SciMLBenchmarks.jl: Benchmarks for Scientific Machine Learning (SciML) and Equation Solvers

SciMLBenchmarks.jl holds webpages, pdfs, and notebooks showing the benchmarks
for the SciML Scientific Machine Learning Software ecosystem, including:

• Benchmarks of equation solver implementations
• Speed and robustness comparisons of methods for parameter estimation / inverse problems
• Training universal differential equations (and subsets like neural ODEs)
• Training of physics-informed neural networks (PINNs)
• Surrogate comparisons, including radial basis functions, neural operators (DeepONets, Fourier Neural Operators), and more

The SciML Bench suite is made to be a comprehensive open source benchmark from the ground up, covering the methods of
computational science and scientific computing all the way to AI for science.

Rules: Optimal, Fair, and Reproducible

These benchmarks are meant to represent good optimized coding style. Benchmarks are preferred to be run on the provided open
benchmarking hardware for full reproducibility (though in some cases, such as with language barriers, this can be difficult).
Each benchmark is documented with the compute devices used along with package versions for necessary reproduction. These
benchmarks attempt to measure in terms of work-precision efficiency, either timing with approximately matching error
or building work-precision diagrams for direct comparison of speed at given error tolerances.

If any of the code from any of the languages can be improved, please open a pull request.

For critiques of benchmarks, please open a pull request that changes the code in the desired manner. Issues with recommended
changes are generally vague and not actionable, while pull requests with code changes are exact. Thus if there is something
you think should be changed in the code, please make the recommended change in the code!

Results

To view the results of the SciML Benchmarks, go to docs.sciml.ai/SciMLBenchmarksOutput. By default, this
will lead to the latest tagged version of the benchmarks. To see the in-development version of the benchmarks, go to
https://docs.sciml.ai/SciMLBenchmarksOutput/dev/.

Static outputs in pdf, markdown, and html reside in SciMLBenchmarksOutput.

Citing

To cite the SciML Benchmarks, please cite the following:

bibCopy
@article{rackauckas2019confederated,
  title={Confederated modular differential equation APIs for accelerated algorithm development and benchmarking},
  author={Rackauckas, Christopher and Nie, Qing},
  journal={Advances in Engineering Software},
  volume={132},
  pages={1--6},
  year={2019},
  publisher={Elsevier}
}

@article{DifferentialEquations.jl-2017,
 author = {Rackauckas, Christopher and Nie, Qing},
 doi = {10.5334/jors.151},
 journal = {The Journal of Open Research Software},
 keywords = {Applied Mathematics},
 note = {Exported from https://app.dimensions.ai on 2019/05/05},
 number = {1},
 pages = {},
 title = {DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia},
 url = {https://app.dimensions.ai/details/publication/pub.1085583166 and http://openresearchsoftware.metajnl.com/articles/10.5334/jors.151/galley/245/download/},
 volume = {5},
 year = {2017}
}

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.
  • Rodas4P and Rodas5P tend to be the most efficient at low tolerances.
• For larger problems (Filament PDE):
  • FBDF and QNDF do the best at all normal 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.
• GeometricIntegrators.jl fails 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 Rodas5P, 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 between
  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.

Interactive Notebooks

To generate the interactive notebooks, first install the SciMLBenchmarks, instantiate the
environment, and then run SciMLBenchmarks.open_notebooks(). This looks as follows:

juliaCopy
]add SciMLBenchmarks#master
]add IJulia
]activate SciMLBenchmarks
]instantiate
using SciMLBenchmarks, IJulia
SciMLBenchmarks.open_notebooks()

The benchmarks will be generated at your pwd() in a folder called generated_notebooks.

Note that when running the benchmarks, the packages are not automatically added. Thus you
will need to add the packages manually or use the internal Project/Manifest tomls to
instantiate the correct packages. This can be done by activating the folder of the benchmarks.
For example,

juliaCopy
using Pkg
Pkg.activate(joinpath(pkgdir(SciMLBenchmarks),"benchmarks","NonStiffODE"))
Pkg.instantiate()

will add all of the packages required to run any benchmark in the NonStiffODE folder.

Contributing

All of the files are generated from the Weave.jl files in the benchmarks folder of the SciMLBenchmarks.jl repository. The generation process runs automatically,
and thus one does not necessarily need to test the Weave process locally. Instead, simply open a PR that adds/updates a
file in the benchmarks folder and the PR will generate the benchmark on demand. Its artifacts can then be inspected in the
GitHub Actions CI as described below before merging. Note that it will use the Project.toml and Manifest.toml of the subfolder, so
any changes to dependencies requires that those are updated.

Handling Method Omissions and Errors in Benchmarks

It is good practice to include sections showing why a method was omitted from a
benchmark — for example, because it errors or is too slow. This gives readers useful
context about solver coverage and limitations.

However, these error demonstrations must be captured so they don't break the build.
By default, any code chunk that throws an uncaught error will fail the benchmark build.
This is intentional: it catches real issues instead of silently including ugly stacktraces
in published output.

