Learning the Limits of Discovery: Distinct Failure Signatures of SINDy, Neural ODEs, and PINNs Under Sparse and Noisy Observations

Abstract Real-world measurements of dynamical systems are often noisy, sparse, or incomplete. We evaluate SINDy, Neural ODEs, and physics-informed neural networks (PINNs) under controlled noise and observation-density grids on Lorenz, Van der Pol, and Duffing systems, using five inde- pendent corruption/subsampling realizations per condition. SINDy retains high recall (≥ 0.886) but loses precision as observations become sparse, with severity strongly affected by threshold and sampling geometry. Regular sampling and stricter thresholds recover much of this precision, while Ensemble-SINDy provides little improvement in the tested conditions. Neural ODE pre- diction error rises with noise on the non-chaotic systems. On Lorenz, a follow-up factorial test showed that apparent fixed-point collapse was largely caused by insufficient optimization rather than training-window length; longer windows required more iterations in the tested configura- tion. PINNs retain low parameter error on Duffing and on Lorenz outside a sparse-observation breakdown, but show training-budget-dependent instability on Van der Pol and strong seed- dependent variance on sparse Lorenz. A Hudson Bay lynx-hare case study shows a related SINDy structural failure in historical data. These results show how noise, sparsity, sampling geometry, and optimization interact to produce different failure signatures across dynamical- system learning formulations.