ABSTRACT Four-panel flowchart summarising the study methodology. Panel (a): Data and event extraction — monthly precipitation is transformed into SPI-γ series at 460 stations, and dry-spell onsets are identified as binary threshold-crossing events. Panel (b): Tempered Fractional Hawkes Process (TFHP) – the conditional intensity is defined with a tempered power-law kernel parameterised by baseline hazard μ, excitation strength κ, fractional order α, and tempering scale τm. Panel (c): Physics-informed inference via tempered Caputo duality – the TFHP intensity satisfies an equivalent fractional integro-differential equation, which is enforced as a physics constraint in a neural network trained with a composite loss. Panel (d): Results, uncertainty quantification, and added value – reconstructed intensity trajectories with ensemble uncertainty bands, spatial parameter fields, and benchmarking against renewal and Markov baselines. Drought risk is governed not only by accumulated precipitation deficits but also by the temporal organisation of dry conditions, including the timing, clustering, and persistence of dry spells. This study introduces a physics-informed stochastic framework to model drought onset dynamics as event-time processes across the heterogeneous hydroclimate of Chile. Daily precipitation records from 460 locations covering the period 1980–2022 are transformed into monthly standardised precipitation index series, from which dry-spell onsets are identified at multiple accumulation scales. Onset sequences are modelled using a self-exciting point-process formulation with a tempered power-law memory kernel, allowing the representation of baseline transition rates, short-term clustering, and long-range persistence with a finite effective horizon. A multi-station physics-informed neural architecture jointly infers spatially coherent parameter fields while enforcing the underlying fractional integro-differential dynamics, ensuring stable and interpretable inference across heterogeneous event densities. The results reveal marked regional contrasts in drought dynamics, with stronger memory and clustering in Mediterranean and transition zones, and weaker long-memory signatures in hyper-arid and humid regimes. By explicitly modelling drought onset timing rather than aggregated deficits, the proposed framework provides an interpretable and scalable alternative to index-based diagnostics, with direct relevance for hydroinformatics applications drought monitoring, spatial comparison, and probabilistic scenario analysis.
Mauricio Herrera-Marín (Mon,) studied this question.