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With the advent of power-meters allowing cyclists to precisely track theirpower outputs throughout the duration of a race, devising optimal power outputstrategies for races has become increasingly important in competitive cycling.To do so, the track, weather, and individual cyclist's abilities must all beconsidered. We propose differential equation models of fatigue and kinematicsto simulate the performance of such strategies, and an innovative optimizationalgorithm to find the optimal strategy.  Our model for fatigue translates a cyclist's power curve (obtained by fittingthe Omni-Power Duration Model to power curve data) into a differential equationto capture which power output strategies are feasible. Our kinematics modelcalculates the forces on the rider, and with power output models the cyclist'svelocity and position via a system of differential equations. Using track data,including the slope of the track and velocity of the wind, the model accuratelycomputes race times given a power output strategy on the exact track beingraced.  To make power strategy optimization computationally tractable, we split thetrack into segments based on changes in slope and discretize the power outputlevels. As the space of possible strategies is large, we vectorize thedifferential equation model for efficient numerical integration of manysimulations at once and develop a parallelized Tree Exploration withMonte-Carlo Evaluation algorithm. The algorithm is efficient, running in$O(ab\sqrt{n})$ time and $O(n)$ space where $n$ is the number of simulationsdone for each choice, $a$ is the number of segments, and $b$ is the number ofdiscrete power output levels.  We present results of this optimization for several different tracks andathletes. As an example, the model's time for Filippo Ganna in Tokyo 2020differs from his real time by just 18%, supporting our model's efficacy.

Pedaling, Fast and Slow: The Race Towards an Optimized Power Strategy