Learning in dynamical systems

QC 20261111

Abstract

This thesis studies modeling and inference in nonlinear dynamical systems using machine learning. Improved data availability and deep learning tools have advanced nonlinear system identification, but real-world applications require models with predictable behavior and effective inference with limited real-time data. We make three contributions: neural models with control-theoretic guarantees, generative models with sampling-theoretic guarantees, and inference with neural surrogates under partially known inputs. 

First, we design neural ordinary differential equations that are locally exponentially stable at the origin with exactly computed regions of attraction (ROAs) and universal approximation guarantees. The main idea is to introduce a learnable projection layer that is defined by a neural Lyapunov function solving a variant of the Zubov equation for the projected dynamics. We prove that the model class universally approximates any sufficiently smooth, locally exponentially stable dynamics over their ROA.

Second, we derive a method for exact, training-free guidance in flow matching. The main idea is that all stochasticity in flow matching comes from the source sampling, so we modify the source density and retain the original flow map. We prove that our method exactly recovers the conditional target distribution. We apply it to an inverse problem in a partial differential equation.

Third, we design a method for state estimation in infinite-dimensional dynamical systems using neural surrogate models. The main idea is to  use Fourier neural operators (FNO) as low-cost surrogates for high-fidelity flow maps, and embed them in a closed-loop algorithm that updates the predicted state estimate with sparse online measurements. We apply the method to traffic density estimation, where the FNO is trained on microscopic traffic simulation data.

Link to DiVA