We propose a novel approach for sequential optimal experimental design (sOED) for Bayesian inverse problems involving expensive models with large-dimensional unknown parameters. The focus of this work is on designs that maximize the expected information gain (EIG) from prior to posterior, which is a computationally challenging task in the non-Gaussian setting. This challenge is amplified in sOED, as the incremental expected information gain (iEIG) must be approximated multiple times in distinct stages, with both prior and posterior distributions often being intractable. To address this, we derive a derivative-based upper bound for the iEIG, which not only guides design placement but also enables the construction of projectors onto likelihood-informed subspaces, facilitating parameter dimension reduction. By combining this approach with conditional measure transport maps for the sequence of posteriors, we develop a unified framework for sOED, together with amortized inference, scalable to high- and infinite-dimensional problems. Numerical experiments for two inverse problems governed by partial differential equations (PDEs) demonstrate the effectiveness of designs that maximize our proposed upper bound.

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