Package: diffeqpy
diffeqpydiffeqpy is a package for solving differential equations in Python. It utilizesDifferentialEquations.jlfor its core routines
to give high performance solving of many different types of differential equations,
including:Discrete equations (function maps, discrete stochastic (Gillespie/Markov)
simulations)Ordinary differential equations (ODEs)Split and Partitioned ODEs (Symplectic integrators, IMEX Methods)Stochastic ordinary differential equations (SODEs or SDEs)Random differential equations (RODEs or RDEs)Differential algebraic equations (DAEs)Delay differential equations (DDEs)Mixed discrete and continuous equations (Hybrid Equations, Jump Diffusions)directly in Python.If you have any questions, or just want to chat about solvers/using the package,
please feel free to chat in theGitter channel. For bug reports, feature requests, etc., please submit an issue.InstallationTo install diffeqpy, use pip:pip install diffeqpyand you're good!Collab Notebook ExamplesSolving the Lorenz equation faster than SciPy+NumbaSolving ODEs on GPUs Fast in Python with diffeqpyGeneral FlowImport and setup the solvers available inDifferentialEquations.jlvia the command:fromdiffeqpyimportdeIn case only the solvers available inOrdinaryDiffEq.jlare required then use the command:fromdiffeqpyimportodeThe general flow for using the package is to follow exactly as would be done
in Julia, except addde.orode.in front. Note thatode.has lesser loading time and a smaller memory footprint compared tode..
Most of the commands will work without any modification. Thusthe DifferentialEquations.jl documentationand theDiffEqTutorialsare the main in-depth documentation for this package. Below we will show how to
translate these docs to Python code.Note about !Python does not allow!in function names, so this is alsoa limitation of pyjuliaTo use functions which on the Julia side have a!, likestep!, replace!by_b, for example:fromdiffeqpyimportdedeff(u,p,t):return-uu0=0.5tspan=(0.,1.)prob=de.ODEProblem(f,u0,tspan)integrator=de.init(prob,de.Tsit5())de.step_b(integrator)is valid Python code for usingthe integrator interface.Ordinary Differential Equation (ODE) ExamplesOne-dimensional ODEsfromdiffeqpyimportdedeff(u,p,t):return-uu0=0.5tspan=(0.,1.)prob=de.ODEProblem(f,u0,tspan)sol=de.solve(prob)The solution object is the same as the one describedin the DiffEq tutorialsand in thesolution handling documentation(note: the array interface is missing). Thus for example the solution time points
are saved insol.tand the solution values are saved insol.u. Additionally,
the interpolationsol(t)gives a continuous solution.We can plot the solution values using matplotlib:importmatplotlib.pyplotaspltplt.plot(sol.t,sol.u)plt.show()We can utilize the interpolation to get a finer solution:importnumpyt=numpy.linspace(0,1,100)u=sol(t)plt.plot(t,u)plt.show()Solve commandsThecommon interface argumentscan be used to control the solve command. For example, let's usesaveatto
save the solution at everyt=0.1, and let's utilize theVern9()9th order
Runge-Kutta method along with low tolerancesabstol=reltol=1e-10:sol=de.solve(prob,de.Vern9(),saveat=0.1,abstol=1e-10,reltol=1e-10)The set of algorithms for ODEs is describedat the ODE solvers page.Compilation withde.jitand JuliaWhen solving a differential equation, it's pertinent that your derivative
functionfis fast since it occurs in the inner loop of the solver. We can
convert the entire ode problem to symbolic form, optimize that symbolic form,
