# Sampling methods for probabilistic approximations of ODEs

Olivier Bouët-Willaumez, Adrien Le Coënt, Benoît Barbot, Nihal Pekergin

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Olivier Bouët-Willaumez, Adrien Le Coënt, Benoît Barbot, Nihal Pekergin. Sampling methods for probabilistic approximations of ODEs. 2026. hal-05440887

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# Sampling methods for probabilistic approximations of ODEs

Olivier Bouët-Willaumez¹, Adrien Le Coënt¹, Benoît Barbot¹, Nihal Pekergin¹

Université Paris-Est Créteil, LACL,
F-94010 Creteil, France

{olivier.bouet-willaumez,adrien.le-coent,benoit.barbot}@u-pec.fr

**Keywords** : *Dynamic Bayesian Networks, Sampling, Parameter Estimation, ODEs*

## 1 Introduction

Ordinary Differential Equations (ODEs) are used to model dynamical systems, notably in biology and biochemistry. In such systems, the estimation of parameters from experimental data is a central challenge, with applications such as drug discovery or clinical diagnosis.

Liu B. et al. proposed to transform ODEs into Dynamic Bayesian Networks (DBNs) [1] to speed up cost function calculations, limit repeated ODEs simulations, and quantify the uncertainty on states and parameters, obtaining better results than classical optimization techniques for systems with less than ≈ 25 variables.

The construction of a DBN requires to discretize the state space and generate a training data set (a set of many ODEs simulations) to fill the Conditional Probability Tables (CPTs). Low-discrepancy sequences, like those of Halton [2], used previously [3], do not guarantee good coverage of discrete intervals [4], certain regions of space being under-represented, as can be seen in Figure 1. This can bias the tables of the DBN and affect the quality of the probabilistic approximation.

<figure><img src="image_2.png" /><figcaption>FIG. 1: 2-D example comparing $L^2$ star discrepancy and spatial coverage with $N = 100$ samples.</figcaption></figure>

In this paper, we propose a structured enumerative sequence ensuring uniform coverage of discrete intervals, allowing more representative tables and a better probabilistic approximation of the dynamic system by the DBN.

## 2 The Enumerative Sequence

As motivated in the section 1, uniform coverage of the discretized state space is crucial for building unbiased CPTs. To ensure this coverage, we replace low-discrepancy sampling with a simple deterministic enumeration.

For the $d$ variables, each domain $[l_i, u_i]$ is uniformly split into $\delta$ sub-intervals of width $h_i = \frac{u_i - l_i}{\delta}$. We then enumerate the mid-point of each sub-interval: $m_{i,k} = l_i + (k + \frac{1}{2})h_i$, $k = 0, \dots, \delta - 1$. The resulting enumerative sequence is the Cartesian product $\mathcal{X}_{enum} = \prod_{i=1}^{d} \{m_{i,k}\}$, providing exact and uniform coverage of every discrete region, with $|\mathcal{X}_{enum}| = \delta^d$.
When under-sampling is desired, we select a number of points $M < |\mathcal{X}_{enum}|$ using a procedure that preserves approximate uniformity: points are first generated using a Halton sequence [2] in $[0, 1]$, mapped to indices in $[0, M-1]$, and duplicates are removed. Any remaining unsampled indices are then randomly chosen without replacement to complete the subset.

When more than $\delta^d$ samples are required, a uniform super-sampling pattern is used.

# 3 Case Study and Results

We compare the Enumerative and Halton sequences on a 7-dimensional (4 species, 3 parameters) Enzyme-Catalyzed reaction [3], discretizing each variable into 5 sub-intervals. The resulting state space has size $|\mathcal{D}| = 5^7 = 78125$.

To evaluate the performance of the DBN-based parameter estimation for different training data set sizes, we use a fixed test set $\mathcal{S}$ consisting of 125 ODE trajectories over 11 time steps, with varying parameter combinations $K_j = [k_{1,j}, k_{2,j}, k_{3,j}] \in [0, 1]^3$, $j = 1, \dots, 125$, and a common species initial condition $\boldsymbol{x}_{\text{init}} \in [0, 15]^4$. Each trajectory in this set is denoted $\text{ODE}_{K_j}$.

For each parameter combination $K_j$, we compute an approximate trajectory derived from the DBN. This trajectory is obtained with the following procedure:

1. Consider this parameter combination as a prior for the DBN.

2. Perform bayesian inference to compute the joint probabilities of every variables at each time step.

3. Weight the distributions of each variable, at each time step, with the mid-point values of the discrete sub-intervals we defined, resulting in a trajectory $\tilde{\text{ODE}}_{K_j}$.

The cost function $\varepsilon$ is the MSE between $\text{ODE}_{K_j}$ and $\tilde{\text{ODE}}_{K_j}$. Figure 2 shows the median $\varepsilon$ values over all 125 combinations, with a box-plot slice for $N = |\mathcal{D}|$. Enumeration gives more concentrated $\varepsilon$ values with a lower median and fewer outliers than Halton sequences.

<figure><img src="image_3.png" /><figcaption>FIG. 2: Median $\varepsilon$ values comparison (**left**) and box-plot slice for $N = |\mathcal{D}|$ (**right**).</figcaption></figure>

# References

[1] Liu, B., Hsu, D., Thiagarajan, P.S. *Probabilistic approximations of odes-based bio-pathway dynamics*. Theoretical Computer Science 412(21), 2188-2206 (2011)

[2] Halton, J. H. *On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals*. Numerische Mathematik, vol.2, pp.84-90 (1960)

[3] Le Coënt, A., Barbot, B., Pekergin, N. *Efficient probabilistic inference in biochemical networks*. Computers in Biology and Medicine 183, 109280 (2024)

[4] Zhiglavsky, A., Noonan, J. *Covering of High-Dimensional Cubes and Quantization*. SN Oper. Res. Forum 1, 18 (2020)