Linear inverse models approximate local dynamics near equilibrium using data correlation functions. They successfully describe many features of El Niño but fundamentally cannot reproduce the asymmetry between warm and cold phases. Nearby work has taken steps toward accounting for noise memory (Colored-LIM) and regression schemes with quadratic terms (XRO), which capture skew but rely on different mathematical techniques. This work follows the same vein: it generalizes the LIM framework itself, adding nonlinearity directly to the inverse problem, thereby expanding the class of models available for analyzing observed time series.
The authors extend the linear stochastic differential equation with quadratic and constant terms, keeping additive noise, either white or Ornstein-Uhlenbeck colored. For white noise (White-nLIM), the dynamics parameters are expressed in terms of the mean and correlation functions of second, third, and fourth order at zero lag, along with their first derivatives. For colored noise (Colored-nLIM), second derivatives and cross-correlations between noise and state are added, and the noise matrix is chosen by minimizing the Frobenius residual subject to a balance relation. Numerical derivatives are computed with forward finite differences, and the three-dimensional tensor of quadratic coefficients is packed into a matrix for ease of estimation.
On a two-dimensional synthetic system, both nLIM variants recover parameters with relative errors ranging from fractions of a percent to tens of percent depending on sample length: at Tf = 5000, errors for B drop to 5–8%, for A to 2–3%, and for C to 6–7%. On the stochastic Lorenz system with colored noise, Colored-nLIM with constrained parameter structure yields σ = 10.0191, ρ = 29.2046, β = −2.6808 against true values 10, 28, and 8/3, while without constraints and with data thinning, errors grow due to inaccurate derivatives. On real El Niño 3.4 data (SST and D20 from ORAS5 for 1979–2021), nLIM reproduces observed correlation functions and, crucially, gives skewed distributions: the Wasserstein distance p=1 for SST drops from 0.0949 for linear models to 0.0713 for White-nLIM and 0.0555 for Colored-nLIM.
This work shows that nonlinearity can be built into inverse modeling directly through high-order correlation functions, without resorting to regression and without losing the physical connection to noise. Unlike neighboring approaches that either remain linear or use regression logic, here a unified framework covers both white and colored noise and explicitly includes a constant term, which is important for anomalies with zero mean. This expands the toolkit for analyzing asymmetric variability regimes and provides a benchmark for comparison with machine learning.
The method can be generalized to polynomial dynamics of arbitrary order, although correlation functions of even higher orders and thus more data will be required. For Colored-nLIM, the automatic selection of the noise correlation time γ remains an open question: currently it is a hyperparameter, and tuning it requires long numerical simulations, but parallelization over γ makes the task feasible. Robust derivative computation schemes could also be developed to reduce sensitivity to sample resolution.
Climate modeling and El Niño forecasting, analysis of other asymmetric atmosphere and ocean regimes, and stochastic modeling of chaotic systems in physics and engineering.
Test nLIM on longer climate records and other indices, and compare predictive skill with XRO and modern neural network models. Develop a robust algorithm for selecting γ in Colored-nLIM.
The asymmetry between El Niño and La Niña remains one of the unsolved problems in climatology: linear stochastic models cannot reproduce it, and nonlinear inverse methods are only beginning to emerge. This work directly attacks that gap by offering a way to extract nonlinear dynamics from observed correlations without a priori symmetry assumptions.
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What the work does
The authors extend linear inverse models (LIMs) with quadratic terms in the stochastic differential equation, estimating parameters from correlation functions up to fourth order. The method is tested on synthetic systems and applied to El Niño indices, where it reproduces skewed distributions and improves the Wasserstein distance compared to linear models.
Where its strength is
The strength of this work is that it embeds nonlinearity directly into inverse modeling through high-order moments, preserving the noise physics (white or colored) and explicitly including a constant term, which provides a unified framework for analyzing asymmetric regimes.
Where it could go next
Worth trying
Why this may matter
Extending LIM to the nonlinear case opens the way to more accurate modeling of asymmetric climate phenomena and can serve as a benchmark for comparison with machine learning approaches, as well as for improving El Niño forecasts.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.51, against a usual 0.64. It looks like few have drilled nearby.
The minimum dimension for chaos in an autonomous differentiable system is three, so the authors first test the method on a two-dimensional system (where chaos is absent) and then on the three-dimensional Lorenz system.