Physics-Aware Learning for Environmental Systems: Surrogate Modeling and Constraint-Driven Machine Intelligence

Understanding and predicting complex environmental and hydrosystem processes is a central challenge in Earth system science. These systems are governed by interacting physical mechanisms across scales, are only partially observed, and are often characterized by limited data and substantial uncertainty. As a result, machine learning (ML) has emerged along two complementary development paths for environmental modeling.

In a first branch, physical process models remain the backbone of simulation, while ML is employed as a surrogate to approximate expensive numerical solvers. Surrogate modeling approaches based on Gaussian process emulators, polynomial chaos expansions, support vector regression, and related probabilistic representations are particularly well suited for data-poor settings. Neural networks are used more selectively in this context, as uncertainty-aware and sample-efficient methods are often preferred. In surrogate modeling, considerable effort is devoted to optimal sampling strategies, including active learning, which adaptively select informative simulations and help preserve scarce computational resources. These surrogate models enable efficient uncertainty quantification, sensitivity analysis, and Bayesian inference, while preserving physical interpretability.

A second, increasingly important branch emerges when physical models are incomplete, unavailable, or deliberately omitted, and ML models replace the governing equations altogether. This branch is most commonly based on neural network representations, but has recently also been explored using Gaussian processes and polynomial chaos–based learning concepts. In this setting, purely data-driven learning is insufficient, as unconstrained models tend to violate physical principles and extrapolate poorly. In this second branch, physical principles such as conservation laws or balance relations are embedded directly into learning architectures. Complementarily, constraint-driven learning strategies enforce physical laws, admissibility conditions, and structural consistency during training. By restricting the hypothesis space, these methods stabilize learning and support robust inference under incomplete physical knowledge.

Taken together, surrogate modeling for physics-based simulations and physics-aware ML for equation-free learning represent two coherent and complementary branches of modern environmental machine learning. We observe a growing convergence between these two branches, as physics-based surrogate modeling and equation-free machine learning increasingly borrow concepts from each other. This convergence is not accidental, but a direct response to fundamental model limitations and the challenge of making reliable predictions under scarce data and knowledge constraints. By integrating physics, probabilistic reasoning and constraints, emerging approaches increasingly focus on robustness and interpretability rather than unconstrained expressive power.