ParetoEnsembles.jl: A Julia Package for Multiobjective Parameter Estimation Using Pareto Optimal Ensemble Techniques

Abstract

Mathematical models of natural and man-made systems often have many adjustable parameters that must be estimated from multiple, potentially conflicting datasets. Rather than reporting a single best-fit parameter vector, it is often more informative to generate an ensemble of parameter sets that collectively map out the trade-offs among competing objectives. This paper presents ParetoEnsembles.jl, an open-source Julia package that generates such ensembles using Pareto Optimal Ensemble Techniques (POETs), a simulated-annealing-based algorithm that requires no gradient information. The implementation corrects the original dominance relation from weak to strict Pareto dominance, reduces the per-iteration ranking cost from $O(n^2 m)$ to $O(nm)$ through an incremental update scheme, and adds multi-chain parallel execution for improved front coverage. We demonstrate the package on a cell-free gene expression model fitted to experimental data and a blood coagulation cascade model with ten estimated rate constants and three objectives. A controlled synthetic-data study reveals parameter identifiability structure, with individual rate constants off by several-fold yet model predictions accurate to 7%. A five-replicate coverage analysis confirms that timing features are reliably covered while peak amplitude is systematically overconfident. Validation against published experimental thrombin generation data demonstrates that the ensemble predicts held-out conditions to within 10% despite inherent model approximation error. By making ensemble generation lightweight and accessible, ParetoEnsembles.jl aims to lower the barrier to routine uncertainty characterization in mechanistic modeling.

Figures (9)

• Figure 1: Ensemble estimation for a cell-free gene expression circuit ($\sigma_{70} \to \text{P70} \to \text{deGFP}$) fitted to experimental data from Adhikari et al. Adhikari2020. (a) mRNA and (b) protein concentration versus time; the blue dashed line is the ensemble mean, the shaded region is the 95% confidence interval, and black points are experimental data with error bars. (c) Pareto front showing the trade-off between mRNA and protein fitting error.
• Figure 2: Training results for the Hockin--Mann blood coagulation model (34 species, 10 estimated rate constants, 3 data-driven objectives, no regularization). (a) Total thrombin concentration versus time at 5 pM (amber), 15 pM (purple), and 25 pM (teal) tissue factor; shaded regions are the ensemble 95% CI, dashed lines are the ensemble mean, and points are noisy synthetic data (15% CV). (b) Parameter recovery in log-space; the dashed red line is the identity and diamonds show the median ensemble estimate; scatter around each parameter reveals the degree of identifiability from thrombin data alone. (c) Pareto front projection ($\varepsilon_1$ vs. $\varepsilon_2$, colored by $\varepsilon_3$), with rank-zero solutions in black.
• Figure 3: Ensemble-based uncertainty characterization for the coagulation model. (a) Held-out thrombin predictions at 10, 20, and 30 pM TF (not used during training); solid lines are true trajectories, dashed lines are ensemble means, and shaded regions are 95% CIs. The ensemble captures the overall shape but shows a systematic positive bias at peak thrombin. (b) Pairwise parameter correlation heatmap revealing extensive compensatory structure; strong negative correlations (e.g., extrinsic Xase $k_{\text{cat}}$ vs. prothrombinase $k_{\text{cat}}$, $r = -0.81$; Xa$\to$IIa vs. IIa$\to$Va, $r = -0.93$) explain why individual parameters can be poorly recovered while model predictions remain accurate. (c) Patient-specific predictions for Factor VIII deficiency (hemophilia A) at 100%, 30%, and 5% of nominal FVIII levels; ensemble prediction bands capture the dose-dependent reduction in peak thrombin, and true trajectories (black dashed) fall within all three bands. (d) TGA feature accuracy at held-out conditions: ensemble-predicted lag time, peak thrombin, and ETP normalized to true values, with 95% CIs; black crosses indicate the true value is covered by the interval, red crosses indicate it is not.
• Figure 4: Ensemble estimation fitted to experimental thrombin generation data from Butenas et al. Butenas1999. Prothrombin was varied from 50% to 150% of its mean plasma concentration in a reconstituted synthetic plasma system initiated by 5 pmol/L TF--VIIa. (a) Training fits at 50% (red), 100% (gray), and 150% (blue) prothrombin; the ensemble captures the experimental data at low and normal prothrombin levels but underestimates the peak at 150%, revealing a model--data tension. (b) Held-out validation at 75% (amber) and 125% (teal) prothrombin; the ensemble predicts peak thrombin to within 1--10% at conditions never used during training. (c) Pareto front projection showing trade-offs among the three training objectives. (d) Parameter estimates versus nominal literature values; deviations reflect compensatory adjustments to fit experimental data with an approximate model.
• Figure S1: Benchmark results for (a,c) Binh--Korn and (b,d) Fonseca--Fleming ($d=3$). Top row: objective space; bottom row: parameter-space projections ($x_1$ vs. $x_2$). Non-dominated solutions (rank $= 0$, dark) define the Pareto front, while near-optimal solutions (blue, colored by rank) form a cloud around it representing the retained ensemble. The dashed red curve in (b) is the theoretical Pareto front. Ten parallel chains were used with $R_{\text{cutoff}} = 12$, $N_{\text{iter}} = 60$, and $\alpha = 0.95$.