and emit efficient native code to simulate it usingde.jitto improve the
efficiency of the solver at the expense of added setup time:fast_prob=de.jit(prob)sol=de.solve(fast_prob)Additionally, you can directly define the functions in Julia. This will also
allow for specialization and could be helpful to increase the efficiency for
repeat or long calls. This is done viaseval:jul_f=de.seval("(u,p,t)->-u")# Define the anonymous function in Juliaprob=de.ODEProblem(jul_f,u0,tspan)sol=de.solve(prob)Limitationsde.jit, uses ModelingToolkit.jl'smodelingtoolkitizeinternally and some
restrictions apply. Not all models can be jitted. See themodelingtoolkitizedocumentationfor more info.Systems of ODEs: Lorenz EquationsTo solve systems of ODEs, simply use an array as your initial condition and
definefas an array function:deff(u,p,t):x,y,z=usigma,rho,beta=preturn[sigma*(y-x),x*(rho-z)-y,x*y-beta*z]u0=[1.0,0.0,0.0]tspan=(0.,100.)p=[10.0,28.0,8/3]prob=de.ODEProblem(f,u0,tspan,p)sol=de.solve(prob,saveat=0.01)plt.plot(sol.t,de.transpose(de.stack(sol.u)))plt.show()or we can draw the phase plot:us=de.stack(sol.u)frommpl_toolkits.mplot3dimportAxes3Dfig=plt.figure()ax=fig.add_subplot(111,projection='3d')ax.plot(us[0,:],us[1,:],us[2,:])plt.show()In-Place Mutating FormWhen dealing with systems of equations, in many cases it's helpful to reduce
memory allocations by using mutating functions. In diffeqpy, the mutating
form adds the mutating vector to the front. Let's make a fast version of the
Lorenz derivative, i.e. mutating and JIT compiled:deff(du,u,p,t):x,y,z=usigma,rho,beta=pdu[0]=sigma*(y-x)du[1]=x*(rho-z)-ydu[2]=x*y-beta*zu0=[1.0,0.0,0.0]tspan=(0.,100.)p=[10.0,28.0,2.66]prob=de.ODEProblem(f,u0,tspan,p)jit_prob=de.jit(prob)sol=de.solve(jit_prob)or using a Julia function:jul_f=de.seval("""function f(du,u,p,t)x, y, z = usigma, rho, beta = pdu[1] = sigma * (y - x)du[2] = x * (rho - z) - ydu[3] = x * y - beta * zend""")u0=[1.0,0.0,0.0]tspan=(0.,100.)p=[10.0,28.0,2.66]prob=de.ODEProblem(jul_f,u0,tspan,p)sol=de.solve(prob)Stochastic Differential Equation (SDE) ExamplesOne-dimensional SDEsSolving one-dimensonal SDEsdu = f(u,t)dt + g(u,t)dW_tis like an ODE except
with an extra function for the diffusion (randomness or noise) term. The steps
follow theSDE tutorial.deff(u,p,t):return1.01*udefg(u,p,t):return0.87*uu0=0.5tspan=(0.0,1.0)prob=de.SDEProblem(f,g,u0,tspan)sol=de.solve(prob,reltol=1e-3,abstol=1e-3)plt.plot(sol.t,de.stack(sol.u))plt.show()Systems of SDEs with Diagonal NoiseAn SDE with diagonal noise is where a different Wiener process is applied to
every part of the system. This is common for models with phenomenological noise.
Let's add multiplicative noise to the Lorenz equation:deff(du,u,p,t):x,y,z=usigma,rho,beta=pdu[0]=sigma*(y-x)du[1]=x*(rho-z)-ydu[2]=x*y-beta*zdefg(du,u,p,t):du[0]=0.3*u[0]du[1]=0.3*u[1]du[2]=0.3*u[2]u0=[1.0,0.0,0.0]tspan=(0.,100.)p=[10.0,28.0,2.66]prob=de.jit(de.SDEProblem(f,g,u0,tspan,p))sol=de.solve(prob)# Now let's draw a phase plotus=de.stack(sol.u)frommpl_toolkits.mplot3dimportAxes3Dfig=plt.figure()ax=fig.add_subplot(111,projection='3d')ax.plot(us[0,:],us[1,:],us[2,:])plt.show()Systems of SDEs with Non-Diagonal NoiseIn many cases you may want to share noise terms across the system. This is
known as non-diagonal noise. TheDifferentialEquations.jl SDE Tutorialexplains how the matrix form of the diffusion term corresponds to the
summation style of multiple Wiener processes. Essentially, the row corresponds
to which system the term is applied to, and the column is which noise term.
