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Faster stiff ODE solve: Robertson's chemical kinetics

Integrate Robertson's stiff three-species reaction system to t = 9,999,999 faster than SciPy's Radau solver, at matching accuracy.

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Faster stiff ODE solve: Robertson's chemical kinetics

Integrate Robertson's three-species reaction system, a classic stiff test problem, out to a final time of about ten million, faster than SciPy's Radau solver and to the same accuracy. Stiff kinetics of this kind arise in combustion, atmospheric chemistry and systems biology.

Task

Solve, for tt from 0 to t1t_1,

y˙1=−k1y1+k3y2y3,y˙2=k1y1−k2y22−k3y2y3,y˙3=k2y22,\dot y_1=-k_1y_1+k_3y_2y_3,\qquad \dot y_2=k_1y_1-k_2y_2^2-k_3y_2y_3,\qquad \dot y_3=k_2y_2^2,

with y(0)=(1,0,0)y(0)=(1,0,0) and t1=n=9,999,999t_1=n=9{,}999{,}999 in the benchmark. Each instance draws k1=0.04 u1k_1=0.04\,u_1, k2=3×107u2k_2=3\times10^{7}u_2 and k3=104u3k_3=10^{4}u_3 with u1,u2,u3u_1,u_2,u_3 uniform on [0.5,2][0.5,2], [0.7,1.5][0.7,1.5] and [0.8,1.3][0.8,1.3]. Output: y(t1)y(t_1). An output is accepted when it is finite, every component is at least −10−10-10^{-10}, ∣y1+y2+y3−1∣≤5×10−6\lvert y_1+y_2+y_3-1\rvert\le5\times10^{-6}, and it agrees with two reference solutions from scipy.integrate.solve_ivp, Radau (rtol 10−1110^{-11}, atol 10−910^{-9}) and BDF (rtol 5×10−105\times10^{-10}, atol 10−910^{-9}), to relative tolerance 3×10−63\times10^{-6} and absolute tolerance 3×10−93\times10^{-9} (loosened to 10−510^{-5} and 10−810^{-8} if the two references disagree). The reference solver is the Radau run.

Score: the reference's run time divided by the candidate's, averaged over the test instances; it is 1 if any output fails the check, and speed-ups below 1 count as 1. Higher is better.

Background

Robertson's 1966 model of an autocatalytic reaction has rate constants that differ by about nine orders of magnitude, so explicit methods need extremely small steps and implicit methods are required; it is a standard test in the literature on stiff solvers (Hairer and Wanner, 1996). The very long horizon, out to 10710^7, probes how well a solver grows its step size once the fast transients have died out.

Data and evaluation

  • Instances: 100 test and 100 development instances from the task's generator at n=9,999,999n=9{,}999{,}999 (that is, t1=9,999,999t_1=9{,}999{,}999), in the AlgoTune dataset; the reference needs about 90 ms per instance on the authors' machine.
  • Evaluation code: the AlgoTune harness (./algotune.sh evaluate in the repository), which runs the task's checker on every output.
  • Hardware: speed-ups depend on the machine, so reference and candidate are timed in the same environment. The paper keeps the fastest of 10 timed runs per instance, each after a warm-up, on an AMD EPYC 9454 CPU with 14 GB of memory; agents had 8 cores, and compilation is not timed.

What is known

  • Best on the AlgoTune leaderboard as of 2026-10-20: 100.81×, by the AlgoTuner agent driven by Claude Opus 4.5, ahead of Claude Opus 4.6 (76.85×) and DeepSeek R1 (68.88×).
  • The task's checker was tightened on 2026-02-08, adding the second (BDF) reference and the sign and mass-conservation tests. The top run was added to the results on 2026-02-02, before that change, and has not been shown to pass the current checker.
  • Press et al. (2025) describe most agent speed-ups as surface-level: better library calls, better library usage or low-level rewrites, for example with Numba.

What would count as progress

  • A solver beating 100.81× that passes the current, tightened checker on all 100 test instances, with its code; anyone can rerun the checker and the timing.
  • An error study comparing the solver's y(t1)y(t_1) with a high-precision reference, showing it is accurate rather than merely close to SciPy's answer.

References

  1. O. Press, B. Amos, H. Zhao, et al. AlgoTune: Can language models speed up general-purpose numerical programs? NeurIPS 2025. arXiv:2507.15887
  2. E. Hairer, G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edition. Springer (1996). doi:10.1007/978-3-642-05221-7
  3. P. Virtanen et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods 17, 261–272 (2020). doi:10.1038/s41592-019-0686-2

Source

Imported from AlgoTune task ode_stiff_robertson, curated by Ori Press and the AlgoTune contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.