Abstract

Fragmentation processes arise in many physical systems, including droplet breakup, aerosols, sprays, comminution, and granular media. A central difficulty in fragmentation modeling is the identification of the breakup law from observed particle-size distributions. In this work, we propose a physics-constrained neural framework for the inverse identification of fragmentation laws. Starting from a size-structured binary fragmentation equation, the kernel is decomposed into a breakup rate B(s), depending on the parent size s, and a daughter-size distribution κ(z), depending on the normalized daughter-size fraction z. The unknown functions B and κ are represented by neural networks designed to satisfy physical constraints, including non-negativity of the breakup rate, non-negativity and normalization of the daughter distribution, and first-moment conservation in the symmetric binary setting. The direct fragmentation equation is solved using a first-moment-conserving quadrature discretization, and the inverse problem is formulated as a regularized optimization problem constrained by the fragmentation dynamics. Synthetic experiments, generated independently of the inverse solver, show that the proposed method can recover B and κ from simulated particle-size distributions, with accurate recovery in the unimodal case and reasonable performance in the more challenging bimodal case. Robustness tests indicate stability with respect to moderate observational noise and highlight the importance of multiple observation times. For experimental droplet-breakup measurements without time-resolved particle-size distributions, the same constrained neural daughter-density representation is used to estimate effective marginal daughter-volume-fraction distributions from normalized daughter-to-parent volume fractions. The resulting distributions distinguish rim and node fragments and different breakup modes, while stratified cluster-bootstrap confidence bands quantify their sampling uncertainty. The experimental analysis is therefore interpreted as constrained density estimation from final fragment measurements rather than as full validation of the PDE-constrained inverse recovery.

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