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

翻译：随着功率计在自行车竞赛中的普及，骑手能精确追踪比赛全程的功率输出，制定最优功率输出策略在竞技自行车领域变得日益重要。这需要综合考虑赛道、天气及个体骑手能力等多重因素。我们提出疲劳动力学与运动学微分方程模型以模拟此类策略表现，并开发创新优化算法寻找最优解。疲劳模型将骑手功率曲线（通过全功率时段模型拟合功率数据获得）转化为微分方程，用以界定可行功率输出策略的边界。运动学模型通过受力分析，结合功率输出解算骑手速度与位置微分方程组。借助包含赛道坡度与风速的实况赛道数据，该模型可精准预测给定功率策略下的完赛时间。为降低优化计算复杂度，我们根据坡度变化将赛道分段并离散化功率等级。由于策略空间规模庞大，我们通过向量化微分方程模型实现多模拟并行数值积分，并开发并行化蒙特卡洛树探索算法。该算法性能高效，时间复杂度为$O(ab\sqrt{n})$，空间复杂度为$O(n)$（$n$为每决策点的模拟次数，$a$为赛道分段数，$b$为离散功率等级数）。我们展示了针对不同赛道与选手的优化结果——以菲利波·甘纳在2020东京奥运会的数据为例，模型预测时间与实际时间误差仅18%，验证了模型有效性。