The two dominant players in North Atlantic climate are the Atlantic Multidecadal Variability, or AMV, a basin-wide fluctuation in sea surface temperatures, and the North Atlantic Oscillation, or NAO, the pressure seesaw between the Azores high and the Icelandic low. Both are major sources of predictability on seasonal to decadal timescales, influencing Atlantic hurricane frequency, shifts in the Intertropical Convergence Zone, drought in the Sahel, and summer climate across Europe and North America. Yet the physical processes and feedback mechanisms linking the oceanic AMV to the atmospheric NAO have remained fiercely debated, with proposed drivers ranging from internal ocean circulation to anthropogenic aerosols, greenhouse gases and stochastic atmospheric forcing. What has been missing, the researchers argue, is a compact mathematical description that captures how these variables actually interact over decades.

To find one, the team employed the Sparse Identification of Nonlinear Dynamics algorithm, known as SINDy, a sparse regression method that learns governing equations directly from time-series data. The researchers trained the algorithm on monthly indices of the AMV, the NAO and North Atlantic precipitation anomalies derived from the ERA5 reanalysis dataset spanning January 1950 to December 2022. A twelve-month running average was applied to filter out high-frequency noise, and the candidate library of functions included constant, linear and quadratic polynomial terms, yielding thirty possible terms across three equations. A LASSO optimiser with sparsity thresholding forced the regression to retain only the most important terms, while a bootstrapping ensemble of one thousand resampled datasets ensured the identified coefficients were robust. Validation and testing used independent segments of the ERA20C reanalysis from 1900 to 1949, data the models had never seen during training.

The procedure generated 344 unique candidate models of varying complexity, and the team faced a subtle selection problem. Simpler models with thirteen or fewer terms either collapsed onto stable fixed points, converged on rigid limit cycles, or blew up entirely, none of which resembles the restless variability of the real North Atlantic. Models with fourteen to twenty terms were stable but produced unnaturally sparse power spectra with delta-like peaks, unlike the broad-band spectra seen in observations. Only models with twenty-one or more terms exhibited chaotic attractors and physically realistic variability. The researchers ultimately selected a twenty-six-term model sitting on the Pareto front, the optimal trade-off between accuracy and complexity, whose fifty-thousand-year simulated trajectories reproduce the irregular, Lorenz-like character of the observed climate system.

Remarkably, the decadal timescales embedded in these equations were never explicitly taught to the model. Because SINDy fits instantaneous tendencies, the emergence of a pronounced quasi-periodic twenty-year oscillation across all three variables is an entirely emergent property. The model’s power spectra show strong peaks near twenty years and thirty-five years, matching significant signals in the observed NAO and precipitation records, along with weaker peaks near 13.5 and 15.5 years. The team interprets the twenty-year mode as a damped oscillatory ocean mode, set by internal ocean dynamics but excited by decadal NAO forcing, consistent with earlier inverse modelling studies that identified oscillatory eigenmodes of North Atlantic sea surface temperature at periods of roughly twenty and thirty-seven years.

The most striking discovery lies in the structure of the equations themselves. Cross-terms multiplying precipitation with either the AMV or the NAO, written as AP and NP products, appear among the largest and most robust coefficients across the entire ensemble of 344 candidate models, persisting even in the sparsest formulations. Because precipitation serves as a proxy for both latent heat release in the atmosphere and freshwater fluxes into the ocean, these terms point to diabatic processes acting as a dynamical bridge between ocean and atmosphere. In the AMV equation, the negative NP term suggests that heavy rainfall over the Labrador Sea region, combined with a positive NAO, freshens the surface ocean, increases stratification, weakens deep convection and thereby dampens the overturning circulation and the AMV itself.

Feedbacks onto the atmosphere appear equally important. In the NAO equation, the positive NP term indicates that enhanced convective precipitation near the southern node of the NAO reinforces a negative NAO state through diabatic heating that lowers surface pressure, echoing recent causal studies of oceanic feedbacks onto the winter NAO. Conversely, the model suggests that the positive feedback of North Atlantic sea surface temperature onto the NAO, in which the familiar SST tripole strengthens the meridional temperature gradient and the zonal winds, is strongest precisely when the precipitation pattern favours a positive NAO state. In other words, the response of the atmosphere to the ocean appears to be conditioned on the sign of rainfall anomalies, a state-dependent coupling that conventional linear analyses would miss entirely.

