Pranavanath Balamurali | Search arXiv

arXiv2026

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

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.

cs.LG↗

arXiv2026

PRISM-UDE: Physics-Regularized Iterative Symbolic Modeling of 3nm FinFETs via Universal Differential Equation

Compact transistor models are the mathematical backbone of circuit simulation. However, at advanced nodes such as 3nm, transport physics becomes too complex for traditional hand-derived equations to capture accurately. Purely data-driven neural surrogates, on the other hand, are numerically unstable inside circuit solvers and offer no physical insight into their own predictions. We introduce PRISM-UDE (Physics-Regularized Iterative Symbolic Modeling via Universal Differential Equations), a framework that embeds a small neural network inside a physics-based transistor model, using the network only to learn the transport behavior that the analytical baseline misses, rather than replacing the physics altogether. Once trained, this neural correction is distilled into a single, interpretable closed-form expression via symbolic regression, making the final model fully analytical and simulator-ready. Applied to a 3nm FinFET benchmark dataset, PRISM-UDE reduces prediction error more than sixfold (70.33% to 11.01%) relative to the standard physics-only baseline. The distilled expression preserves this accuracy almost exactly while eliminating the neural network entirely. We further validate the extracted expression directly inside a SPICE circuit simulator, confirming stable, physically consistent behavior under both static bias sweeps and dynamic switching conditions.

cs.ET↗