torchdiffeq (Python) vs DifferentialEquations.jl (Julia) ODE Benchmarks

diffeqflux_differentialequations_vs_torchdiffeq_results.md

Torchdiffeq vs DifferentialEquations.jl (/ DiffEqFlux.jl) Benchmarks

Benchmark: Solve the Lorenz equations from 0 to 100 with abstol=reltol=1e-8

Absolute Timings

• DifferentialEquations.jl: 1.675 ms
• diffeqpy (DifferentialEquations.jl called from Python): 3.473 ms
• SciPy+Numba: 50.99 ms
• SciPy: 110.6 ms
• torchdiffeq: 48 seconds
• torchscript torchdiffeq: 48 seconds

Timings Relative to DifferentialEquations.jl

• DifferentialEquations.jl: 1x
• diffeqpy (DifferentialEquations.jl called from Python): 2.07x Slower
• SciPy+Numba: 30x Slower
• SciPy: 66x Slower
• torchdiffeq: 30,000x Slower
• torchscript torchdiffeq: 30,000x Slower

The torchscript versions are kept as separate scripts to allow for the JITing process to occur, and are called before timing to exclude JIT timing, as per the PyTorch documentation suggestions. Python results were scaled by the number of times ran in timeit.

diffeqpy.py

import numpy as np
from julia import Main
from diffeqpy import de
jul_f = Main.eval("""
function f(du,u,p,t)
  x, y, z = u[1], u[2], u[3]
  sigma, rho, beta = p[1], p[2], p[3]
  du[1] = sigma * (y - x)
  du[2] = x * (rho - z) - y
  du[3] = x * y - beta * z
end""")
u0 = [1.0,0.0,0.0]
tspan = (0., 100.)
t = np.linspace(0, 100, 1001)
p = [10.0,28.0,8/3]
prob = de.ODEProblem(jul_f, u0, tspan, p)
sol = de.solve(prob,saveat=t,abstol=1e-8,reltol=1e-8)
 
def time_func():
    sol = de.solve(prob,saveat=t,abstol=1e-8,reltol=1e-8)
 
timeit.Timer(time_func).timeit(number=100) # 0.347320199999956 seconds

differentialequations.jl

using OrdinaryDiffEq, StaticArrays, BenchmarkTools
function lorenz_static(u,p,t)
    @inbounds begin
        dx = p[1]*(u[2]-u[1])
        dy = u[1]*(p[2]-u[3]) - u[2]
        dz = u[1]*u[2] - p[3]*u[3]
    end
    @SVector [dx,dy,dz]
end
u0 = @SVector [1.0,0.0,0.0]
p  = @SVector [10.0,28.0,8/3]
tspan = (0.0,100.0)
prob = ODEProblem(lorenz_static,u0,tspan,p)
@btime solve(prob,Tsit5(),saveat=0.1,reltol=1e-8,abstol=1e-8) # 1.879 ms (56 allocations: 60.19 KiB)
@btime solve(prob,DP5(),saveat=0.1,reltol=1e-8,abstol=1e-8) # 1.675 ms (56 allocations: 59.83 KiB)

scipy_numba.py

import numpy as np
from scipy.integrate import odeint
import timeit
import numba
 
def f(u, t, sigma, rho, beta):
    x, y, z = u
    return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]
 
u0 = [1.0,0.0,0.0]
tspan = (0., 100.)
t = np.linspace(0, 100, 1001)
sol = odeint(f, u0, t, args=(10.0,28.0,8/3))
 
def time_func():
    odeint(f, u0, t, args=(10.0,28.0,8/3),rtol = 1e-8, atol=1e-8)
 
timeit.Timer(time_func).timeit(number=100) # 11.0623657 seconds
 
numba_f = numba.jit(f,nopython=True)
def time_func():
    odeint(numba_f, u0, t, args=(10.0,28.0,8/3),rtol = 1e-8, atol=1e-8)
 
timeit.Timer(time_func).timeit(number=100) # 5.099028400000009 seconds

torchdiffeq_jit1.py

import torch
from torchdiffeq import odeint
import timeit

@torch.jit.script
def f_script(t, u):
    x, y, z = u[0],u[1],u[2]
    du1 = 10.0 * (y - x)
    du2 = x * (28.0 - z) - y
    du3 = x * y - 2.66 * z
    return torch.stack([du1, du2, du3])

u0 = torch.tensor([1.0,0.0,0.0])
t = torch.linspace(0, 100, 1001)
sol = odeint(f_script, u0, t)

def time_func():
    odeint(f_script, u0, t, rtol = 1e-8, atol=1e-8)

time_func()

_t = timeit.Timer(time_func).timeit(number=2)
print(_t) # 106.84723600000001 seconds

torchdiffeq_jit2.py

import torch
from torchdiffeq import odeint
import timeit

@torch.jit.script
class LorenzODE(torch.nn.Module):

    def __init__(self):
        super(LorenzODE, self).__init__()

    def forward(self, t, u):
        x, y, z = u[0],u[1],u[2]
        du1 = 10.0 * (y - x)
        du2 = x * (28.0 - z) - y
        du3 = x * y - 2.66 * z
        return torch.stack([du1, du2, du3])

u0 = torch.tensor([1.0,0.0,0.0])
t = torch.linspace(0, 100, 1001)
sol = odeint(LorenzODE(), u0, t)

def time_func():
    odeint(LorenzODE(), u0, t, rtol = 1e-8, atol=1e-8)

time_func()

_t = timeit.Timer(time_func).timeit(number=2)
print(_t) # 96.36945809999997 seconds

torchdiffeq_nojit.py

import torch
from torchdiffeq import odeint
import timeit

def f(t, u):
    x, y, z = u[0],u[1],u[2]
    du1 = 10.0 * (y - x)
    du2 = x * (28.0 - z) - y
    du3 = x * y - 2.66 * z
    return torch.stack([du1, du2, du3])

u0 = torch.tensor([1.0,0.0,0.0])
t = torch.linspace(0, 100, 1001)
sol = odeint(f, u0, t)

def time_func():
    odeint(f, u0, t, rtol = 1e-8, atol=1e-8)

time_func()

_t = timeit.Timer(time_func).timeit(number=2)
print(_t) # 94.1328381000003 seconds

class LorenzODE(torch.nn.Module):

    def __init__(self):
        super(LorenzODE, self).__init__()

    def forward(self, t, u):
        x, y, z = u[0],u[1],u[2]
        du1 = 10.0 * (y - x)
        du2 = x * (28.0 - z) - y
        du3 = x * y - 2.66 * z
        return torch.stack([du1, du2, du3])

sol = odeint(LorenzODE("cpu"), u0, t)

def time_func():
    odeint(LorenzODE("cpu"), u0, t, rtol = 1e-8, atol=1e-8)

time_func()

_t = timeit.Timer(time_func).timeit(number=2)
print(_t) # 96.36945809999997 seconds


import torch
from torchdiffeq import odeint
import timeit

@torch.jit.script
def f_script(t, u):
    x, y, z = u[0],u[1],u[2]
    du1 = 10.0 * (y - x)
    du2 = x * (28.0 - z) - y
    du3 = x * y - 2.66 * z
    return torch.stack([du1, du2, du3])

u0 = torch.tensor([1.0,0.0,0.0])
t = torch.linspace(0, 100, 1001)
sol = odeint(f_script, u0, t)

def time_func():
    odeint(f_script, u0, t, rtol = 1e-8, atol=1e-8)

time_func()

_t = timeit.Timer(time_func).timeit(number=2)
print(_t) # 106.84723600000001 seconds