Abstract
<jats:p>This work proposes a compositional operator framework in which local exit operators represent particle transport within individual microfluidic modules. The operator maps an encoded representation of the inlet interface state and module context to an exit descriptor containing the outlet branch, normalized outlet position, velocity, and local transit time. Four task-specific neural networks are trained to approximate these quantities, and network-scale transport is obtained by recursively evaluating the learned exit operator together with topology-guided interface transitions. A fluid-circuit analogy is further incorporated to determine the branch-flow environment under prescribed boundary conditions. Single-module experiments and multi-module network tests demonstrate that the learned operators reproduce local exit states and can be recursively evaluated to recover multi-module particle paths. The framework is subsequently applied to single-module interface focusing, network-level path optimization, reinforcementlearning-based path control, and structural inverse design. Repeated numerical evaluations demonstrate consistent attainment of the prescribed objectives across these application tasks, while forward validation over a domain of inlet positions and inlet velocities confirms that the optimized structure preserves the prescribed path over a subset of the tested operating range. The proposed framework provides a unified and reusable representation for particle transport prediction, control, and inverse design in modular microfluidic networks.</jats:p>