There are two ways to safely demonstrate errors:

Option 1: Use a try/catch block (preferred when you want to show a friendly message):

markdownCopy
```julia
try
    sol = solve(prob, SomeMethod())
catch e
    println("SomeMethod failed: ", e)
end
```

Option 2: Use the error=true chunk option (when you want to show the actual error output):

markdownCopy
```julia; error=true
sol = solve(prob, SomeMethod())
```

Do not leave raw uncaught errors in benchmark code. They will:

1. Fail the CI build
2. Produce ugly stacktrace output if the chunk somehow runs

When adding a new solver comparison that you expect to fail, always wrap it in one of the
above patterns.

Reporting Bugs and Issues

Report any bugs or issues at the SciMLBenchmarks repository.

Inspecting Benchmark Results

To see benchmark results before merging, click into the GitHub Actions CI results, click into
"Upload Benchmark Artifacts", and then investigate the trained results by clicking the link to download the zip file
with all of the results.

CI Architecture

Benchmarks run on self-hosted GitHub Actions runners. Each benchmark directory can customize its CI behavior via two optional configuration files.

Runner Configuration (benchmark_config.toml)

Each benchmark directory can contain a benchmark_config.toml to specify which runner to use and the job timeout. If absent, defaults from .github/benchmark_defaults.toml apply.

tomlCopy
# CPU benchmark (default — no config file needed)
runner = ["self-hosted", "benchmark"]
timeout = 12000  # minutes

# GPU benchmark (e.g. benchmarks/PINNErrorsVsTime/benchmark_config.toml)
runner = ["self-hosted", "gpu", "exclusive"]
timeout = 12000

Runner Type	Labels	Hardware	Use Case
CPU	["self-hosted", "benchmark"]	amdci1, amdci3 (AMD EPYC, stable hardware)	Most benchmarks. Stable hardware for regression checking.
GPU	["self-hosted", "gpu", "exclusive"]	demeter3 (AMD EPYC + 2x Tesla V100)	NeuralPDE/PINN benchmarks and other GPU workloads.

Setup Scripts (setup.sh)

If a benchmark directory contains an executable setup.sh, it runs before Pkg.instantiate() and the benchmark itself. Use this for installing system dependencies, configuring package registries, downloading data, etc.

benchmarks/
  BayesianInference/
    setup.sh            # Installs CmdStan
    Project.toml
    DiffEqBayesLorenz.jmd
  ModelingToolkit/
    setup.sh            # Configures JuliaHubRegistry
    Project.toml
    ...
  NonStiffODE/
    Project.toml        # No setup.sh needed — just uses defaults
    linear_wpd.jmd

Setup scripts can export environment variables to the benchmark process by writing to $BENCHMARK_ENV_FILE:

bashCopy
#!/bin/bash
set -euo pipefail
# Install something...
export MY_VAR="/path/to/thing"
echo "export MY_VAR=\"${MY_VAR}\"" >> "${BENCHMARK_ENV_FILE}"

The CI pipeline for each benchmark is: setup.sh (if present) → Pkg.instantiate() → julia benchmark.jl <target>.

Manually Generating Files

All of the files are generated from the Weave.jl files in the benchmarks folder. To run the generation process, do for example:

juliaCopy
]activate SciMLBenchmarks # Get all of the packages
using SciMLBenchmarks
SciMLBenchmarks.weave_file(joinpath(pkgdir(SciMLBenchmarks),"benchmarks","NonStiffODE"),"linear_wpd.jmd")

To generate all of the files in a folder, for example, run:

juliaCopy
SciMLBenchmarks.weave_folder(joinpath(pkgdir(SciMLBenchmarks),"benchmarks","NonStiffODE"))

To generate all of the notebooks, do:

juliaCopy
SciMLBenchmarks.weave_all()

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
AMD EPYC 7502 32-Core Processor @ 2.50GHz to match the performance characteristics of
a standard node in a high performance computing (HPC) cluster or cloud computing
setup.

Choosing a Reference Solution

For almost all equations, there is no analytical solution. A low tolerance
reference solution is required in order to compute the error. However, there
are many questions as to the potential of biasing the results via a reference
computed from a given program. If we use a reference solution from Julia, does
that make our errors lower?

The answer is no because all of the equation solvers should be convergent to
the same solution. Because of this, it does not matter which solver is used
to generate the reference solution. However, caution is required to ensure
that the reference solution is sufficiently accurate.

Thankfully, there's a very clear indicator of when a reference solution is
not sufficiently correct. Because all other methods will be converging to a
different solution, there will be a digit of accuracy at which all other
solutions stop converging to the reference. If this occurs, all solutions will
give a straight line, you can see there here:

In this image (taken from the TransistorAmplifierDAE benchmark),
the second Rodas5P and Rodas4 are from a different problem implementation, and
you can see they hit lower errors. But all of the others use the same
reference solution and seem to "hit a wall" at around 1e-5. This is because
the chosen reference solution was only 1e-5 accurate. Changing to a
different reference solution makes them all converge:

This shows that all that truly matters is that the chosen reference is
sufficiently accurate, and any walling behavior is an indicator that some
method in the benchmark set is more accurate than the reference (in which
case the benchmark should be updated to use the more accurate reference).