Sodu[i,j]is the amount of noise due to thejth Wiener process that's
applied tou[i]. We solve the Lorenz system with correlated noise as follows:deff(du,u,p,t):x,y,z=usigma,rho,beta=pdu[0]=sigma*(y-x)du[1]=x*(rho-z)-ydu[2]=x*y-beta*zdefg(du,u,p,t):du[0,0]=0.3*u[0]du[1,0]=0.6*u[0]du[2,0]=0.2*u[0]du[0,1]=1.2*u[1]du[1,1]=0.2*u[1]du[2,1]=0.3*u[1]u0=[1.0,0.0,0.0]tspan=(0.0,100.0)p=[10.0,28.0,2.66]nrp=numpy.zeros((3,2))prob=de.SDEProblem(f,g,u0,tspan,p,noise_rate_prototype=nrp)jit_prob=de.jit(prob)sol=de.solve(jit_prob,saveat=0.005)# Now let's draw a phase plotus=de.stack(sol.u)frommpl_toolkits.mplot3dimportAxes3Dfig=plt.figure()ax=fig.add_subplot(111,projection='3d')ax.plot(us[0,:],us[1,:],us[2,:])plt.show()Here you can see that the warping effect of the noise correlations is quite visible!Differential-Algebraic Equation (DAE) ExamplesA differential-algebraic equation is defined by an implicit functionf(du,u,p,t)=0. All of the controls are the same as the other examples, except
here you define a function which returns the residuals for each part of the
equation to define the DAE. The initial valueu0and the initial derivativedu0are required, though they do not necessarily have to satisfyf(known
as inconsistent initial conditions). The methods will automatically find
consistent initial conditions. In order for this to occur,differential_varsmust be set. This vector states which of the variables are differential (have a
derivative term), withfalsemeaning that the variable is purely algebraic.This example shows how to solve the Robertson equation:deff(du,u,p,t):resid1=-0.04*u[0]+1e4*u[1]*u[2]-du[0]resid2=+0.04*u[0]-3e7*u[1]**2-1e4*u[1]*u[2]-du[1]resid3=u[0]+u[1]+u[2]-1.0return[resid1,resid2,resid3]u0=[1.0,0.0,0.0]du0=[-0.04,0.04,0.0]tspan=(0.0,100000.0)differential_vars=[True,True,False]prob=de.DAEProblem(f,du0,u0,tspan,differential_vars=differential_vars)sol=de.solve(prob)and the in-place JIT compiled form:deff(resid,du,u,p,t):resid[0]=-0.04*u[0]+1e4*u[1]*u[2]-du[0]resid[1]=+0.04*u[0]-3e7*u[1]**2-1e4*u[1]*u[2]-du[1]resid[2]=u[0]+u[1]+u[2]-1.0prob=de.DAEProblem(f,du0,u0,tspan,differential_vars=differential_vars)jit_prob=de.jit(prob)# Error: no method matching matching modelingtoolkitize(::SciMLBase.DAEProblem{...})sol=de.solve(jit_prob)Mass Matrices, Sparse Arrays, and MoreMass matrix DAEs, along with many other forms, can be handled by doing an explicit conversion to the Julia types.
Seethe PythonCall module's documentation for more details.As an example, let's convertthe mass matrix ODE tutorial in diffeqpy.
To do this, the one aspect we need to handle is the conversion of the mass matrix in to a Julia array object. This is done as follows:fromdiffeqpyimportdefromjuliacallimportMainasjlimportnumpyasnpdefrober(du,u,p,t):y1,y2,y3=uk1,k2,k3=pdu[0]=-k1*y1+k3*y2*y3du[1]=k1*y1-k3*y2*y3-k2*y2**2du[2]=y1+y2+y3-1returnM=np.array([[1.0,0.0,0.0],[0.0,1.0,0.0],[0.0,0.0,0.0]])f=de.ODEFunction(rober,mass_matrix=jl.convert(jl.Array,M))prob_mm=de.ODEProblem(f,[1.0,0.0,0.0],(0.0,1e5),(0.04,3e7,1e4))sol=de.solve(prob_mm,de.Rodas5P(),reltol=1e-8,abstol=1e-8)Notice the only addition is to create thenp.arrayobject and to perform the manual conversion viajl.convert(jl.Array,M)to receive the
JuliaArrayobject. This can be done in any case where diffeqpy is not adequately auto-converting to the right Julia type. In some cases this
can be used to improve performance, though here we do it simply for compatability.Similarly, sparse matrices can be passed in much the same way. For example:importscipyspM=scipy.sparse.csr_array(M)jl.seval("using SparseArrays")f=de.ODEFunction(rober,mass_matrix=jl.convert(jl.SparseMatrixCSC,M))prob_mm=de.ODEProblem(f,[1.0,0.0,0.0],(0.0,1e5),(0.04,3e7,1e4))sol=de.solve(prob_mm,de.Rodas5P(),reltol=1e-8,abstol=1e-8)Delay Differential EquationsA delay differential equation is an ODE which allows the use of previous values.
In this case, the function needs to be a JIT compiled Julia function. It looks
just like the ODE, except in this case there is a functionh(p,t)which allows
you to interpolate and grab previous values.We must provide a history functionh(p,t)that gives values forubeforet0.
