Inference on unknown quantities in dynamical systems via observational data is essential for providing meaningful insight, furnishing accurate predictions, enabling robust control, and establishing appropriate designs for future experiments. Merging mathematical theory with empirical measurements in a statistically coherent way is critical and challenges abound, e.g., ill-posedness of the parameter estimation problem, proper regularization and incorporation of prior knowledge, and computational limitations. To address these issues, we propose a new method for learning parameterized dynamical systems from data. We first customize and fit a surrogate stochastic process directly to observational data, front-loading with statistical learning to respect prior knowledge (e.g., smoothness), cope with challenging data features like heteroskedasticity, heavy tails, and censoring. Then, samples of the stochastic process are used as "surrogate data" and point estimates are computed via ordinary point estimation methods in a modular fashion. Attractive features of this two-step approach include modularity and trivial parallelizability. We demonstrate its advantages on a predator-prey simulation study and on a real-world application involving within-host influenza virus infection data paired with a viral kinetic model, with comparisons to a more conventional Markov chain Monte Carlo (MCMC) based Bayesian approach.
Keywords: 60G15; 62F10; 62F15; 65L05; 65L09; 92-08; Gaussian process; dynamical systems; inverse problems; parameter estimation; uncertainty estimation; viral kinetic model.