SciMLBenchmarksOutput.jl: Benchmarks for Scientific Machine Learning (SciML) and Differential Equation Solver Software
SciMLBenchmarksOutput.jl holds webpages, pdfs, and notebooks showing the benchmarks
for the SciML Scientific Machine Learning Software ecosystem, including cross-language
benchmarks of differential equation solvers and methods for parameter estimation,
training universal differential equations (and subsets like neural ODEs), and more.
Currently, only the html, pdf, and markdown versions should be viewed here. To use
the interactive notebooks and scripts, see https://github.com/SciML/SciMLBenchmarks.jl,
since the interactive components require the environments contained in that repo.
Table of Contents
- Multi-Language Wrapper Benchmarks
- Non-stiff ODEs
- Stiff ODEs
- Method of Lines PDEs
- Dynamical ODEs
- N-body problems
- Nonstiff SDEs
- Stiff SDEs
- Nonstiff DDEs
- Stiff DDEs
- Jump Equations
- Parameter Estimation
The following tests were developed for the paper Adaptive Methods for Stochastic Differential Equations via Natural Embeddings and Rejection Sampling with Memory. These notebooks track their latest developments.
Current Summary
The following is a quick summary of the benchmarks. These paint broad strokes
over the set of tested equations and some specific examples may differ.
Non-Stiff ODEs
- OrdinaryDiffEq.jl’s methods are the most efficient by a good amount
- The
Vern
methods tend to do the best in every benchmark of this category
- At lower tolerances,
Tsit5
does well consistently.
- ARKODE and Hairer’s
dopri5
/dop853
perform very similarly, but are both
far less efficient than the Vern
methods.
- The multistep methods,
CVODE_Adams
and lsoda
, tend to not do very well.
- The ODEInterface multistep method
ddeabm
does not do as well as the other
multistep methods.
- ODE.jl’s methods are not able to consistently solve the problems.
- Fixed time step methods are less efficient than the adaptive methods.
Stiff ODEs
- In this category, the best methods are much more problem dependent.
- For smaller problems:
Rosenbrock23
, lsoda
, and TRBDF2
tend to be the most efficient at high
tolerances.
Rodas4
and Rodas5
tend to be the most efficient at low tolerances.
- For larger problems (Filament PDE):
CVODE_BDF
does the best at all tolerances.
- The ESDIRK methods like
TRBDF2
and KenCarp4
can come close.
radau
is always the most efficient when tolerances go to the low extreme
(1e-13
)
- Fixed time step methods tend to diverge on every tested problem because the
high stiffness results in divergence of the Newton solvers.
- ARKODE is very inconsistent and requires a lot of tweaking in order to not
diverge on many of the tested problems. When it doesn’t diverge, the similar
algorithms in OrdinaryDiffEq.jl (
KenCarp4
) are much more efficient in most
cases.
- ODE.jl and GeometricIntegrators.jl fail to converge on any of the tested
problems.
Dynamical ODEs
- Higher order (generally order >=6) symplectic integrators are much more
efficient than the lower order counterparts.
- For high accuracy, using a symplectic integrator is not preferred. Their extra
cost is not necessary since the other integrators are able to not drift simply
due to having low enough error.
- In this class, the
DPRKN
methods are by far the most efficient. The Vern
methods do well for not being specific to the domain.
Non-Stiff SDEs
- For simple 1-dimensional SDEs at low accuracy, the
EM
and RKMil
methods
can do well. Beyond that, they are simply outclassed.
- The
SRA
and SRI
methods both are very similar within-class on the simple
SDEs.
SRA3
is the most efficient when applicable and the tolerances are low.
- Generally, only low accuracy is necessary to get to sampling error of the mean.
- The adaptive method is very conservative with error estimates.
Stiff SDEs
- The high order adaptive methods (
SRIW1
) generally do well on stiff problems.
- The “standard” low-order implicit methods,
ImplicitEM
and ImplicitRK
, do
not do well on all stiff problems. Some exceptions apply to well-behaved
problems like the Stochastic Heat Equation.
Non-Stiff DDEs
- The efficiency ranking tends to match the ODE Tests, but the cutoff from
low to high tolerance is lower.
Tsit5
does well in a large class of problems here.
- The
Vern
methods do well in low tolerance cases.
Stiff DDEs
- The Rosenbrock methods, specifically
Rodas5
, perform well.
Parameter Estimation
- Broadly two different approaches have been used, Bayesian Inference and Optimisation
algorithms.
- In general it seems that the optimisation algorithms perform more accurately but that can be
attributed to the larger number of data points being used in the optimisation cases, Bayesian
approach tends to be slower of the two and hence lesser data points are used, accuracy can
increase if proper data is used.
- Within the different available optimisation algorithms, BBO from the BlackBoxOptim package and GN_CRS2_LM
for the global case while LD_SLSQP,LN_BOBYQA and LN_NELDERMEAD for the local case from the NLopt package
perform the best.
- Another algorithm being used is the QuadDIRECT algorithm, it gives very good results in the shorter problem case
but doesn’t do very well in the case of the longer problems.
- The choice of global versus local optimization make a huge difference in the timings. BBO tends to find
the correct solution for a global optimization setup. For local optimization, most methods in NLopt,
like :LN_BOBYQA, solve the problem very fast but require a good initial condition.
- The different backends options available for Bayesian method offer some tradeoffs beteween
time, accuracy and control. It is observed that sufficiently high accuracy can be observed with
any of the backends with the fine tuning of stepsize, constraints on the parameters, tightness of the
priors and number of iterations being passed.
Notes
Each of the benchmarks displays the computer characteristics at the bottom of
the benchmark. Since performance-necessary computations are normally performed on
compute clusters, the official benchmarks use a workstation with an
Intel Xeon CPU E5-2680 v4 @ 2.40GHz to match the performance characteristics of
a standard node in a high performance computing (HPC) cluster or cloud computing
setup.