Andrea Arnold

Caitlin Ho, Andrea Arnold

Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this work, we combine deep learning techniques and classic numerical methods for differential equations to address two challenging missing physics problems in dynamical systems theory: dynamics discovery and parameter estimation. The presented methods encode available information relating to the system dynamics into deep learning architectures, incorporating different assumptions on the known inputs and desired outputs in each case. Results demonstrate the effectiveness of the proposed approaches in making data-driven model predictions given corrupt system observations on a suite of test problems exhibiting oscillatory and chaotic dynamics. When comparing the performance of various numerical schemes, such as the Runge-Kutta and linear multistep families of methods, we observe promising results in predicting the system dynamics and estimating physical parameters, given appropriate choices of spatial and temporal discretization schemes and numerical method orders.

Alexander DeLise, Kyle Loh, Krish Patel et al.

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.