Recovering Low-Order Nonlinear Dynamics of ENSO from NOAA Observations via Sparse Equation Discovery

This repository contains the data, code, and results associated with the study: "Recovering Low-Order Nonlinear Dynamics of ENSO from NOAA Observations via Sparse Equation Discovery" We apply sparse equation discovery techniques (WSINDy combined with Lasso regularization) to the Oceanic Niño Index (ONI), derived from NOAA observations, with the goal of identifying a low-order dynamical representation of ENSO variability directly from observational data. Our results show that a one-dimensional formulation fails to produce stable models under regularization, indicating that ENSO dynamics cannot be adequately described as a simple relaxation process. In contrast, a second-order state-space representation consistently reveals a robust and interpretable nonlinear structure of the form: d²x/dt² = -ax - b x³ This equation corresponds to a nonlinear oscillator with amplitude-limiting feedback, capturing the intrinsic dynamical backbone of ENSO variability. Beyond reproducing oscillatory behavior, the recovered model provides a key structural insight: deviations from the deterministic oscillator are not purely random, but instead reflect unresolved processes, external forcing, and multiscale interactions. In this sense, the model can be interpreted as a reduced-order projection of a higher-dimensional dynamical system, with residuals encoding additional dynamical structure. The methodology includes: Data preprocessing using Savitzky–Golay filtering and normalization Weak-form sparse regression (WSINDy) for noise-robust system identification Model selection via sparsity analysis across regularization parameters Out-of-sample validation (train/test split) Statistical robustness tests (bootstrap and surrogate analysis) Phase-space reconstruction and dynamical interpretation Numerical simulation and forward integration Spectral analysis comparing observed and simulated variability The recovered model reproduces a coherent oscillatory timescale consistent with ENSO variability, while differences in spectral structure indicate the influence of stochastic forcing, unresolved coupling, and multiscale dynamics. This work establishes a bridge between data-driven equation discovery and interpretable dynamical systems, demonstrating that meaningful nonlinear climate dynamics can be extracted directly from observations. The identified structure is robust and generalizes across different modes of climate variability (e.g., ENSO and PDO), connecting to broader efforts in interpretable, data-driven modeling of complex systems. Keywords: ENSO, ONI, nonlinear dynamics, sparse identification, SINDy, WSINDy, climate variability, dynamical systems, data-driven modeling