Deep Adaptive Model-Based Design of Experiments

Arno Strouwen; Sebastian Micluţa-Câmpeanu

stat.MLMar 17, 2026

Deep Adaptive Model-Based Design of Experiments

Authors: Arno Strouwen, Sebastian Micluţa-Câmpeanu

Organizations: Strouwen Statistics, Hasselt, Belgium · Biosystems Department, KU Leuven, Belgium · JuliaHub, Massachusetts, USA · Faculty of Physics, University of Bucharest, Romania

Abstract

Model-based design of experiments (MBDOE) is essential for efficient parameter estimation in nonlinear dynamical systems. However, conventional adaptive MBDOE requires costly posterior inference and design optimization between each experimental step, precluding real-time applications. We address this by combining Deep Adaptive Design (DAD), which amortizes sequential design into a neural network policy trained offline, with differentiable mechanistic models. For dynamical systems with known governing equations but uncertain parameters, we extend sequential contrastive training objectives to handle nuisance parameters and propose a transformer-based policy architecture that respects the temporal structure of dynamical systems. We demonstrate the approach on four systems of increasing complexity: a fed-batch bioreactor with Monod kinetics, a Haldane bioreactor with uncertain substrate inhibition, a two-compartment pharmacokinetic model with nuisance clearance parameters, and a DC motor for real-time deployment.

Figures & tables

Fig. 1: Design comparison for the Monod bioreactor (20 rollouts per strategy, shared prior samples). Left to right: substrate $C_{s}$ , biomass $C_{x}$ , and feed rate $Q_{in}$ over 14 hours. The adaptive policy varies its design per rollout, keeping the feed near zero during initial growth before ramping up. All three strategies concentrate feeding towards the end; the adaptive policy adapts the timing and magnitude to each realization.

Design strategy	sPCE score	RMSE ( $\times 10^{3}$ )
		$\mu_{\max}$	$K_{s}$
Adaptive policy	$\mathbf{3.68\pm 0.04}$	$\mathbf{3.1}$	$\mathbf{24.9}$
Static (sPCE-opt)	$3.34\pm 0.04$	$5.6$	$41.5$
Static (BIM)	$3.45\pm 0.04$	$4.4$	$34.3$

TABLE I: Monod bioreactor: targeted sPCE scores (mean

\pm

SEM,

n=1000

; higher is better) and posterior RMSE (

\times 10^{3}

n=5000

; lower is better).

Fig. 2: Posterior mean estimates ( $\hat{\mu}_{\max}$ , $\hat{K}_{s}$ ) across 5000 trials for three design strategies. Convex hulls (90% closest to median) indicate the spread of posterior means; the adaptive policy yields the most concentrated estimates around the true values.

Fig. 3: Haldane bioreactor rollouts (20 per scenario, shared nuisance samples). Blue: no inhibition ( $\alpha=0$ ); red: strong inhibition ( $\alpha=0.15$ ). The policy first applies the same probing feed pulse in both cases, pushing the reactor into a regime where inhibition can become visible. It then adapts to the true target value: without inhibition the policy resumes feeding earlier to avoid returning to a low-substrate, non-inhibitory regime, whereas under strong inhibition the final ramp to maximal feed is delayed.

Fig. 4: PK model rollouts (20 per scenario, shared absorption parameters $k_{a}$ , $k_{tr}$ at midrange). Blue: slow elimination ( $CL=1.0$ L/hr, $Q_{d}=0.5$ L/hr); red: fast elimination ( $CL=5.0$ L/hr, $Q_{d}=3.0$ L/hr). Left to right: central concentration $C_{c}=A_{c}/V_{C}$ , peripheral concentration $C_{p}=A_{p}/V_{P}$ , and infusion rate $R_{\text{inf}}$ (design). After a common initial infusion pulse, the policy resumes dosing earlier under fast elimination, whereas under slow elimination it delays redosing because the drug persists longer.

	sPCE	RMSE ( $\times 10^{3}$ )		Time/step
		$k$	$J$
Adaptive (sPCE)	$\mathbf{2.96\pm 0.06}$	$\mathbf{20.6}$	$\mathbf{2.9}$	$24\,\mu$ s
Adaptive (BIM)	$2.56\pm 0.06$	$35.1$	$4.6$	$7.4\,$ ms

TABLE II: DC motor: adaptive sPCE policy vs. adaptive BIM (online re-optimization). sPCE scores (mean

\pm

SEM,

n=500

; higher is better), posterior RMSE (

\times 10^{3}

n=500

; lower is better), and per-step wall time.

