Experimental data play a crucial role in developing and refining models of physical systems. Some experiments can be more valuable than others, however. Well-chosen experiments can save substantial resources, and hence optimal experimental design seeks to quantify and maximize the value of experimental data. Common current practice for designing a sequence of experiments uses suboptimal approaches: open-loop design that chooses all experiments simultaneously with no feedback of information, or greedy design that optimally selects the next experiment without accounting for future observations and dynamics. By contrast, an optimal sequential design is obtained via closed-loop dynamic programming (DP). This approach accounts for future experiments while making each design decision as late as possible, thus allowing the result of each experiment to guide the design of all remaining future experiments.

With the goal of acquiring experimental data that are optimal for model parameter inference, we develop new numerical tools to make solution of the DP design problem computationally feasible. Our approach uses a design objective that incorporates a measure of information gain, and accommodates nonlinear models with continuous (often unbounded) parameter, design, and observation spaces. First, an adaptive one-step look-ahead value function approximation strategy is used to find the optimal policy. This approximate dynamic programming method iteratively generates scenarios via exploration and exploitation, and uses them to create better approximations of value functions in frequently visited regions of the state space. Second, transport maps are created to represent belief states, which are the intermediate posterior random variables within the sequential design process. Not only do these maps provide a finite-dimensional representation of these generally non-Gaussian random variables, they also enable fast approximate Bayesian inference, which must be performed millions of times under nested combinations of optimization and Monte Carlo sampling.

 

These methods are demonstrated on a simple linear-Gaussian model and on a nonlinear application of optimal sequential sensing: inferring contaminant source location from an airborne sensor in a time-dependent convection-diffusion system.