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Predicting time-to-event outcomes when event times are interval censored is challenging because the exact event time is unobserved. Many existing survival analysis approaches for interval-censored data rely on strong model assumptions or cannot handle high-dimensional predictors. We develop ICODEN, an ordinary differential equation-based neural network for interval-censored data that models the hazard function through deep neural networks and obtains the cumulative hazard by solving an ordinary differential equation. ICODEN does not require the proportional hazards assumption or a prespecified parametric form for the hazard function, thereby permitting flexible survival modeling. Across simulation settings with proportional or non-proportional hazards and both linear and nonlinear covariate effects, ICODEN consistently achieves satisfactory predictive accuracy and remains stable as the number of predictors increases. Applications to data from multiple phases of the Alzheimer's Disease Neuroimaging Initiative (ADNI) and to two Age-Related Eye Disease Studies (AREDS and AREDS2) for age-related macular degeneration (AMD) demonstrate ICODEN's robust prediction performance. In both applications, predicting time-to-AD or time-to-late AMD, ICODEN effectively uses hundreds to more than 1,000 SNPs and supports data-driven subgroup identification with differential progression risk profiles. These results establish ICODEN as a practical assumption-lean tool for prediction with interval-censored survival data in high-dimensional biomedical settings.