Data-efficient inverse design of inlet boundary conditions in hexagonal microfluidic arrays for concentration...

Complex concentration fields in microfluidic arrays are often designed through repeated forward simulation or experiment, which becomes inefficient when a prescribed target field is given in advance. Here, we present a Bayesian optimization framework for the inverse design of inlet boundary conditions in close-packed hexagonal microfluidic arrays. The framework combines Gaussian process regression with a Matérn 2.5 kernel and a log-scale loss function to mitigate the vanishing sensitivity in low-error regimes. To address user-defined target fields that may contain strongly incompatible local features due to diffusion-limited transport, we introduce an adaptive constraint optimization strategy that reduces the influence of persistently high-error constraints. For larger arrays, we further incorporate a sparse sampling strategy motivated by the low-dimensional structure of the concentration field with respect to the 12 inlet boundary-condition variables. The framework was evaluated on hexagonal arrays containing 60 and 468 chambers. In the 60-chamber self-consistency test, the inverse solver recovered target fields with high accuracy. Across nine additional physically generated targets, the 60-chamber model achieved a median R² of 0.9572. For empirically prescribed targets, the adaptive constraint strategy improved convergence toward physically realizable solutions. In the 468-chamber array, optimizing 54 representative chambers (corresponding to ≈11.5% of the total) recovered the dominant spatial structure of the full concentration field, yielding an R² of 0.8457 when evaluated against all 468 chambers. These results show that Bayesian inverse design can provide a practical and data-efficient route for concentration-field programming in large microfluidic networks and support the automated generation of complex chemical microenvironments.