Title: Assumption-Free Survival Analysis Under Local Smoothness Prior. Abstract: Survival analysis appears in various fields such as medicine, economics, engineering, and business. Due to the difficulty of integration that naturally arises in continuous-time modeling, previous works either made a strong assumption or discretized the time domain, thus limits their practical usages. In this paper, we propose assumption-free survival analysis, which models continuous-time survival function without any assumption. Our model obtains an assumption-free survival function by integrating an assumption-free hazard function using Neural Ordinary Differential Equations. Inspired by smoothness prior from semi-supervised learning literature, we further propose a regularizer that encourages the survival function to be locally smooth by minimizing the variation of the survival function in the covariate space. We found this regularizer increases the predictive power of the survival function as it propagates high-quality local information to the neighborhoods of data points. Experimental results on three public benchmarks show that our approach has better predictive power and is well-calibrated compared to strong baselines. Moreover, the proposed regularizer is superior to global regularizers and insensitive to hyperparameters.