Ensemble Extreme Learning Machines for Uncertainty Quantification in Ordinary Differential Equations

Abstract

This paper proposes an uncertainty-quantified surrogate framework for ordinary differential equations (ODEs) based on ensembles of extreme learning machines (ELMs). The method constructs M=100 independently randomized ELM surrogates for a given ODE trajectory, uses the ensemble mean as the predictor, and defines a pointwise ensemble spread as a preliminary uncertainty measure. To obtain statistically valid prediction intervals, a split-conformal calibration procedure is applied to the ensemble spread, yielding finite-sample marginal coverage under exchangeability while preserving computational efficiency, since each ELM is trained via a single ridge-regression solve. Theoretical results establish exact satisfaction of the prescribed initial condition, stability of the ensemble mean and variance with respect to perturbations in the training data, and almost-sure convergence of the empirical ensemble variance to the corresponding random-feature prediction variance. The hidden-layer sampling hypothesis is made explicit: the experiments use bounded hyperbolic-tangent features with independent uniform draws as the default and independent normal draws in sensitivity tests, both of which satisfy the finite-moment assumptions required by the convergence theorem. Comparisons with capacity-matched Bayesian random-feature neural surrogates and Monte Carlo dropout clarify differences in uncertainty representation. Numerical experiments on exponential, logistic, and damped oscillator dynamics demonstrate accurate reconstruction and calibrated uncertainty quantification in sparse and noisy regimes. Additional ablation studies quantify the effect of the denominator safeguard, calibration-sample size, ensemble size, training time, noise level, and hidden-parameter distribution.