# TinyUDE: Solver-Free Universal Differential Equations on Microcontrollers via Lie-Taylor Jet Matching

arXiv: https://arxiv.org/abs/2609.26972 · HTML: https://mlsanity.com/paper/2609.26972/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 58 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [MCUNet: Tiny Deep Learning on IoT Devices](https://mlsanity.com/paper/2007.10319/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [TensorFlow Lite Micro: Embedded Machine Learning on TinyML Systems](https://mlsanity.com/paper/2010.08678/) (2020)
- [On-Device Training Under 256KB Memory](https://mlsanity.com/paper/2206.15472/) (2022)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [MLPerf Tiny Benchmark](https://mlsanity.com/paper/2106.07597/) (2021)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [TinyOL: TinyML with Online-Learning on Microcontrollers](https://mlsanity.com/paper/2103.08295/) (2021)
- [Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows](https://mlsanity.com/paper/2005.13420/) (2020)
- [Numerical differentiation of noisy data: A unifying multi-objective optimization framework](https://mlsanity.com/paper/2009.01911/) (2020)
- [Multiple shooting for training neural differential equations on time series](https://mlsanity.com/paper/2109.06786/) (2021)

## Abstract (arXiv)

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.