Fig. 5: Per-step computation time (500 rollouts, single-threaded, AMD Ryzen 9 5900X). Boxes show IQR; whiskers the 0.1–99.9th percentile. Adaptive BIM (blue) grows from ${\sim}1\,$ ms to ${\sim}13\,$ ms, exceeding the $\Delta t=10\,$ ms sampling interval (dashed red). The amortized sPCE policy (purple) evaluates in ${\sim}24\,\mu$ s regardless of step.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

	Monod	Haldane	Weibull	Motor
Iterations	1000	1000	1000	1000
$\mathcal{C}_{\text{traj}}$	6.25M	2.12M	24M	56M
$G$ (accum. steps)	8	16	8	8
$L$ (contrastive)	464	161	725	960
$M$ (nuisance)	466	162	727	962
$B_{\text{micro}}$	6706	408	16506	29106

Appendix

TABLE III: Per-example training settings used in the reported runs. The main examples share the architecture and optimizer described above.

Fig. 6: Training loss curve (targeted sPCE objective) over iterations for the Monod bioreactor. The objective converges within approximately 1000 iterations.

Architecture	sPCE (mean $\pm$ SEM)	Paired $t$ vs. Full
Full Transformer	$1.709\pm 0.031$	—
Flat MLP	$1.679\pm 0.031$	$t=2.24$ ( $p<0.05$ )
Transformer (no PE)	$1.364\pm 0.030$	$t=12.09$ ( $p<0.001$ )

Appendix

TABLE IV: Architecture ablation on the Weibull PK model (1,000 paired trials,

L{=}M{=}5{,}000

). Higher sPCE is better.

Bayesian Experimental Design

Bayesian experimental design uses probabilistic models to choose measurements that improve learning or support better decisions.

See all 63 papers

JADAI: Jointly Amortizing Adaptive Design and Bayesian Inference
Dec 28, 2025Niels Bracher, Lars Kühmichel, Desi R. Ivanova +3Parameter Estimation Amortized Inference
1Rensselaer Polytechnic Institute, USA · 2TU Dortmund University, Germany · University of Oxford, UK.
Constrained Bayesian Experimental Design via Online Planning
May 26, 2026Yujia Guo, Daolang Huang, Xinyu Zhang +3Bayesian Experimental Design Sequential Experimental Design
ELLIS Institute Finland · Department of Computer Science, Aalto University, Finland · Department of Computer Science, University of Manchester, UK
Using the SEKF to Transfer NN Models of Dynamical Systems with Limited Data
Mar 2, 2026Joshua E. Hammond, Tyler A. Soderstrom, Brian A. Korgel +1System Identification Neural Surrogate Modeling
McKetta Department of Chemical Engineering, The University of Texas at Austin, 200 East Dean Keeton St., Stop C0400, Austin, 78712, TX, United States · Energy Institute, The University of Texas at Austin, 2304 Whitis Ave., Stop C2400, Austin, 78712, TX, United States · Texas Materials Institute, The University of Texas at Austin, 204 E. Dean Keeton St., Stop C2201, Austin, 78712, TX, United States +2

Dec 28, 2025J/K move · Enter open · S save

JADAI: Jointly Amortizing Adaptive Design and Bayesian Inference

Niels Bracher, Lars Kühmichel, Desi R. Ivanova, Xavier Intes+2

1Rensselaer Polytechnic Institute, USA · 2TU Dortmund University, Germany · University of Oxford, UK.

Parameter Estimation Amortized Inference Bayesian Inference Bayesian Experimental Design

Open paper AskPDF arXiv

We consider problems of parameter estimation where design variables can be actively optimized to maximize information gain. To this end, we introduce JADAI, a framework that jointly amortizes Bayesian adaptive design and inference by training a policy, a history network, and an inference network end-to-end. The networks minimize a generic loss that aggregates incremental reductions in posterior error along experimental sequences. Inference networks are instantiated with diffusion-based posterior estimators that can approximate high-dimensional and multimodal posteriors at every experimental step. Across standard adaptive design benchmarks, JADAI achieves superior or competitive performance.

Abstract

Bayesian Experimental Design

Explore similar work

JADAI: Jointly Amortizing Adaptive Design and Bayesian Inference

Constrained Bayesian Experimental Design via Online Planning

Using the SEKF to Transfer NN Models of Dynamical Systems with Limited Data