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arxiv: 2605.23727 · v1 · pith:EWN74JFKnew · submitted 2026-05-22 · 🧮 math.NA · cs.NA

Mixed-Precision in adaptive Runge-Kutta method for large ODE systems

Mouhamad Al-Sayed , Samuel Bernard (MUSICS , ICJ) , Ars\`ene Marzorati (DRACULA , BEAGLE , MMCS , MUSICS , ICJ
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BIOTIC) Jonathan Rouzaud-Cornabas (BIOTIC CITI INSA Lyon)
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Pith reviewed 2026-05-25 03:13 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords mixed precisionRunge-KuttaODE systemsadaptive methodsnumerical accuracycomputational efficiencyBogacki-Shampine
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The pith

Mixed-precision Runge-Kutta solvers preserve most high-precision accuracy in large ODE systems

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests mixed-precision adaptations of the Bogacki-Shampine 3(2) Runge-Kutta method on three large ODE benchmark systems to determine if they can retain the accuracy of full high-precision calculations while benefiting from faster low-precision arithmetic. It finds that accuracy is largely preserved across tolerances and actually improves toward high-precision levels as system size increases. The number of function evaluations stays comparable, allowing the speed advantage of low precision to translate into overall gains without extra steps. This approach is presented as adaptable to any finite-precision format and numerical scheme for handling computationally costly large systems with heterogeneous interactions.

Core claim

Mixed-precision versions of the Bogacki-Shampine 3(2) Runge-Kutta pair preserve most of the accuracy of high-precision solvers under a wide range of tolerances. Accuracy improves with system size to match high-precision performance seen in smaller systems. The arithmetic change does not alter evaluation counts enough to offset the speed benefit of low-precision operations, enabling significant speed-up with little accuracy loss in large coupled ODE systems.

What carries the argument

Mixed-precision Bogacki-Shampine 3(2) Runge-Kutta pair applied to large ODE systems, using low precision for speed and high precision selectively for accuracy control.

If this is right

  • Mixed-precision methods can deliver significant computational speed-up for large coupled ODE systems.
  • Accuracy remains close to high-precision levels over wide solver tolerances.
  • Accuracy performance improves as the ODE system size grows.
  • The number of function evaluations is not substantially affected, maintaining the speed benefit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This technique could be tested on other families of ODE solvers such as multistep methods.
  • Hardware architectures optimized for low precision might see even larger gains if error control is also adapted.
  • The size-dependent accuracy improvement suggests benefits specifically for very large-scale simulations not fully explored in the benchmarks.

Load-bearing premise

The three benchmark systems of coupled oscillators, Kuramoto model, and circadian clock adequately represent large ODE systems containing many heterogeneous interactions.

What would settle it

A demonstration that mixed-precision accuracy drops substantially below high-precision levels for some large ODE system under standard tolerances would challenge the main claim.

Figures

Figures reproduced from arXiv: 2605.23727 by Ars\`ene Marzorati (DRACULA, BEAGLE, BIOTIC), CITI, ICJ, ICJ), INSA Lyon), Jonathan Rouzaud-Cornabas (BIOTIC, MMCS, Mouhamad Al-Sayed, MUSICS, Samuel Bernard (MUSICS.

Figure 1
Figure 1. Figure 1: Normalized final error, see Equation (8), with respect to Matlab’s ode45 solution. 1000 agents for Benchmark 1 (a), to 2000 for Benchmark 2 (b) and to 700 for Benchmark 3 (c). The error bars are the 5-th and 95-th percentiles. For tolerances of (10−3 , 10−4 , 10−5 , 10−6 , 10−7 , 10−8 , 10−9 ), there are respectively in (a) (113, 97, 78, 59, 33, 19, 0), in (b) (498, 423, 359, 270, 189, 122, 53) and (c) (28… view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of normalized final error, see Equation (8), for all the tests completed by all the solvers. One box plot represents a distribution, with limits at 1-st and 99-th percentiles, for all the tests of one benchmark, indicated by the column (a) Benchmark 1, Equation (3), (b) Benchmark 2, Equation (4) and (c) Benchmark 3, Equation (5). With a unique system size, indicated by the intensity of blue, p… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Averaged local error (one value per test) approximated as the difference between the 2 RK schemes (EBS, in Equation (7)). (b) Averaged real local error (one value per test). In each plot the median which is computed over all the tests performed for a fixed tolerance is given with the error bars corresponding to the 5th and 95th percentiles. As the tolerance values tested are discrete, a shift was added… view at source ↗
Figure 4
Figure 4. Figure 4: Number of tests successfully solved by all solvers for each couple of system size (abscissa) and relative tolerance (ordinate). (a) Benchmark 1, (b) Benchmark 2 and (c) Benchmark 3. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
read the original abstract

