Fig. 3: Explaining the fidelity of artificial neural networks (ANNs) from the perspective of El Niño-Southern Oscillation (ENSO) dynamics.

a, b Relationship between ANN performance and bias in the Bjerknes stability (BJ) index. DD (dynamic damping), TD (thermodynamic damping), ZA (zonal advective feedback), MA (meridional advective feedback), VA (vertical advective feedback), and TH (thermocline feedback) represent the six components of the BJ index. Compared to ACCESS-ESM1-5, the GISS-E2-1-H model, whose ANN achieves the highest performance on observational data, exhibits BJ index components closer to those of the SODA reanalysis. BJ bias is quantified as the root mean square error of the six BJ index components between each climate model and SODA reanalysis. ANN performance is measured by the correlation coefficient between ANN outputs and observations. High ANN performance is associated with a low BJ bias. c, d Relationship between ANN performance and modeled ENSO nonlinearity. Nonlinearity is quantified using the leading coefficient α, obtained by fitting a quadratic curve to the first and second principal components (PC2(t) = α[PC1(t)]² + β[PC1(t)] + γ), as determined by empirical orthogonal function (EOF) analysis of monthly SST anomalies in the tropical Pacific (1950–2014, 15°S-15°N and 140°E-80°W). Nonlinearity bias (NL bias) is defined as the absolute difference between the α estimated from each climate model and from three observational datasets. High ANN performance is associated with a close agreement with the observed nonlinearity. Together, these results confirm that ANNs achieve higher fidelity when trained on climate models that better replicate observed ENSO physics, including both linear stability and nonlinear dynamics.