Studying Uncertainty: 5 Questions with UAlbany Mathematician Felix Ye
ALBANY, N.Y. (Sept. 17, 2026) — Felix Ye is an assistant professor in UAlbany’s Department of Mathematics and Statistics whose research sits at the intersection of scientific computing and machine learning.
Ye studies stochastic dynamical systems — complex processes that change over time but whose behavior is influenced in part by randomness or uncertainty. These can include everything from the motion of molecules and chemical reactions to biological and atmospheric processes. Ye uses mathematics and data-driven methods to uncover patterns within these complex systems and develop more efficient ways to model and predict their long-term behavior.
This fall, Ye received a three-year, $200,000 grant from the National Science Foundation for a project developing new data-driven methods to make it faster and more efficient to simulate these complex systems. The project, conducted with UAlbany co-principal investigator and assistant professor Barbara Giunti, will also support a doctoral student.
What first drew you to mathematics?
I have always been drawn to applied mathematics because it gives us a way to translate complicated real-world questions into models that we can analyze and use to make predictions. As I learned more about problems in physics, I became increasingly interested in systems in which uncertainty is not simply measurement error, but an intrinsic part of the dynamics.
That naturally led me to stochastic systems. Many important scientific processes, from the motion of molecules to changes in the atmosphere, are influenced by random fluctuations and interactions across many different time scales. I was fascinated by the possibility that mathematics could help us identify structure within that apparent randomness and make meaningful predictions about how these systems behave over long periods of time.
What's interesting about stochastic systems?
The stochastic systems I study are usually high-dimensional and inherently multiscale. They contain many interacting variables, some changing extremely quickly and others evolving much more slowly. Although individual trajectories may appear unpredictable, their long-term statistical behavior can often have a surprisingly clear structure.
What continues to interest me is the challenge of finding that hidden structure. We want to answer questions such as: Which states will a system spend most of its time in? How often will it transition from one state to another? How long will that transition take? Direct simulation can require enormous amounts of time and memory, so there is both a deep mathematical challenge and a practical computational need for better methods.
Why is this important?
It helps us understand complex systems whose exact behavior cannot be predicted one step at a time, but whose patterns can still be described statistically. Examples include molecular motion, chemical reactions, biological systems and atmospheric processes.
A central idea is related to what is known in machine learning as the “manifold hypothesis.” A system may be described using thousands or even millions of variables, but its most important long-term behavior may be governed by only a small number of hidden variables. In other words, the data occupy a much simpler structure inside an enormous mathematical space. If we can discover that structure, we can build a smaller model that is much less expensive to simulate while still capturing the behavior scientists care about.
Why are systems involving uncertainty so difficult to study?
Each of those features creates a different difficulty. Uncertainty means that running the same experiment twice may produce different trajectories, so one simulation is not enough; we need enough samples to understand an entire distribution of possible outcomes. Having many variables makes both computation and data storage expensive. Multiple time scales make the problem even harder because a simulation must use very small time steps to resolve the fastest motion, even when our real interest is in behavior occurring much later.
Rare events create perhaps the most severe challenge. A scientifically important transition may occur only after millions or billions of simulation steps. For example, a molecule may remain in one configuration for a very long time before suddenly changing to another. A conventional simulation spends most of its effort waiting for that transition to happen, yet scientists may need many such transitions to estimate their probabilities accurately.
What's the main idea behind this project?
The central idea is that instead of simulating the full, high-dimensional system directly for an extremely long time, we will run many short simulations in parallel and use the resulting data to learn a reduced model of its essential dynamics. These short simulations reveal local information about the system. We will then connect that local information to construct an efficient global simulator.
Ultimately, this could allow scientists to estimate stable states, transition rates and other long-term statistics without storing and simulating every variable at every tiny time step. The goal is not merely to compress the data, but to preserve the scientifically meaningful behavior of the original system.