Nia BowersA self-driving car, an operating power grid, a chemical manufacturing process: each is a cyber-physical system, a physical process run by software that reads sensors and issues commands. Increasingly, the models behind them are learned from data. Before such a model can control a real system, engineers need to know whether it captures properties such as stability, how large its errors can be, and whether its data can be trusted. “Most of the time, a machine-learning model has no idea whether the data it learned from is wrong,” said Dr. Xiuzhen Ye, a leading postdoctoral research scholar at North Carolina State University. “My research asks how much harm bad data can do to a physical system, and how much we can trust a model learned from data.” She has spent the better part of a decade on exactly those questions, first on the security of the data that power grids depend on, and now on the mathematics of certifying what a machine has learned. Her answers have appeared in the leading journals of two engineering disciplines, have been taken up by researchers on four continents, and have earned her a reputation as someone who brings mathematical rigor to problems the field had been handling by rule of thumb.
Seeing nonlinear systems through a linear lens
Dr. Ye’s work addresses a core question of trust: when is a data-driven model reliable? One of the most active approaches is Koopman learning, which builds on an idea proposed by mathematician Bernard Koopman in 1931: track how functions of a system’s state evolve, rather than the state itself, and even highly nonlinear dynamics become linear, though in a much larger, often infinite-dimensional, space. Linear models are well understood and efficient to compute, which is what makes the approach attractive. But the Koopman approach has well-known gaps: results hinge on the choice of function space, the learned model does not reveal whether the real system is stable, and systems with control inputs do not fit the linear picture at all.
At North Carolina State University, Dr. Ye and her collaborators address these gaps with a kernel-based theory, supported by the U.S. National Science Foundation (CBET Award #2414369) and the American Chemical Society Petroleum Research Fund (#66911-DNI9). Their key idea is a “linear–radial product” kernel that builds knowledge of the system’s equilibrium point directly into the model. In 2026 alone, five papers from this work have been presented, or are scheduled for presentation, at four conferences: the Conference on Learning for Dynamics and Control (L4DC), a leading venue for machine learning and control; the Chinese Control Conference (CCC); the Allerton Conference on Communication, Control, and Computing; and the IEEE Conference on Decision and Control (CDC), the flagship conference of the IEEE Control Systems Society. Producing five papers at that level in a single year is a remarkable pace, and Dr. Ye has been a main contributor across the series.
From modeling and analysis to design
Dr. Ye co-authored the paper that establishes the foundation, which was selected as one of 16 oral presentations among 167 accepted papers at L4DC 2026 and published in the Proceedings of Machine Learning Research. It addresses a central question for Koopman learning: how to define the operator on a function space so that its spectrum reflects the stability of the underlying nonlinear system. The model learned from data, with a provable error bound, can serve as a stability certificate. A companion paper at CCC 2026 extends this to learning Lyapunov functions, the classic mathematical certificate that a system will return to equilibrium, directly from data. It does so by solving an operator version of the Lyapunov equation from classical linear control, with an error that provably shrinks as the learned model improves.
Dr. Ye then led two studies that move the framework from modeling and analysis toward design, both of which will be presented at CDC 2026 in Honolulu this December. In the first, she developed a new way to estimate dissipativity, which describes how a system stores and releases energy and underlies the analysis of robotic, electrical, and chemical systems, directly from data, with a statistical guarantee on its accuracy.
In the second, she addressed a common practical limitation: engineers can rarely measure every variable in a system. She designed state observers, algorithms that infer unmeasured variables from available measurements, using models learned from data. The design accounts for two sources of error, one from the finite set of functions used to build the model and one from the randomness of the data samples, so that the observer remains reliable despite an imperfect model.
Dr. Ye also contributed to extending the theory to systems with control inputs, in a paper presented at Allerton 2026 in Urbana in September. She and her co-authors proved that, under suitable conditions, nonlinear control systems can be represented exactly as bilinear dynamics, and built a stabilization method on that result. Her work is also reaching practitioners: Dr. Ye co-authored the chapter “Data-Driven State-Space Modeling and State Observers for Nonlinear Dynamics” in Digital Twins in Manufacturing, published by Elsevier, the world’s largest scientific publisher, and edited by Masoud Soroush at Drexel University and Richard D. Braatz at MIT, two of the leading experts in process systems engineering.
Where Koopman learning goes next
The results point toward a bigger ambition: moving from “the model seems to work” to “the model is proven to work,” from raw sensor data to safe, stable control. Stability certificates establish when a learned model can be trusted, observers recover the states sensors cannot see, and the bilinear representation handles control inputs. Combined, they open the way to output-feedback controllers designed entirely from data, with guarantees carried through each step. Because dissipativity describes how subsystems exchange energy, the framework could scale to large interconnected systems, such as multi-unit chemical plants or power networks, by certifying each component from its own data. For industries that run complex physical processes, that means deploying learned models with quantified confidence rather than trial and error. Tighter error bounds also leave less room for stealth attacks, bringing Dr. Ye’s research full circle back to the power-grid security problems where it began.