Zhewen Hou

Zhewen Hou, Tian Zheng

Distribution shift between training and deployment is a pervasive challenge for modern AI systems. In many cases, the target marginals of covariates and response are known or specified through population-level observations, boundary conditions, properties of simulator configurations, or alignment-time distributional constraints. Such knowledge may provide valuable side information for regression estimation. We study this problem in the multivariate linear regression setting with a stable conditional mean $E[Y\mid X]$ across source and target, and identify the hybrid-loss estimator, which jointly incorporates both target marginals, as a benchmark target-aware estimator. Its direct computation, however, requires solving a coupled nonlinear optimization that is expensive at scale. Our main contribution is to develop and evaluate two computationally tractable alternatives: a constrained moment-matching estimator and a two-stage estimator that augments ordinary least squares with a calibration step. For all three estimators, we derive and compare closed-form asymptotic mean squared errors, yielding conditions under which the tractable alternatives match or closely approximate the hybrid benchmark, and regimes in which they do not. Monte Carlo experiments across three controlled shift regimes validate the theoretical results, investigate the accuracy-runtime tradeoffs among the three estimators, and translate into guidance on estimator choice. In particular, the two-stage estimator nearly matches the hybrid benchmark in the high signal-to-noise regime at essentially no additional cost, providing theoretical grounding for empirical observations in nonlinear settings.

Zhewen Hou, Jiajin Sun, Subashree Venkatasubramanian et al.

Machine learning (ML) 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 work, 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 (KSD) to enhance ML predictions. The method here employs KSD within a reproducing kernel Hilbert space to calibrate model outputs, improving their fidelity to known physical distributions. This not only sharpens pointwise predictions but also enforces consistency with non-local statistical structures rooted in physical principles. Through synthetic experiments-spanning offline climatological CO2 fluxes and online quasi-geostrophic flow simulations-we demonstrate the robustness and broad utility of the proposed framework.

Tian Zheng, Subashree Venkatasubramanian, Shuolin Li et al.

Machine learning has been increasingly applied in climate modeling on system emulation acceleration, data-driven parameter inference, forecasting, and knowledge discovery, addressing challenges such as physical consistency, multi-scale coupling, data sparsity, robust generalization, and integration with scientific workflows. This paper analyzes a series of case studies from applied machine learning research in climate modeling, with a focus on design choices and workflow structure. Rather than reviewing technical details, we aim to synthesize workflow design patterns across diverse projects in ML-enabled climate modeling: from surrogate modeling, ML parameterization, probabilistic programming, to simulation-based inference, and physics-informed transfer learning. We unpack how these workflows are grounded in physical knowledge, informed by simulation data, and designed to integrate observations. We aim to offer a framework for ensuring rigor in scientific machine learning through more transparent model development, critical evaluation, informed adaptation, and reproducibility, and to contribute to lowering the barrier for interdisciplinary collaboration at the interface of data science and climate modeling.