We thank the reviewers for their insightful feedback. R1 feels the paper does "a good job of including reasonable baselines" while R2 and R4 prefer (Supplement Sec 3). R4 considers the comparisons to be "a variant of proposed methods", which we disagree with R3 is concerned that the experiments are 1D designs. Eqn. 4) are typically used as proxies for the true MI (Eqn. One could use (non-nested) alternatives (Supp.

We introduce Neural Optimal Design of Experiments, a learning-based framework for optimal experimental design in inverse problems that avoids classical bilevel optimization and indirect sparsity regularization. NODE jointly trains a neural reconstruction model and a fixed-budget set of continuous design variables representing sensor locations, sampling times, or measurement angles, within a single optimization loop. By optimizing measurement locations directly rather than weighting a dense grid of candidates, the proposed approach enforces sparsity by design, eliminates the need for l1 tuning, and substantially reduces computational complexity. We validate NODE on an analytically tractable exponential growth benchmark, on MNIST image sampling, and illustrate its effectiveness on a real world sparse view X ray CT example. In all cases, NODE outperforms baseline approaches, demonstrating improved reconstruction accuracy and task-specific performance.

We revisit the classical problem of optimal experimental design (OED) under a new mathematical model grounded in a geometric motivation. Specifically, we introduce models based on elementary symmetric polynomials; these polynomials capture partial volumes and offer a graded interpolation between the widely used A-optimal and D-optimal design models, obtaining each of them as special cases. We analyze properties of our models, and derive both greedy and convex-relaxation algorithms for computing the associated designs. Our analysis establishes approximation guarantees on these algorithms, while our empirical results substantiate our claims and demonstrate a curious phenomenon concerning our greedy algorithm. Finally, as a byproduct, we obtain new results on the theory of elementary symmetric polynomials that may be of independent interest.

We introduce a novel geometric framework for optimal experimental design (OED). Traditional OED approaches, such as those based on mutual information, rely explicitly on probability densities, leading to restrictive invariance properties. To address these limitations, we propose the mutual transport dependence (MTD), a measure of statistical dependence grounded in optimal transport theory which provides a geometric objective for optimizing designs. Unlike conventional approaches, the MTD can be tailored to specific downstream estimation problems by choosing appropriate geometries on the underlying spaces. We demonstrate that our framework produces high-quality designs while offering a flexible alternative to standard information-theoretic techniques.

We thank the reviewers for their insightful feedback. R1 feels the paper does "a good job of including reasonable baselines" while R2 and R4 prefer (Supplement Sec 3). R4 considers the comparisons to be "a variant of proposed methods", which we disagree with R3 is concerned that the experiments are 1D designs. Eqn. 4) are typically used as proxies for the true MI (Eqn. One could use (non-nested) alternatives (Supp.

The Design of Experiments (DOEs) is a fundamental scientific methodology that provides researchers with systematic principles and techniques to enhance the validity, reliability, and efficiency of experimental outcomes. In this study, we explore optimal experimental design within a Bayesian framework, utilizing Bayes' theorem to reformulate the utility expectation--originally expressed as a nested double integral--into an independent double integral form, significantly improving numerical efficiency. To further accelerate the computation of the proposed utility expectation, conditional density estimation is employed to approximate the ratio of two Gaussian random fields, while covariance serves as a selection criterion to identify informative data-set during model fitting and integral evaluation. In scenarios characterized by low simulation efficiency and high costs of raw data acquisition, key challenges such as surrogate modeling, failure probability estimation, and parameter inference are systematically restructured within the Bayesian experimental design framework. The effectiveness of the proposed methodology is validated through both theoretical analysis and practical applications, demonstrating its potential for enhancing experimental efficiency and decision-making under uncertainty.

We introduce a gradient-free framework for Bayesian Optimal Experimental Design (BOED) in sequential settings, aimed at complex systems where gradient information is unavailable. Our method combines Ensemble Kalman Inversion (EKI) for design optimization with the Affine-Invariant Langevin Dynamics (ALDI) sampler for efficient posterior sampling--both of which are derivative-free and ensemble-based. To address the computational challenges posed by nested expectations in BOED, we propose variational Gaussian and parametrized Laplace approximations that provide tractable upper and lower bounds on the Expected Information Gain (EIG). These approximations enable scalable utility estimation in high-dimensional spaces and PDE-constrained inverse problems. We demonstrate the performance of our framework through numerical experiments ranging from linear Gaussian models to PDE-based inference tasks, highlighting the method's robustness, accuracy, and efficiency in information-driven experimental design.

Conventional Bayesian optimal experimental design seeks to maximize the expected information gain (EIG) on model parameters. However, the end goal of the experiment often is not to learn the model parameters, but to predict downstream quantities of interest (QoIs) that depend on the learned parameters. And designs that offer high EIG for parameters may not translate to high EIG for QoIs. Goal-oriented optimal experimental design (GO-OED) thus directly targets to maximize the EIG of QoIs. We introduce LF-GO-OED (likelihood-free goal-oriented optimal experimental design), a computational method for conducting GO-OED with nonlinear observation and prediction models. LF-GO-OED is specifically designed to accommodate implicit models, where the likelihood is intractable. In particular, it builds a density ratio estimator from samples generated from approximate Bayesian computation (ABC), thereby sidestepping the need for likelihood evaluations or density estimations. The overall method is validated on benchmark problems with existing methods, and demonstrated on scientific applications of epidemiology and neural science.

We introduce a new method to jointly reduce the dimension of the input and output space of a high-dimensional function. Choosing a reduced input subspace influences which output subspace is relevant and vice versa. Conventional methods focus on reducing either the input or output space, even though both are often reduced simultaneously in practice. Our coupled approach naturally supports goal-oriented dimension reduction, where either an input or output quantity of interest is prescribed. We consider, in particular, goal-oriented sensor placement and goal-oriented sensitivity analysis, which can be viewed as dimension reduction where the most important output or, respectively, input components are chosen. Both applications present difficult combinatorial optimization problems with expensive objectives such as the expected information gain and Sobol indices. By optimizing gradient-based bounds, we can determine the most informative sensors and most sensitive parameters as the largest diagonal entries of some diagnostic matrices, thus bypassing the combinatorial optimization and objective evaluation.