackend()),trajectories=10000,saveat=0.01)
```

For the full list of choices for specialized GPU solvers, see 
[the DiffEqGPU.jl documentation](https://docs.sciml.ai/DiffEqGPU/stable/manual/ensemblegpukernel/).

Note that `EnsembleGPUArray` can be used as well, like:

```R
sol <- de$solve(ensembleprob,de$Tsit5(),degpu$EnsembleGPUArray(degpu$CUDABackend()),trajectories=10000,saveat=0.01)
```

though we highly recommend the `EnsembleGPUKernel` methods for more speed. Given
the way the JIT compilation performed will also ensure that the faster kernel
generation methods work, `EnsembleGPUKernel` is almost certainly the
better choice in most applications.

### Benchmark

To see how much of an effect the parallelism has, let's test this against R's
deSolve package. This is exactly the same problem as the documentation example
for deSolve, so let's copy that verbatim and then add a function to do the
ensemble generation:

```R
library(deSolve)
Lorenz <- function(t, state, parameters) {
  with(as.list(c(state, parameters)), {
    dX <-  a * X + Y * Z
    dY <-  b * (Y - Z)
    dZ <- -X * Y + c * Y - Z
    list(c(dX, dY, dZ))
  })
}

parameters <- c(a = -8/3, b = -10, c = 28)
state      <- c(X = 1, Y = 1, Z = 1)
times      <- seq(0, 100, by = 0.01)
out <- ode(y = state, times = times, func = Lorenz, parms = parameters)

lorenz_solve <- function (i){
  state      <- c(X = runif(1), Y = runif(1), Z = runif(1))
  parameters <- c(a = -8/3 * runif(1), b = -10 * runif(1), c = 28 * runif(1))
  out <- ode(y = state, times = times, func = Lorenz, parms = parameters)
}
```

Using `lapply` to generate the ensemble we get:

```
> system.time({ lapply(1:1000,lorenz_solve) })
   user  system elapsed
 225.81    0.46  226.63
```

Now let's see how the JIT-accelerated serial Julia version stacks up against that:

```
> system.time({ de$solve(ensembleprob,de$Tsit5(),de$EnsembleSerial(),trajectories=1000,saveat=0.01) })
   user  system elapsed
   2.75    0.30    3.08
```

Julia is already about 73x faster than the pure R solvers here! Now let's add
GPU-acceleration to the mix:

```
> system.time({ de$solve(ensembleprob,degpu$GPUTsit5(),degpu$EnsembleGPUKernel(degpu$CUDABackend()),trajectories=1000,saveat=0.01) })
   user  system elapsed 
   0.11    0.00    0.12
```

Already 26x times faster! But the GPU acceleration is made for massively
parallel problems, so let's up the trajectories a bit. We will not use more
trajectories from R because that would take too much computing power, so let's
see what happens to the Julia serial and GPU at 10,000 trajectories:

```
> system.time({ de$solve(ensembleprob,de$Tsit5(),de$EnsembleSerial(),trajectories=10000,saveat=0.01) })
   user  system elapsed
  35.02    4.19   39.25
```

```
> system.time({ de$solve(ensembleprob,degpu$GPUTsit5(),degpu$EnsembleGPUKernel(degpu$CUDABackend()),trajectories=10000,saveat=0.01) })
   user  system elapsed 
   1.22    0.23    1.50 
```

To compare this to the pure Julia code:

```julia
using OrdinaryDiffEq, DiffEqGPU, CUDA, StaticArrays
function lorenz(u, p, t)
    σ = p[1]
    ρ = p[2]
    β = p[3]
    du1 = σ * (u[2] - u[1])
    du2 = u[1] * (ρ - u[3]) - u[2]
    du3 = u[1] * u[2] - β * u[3]
    return SVector{3}(du1, du2, du3)
end

u0 = SA[1.0f0; 0.0f0; 0.0f0]
tspan = (0.0f0, 10.0f0)
p = SA[10.0f0, 28.0f0, 8 / 3.0f0]
prob = ODEProblem{false}(lorenz, u0, tspan, p)
prob_func = (prob, i, repeat) -> remake(prob, p = (@SVector rand(Float32, 3)) .* p)
monteprob = EnsembleProblem(prob, prob_func = prob_func, safetycopy = false)
@time sol = solve(monteprob, GPUTsit5(), EnsembleGPUKernel(CUDA.CUDABackend()),
    trajectories = 10_000,
    saveat = 1.0f0);

# 0.015064 seconds (257.68 k allocations: 13.132 MiB)
```

which is about two orders of magnitude faster for computing 10,000 trajectories,
note that the major factors are that we cannot define 32-bit floating point values
from R and the `prob_func` for generating the initial conditions and parameters
is a major bottleneck since this function is written in R.

To see how this scales in Julia, let's take it to insane heights. First, let's
reduce the amount we're saving:

```julia
@time sol = solve(monteprob,GPUTsit5(),EnsembleGPUKernel(CUDA.CUDABackend()),trajectories=10_000,saveat=1.0f0)
0.015040 seconds (257.64 k allocations: 13.130 MiB)
```

This highlights that controlling memory pressure is key with GPU usage: you will
get much better performance when requiring less saved points on the GPU.

```julia
@time sol = solve(monteprob,GPUTsit5(),EnsembleGPUKernel(CUDA.CUDABackend()),trajectories=100_000,saveat=1.0f0)
# 0.150901 seconds (2.60 M allocations: 131.576 MiB)
```

compared to serial:

```julia
@time sol = solve(monteprob,Tsit5(),EnsembleSerial(),trajectories=100_000,saveat=1.0f0)
# 22.136743 seconds (16.40 M allocations: 1.628 GiB, 42.98% gc time)
```

And now we start to see that scaling power! Let's solve 1 million trajectories:

```julia
@time sol = solve(monteprob,GPUTsit5(),EnsembleGPUKernel(CUDA.CUDABackend()),trajectories=1_000_000,saveat=1.0f0)
# 1.031295 seconds (3.40 M allocations: 241.075 MiB)
```

For reference, let's look at deSolve with the change to only save that much:

```R
times      <- seq(0, 100, by = 1.0)
lorenz_solve <- function (i){
  state      <- c(X = runif(1), Y = runif(1), Z = runif(1))
  parameters <- c(a = -8/3 * runif(1), b = -10 * runif(1), c = 28 * runif(1))
  out <- ode(y = state, times = times, func = Lorenz, parms = parameters)
}

system.time({ lapply(1:1000,lorenz_solve) })
```

```
   user  system elapsed 
  49.69    3.36   53.42 
```

The GPU version is solving 1000x as many trajectories, 50x as fast! So conclusion,
if you need the most speed, you may want to move to the Julia version to get the
most out of your GPU due to Float32's, and when using GPUs make sure it's a problem
with a relatively average or low memory pressure, and these methods will give
orders of magnitude acceleration compared to what you might be used to.
