REDUCED-PRECISION STOCHASTIC SIMULATION FOR MATHEMATICAL BIOLOGY A PREPRINT

Introduction

Stochastic simulation is a central tool in computational systems biology. The stochastic simulation algorithm and its variants provide exact forward simulation of discrete stochastic processes.

The computational cost of realistic models remains substantial.

Traditional acceleration strategies have focused on algorithmic modifications.

Methodology

In the Direct Method formulation, each step proceeds as follows. Three operations in the Direct Method are particularly sensitive to floating-point precision.

Study Design

For deterministic solvers, rounding error bounds would need to be established through traditional numerical analysis.

Results & Findings

These formats cut memory traffic and reduce energy consumption. No systematic study has examined whether SSA can achieve similar gains in mathematical biology.

• These formats cut memory traffic and reduce energy consumption.
• No systematic study has examined whether SSA can achieve similar gains in mathematical biology.
• We evaluate mixed-precision SSA on canonical model systems.
• These test cases allow us to assess precision degradation and identify safe precision maps.
• Section 5 discusses the implications of these results.

Practical Applications

Intrinsic noise and uncertainty may exceed any bias from low-precision arithmetic. Deterministic reaction-diffusion PDE solvers could benefit from halved data traffic.

Computing the propensity sum involves additions that may span several orders of magnitude.

The practical implication is that FP16 is materially safer than BF16 for stiff oscillatory systems.

The Stochastic Simulation Algorithm

This section describes the SSA, which generates exact realizations of chemical systems through a series of computational steps. It outlines the computational costs associated with the algorithm and the challenges posed by floating-point precision.

Reduced precision and stochastic rounding

This section explains the concept of reduced precision in digital computing, focusing on the implications of rounding errors when using lower precision formats. It discusses the trade-offs involved in using reduced-precision formats for scientific computing.

Half-precision floating-point formats

This section compares different floating-point formats, particularly focusing on binary16 (fp16) and bfloat16 (bf16). It discusses their properties, advantages, and disadvantages in the context of SSA and their impact on memory usage.

Figures Explained

The paper’s visual material highlights the workflow and the main system components.

• \u2022: FP64: 64-bit floating point (reference baseline) \u2022 FP32: 32-bit floating point \u2022 FP16 RTN: IEEE half precision with round-to-nearest \u2022 FP16+SR: IEEE half precision with SR \u2022 BF16 RTN: Brain float with round-to-nearest \u2022 BF16+SR: Brain float with SR.
• Figure 1 :: Figure 1: Steady-state distribution of the birth-death process (N = 50,000 realisations each), compared to the analytical Poisson distribution (black dashed line). (a) All six mixed-precision configurations produce indistinguishable results. (b) Uniform-mode comparison: SR modes closely match the FP64 reference, while BF16 RTN (pink) shifts left to mean \u2248 16, where time stagnation terminates the simulation prematurely.
• Figure 3: The FP16 exclusion above is a parameter representation limitation, not a fundamental arithmetic precision problem. The SSA dynamics depend only on the products k 1 A and k 3 B, not on the individual values. By reparameterising (setting k 1 = 1.5 \u00d7 10 -4 , A = 100, k 3 = 1.0, B = 200 while preserving k 1 A = 0.015 and k 3 B = 200), all parameter values and intermediate propensities remain within FP16’s representable range (verified: a 1 = 4696, a 2 = 2914, a 3 = 200, a 4 = 1960 at n = 560, all \u226a 65,504). This nondimensionalisation preserves identical peak positions and transition dynamics.
• Figure 2 :: Figure 2: Bimodal stationary distribution of the Schl\u00f6gl system (N = 10,000 realisations each). Top row: original parameterisation. Bottom row: rescaled parameterisation. (a) FP32 is indistinguishable from FP64; BF16 modes show minor shifts in peak occupancy. FP16 modes are excluded (cubic propensity overflow). (b) BF16+SR slightly favours the high peak; BF16 RTN shifts further toward the high peak. FP16 uniform modes overflow identically to their mixed counterparts. (c) All six modes reproduce the bimodal structure; FP16+SR is nearly indistinguishable from FP64, confirming the original failure was due to parameter range. (d) FP16+SR (W 1 = 2.76, KS p = 0.81) closely matches FP64, while BF16 uniform modes shift toward the high peak.
• Figure 5: The steady-state distribution of n M is governed by the dimensionless ratios a = k on \/\u03b2, b = k off \/\u03b2, and \u03bb = \u03b1\/\u03b2. Depending on these parameters, the model exhibits qualitatively different behaviour, ranging from unimodal Poissonlike distributions when switching is fast (a, b \u226b 1) to bimodal distributions reflecting the two promoter states when switching is slow (a, b \u226a 1). We examine two slow-switching parameterisations that probe different aspects of precision sensitivity.We first consider k on = k off = 0.05, \u03b1 = 20, \u03b2 = 1, giving dimensionless parameters a = b = 0.05 and \u03bb = 20. The gene spends equal time in the ON and OFF states, and the slow switching produces a bimodal mRNA distribution with mean \u2248 10. Because mRNA decay (\u03b2 = 1) is much faster than promoter switching (k on = 0.05), the molecule count.