中文赛题：智能手机电池消耗建模智能手机（Smartphones）是现代生活中不可或缺的工具，然而其电池行为往往显得不可预测。有时手机可以续航一整天；而有时在午餐前电量就会迅速耗尽。虽然有些用户将此归因于“高强度使用”，但电池耗尽的真正驱动因素要复杂得多。
功耗（Power consumption）取决于屏幕尺寸与亮度、处理器负载（Processor Load）、网络活动以及即使设备看似闲置时仍在消耗能量的后台应用程序之间的相互作用。环境条件（如温度）进一步使问题复杂化：某些电池在寒冷天气下会失去有效容量，而在持续高强度使用下可能会过热。此外，电池的行为还受其历史记录以及在其使用寿命期间的充电方式的影响。
你的任务是开发一个智能手机电池的连续时间数学模型（continuous-time mathematical model），该模型需在现实使用条件下返回作为时间函数的荷电状态（SOC）。这将用于预测在不同条件下的放空时间（Time-to-Empty）。你应该假设手机使用的是锂离子电池。
要求：
连续时间模型：开发一个模型，使用连续时间方程或方程组来表示荷电状态（SOC）。
你可以从最合理的电池消耗简单描述开始，然后将其扩展以包含其他影响因素，如屏幕使用、处理器负载、网络连接、GPS 使用和其他后台任务。
数据作为支持，而非替代品：你可以收集或使用数据进行参数估计和验证。如果公开数据集有限，你可以使用已发表的测量数据或规格说明（需正确引用），前提是参数必须经过明确论证并验证其合理性。
重要提示：仅基于离散曲线拟合、时间步长回归或黑盒机器学习且缺乏明确连续时间模型的项目，不满足本题要求。所有使用的数据必须记录良好且可免费获得，数据必须在开放许可下免费使用。
放空时间预测：使用你的模型计算或近似在各种初始电量水平和使用场景下的放空时间（Time-to-Empty）。
将预测结果与观察到的或合理的行为进行比较，量化不确定性，并确定模型表现良好或不佳的地方。
展示你的模型如何解释这些结果中的差异，并确定每种情况下电池快速耗尽的具体驱动因素。
哪些活动 or 条件会导致电池寿命最大程度的减少？哪些因素对模型的改变出人意料地小？
灵敏度与假设：检查在改变建模假设、参数值和使用模式波动后，你的预测会发生怎样的变化。
建议：将你的发现转化为给手机用户的切实建议。
例如，哪些用户行为（如降低亮度、禁用后台任务或切换网络模式）能最大程度地提高电池寿命？
操作系统如何根据你模型的见解实施更有效的省电策略？
考虑电池老化如何降低有效容量，或者你的建模框架如何推广到其他便携式设备。
您的报告应包含：
对模型和控制方程的清晰描述。
设计选择背后的假设和理由。
参数估计方法和验证结果。
对优势、局限性和可能的扩展的讨论。
一份行政摘要（Executive Summary），重点介绍主要结果、见解和建议。
重要提示：您的模型必须建立在明确定义的物理或机械推理之上；与电池行为的明确连续时间描述脱节的离散曲线拟合或其他数学形式将无法满足要求。仅依赖离散曲线拟合或统计回归而没有明确制定的连续时间模型的项目将不满足本题要求。
您的 PDF 解决方案总页数不超过 25 页，应包括：
一页摘要页。
目录。
完整的解决方案。
文中引用和参考文献列表。
AI 使用报告（如果使用了 AI，该部分不计入 25 页限制）。
术语表
Smartphone (智能手机)：一种结合了传统手机功能与先进计算能力的移动设备。
Power Consumption (功耗)：设备从电池或电源使用电能的速率。
Processor Load (处理器负载)：处理器在给定时刻正在完成的实际工作量。
State of Charge / SOC (荷电状态)：衡量电池相对于其满容量剩余多少能量的指标，以百分比表示。
Time-to-Empty (放空时间)：电池完全放电前的估计剩余时间。
Problem: Modeling Smartphone Battery Drain
Smartphonesare indispensable tools in modern life, yet their battery behavior often seems unpredictable. On some days a phone may last the whole day; on other days it drains rapidly before lunch. Although some users attribute this to "heavy use," the true drivers of battery depletion are more complex.
Power consumptiondepends on the interplay of screen size and brightness,processor load, network activity, and background applications that continue drawing energy even when the device appears idle. Environmental conditions such as temperature further complicate matters: some batteries lose effective capacity in cold weather and may overheat under sustained heavy use. A battery's behavior is also influenced by its history and how it has been charged during its lifetime.
Your task is to develop acontinuous-time mathematical modelof a smartphone's battery that returns thestate of charge (SOC)as a function of time under realistic usage conditions. This will be used to predict the remainingtime-to-emptyunder different conditions. You should assume that the phone has a lithium-ion battery.
Requirements:
Continuous-Time Model:Develop a model to represent the state of charge using a continuous-time equation or system of equations.
You may want to begin with the simplest reasonable description of battery drain and then extend it to incorporate additional contributors such as screen usage, processor load, network connections, GPS usage, and other background tasks.
Data as support, not substitute:You may collect or use data for parameter estimation and validation. If open datasets are limited, you may use published measurements or specifications (with proper citation), provided parameters are clearly justified and validated for plausibility.
Constraint:However, projects based solely on discrete curve fitting, time-step regression, or black-box machine learning without an explicit continuous-time model will not satisfy this problem's requirements. All data used must be well documented and freely available, and the data must be free for use under an open license.
Time-to-Empty predictions:Use your model to compute or approximate thetime-to-emptyunder various initial charge levels and usage scenarios.
Compare predictions to observed or plausible behavior, quantify uncertainty, and identify where the model performs well or poorly.
Show how your model explains differences in these outcomes and identify the specific drivers of rapid battery drain in each case.
Which activities or conditions produce the greatest reductions in battery life? Which ones change the model surprisingly little?
Sensitivity and Assumptions:Examine how your predictions vary after making changes in your modeling assumptions, parameter values, and fluctuations in usage patterns.
Recommendations:Translate your findings into practical recommendations for a cellphone user.
For example, which user behaviors such as reducing brightness, disabling background tasks, or switching network modes—yield the largest improvements in battery life?
How might an operating system implement more effective power-saving strategies based on insights from your model?
Consider how battery aging reduces effective capacity or how your modeling framework could generalize to other portable devices.
Your report should present:
A clear description of your model and governing equations.
The assumptions and rationale behind your design choices.
Parameter estimation methods and validation results.
A discussion of strengths, limitations, and possible extensions.
An executive-style summary highlighting main results, insights, and recommendations.
Important:Your model must be grounded in clearly defined physical or mechanical reasoning; discrete curve fitting or other mathematical forms that are disconnected from an explicit continuous-time description of battery behavior will not satisfy the requirements. Projects that rely solely on discrete curve fitting or statistical regression without a clearly formulated continuous-time model will not satisfy the requirements of this problem.
Submission Guidelines:
Your PDF solution of no more than 25 total pages should include:
One-page Summary Sheet.
Table of Contents.
Your complete solution.
In-text Citations and A Reference List.
AI Use Report (If used does not count toward the 25-page limit.)
Glossary
Smartphone: is a mobile device that combines the functionality of a traditional cell phone with advanced computing capabilities.
Power Consumption: the rate at which a device uses electrical energy from its battery or power source.
Processor Load: the actual amount of work being done by the processor at a given moment.
State of Charge (SOC): a measure of how much energy remains in a battery compared to its full capacity, expressed as a percentage.
Time-to-Empty: the estimated amount of time remaining before a battery is completely discharged.