The predictive credentials of the twenty-six-term model are substantial. Initialised every six months from observed data and integrated forward with a fourth-order Runge-Kutta scheme, the model achieves anomaly correlation coefficients of 0.86 for the AMV and 0.54 for the NAO at one-year lead times on training data, and 0.93 and 0.52 respectively on the unseen early-twentieth-century test data. Most impressively, the model shows significant skill in predicting North Atlantic precipitation anomalies out to a decade ahead, with correlations reaching 0.7 at one-year leads on test data. The authors attribute this precipitation skill to the fact that the decadal precipitation index varies in phase with the damped twenty-year ocean oscillation, so that slow, predictable ocean responses to NAO forcing propagate directly into rainfall patterns over the ocean and adjacent continents.

The researchers are careful to note the limits of their framework. The equations encode dynamical couplings but do not by themselves distinguish cause from effect, and the mechanistic pathways they propose are supported by consistency with existing theory rather than uniquely proven. The characteristic fifty-to-seventy-year signal of the AMV is not well captured, likely because the training record is too short, and external influences such as volcanic eruptions and lunar cycles were not considered. The authors also acknowledge that stochastic processes present in the real system are only implicitly represented through the nonlinear terms, and they propose comparing their models against Linear Inverse Models and coupled climate simulations as future work.

Even so, the study represents a compelling demonstration that machine learning can do more than forecast: it can illuminate physics. As the journal’s editorial statement observes, the paper exemplifies how equation discovery transforms AI from a black box into a hypothesis-generating scientific instrument. If the identified feedbacks hold up under scrutiny, the implications extend beyond academic understanding. Correctly initialised versions of these low-order models could sharpen decadal forecasts of rainfall over the North Atlantic and neighbouring regions such as Europe and North America, offering societies a rare glimpse of climate conditions years in advance. In an era when decadal prediction remains one of climate science’s hardest challenges, three short equations learned by an algorithm may prove to be an unexpectedly powerful key.

Subject of Research: Data-driven discovery of coupled ocean-atmosphere equations governing decadal North Atlantic climate variability

Article Title: New insights into decadal climate variability in the North Atlantic revealed by data-driven dynamical models

Article References: Nicoll, A. J., Christensen, H. M., Huntingford, C., & Smith, D. (2026). New insights into decadal climate variability in the North Atlantic revealed by data-driven dynamical models. Earth System Dynamics, 17(4), 1061-1079. https://doi.org/10.5194/esd-17-1061-2026

Image Credits: AI Generated

DOI: 10.5194/esd-17-1061-2026

Keywords: North Atlantic, Atlantic Multidecadal Variability, North Atlantic Oscillation, equation discovery, SINDy, machine learning, decadal prediction, precipitation feedbacks, climate variability, ocean-atmosphere coupling, ERA5 reanalysis, latent heat

The method, called Equation Discovery for Classification, or EDC, was developed by Guus Toussaint and Arno Knobbe of Leiden University. It extends a research tradition that has long flourished in regression: symbolic regression, the automated search for analytical formulas that fit numerical data. Theauthors of the study point out that equation discovery has historically been applied almost exclusively to problems where the target is a continuous number, such as recovering physical laws like the relationship between the pressure, volume, and temperature of a gas. Their contribution is to redirect that machinery toward binary classification, where the goal is to separate two classes cleanly and explainably.

The core idea is elegantly simple in concept. Rather than learning an opaque scoring function, EDC searches for a concise mathematical expression f(x) and a threshold theta, such that a data point is assigned to the positive class whenever f(x) meets or exceeds theta. For a linear equation, this is essentially the geometry underlying logistic regression or a linear support vector machine: a hyperplane slicing the feature space into two halves. What sets EDC apart is that the search is not confined to straight lines. By allowing nonlinear building blocks such as products of features and exponential terms, the algorithm can trace curved, interaction-driven boundaries while still producing an expression short enough for a domain expert to read, critique, and even correct by hand.

Technically, the framework rests on two pillars: a structured search and a dedicated numerical optimizer. The search space of candidate equations is defined by a configurable context-free grammar, a design choice inherited from classic work on declarative bias in equation discovery. The grammar constrains equations to sums of simple summands, including linear terms, products of two features, and exponential expressions, each parameterized by constants. Crucially, the grammar is redundancy-aware: constructions that would produce syntactically different but semantically identical equations are pruned in advance, since, for example, subtraction between summands is unnecessary when a constant coefficient can simply be negative. Traversing this space exhaustively is impossible for all but the smallest problems, so the algorithm employs beam search, iteratively refining the most promising candidate equations level by level while keeping only a fixed number of survivors at each stage.

