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Understanding how pre-eruptive processes manifest in geophysical observables remains a central challenge in volcanology and volcanic hazard assessment. Among these observables, seismic tremor, a persistent ground vibration commonly recorded at active volcanoes, holds strong potential for eruption forecasting, yet its temporal evolution is notoriously difficult to interpret. Bridging tremor observations with eruption forecasting therefore requires computational frameworks that explicitly link tremor characteristics to the degree of volcanic unrest and the likelihood of eruption. Here, we present two complementary computational frameworks for eruption forecasting from continuous seismic tremor data that integrate physics-based forward modeling, inverse methods, and machine learning. Both approaches are tested using the 13 paroxysms of Shishaldin Volcano (Alaska) that occurred between July and November 2023. The first framework is physics-informed and relies on data assimilation to invert tremor observations and retrieve subsurface pressure evolution. It couples a physical model of tremor generation, rooted in multiphase gas accumulation and porous-media flow within the upper conduit, with genetic algorithm optimization and Monte Carlo simulations. This approach captures the effects of magma ascent, volatile exsolution, partial conduit sealing, and gas transport on transient tremor signals, revealing pressure increases of several MPa and a systematic rise in eruption probability hours before each paroxysm. The second framework is data-driven and applies pattern-recognition techniques to extract physically motivated seismic features (e.g., dominant frequency, amplitude, kurtosis, entropy), which are combined with a supervised machine-learning classifier (random forest) to estimate eruption probabilities. Despite their differing philosophies, both frameworks consistently relate pre-eruptive tremor evolution to probabilistic eruption forecasts. Together, these results demonstrate how computational approaches can enhance the interpretation of seismic tremor, provide quantitative insight into magma–volatile interactions, and advance eruption forecasting and volcanic hazard assessment strategies.