2026 Theses Doctoral

On Distribution-Informed Learning for Dynamical Systems

Hou, Zhewen

Machine learning has shown significant promise in studying complex geophysical dynamical systems, including turbulence and climate processes. Such systems often display sensitive dependence on initial conditions, reflected in positive Lyapunov exponents, where even small perturbations in short-term forecasts can lead to large deviations in long-term outcomes. Thus, meaningful inference requires not only accurate short-term predictions, but also consistency with the system’s long-term attractor that is captured by the marginal distribution of state variables. Existing approaches attempt to address this challenge by incorporating spatial and temporal dependence, but these strategies become impractical when data are extremely sparse.

In this dissertation, we show that prior knowledge of marginal distributions offers valuable complementary information to short-term observations, motivating a distribution-informed learning framework. We introduce a calibration algorithm based on normalization and the Kernelized Stein Discrepancy to enhance machine learning predictions. The method here employs Kernelized Stein Discrepancy within a reproducing kernel Hilbert space to calibrate model outputs, improving their fidelity to known physical distributions. We further establish a theoretical framework for linear regression with knowledge distribution. It systematically investigates the properties of distribution-informed estimators. We also apply the proposed framework on two scientific problems: spanning offline climatological carbon dioxide fluxes and online quasi-geostrophic flow simulations.

The results indicate that knowledge distribution can serve as an important information source. We prove that the distribution information provides constraints complementary to sample information. The Kernelized Stein Discrepancy calibration not only sharpens pointwise predictions but also enforces consistency with non-local statistical structures rooted in physical principles. In brief, we demonstrate the robustness and broad utility of the proposed framework in scientific problems.