UNED: One-shot Uncertainty-aware Neural Experimental Design for Transient PDEs

Mahdi Masmoudi, Xuyang Li, Rami Gharbi, Nizar Lajnef, Vishnu Boddeti

Keywords: Experimental design, PDE inverse problems, uncertainty quantification, conformal prediction, sensor placement.

TL;DR: UNED is a differentiable framework for one-shot experimental design in PDE inverse problems. It optimizes sensor locations to minimize expected posterior uncertainty, using joint conformal prediction calibrated over model and measurement noise.

Abstract:

Partial differential equations (PDEs) are foundational tools for modeling complex physical systems. A central challenge is inferring spatially or temporally varying PDE parameters from limited observations, a task made difficult by scarce and noisy data. Since running physical experiments is often costly, experimental design (ED) is commonly performed using PDE simulations to select the most informative sensor configurations before data collection. This is particularly important under strict budgets that allow only one-shot deployment, where uncertainty from measurement noise or surrogate models can significantly affect parameter recovery. Existing ED methods often optimize sensor placement assuming a fixed model realization or noise structure, and rarely aim to explicitly minimize uncertainty or ensure robustness to model and noise variability.
We introduce UNED, a differentiable experimental design framework that enables efficient gradient-based optimization of sensor locations to minimize expected posterior uncertainty for one-shot deployment. Rather than tuning designs for a single realization, UNED accounts for randomness in model initialization and observation noise, enabling robust sensor placement in settings with spatially distributed parameters. We validate UNED through experiments based on noisy simulated data and real-world experimental data. By explicitly minimizing uncertainty using an unsupervised uncertainty quantification (UQ) model, it outperforms existing ED methods in solving inverse problems.
Our results demonstrate that explicitly minimizing expected uncertainty under model and noise randomness leads to robust sensor placements whose performance generalizes across realizations, with improved parameter recovery emerging as a direct consequence rather than a design objective.

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Submission Number: 114