This study develops a leakage-free hybrid Markov–Random Forest framework for cryptocurrency price forecasting and evaluates it on Bitcoin and Ethereum. Daily OHLCV features are lagged by one trading day to prevent look-ahead bias, while regime labels are assigned from observed price changes using a two-state Markov chain with increasing and decreasing states. Regime-specific Random Forest models are then tuned independently via time-series cross-validation, allowing the predictive structure to adapt to regime-specific market conditions. The empirical results exhibit clear asset dependence. For Ethereum, the hybrid model outperforms the standalone Random Forest on magnitude-based metrics, attaining lower MAE and RMSE while also delivering a modest improvement in directional accuracy. Regime-specific tuning further identifies distinct optimal hyperparameter configurations across the increasing and decreasing states, suggesting that Ethereum’s upward and downward dynamics are structurally heterogeneous and can be better captured through regime-aware learning. By contrast, for Bitcoin, the standalone Random Forest delivers superior magnitude forecasting performance, while the regime-specific models differ only in tree depth and share the remaining tuning parameters, indicating that regime conditioning adds limited incremental value in a more persistent market. Statistical tests reinforce these findings. For Ethereum, Diebold–Mariano tests show that the hybrid significantly outperforms the standalone Random Forest under squared loss, while the absolute-loss comparison is only marginal. Across both assets, directional accuracy remains close to random chance, confirming the limited predictability of next-day price direction from lagged OHLCV features. Overall, the hybrid framework is most valuable when regime-specific dynamics are sufficiently distinct, offering improved forecasting performance and greater interpretability than a single global model.

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 $B\left(s\right)$, depending on the parent size s, and a daughter-size distribution $\kappa \left(z\right)$, depending on the normalized daughter-size fraction z. The unknown functions B and $\kappa$ 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 $\kappa$ 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.