Fitting the constants inside each candidate equation proved surprisingly subtle. Although every equation in the grammar is differentiable, gradient descent performed poorly in practice, largely because of the exponential terms that pepper the search space. The authors instead adopted a tailored hill-climbing procedure that first samples a large pool of random constant configurations, then concentrates its remaining budget on refining the best ones. In a systematic comparison against off-the-shelf optimizers from the SciPy library, including Powell, Cobyqa, Cobyla, Nelder-Mead, and stochastic gradient descent, this hill climber achieved the highest mean area under the ROC curve across one hundred randomly generated test problems. Perhaps most strikingly, a simple random-sampling baseline already reached a mean AUC of 0.9974 with a budget of one thousand evaluations, revealing that the inner optimization problem is more tractable than one might fear.

The experiments on synthetic data produced one of the study’s most intriguing findings. When Gaussian noise was injected into datasets whose generating decision boundaries were known, EDC did not merely match the original boundary; beyond a certain noise level it outperformed it. The explanation is a phenomenon the authors describe carefully: noise pushes data points near a curved boundary across it, and concave sections of that boundary collect more stray points than they lose, causing the effective boundary embedded in the data to drift away from its original position and gradually straighten. EDC, fitting the data rather than the hidden formula, tracks this shifted, smoothed boundary, achieving higher AUC than the very equation that generated the data. As noise grew from negligible to substantial across seventeen hundred artificial datasets, even a restricted linear version of EDC eventually beat the original nonlinear boundary.

The framework also proved capable of reconstructing notoriously difficult structures, including XOR-like and interaction-driven boundaries, and of fitting data produced by Gaussian clusters where no explicit target equation exists at all. On these cluster-based problems, which the authors describe as closest to real-world conditions among their artificial experiments, EDC outperformed existing symbolic classification approaches while landing near state-of-the-art black-box methods such as random forests, multi-layer perceptrons, and radial-basis-function support vector machines, all of which posted mean AUC values around 0.97.

Real-world benchmarks reinforced the pattern. Across nine binary classification datasets from the UCI repository, spanning tasks from banknote authentication to income prediction, EDC achieved a higher AUC than every competing equation-discovery-based classifier on every dataset, and beat simple decision trees across the board. A critical distance analysis showed that random forests, neural networks, and SVMs did not statistically significantly outperform EDC, even though those black-box methods won on several datasets, particularly ionosphere and sonar, where feature-class relationships appear to lie outside EDC’s current set of building blocks. On the Adult income dataset, the discovered equation offered a vivid demonstration of interpretability in action: one term acted as a penalty on years of education only when the individual appeared as a child in the household, while an exponential term over marital status effectively functioned as an if-else statement, adding roughly 28,586 to the score for married individuals living with a spouse and a negligible 8 otherwise.

The method’s main weakness is computational cost. In its default configuration, with a search depth of six and a beam width of ten, EDC takes dramatically longer than the sub-second runtimes of conventional classifiers, largely because pairwise interaction terms grow quadratically with the number of features and because one-hot encoding of categorical variables inflates the feature count. The authors show, however, that the expense is largely optional. Restricting the search depth, narrowing the beam, or replacing interaction terms with simple quadratic terms produced speed-ups of up to a factor of fifty at only a marginal loss of accuracy, with no statistically significant difference in performance across the benchmark suite. The authors also note that the framework is not limited to two classes: standard one-versus-rest schemes can extend EDC to multi-class problems, with each of the resulting equations explaining when a particular class label prevails.

The work arrives amid a broader movement toward explainable machine learning, driven by domains such as medicine, finance, and criminal justice where decisions carry real consequences and regulators increasingly demand justification. EDC offers a principled bridge between symbolic regression and classification, delivering models that are simultaneously compact, expressive, and auditable. While it may not dethrone random forests or deep networks on raw accuracy, it demonstrates that the gap is small enough, and the interpretability dividend large enough, that equations once again deserve a seat at the machine learning table. The authors have released their code, experimental setup, and datasets openly so that the entire study can be reproduced and the grammar extended by practitioners in new domains.

Subject of Research: Equation discovery for binary classification using interpretable symbolic decision boundaries

Article Title: Equation Discovery for Classification: Finding Interpretable Symbolic Specifications of the Decision Boundary

Article References: Toussaint, G., & Knobbe, A. (2026). Equation Discovery for Classification: Finding Interpretable Symbolic Specifications of the Decision Boundary. Machine Learning, 115(9), Article 215. https://doi.org/10.1007/s10994-026-07156-1

Image Credits: AI Generated

DOI: 10.1007/s10994-026-07156-1

Keywords: equation discovery, binary classification, symbolic regression, decision boundary, interpretable machine learning, beam search, symbolic classification, explainable AI, context-free grammar, UCI datasets, machine learning, Equation