Here we assume that the solution was constant before the initial time point.
Additionally, we passconstant_lags = [20.0]to tell the solver that only
constant-time lags were used and what the lag length was. This helps improve
the solver accuracy by accurately stepping at the points of discontinuity.
Together this is:f=de.seval("""function f(du, u, h, p, t)du[1] = 1.1/(1 + sqrt(10)*(h(p, t-20)[1])^(5/4)) - 10*u[1]/(1 + 40*u[2])du[2] = 100*u[1]/(1 + 40*u[2]) - 2.43*u[2]end""")u0=[1.05767027/3,1.030713491/3]h=de.seval("""function h(p,t)[1.05767027/3, 1.030713491/3]end""")tspan=(0.0,100.0)constant_lags=[20.0]prob=de.DDEProblem(f,u0,h,tspan,constant_lags=constant_lags)sol=de.solve(prob,saveat=0.1)u1=[sol.u[i][0]foriinrange(0,len(sol.u))]u2=[sol.u[i][1]foriinrange(0,len(sol.u))]importmatplotlib.pyplotaspltplt.plot(sol.t,u1)plt.plot(sol.t,u2)plt.show()Notice that the solver accurately is able to simulate the kink (discontinuity)
att=20due to the discontinuity of the derivative at the initial time point!
This is why declaring discontinuities can enhance the solver accuracy.GPU-Accelerated ODE Solving of EnsemblesIn many cases one is interested in solving the same ODE many times over many
different initial conditions and parameters. In diffeqpy parlance this is called
an ensemble solve. diffeqpy inherits the parallelism tools of theSciML ecosystemthat are used for things likeautomated equation discovery and acceleration.
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:fromdiffeqpyimportdedeff(u,p,t):x,y,z=usigma,rho,beta=preturn[sigma*(y-x),x*(rho-z)-y,x*y-beta*z]u0=[1.0,0.0,0.0]tspan=(0.,100.)p=[10.0,28.0,8/3]prob=de.ODEProblem(f,u0,tspan,p)fast_prob=de.jit32(prob)sol=de.solve(fast_prob,saveat=0.01)Note that here we usedde.jit32to JIT-compile the problem into aFloat32form in order to make it more
efficient on most GPUs.Now we use theEnsembleProblemas defined on theensemble parallelism page of the documentation:
Let's build an ensemble by utilizing uniform random numbers to randomize the
initial conditions and parameters:importrandomdefprob_func(prob,i,rep):de.remake(prob,u0=[random.uniform(0,1)*u0[i]foriinrange(0,3)],p=[random.uniform(0,1)*p[i]foriinrange(0,3)])ensembleprob=de.EnsembleProblem(fast_prob,safetycopy=False)Now we solve the ensemble in serial:sol=de.solve(ensembleprob,de.Tsit5(),de.EnsembleSerial(),trajectories=10000,saveat=0.01)To add GPUs to the mix, we need to bring inDiffEqGPU.
The commandfrom diffeqpy import cudawill install CUDA for you and bring all of the bindings into the returned object:Note:from diffeqpy import cudacan take awhile to run the first time as it installs the drivers!Now we simply useEnsembleGPUKernel(cuda.CUDABackend())with a
GPU-specialized ODE solvercuda.GPUTsit5()to solve 10,000 ODEs on the GPU in
parallel:sol=de.solve(ensembleprob,cuda.GPUTsit5(),cuda.EnsembleGPUKernel(cuda.CUDABackend()),trajectories=10000,saveat=0.01)For the full list of choices for specialized GPU solvers, seethe DiffEqGPU.jl documentation.Note thatEnsembleGPUArraycan be used as well, like:sol=de.solve(ensembleprob,de.Tsit5(),cuda.EnsembleGPUArray(cuda.CUDABackend()),trajectories=10000,saveat=0.01)though we highly recommend theEnsembleGPUKernelmethods for more speed. Given
the way the JIT compilation performed will also ensure that the faster kernel
generation methods work,EnsembleGPUKernelis almost certainly the
better choice in most applications.BenchmarkTo 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:importnumpyasnpfromscipy.integrateimportodeintdeflorenz(state,t,sigma,beta,rho):x,y,z=statedx=sigma*(y-x)dy=x*(rho-z)-ydz=x*y-beta*zreturn[dx,dy,dz]sigma=10.0beta=8.0/3.0rho=28.0p=(sigma,beta,rho)y0=[1.0,1.0,1.0]t=np.arange(0.0,100.0,0.01)result=odeint(lorenz,y0,t,p)Usinglapplyto generate the ensemble we get:importtimeitdeftime_func():foritrinrange(1,1001):result=odeint(lorenz,y0,t,p)timeit.Timer(time_func).timeit(number=1)# 38.08861699999761 secondsNow let's see how the JIT-accelerated serial Julia version stacks up against that:deftime_func():sol=de.solve(ensembleprob,de.Tsit5(),de.EnsembleSerial(),trajectories=1000,saveat=0.01)timeit.Timer(time_func).timeit(number=1)# 3.1903300999983912Julia is already about 12x faster than the pure Python solvers here! Now let's add
