Cumulative hazard functions using monotonic neural networks in transformation models for right-censored data

Deep Cox-type survival models have been widely used to extend classical survival analysis by capturing nonlinear covariate effects and complex risk patterns. However, many existing deep Cox-type models retain the proportional hazards structure and use the Cox partial likelihood to estimate covariate-dependent relative-risk scores. Because this procedure leaves the baseline cumulative hazard unspecified, the fitted models do not directly yield individualized survival distributions. To address these limitations, we developed a neural network-based cumulative hazard modeling framework for continuous-time right-censored survival data under transformation models. We proposed DeepMonoCum, a deep transformation survival model that parameterizes the conditional cumulative hazard function through a transformation of a monotone neural-network baseline cumulative hazard and a neural-network covariate effect. The monotone neural network was used to ensure that the estimated baseline cumulative hazard function is nonnegative and nondecreasing over time. The corresponding baseline hazard function was obtained through automatic differentiation. The proposed method was evaluated through simulation studies and analyses of publicly available right-censored survival datasets. Predictive performance was assessed using the concordance index, integrated Brier score, integrated negative log-likelihood and Calibration curve. In simulation studies, DeepMonoCum achieved estimation accuracy comparable to the neural frailty model overall and showed lower relative error in several settings across different transformation structures, sample sizes, and censoring rates. Across four real-world survival datasets, DeepMonoCum performed competitively against established survival models, achieving the best or second-best integrated Brier score and generally attaining C-index and INBLL values close to those of the best-performing methods. Directly modeling the cumulative hazard function with monotonicity constraints provides a flexible likelihood-based approach for continuous-time survival analysis. The proposed DeepMonoCum method yields valid individualized survival functions and avoids repeated numerical integration of neural network-based hazard functions during likelihood evaluation. The empirical results suggest that the method is a competitive alternative for right-censored biomedical survival prediction.