Physics-Informed Neural Mortality: Embedding Biological Mortality Laws in Neural Differential Equations for Life-Table Closure and Longevity Uncertainty Quantification

Deep learning has become the default toolkit for mortality forecasting, yet essentially all existing models are purely data-driven: they fit well in-sample but can produce biologically implausible surfaces and extrapolate poorly to the oldest ages, and rarely deliver calibrated uncertainty. Classical parametric laws (Gompertz–Makeham, Kannisto) have the opposite profile—smooth and well-behaved under extrapolation, but rigid. We introduce Physics-Informed Neural Mortality (PINM), a continuous neural model of the log force of mortality whose training objective couples the Poisson likelihood of observed deaths with (i) a physics residual that softly imposes a mortality law in the age dimension, including a Kannisto-type deceleration penalty for old-age closure, and (ii) an ODE-structured, learned improvement field in the time dimension (a state-independent special case of a neural ODE), so that forecasts are produced by integrating the learned dynamics forward rather than by a separate time-series step. We give a compact mathematical foundation—existence of minimisers, well-posed forecast dynamics, constrained approximation, and consistency of the fitted rates—and evaluate the method candidly on controlled ground-truth simulation and on real data for six European populations (Eurostat). On the simulation, PINM leads at the older, annuity-relevant ages (RMSE 0.131 versus 0.143 for Lee–Carter) and retains substantially higher interval coverage under the specific misspecification scenario studied (44.8% versus 12.2% for Lee–Carter), though both remain materially below the nominal 95% level; in a held-out closure experiment on the same simulated surface, continuous neural surfaces extrapolate mortality from age 90 to age 110 with RMSE 0.196 against 0.301 for Lee–Carter completed with a Kannisto tail. On the six-population backtest (2010–2019), classical Lee–Carter and Renshaw–Haberman remain the most accurate forecasters overall (mean RMSE 0.123 versus 0.170 for PINM), with the neural surfaces competitive at older ages and on the longest series; PINM matches or improves on an unconstrained network of equal capacity in all six cases (outright in five). PINM’s contribution is a novel, unified differential-equation framework for fitting, structured forecasting, structured old-age closure, and empirically evaluated uncertainty quantification in one model, together with an explicit discussion of the method’s robustness and tail-estimation limitations.