GPU-acceleration to the mix:deftime_func():sol=de.solve(ensembleprob,cuda.GPUTsit5(),cuda.EnsembleGPUKernel(cuda.CUDABackend()),trajectories=1000,saveat=0.01)timeit.Timer(time_func).timeit(number=1)# 0.013322799997695256Already 2900x faster than SciPy! 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:deftime_func():sol=de.solve(ensembleprob,de.Tsit5(),de.EnsembleSerial(),trajectories=10000,saveat=0.01)timeit.Timer(time_func).timeit(number=1)# 68.80795999999827deftime_func():sol=de.solve(ensembleprob,cuda.GPUTsit5(),cuda.EnsembleGPUKernel(cuda.CUDABackend()),trajectories=10000,saveat=0.01)timeit.Timer(time_func).timeit(number=1)# 0.10774460000175168To compare this to the pure Julia code:usingOrdinaryDiffEq,DiffEqGPU,CUDA,StaticArraysfunctionlorenz(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]returnSVector{3}(du1,du2,du3)endu0=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=(@SVectorrand(Float32,3)).*p)monteprob=EnsembleProblem(prob,prob_func=prob_func,safetycopy=false)@timesol=solve(monteprob,GPUTsit5(),EnsembleGPUKernel(CUDA.CUDABackend()),trajectories=10_000,saveat=0.01);# 0.014481 seconds (257.64 k allocations: 13.130 MiB)which is about an order of magnitude faster for computing 10,000 trajectories,
note that the major factors are that we cannot define 32-bit floating point values
from Python and theprob_funcfor generating the initial conditions and parameters
is a major bottleneck since this function is written in Python.To see how this scales in Julia, let's take it to insane heights. First, let's
reduce the amount we're saving:@timesol=solve(monteprob,GPUTsit5(),EnsembleGPUKernel(CUDA.CUDABackend()),trajectories=10_000,saveat=1.0f0)0.015040seconds(257.64kallocations:13.130MiB)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.@timesol=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:@timesol=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:@timesol=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:t=np.arange(0.0,100.0,1.0)deftime_func():foritrinrange(1,1001):result=odeint(lorenz,y0,t,p)timeit.Timer(time_func).timeit(number=1)# 37.42609280000033The GPU version is solving 1000x as many trajectories, 37x 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.GPU Backend ChoicesJust like DiffEqGPU.jl, diffeqpy supports many different GPU venders.from diffeqpy import cudais just for cuda, but the following others are supported:from diffeqpy import cudawithcuda.CUDABackendfor NVIDIA GPUs via CUDAfrom diffeqpy import amdgpuwithamdgpu.AMDGPUBackendfor AMD GPUsfrom diffeqpy import oneapiwithoneapi.oneAPIBackendfor Intel's oneAPI GPUsfrom diffeqpy import metalwithmetal.MetalBackendfor Apple's Metal GPUs (on M-series processors)For more information, seethe DiffEqGPU.jl documentation.Known LimitationsAutodiff does not work on Python functions. When applicable, either define the derivative function
as a Julia function or set the algorithm to use finite differencing, i.e.Rodas5(autodiff=false).
All default methods use autodiff.Delay differential equations have to use Julia-defined functions otherwise the history function is
not appropriately typed with the overloads.TestingUnit tests can be run bytox.toxTroubleshootingIn case you encounter silent failure fromtox, try running it with-- -s(e.g.,tox -e py36 -- -s) where-soption (--capture=no,
i.e., don't capture stdio) is passed topy.test.  It may show an
error message"error initializing LibGit2 module".  In this case,
setting environment variableSSL_CERT_FILEmay help; e.g., try:SSL_CERT_FILE=PATH/TO/cert.pemtox-epy36See also:julia#18693.