Title: Interpreting Neural Networks Through the Lens of Heat Flow. Abstract: Machine learning models are often developed in a way that prioritizes task-specific performance but defers the understanding of how they actually work. This is especially true nowadays for deep neural networks. In this paper, we step back and consider the basic problem of understanding a learned model represented as a smooth scalar-valued function. We introduce HeatFlow, a framework based upon the heat diffusion process for interpreting the multi-scale behavior of the model around a test point. At its core, our approach looks into the heat flow initialized at the function of interest, which generates a family of functions with increasing smoothness. By applying differential operators to these smoothed functions, summary statistics (i.e., explanations) characterizing the original model on different scales can be drawn. We place an emphasis on studying the heat flow on data manifold, where the model is trained and expected to be well behaved. Numeric approximation procedures for implementing the proposed method in practice are discussed and demonstrated on image recognition tasks.