Mixed-precision methods combine low and high precision arithmetics to exploit low precision computational speed and high precision accuracy. Large ODE systems that contain many heterogeneous interactions lead to a high computational cost that could be tackled with mixed-precision solvers. We tested mixedprecision versions of the Bogacki-Shampine 3(2) Runge-Kutta pair over three benchmark systems: coupled linear oscillators, the Kuramoto model and a circadian clock model. Our study is performed in a way that can be adapted to any finite-precision format, software architecture and numerical scheme. We found that mixed-precision solvers can preserve most of the high-precision solver accuracy under a wide range of solver tolerances. Moreover, mixed-precision solver accuracy improves with system size, reaching levels equivalent to high-precision solvers in small system size. We also observed that mixed-precision arithmetic does not impact the number of evaluation in a way that balances the benefit of fast operations in low precision. Taken together, these results show that mixed-precision methods can offer significant computational speed-up at little or no loss of accuracy in large coupled ODE systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents an empirical study of mixed-precision arithmetic applied to the adaptive Bogacki-Shampine 3(2) Runge-Kutta method for solving large systems of ODEs. It tests this approach on three benchmark systems—coupled linear oscillators, the Kuramoto model, and a circadian clock model—and reports that mixed-precision solvers maintain most of the accuracy of full high-precision solvers across a range of tolerances, with accuracy improving as system size grows, while not significantly affecting the number of function evaluations, thereby offering computational speed-up at little accuracy loss.

Significance. If the empirical findings prove robust upon detailed verification and generalize beyond the tested models, the work could enable meaningful speed-ups in simulating large coupled ODE systems with minimal accuracy trade-offs, which would be valuable for computational applications in physics, biology, and engineering involving high-dimensional dynamical systems.

major comments (2)
  1. [Abstract] Abstract: the abstract states empirical findings but supplies no implementation details, error-bar methodology, statistical tests, or description of how mixed-precision arithmetic was realized inside the adaptive step-size controller; claims such as 'accuracy improves with system size' therefore cannot be verified.
  2. [Numerical experiments] Benchmark systems: the three chosen benchmark systems (coupled oscillators, Kuramoto, circadian clock) have relatively uniform or low-dimensional coupling structures; none is a genuinely large (N ≫ 10^3), stiff, or randomly heterogeneous network. This raises concerns about whether the reported size-dependent accuracy gain is a general property of mixed-precision Runge-Kutta or model-specific.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the recommendation of major revision. We address each major comment below, agreeing to expand the abstract and to add discussion of benchmark limitations while defending the relevance of the chosen systems.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the abstract states empirical findings but supplies no implementation details, error-bar methodology, statistical tests, or description of how mixed-precision arithmetic was realized inside the adaptive step-size controller; claims such as 'accuracy improves with system size' therefore cannot be verified.

    Authors: We agree the abstract is too concise for full verification. In the revised manuscript we will add a sentence describing the mixed-precision realization (low-precision arithmetic applied only to the function evaluations inside the Bogacki-Shampine pair while retaining high-precision error estimation and step-size control), specify that accuracy is measured by direct comparison of solution trajectories against a reference high-precision run at each tolerance, and note that the size-dependent trend is shown by systematic variation of N in the numerical experiments section rather than by formal statistical tests. revision: yes

  2. Referee: [Numerical experiments] Benchmark systems: the three chosen benchmark systems (coupled oscillators, Kuramoto, circadian clock) have relatively uniform or low-dimensional coupling structures; none is a genuinely large (N ≫ 10^3), stiff, or randomly heterogeneous network. This raises concerns about whether the reported size-dependent accuracy gain is a general property of mixed-precision Runge-Kutta or model-specific.

    Authors: The three systems were selected because they are standard large-scale coupled ODE benchmarks in the literature and exhibit heterogeneous interactions (phase differences in Kuramoto, multiple regulatory loops in the circadian model). Experiments were performed with system sizes reaching several thousand equations, and the accuracy gain with N is observed consistently. We acknowledge that the models are not stiff or randomly heterogeneous; the revised manuscript will include an explicit limitations paragraph stating that generalization to such networks remains to be verified and suggesting this as future work. revision: partial

Circularity Check

0 steps flagged

No circularity: purely empirical benchmark study with no derivations or fitted predictions

full rationale

The manuscript reports direct numerical experiments comparing mixed-precision and high-precision Bogacki-Shampine 3(2) integrators on three concrete ODE systems (coupled oscillators, Kuramoto, circadian clock). No equations, ansatzes, uniqueness theorems, or parameter-fitting steps appear in the abstract or described content. Claims about accuracy preservation and size-dependent improvement are presented as observed outcomes of those runs, not as predictions derived from prior fitted quantities or self-citations. The central result therefore does not reduce to its own inputs by construction; the derivation chain is empty and the work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

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