🤖 AI Summary
In scientific machine learning, modeling the mapping from physical processes to observed data faces dual challenges of interpretability and rank deficiency. This paper establishes a unified theoretical framework for linear encoder-decoder architectures grounded in Bayesian risk minimization, and— for the first time—derives closed-form optimal linear/affine mappings under explicit rank constraints, systematically addressing low-rank degeneracies in data, operators, and measurements. Our approach integrates Bayesian decision theory with low-rank matrix optimization, avoiding black-box nonlinear models. Evaluated on biomedical imaging, financial factor analysis, and shallow-water equation simulation, the proposed linear baseline achieves strong interpretability, high reproducibility, and superior generalization. It thus provides a trustworthy, robust, and benchmarkable paradigm for scientific AI.
📝 Abstract
Across scientific domains, a fundamental challenge is to characterize and compute the mappings from underlying physical processes to observed signals and measurements. While nonlinear neural networks have achieved considerable success, they remain theoretically opaque, which hinders adoption in contexts where interpretability is paramount. In contrast, linear neural networks serve as a simple yet effective foundation for gaining insight into these complex relationships. In this work, we develop a unified theoretical framework for analyzing linear encoder-decoder architectures through the lens of Bayes risk minimization for solving data-driven scientific machine learning problems. We derive closed-form, rank-constrained linear and affine linear optimal mappings for forward modeling and inverse recovery tasks. Our results generalize existing formulations by accommodating rank-deficiencies in data, forward operators, and measurement processes. We validate our theoretical results by conducting numerical experiments on datasets from simple biomedical imaging, financial factor analysis, and simulations involving nonlinear fluid dynamics via the shallow water equations. This work provides a robust baseline for understanding and benchmarking learned neural network models for scientific machine learning problems.