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Christopher Rackauckas ChrisRackauckas

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ChrisRackauckas /
Last active Sep 28, 2022
torchsde vs DifferentialEquations.jl / DiffEqFlux.jl (Julia) benchmarks

torchsde vs DifferentialEquations.jl / DiffEqFlux.jl (Julia)

This example is a 4-dimensional geometric brownian motion. The code for the torchsde version is pulled directly from the torchsde README so that it would be a fair comparison against the author's own code. The only change to that example is the addition of a dt choice so that the simulation method and time step matches between the two different programs.

The SDE is solved 100 times. The summary of the results is as follows:

ChrisRackauckas /
Last active Sep 28, 2022
torchdiffeq vs Julia DiffEqflux Neural ODE Training Benchmark

torchdiffeq vs Julia DiffEqFlux Neural ODE Training Benchmark

The spiral neural ODE was used as the training benchmark for both torchdiffeq (Python) and DiffEqFlux (Julia) which utilized the same architecture and 500 steps of ADAM. Both achived similar objective values at the end. Results:

  • DiffEqFlux defaults: 7.4 seconds
  • DiffEqFlux optimized: 2.7 seconds
  • torchdiffeq: 288.965871299999 seconds
ChrisRackauckas /
Last active Sep 28, 2022
torchdiffeq (Python) vs DifferentialEquations.jl (Julia) ODE Benchmarks (Neural ODE Solvers)

Torchdiffeq vs DifferentialEquations.jl (/ DiffEqFlux.jl) Neural ODE Compatible Solver Benchmarks

Only non-stiff ODE solvers are tested since torchdiffeq does not have methods for stiff ODEs. The ODEs are chosen to be representative of models seen in physics and model-informed drug development (MIDD) studies (quantiative systems pharmacology) in order to capture the performance on realistic scenarios.


Below are the timings relative to the fastest method (lower is better). For approximately 1 million ODEs and less, torchdiffeq was more than an order of magnitude slower than DifferentialEquations.jl

ChrisRackauckas /
Last active Sep 26, 2022
SciPy+Numba odeint vs Julia ODE vs NumbaLSODA: 50x performance difference on stiff ODE

SciPy+Numba odeint vs Julia DifferentialEquations.jl vs NumbaLSODA Summary

All are solved at reltol=1e-3, abstol=1e-6 using the fastest ODE solver of the respective package for the given problem.

Absolute Performance Numbers:

  • SciPy LSODA through odeint takes ~489μs
  • SciPy LSODA through odeint with Numba takes ~257μs
  • NumbaLSODA takes ~25μs
  • DifferentialEquations.jl Rosenbrock23 takes ~9.2μs
ChrisRackauckas /
Last active Sep 21, 2022
DiffEqFlux.jl (Julia) vs Jax on an Epidemic Model

DiffEqFlux.jl (Julia) vs Jax on an Epidemic Model

The Jax developers optimized a differential equation benchmark in this issue which used DiffEqFlux.jl as a performance baseline. The Julia code from there was updated to include some standard performance tricks and is the benchmark code here. Thus both codes have been optimized by the library developers.


Forward Pass

View sparsity_reaction_diffusion.jl
using OrdinaryDiffEq, RecursiveArrayTools, LinearAlgebra, Test, SparseArrays, SparseDiffTools, Sundials
# Define the constants for the PDE
const α₂ = 1.0
const α₃ = 1.0
const β₁ = 1.0
const β₂ = 1.0
const β₃ = 1.0
const r₁ = 1.0
const r₂ = 1.0
View stacktrace_after.jl
ERROR: MethodError: Cannot `convert` an object of type Float32 to an object of type Vector{Float32}
Closest candidates are:
convert(::Type{Array{T, N}}, ::SizedArray{S, T, N, N, Array{T, N}}) where {S, T, N} at C:\Users\accou\.julia\packages\StaticArrays\0T5rI\src\SizedArray.jl:121
convert(::Type{Array{T, N}}, ::SizedArray{S, T, N, M, TData} where {M, TData<:AbstractArray{T, M}}) where {T, S, N} at C:\Users\accou\.julia\packages\StaticArrays\0T5rI\src\SizedArray.jl:115
convert(::Type{<:Array}, ::LabelledArrays.LArray) at C:\Users\accou\.julia\packages\LabelledArrays\lfn1b\src\larray.jl:133
[1] setproperty!(x::OrdinaryDiffEq.ODEIntegrator{Tsit5, false, Vector{Float32}, Float32}, f::Symbol, v::Float32)
@ Base .\Base.jl:43
[2] initialize!(integrator::OrdinaryDiffEq.ODEIntegrator{Tsit5, false, Vector{Float32}, Float32})
View gist:d0d0324c5c7bcef6012ed12a03e35859
This file has been truncated, but you can view the full file.
function (ˍ₋out, ˍ₋arg1, ˍ₋arg2, t)
#= C:\Users\accou\.julia\packages\SymbolicUtils\v2ZkM\src\code.jl:349 =#
#= C:\Users\accou\.julia\packages\SymbolicUtils\v2ZkM\src\code.jl:350 =#
#= C:\Users\accou\.julia\packages\SymbolicUtils\v2ZkM\src\code.jl:351 =#
#= C:\Users\accou\.julia\packages\Symbolics\vQXbU\src\build_function.jl:452 =#
#= C:\Users\accou\.julia\packages\SymbolicUtils\v2ZkM\src\code.jl:398 =# @inbounds begin
#= C:\Users\accou\.julia\packages\SymbolicUtils\v2ZkM\src\code.jl:394 =#
ChrisRackauckas / automatic_differentiation_done_quick.html
Created Mar 27, 2022
Automatic Differentiation Done Quick: Forward and Reverse Mode Differentiable Programming
View automatic_differentiation_done_quick.html
<!DOCTYPE html>
<HTML lang = "en">
<meta charset="UTF-8"/>
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<title>Forward and Reverse Automatic Differentiation In A Nutshell</title>
<script type="text/x-mathjax-config">
ChrisRackauckas /
Last active Jan 11, 2022
Brusselator Stiff Partial Differential Equation Benchmark: Julia DifferentialEquations.jl vs Python SciPy. Julia 935x faster

Brusselator Stiff Partial Differential Equation Benchmark: Julia DifferentialEquations.jl vs Python SciPy

Tested is DifferentialEquations.jl vs Python's SciPy ODE solvers. Notes:

  • Stiff ODE solvers are used since they are required to solve this problem effectively.
  • The Python code is vectorized with for maximum performance
  • All of the performance features are tried: automatic sparsity detection, preconditioners, etc.