Deep Optimal Experimental Design for Parameter Estimation Problems

TL;DR

The paper tackles parameter estimation for nonlinear ODEs under experimentally constrained design by replacing costly bilevel optimization with a deep likelihood-free estimator (LFE). It learns a direct mapping from measured data to parameters while jointly optimizing an experimental design vector, using self-supervised training. Two design strategies are proposed: continuous weights with soft-threshold sparsity and binary weights with Tabu search, both integrated into a single training framework. Demonstrations on a 3-Tissue Compartment model and a Lotka-Volterra system show that the LFE yields lower recovery and data-fit risk than random designs and substantially reduces data-collection costs, offering a scalable path for optimal experimental design in nonlinear dynamical systems.

Abstract

Optimal experimental design is a well studied field in applied science and engineering. Techniques for estimating such a design are commonly used within the framework of parameter estimation. Nonetheless, in recent years parameter estimation techniques are changing rapidly with the introduction of deep learning techniques to replace traditional estimation methods. This in turn requires the adaptation of optimal experimental design that is associated with these new techniques. In this paper we investigate a new experimental design methodology that uses deep learning. We show that the training of a network as a Likelihood Free Estimator can be used to significantly simplify the design process and circumvent the need for the computationally expensive bi-level optimization problem that is inherent in optimal experimental design for non-linear systems. Furthermore, deep design improves the quality of the recovery process for parameter estimation problems. As proof of concept we apply our methodology to two different systems of Ordinary Differential Equations.

Figures (6)

• Figure 1: A schematic process of parameter estimation and experimental design. Given a set of plausible parameters and a particular experimental setting the data of the forward process is simulated. One then use some parameter estimation routine and measures the quality of the estimated parameters. The design is changed to have better estimation of the parameters.
• Figure 2: Synthetic Time Activity Curves for the 3-TC model at various (multiplicative) Gaussian noise levels. The vertical dotted lines correspond to an optimal data sampling scheme for $sparsity=6$ that minimizes $\ell_{T}({\bf w})$, obtained using binary design variables ${\bf w}$ (Method 2). $400$ time points with logarithmic spacing are considered.The design weight vector ${\bf w}$ has the value $1$ at the optimal time points and $0$ for the others.
• Figure 3: A comparison of the performance of the estimator trained using continuous ${\bf w}$ (Method 1) and binary ${\bf w}$ (Method 2) for the 3-Tissue Compartment (3-TC) model. Networks trained for optimal designs ${\bf w}_{\rm opt}$ at the sparsities shown were evaluated for $\ell_{T}({\bf w})$ over $175K$ unseen samples of ${\bf q}$ (and corresponding ${\bf d}$). Error bars showcase the Standard Error of the Mean (SEM) for the mean risk $\ell_{T}({\bf w})$ exhibited by each network at its corresponding sparsity. See Appendix \ref{['appendix_B']} for details.
• Figure 4: Synthetic prey population data ${\bf d} = x(t)$ for the Predator-Prey system at noise levels $\sigma = 0\%, 1\%, 2\%, 3\%...10\%$. 200 sampled parameter sets $[\alpha, \beta, \gamma, \delta]$ were used to generate ${\bf d}$ which are plotted for $t=0$ to $t=30$ years. $200$ equally spaced time points are considered. The four dotted lines indicate an optimal sampling scheme of $sparsity=4$ at $t = 5.1, 8.9, 26.1, 29.9$ years obtained using continuous design variables ${\bf w}$ (Method 1).
• Figure 5: A comparison of the performance of the estimator trained using continuous ${\bf w}$ (Method 1) and binary ${\bf w}$ (Method 2) for the Predator-Prey model (PPM). Networks trained for optimal designs ${\bf w}_{\rm opt}$ at the sparsities shown were evaluated for $\ell_{T}({\bf w})$ over $175K$ unseen samples of ${\bf q}$ (and corresponding ${\bf d}$). Error bars that appear point-like are plotted showcasing the Standard Error of the Mean (SEM) for the mean risk $\ell_{T}({\bf w})$ exhibited by each network at its corresponding sparsity. See Appendix \ref{['appendix_B']} for details.