TinyUDE：基于Lie-Taylor射流匹配的微控制器上免求解器通用微分方程
TinyUDE: Solver-Free Universal Differential Equations on Microcontrollers via Lie-Taylor Jet Matching
- The University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出Lie-Taylor射流匹配的免求解器训练框架，直接在微控制器上拟合UDEs，通过噪声自适应机制在精度上匹配或超越传统基线，实现实时设备上训练。
AI中文摘要：
训练通用微分方程（UDEs）传统上依赖于通过数值ODE求解器进行反向传播，这产生的内存占用远超边缘微控制器的能力。我们提出Lie-Taylor射流匹配，一种免求解器的训练框架，直接将混合向量场拟合到观测系统状态的一阶和二阶时间导数上。这些导数（截断的Lie-Taylor射流）通过Savitzky-Golay滤波在线估计，无需自动微分软件即可获得完全解析的梯度。我们评估了消除求解器是否会在与常规基线（固定步长RK4积分、多重打靶、精确离散伴随、Adam）共享相同动力学、噪声模型、网络架构和指标的情况下损害精度。虽然朴素导数匹配在传感器噪声下性能下降，但我们的噪声自适应机制缩小并逆转了这一差距：全速率相移采样、储备缓冲器、余弦退火优化与权重平均、设备上噪声估计以及多项式失配质量门控。在阻尼摆和混沌双摆上，我们的方法在匹配数据窗口下达到或超过基线精度，并恢复未建模的阻尼系数。在0%至5%的噪声水平下，它在108 kB静态内存内实现了几何平均相对场误差为基线的0.65倍，而求解器磁带则需要数兆字节。在ESP32微控制器上，设备上运行达到场误差0.0020，并在61.3 kB静态内存和每次更新7.24毫秒（25 Hz下18.1%占空比）内将阻尼系数恢复至c = 0.400（真实值0.400），证实了无需数值求解器的实时设备上训练是可行的。
英文摘要：
Training Universal Differential Equations (UDEs) traditionally relies on backpropagating through numerical ODE solvers, creating memory footprints far exceeding the capabilities of edge microcontrollers. We present Lie-Taylor jet matching, a solver-free training framework that fits a hybrid vector field directly to the first and second time-derivatives of observed system states. These derivatives, the truncated Lie-Taylor jet, are estimated online via Savitzky-Golay filtering, yielding fully analytic gradients without automatic differentiation software. We evaluate whether eliminating the solver compromises accuracy against a conventional baseline (fixed-step RK4 integration, multiple shooting, exact discrete adjoints, Adam) sharing identical dynamics, noise models, network architectures, and metrics. While naive derivative matching degrades under sensor noise, our noise-adaptive mechanisms close and reverse this gap: full-rate phase-shifted sampling, a reservoir buffer, cosine-annealed optimization with weight averaging, on-device noise estimation, and polynomial-misfit quality gating. On a damped pendulum and chaotic double pendulum, our method matches or exceeds baseline accuracy at matched data windows and recovers unmodeled damping coefficients. Across noise levels from 0% to 5%, it attains a geometric-mean relative field error of 0.65x that of the baseline within 108 kB of static memory, compared with megabytes of solver tape. On an ESP32 microcontroller, the on-device run reaches a field error of 0.0020 and recovers the damping coefficient to c = 0.400 (true 0.400) within 61.3 kB of static memory and 7.24 ms per update (18.1% duty cycle at 25 Hz), confirming real-time on-device training is feasible without a numerical solver.
