Deep Optimal Sensor Placement for Black Box Stochastic Simulations

TL;DR

The paper tackles sensor-placement for black-box stochastic simulations by introducing Functional Neural Couplings, a joint probabilistic surrogate that models the latent relationship between functional inputs κ and their stochastic outputs u via an implicit neural representation (INR) and a joint energy-based model (EBM) with pθ(zκ, zu) ∝ exp(−Eθ(zκ, zu)). This resolution-independent framework enables efficient Bayesian experimental design using a prior-contrastive bound, ŨPCE, and adaptive sequential sensor placement to maximize information gain about κ and u. Empirical results on 1D boundary-value problems, 2D Darcy flow, and 2D Navier–Stokes show that adaptive sensor locations derived from FNC yield superior posterior accuracy and lower computational cost than Fourier Neural Operator baselines, closely approaching the performance of FNO with oracle noise. The approach offers practical, scalable sensor design for complex stochastic PDEs without requiring full access to the driving noise, enabling reliable inference in challenging inverse problems.

Abstract

Selecting cost-effective optimal sensor configurations for subsequent inference of parameters in black-box stochastic systems faces significant computational barriers. We propose a novel and robust approach, modelling the joint distribution over input parameters and solution with a joint energy-based model, trained on simulation data. Unlike existing simulation-based inference approaches, which must be tied to a specific set of point evaluations, we learn a functional representation of parameters and solution. This is used as a resolution-independent plug-and-play surrogate for the joint distribution, which can be conditioned over any set of points, permitting an efficient approach to sensor placement. We demonstrate the validity of our framework on a variety of stochastic problems, showing that our method provides highly informative sensor locations at a lower computational cost compared to conventional approaches.

Figures (8)

• Figure 1: Workflow for the training of the INR and joint energy-based model. Layout based on serrano2023operator.
• Figure 2: Sensor placement task for the BVP using our Functional Neural Coupling surrogate. Posterior samples based on two optimal design points, adaptive (top) and batch non-adaptive (middle) versus two points from a random sequence (bottom). True values $a=-2$, $b=1.5$.
• Figure 3: Top left: Observed solution. Posterior mean solutions from observations at adaptively chosen locations (top right), batch non-adaptively selected locations (bottom left) and Quasi Monte Carlo points (bottom right).
• Figure 4: Top left: True diffusion coefficient. Posterior mean coefficients from observations at adaptively chosen locations (top right), batch non-adaptively selected locations (bottom left) and Quasi Monte Carlo points (bottom right).
• Figure 5: Top: ground truth solution. Bottom: posterior mean based on observations at adaptively chosen sensor locations using our Functional Neural Coupling surrogate.

Theorems & Definitions (2)

• Proposition 1
• proof