Mortality Prediction as Boundary Value Problem

Abstract

We present a mathematical framework for mortality prediction from discrete medical event sequences, formulating the problem as a boundary value problem on causal path space. Discrete event sequences lift to rough path space; signatures provide coordinate-free trajectory analysis. The transformer architecture emerges as computing weighted projections of path signatures, with Time2Vec temporal encoding providing spectral parameterization of the rough path lift. We establish connections to tropical geometry: ReLU networks compute tropical rational functions, and the decision boundary is a tropical hypersurface interpretable as the viscosity solution to a Hamilton-Jacobi equation on trajectory space. Mortality acts as absorbing boundary condition, with gradients propagating retroactively to reshape early event representations - the computational realization of ”boundary conditions determine interior.” The framework unifies perspectives from causal set theory (discrete causal ordering as primitive), rough path theory (coordinate-free trajectory invariants), tropical geometry (piecewise-linear structure from max-plus algebra), and viscosity solutions (weak solutions selected by vanishing diffusion). The architecture’s parameter efficiency is explained by alignment with the problem’s intrinsic mathematical structure: tropical decision boundaries, Lie algebraic dimension reduction, and binary boundary factorization