Learning Nonlinear Dynamics: Methods and Applications

Abstract

Accurate modeling of dynamical systems through differential equations is essential for scientific prediction and prescriptive control. Traditional model development, which relies on expert knowledge, parameter fitting and validation, is often iterative, time-consuming, and complicated by real-world data complexities such as noise and missing observations. This thesis addresses these challenges by developing robust, scalable, and interpretable methods for learning nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) directly from data, with a particular emphasis on applications in fluid dynamics.

In Chapter 2, we introduce a novel methodology for learning arbitrary nonlinear ODEs using collocation methods combined with interpolation. This approach demonstrates enhanced robustness to noise and significant computational speed-ups compared to classical system identification techniques, including the popular SINDy framework. It also provides a constructive method for reconstructing unobserved system components, making it applicable to partially observed systems, and offers theoretical guarantees on accuracy traditionally absent in strong-form identification.

In Chapter 3, we combine the approach from Chapter 2 with sparse regression to derive sparse ODEs from data, demonstrating enhanced robustness to observational noise. Our method shows improved performance in recovering the true structures and coefficients on canonical benchmark tests under significant noise, while the performance of traditional surrogate methods deteriorates even with minimal noise.

In Chapter 4, we extend this methodology to Partial Differential Equations (PDEs) using the method of lines, addressing issues related to data scale and interpolation ill-posedness. With a focus on Computational Fluid Dynamics (CFD), we show that our method goes beyond recovering complex nonlinear PDEs, such as the Navier-Stokes equations, from simulation data. The method can also be used as an a-posteriori indicator of simulation quality, providing insights into the effective PDEs represented by a given simulation, and pinpointing error-generating areas to inform adaptive mesh techniques.

Lastly, in Chapter 5, we introduce a novel data-driven framework for modeling turbulent phenomena, a long-standing challenge in aerospace and climate science. Our approach addresses the Reynolds-Averaged Navier-Stokes (RANS) closure problem, which involves modeling the unobserved eddy viscosity field. We tackle two interconnected inverse problems: reconstructing the eddy viscosity from flow data and discovering its governing partial differential equations (PDEs), thereby proposing a new pathway to uncover new or refined RANS closure models directly from high-fidelity simulations. This chapter establishes a tractable baseline using a composite loss function, which we evaluate on canonical turbulent flows. Our results demonstrate that while the approach can recover governing equations when the ground truth eddy viscosity is known, significant challenges remain due to noise and numerical errors. We conclude that a more advanced reconstruction methodology is essential for robustly discovering these models, underscoring the potential of this data-driven approach and identifying critical areas for future research.