Adjoint-Compatible Surrogates of the Expected Information Gain for Optimal Experimental Design in Controlled Dynamical Systems

Abstract

We consider optimal experimental design for parameter estimation in dynamical systems governed by controlled ordinary differential equations. In such problems, Fisher-based criteria are attractive because they lead to time-additive objectives compatible with adjoint-based optimal control, but they remain intrinsically local and may perform poorly under strong nonlinearities or non-Gaussian prior uncertainty. By contrast, the expected information gain (EIG) provides a principled Bayesian objective, yet it is typically too costly to evaluate and does not naturally admit an adjoint-compatible formulation. In this work, we introduce adjoint-compatible surrogates of the EIG based on an exact chain-rule decomposition and tractable approximations of the posterior distribution of the unknown parameter. This leads to two surrogate criteria: an instantaneous surrogate, obtained by replacing the posterior with the prior, and a Gaussian tilting surrogate, obtained by reweighting the prior through a design-driven quadratic information factor. We also propose a multi-center tilting surrogate to improve robustness for complex or multimodal priors. We establish theoretical properties of these surrogates, including exactness of the Gaussian tilting surrogate in the linear-Gaussian setting, and illustrate their behavior on benchmark controlled dynamical systems. The results show that the proposed surrogates remain competitive in nearly Gaussian regimes and provide clearer benefits over Fisher-based designs when prior uncertainty is non-Gaussian or multimodal.

Figures (4)

• Figure 1: Empirical distributions of the parameter-estimation errors for the harmonic oscillator test case with similar observability over 1000 Monte Carlo runs. The orange line denotes the median and the triangle the mean. All EIG-based surrogates perform similarly in this balanced regime, while the multi-center tilting surrogate achieves the best overall accuracy
• Figure 2: Empirical distributions of the parameter-estimation errors for the harmonic oscillator test case with uneven observability over 1000 Monte Carlo runs. The orange line denotes the median and the triangle the mean. The instantaneous surrogate strongly favors the first component, whereas the tilting surrogates provide more balanced reconstructions.
• Figure 3: Empirical distributions of the parameter-estimation errors for the Lotka-Volterra test case with log-normal prior over 1000 Monte Carlo runs. The orange line denotes the median and the triangle the mean. All designs achieve comparable performance in this case, although the EIG-based surrogates remain slightly better overall and the multi-center tilting surrogate yields the best results.
• Figure 4: Empirical distributions of the parameter-estimation errors for the Lotka-Volterra test case with log-normal mixture prior over 1000 Monte Carlo runs. The orange line denotes the median and the triangle the mean. The Fisher-based designs deteriorate significantly in this bimodal setting, whereas the EIG-based surrogates remain effective and better adapt to the multimodal prior structure.

Theorems & Definitions (15)

• Remark 1: Distinction from the Laplace approximation
• Remark 2: Multi-center extension
• Lemma 1: Redundancy of the instantaneous surrogate
• proof
• Proposition 1: Instantaneous surrogate bounds
• proof
• Proposition 2: Consistency in the Linear--Gaussian setting
• proof
• Remark 3: Avoiding nested model simulations
• Remark 4: Bang--bang optimality of the sampling policy