Approximating nonlinear dynamic by linear systems via Koopman operator (Funded by AI Institute in Dynamic Systems)

We are often interested in modeling dynamical systems from observed measurements. This is especially challenging when the system exhibits nonlinear dynamics. Koopman theory tells us that we can analyze an nonlinear dynamical systems using linear operators on functions of the states, rather than the states themselves.

In practice, however, it is difficult to derive these functions analytically. To address this, we turn to neural networks, which serve as powerful universal function approximators capable of learning Koopman functions directly from data.

The challenge is twofold: we need models that not only achieve strong performance on high-dimensional nonlinear systems (such as fluid dynamics), but also provide a degree of explainability—a property that deep learning methods often lack.

Detecting pediatric heart diseases using machine learning

Electrocardiogram (ECG) data are widely used to diagnose heart diseases. Machine learning methods hold great promise for improving diagnostic accuracy, but several challenges remain. One key issue is data imbalance: most recordings come from healthy individuals, while cases of disease are relatively rare. Moreover, disease diagnosis is not purely a classification problem. It also resembles anomaly detection, since there is always a chance of encountering patients with previously unseen conditions.

To address these challenges, we are exploring the use of transformer models to learn useful representations for both anomaly detection and disease classification.

Paremeter Estimation and Model Selection of Interacting Particle Systems

Interacting particle systems are continuous-time Markov processes that describe the stochastic evolution of node states on a graph structure. They are the continuous-time analog of stochastic cellular automata systems. Interacting particle systems have also been studied as dynamic Bayesian networks and dynamic Markov networks.

Complex system involving interacting particles/agents/states occurs separately in many different areas (e.g., quantum systems). The commonality is these systems are challenging to analyze. My current interest is in a feasible approach to parameter estimation and model selection of such a system.