Student Name: Tomoki Koike

Advisor: Dr. Elizabeth Qian

Milestone: PhD Thesis Final Examination (Defense)

Degree Program: Aerospace Engineering

Title: Data-Driven Nonlinear Reduced Model Learning with Stability Guarantee and Large-Scale Adaptation

Abstract: High-fidelity numerical models of engineering systems have become increasingly accurate, but their computational and memory costs remain a barrier to many-query tasks such as design optimization, uncertainty quantification, and control. The field of scientific machine learning (SciML) aims to address these costs by combining data-driven learning with physical knowledge to construct accurate, efficient surrogate models. This thesis develops new SciML methods based on Operator Inference (OpInf), a method that learns low-dimensional surrogate models from trajectory data without access to the underlying simulation code by fitting the operators of a prescribed polynomial model through least-squares regression. Although this structure facilitates analysis, it does not guarantee that the learned model inherits the stability of the original system. Moreover, the batch formulation requires that all training data be loaded into memory, limiting its applicability to large-scale systems. This thesis addresses these two challenges: stability and scalability. For stability, we propose Energy-Preserving OpInf, which constrains the learned quadratic operator to conserve energy, yielding provably bounded reduced dynamics when the linear dynamics are dissipative. We further propose a complementary method called Lyapunov Function Inference, which uses trajectory data alone to approximate solutions of Zubov’s equation, a nonlinear partial differential equation whose solution characterizes the stability region of an asymptotically stable equilibrium. By minimizing the equation’s residual over polynomial candidates, the method constructs Lyapunov functions to certify stability and estimate this region. For scalability, we develop Streaming OpInf, a method that combines streaming singular value decomposition with recursive least-squares to learn reduced models from incoming data without storing the full dataset. An accompanying error analysis bounds errors in the learned operators in terms of the incrementally updated subspaces. Beyond benchmark problems, we demonstrate Streaming OpInf on three large-scale turbulent flows that present significant memory challenges for batch OpInf: a compressible magnetohydrodynamic flow, a turbulent channel flow, and an atmospheric boundary layer flow with nearly 24 million states. We achieve models that reproduce relevant flow statistics while reducing memory requirements by 60–90%. Together, these contributions advance SciML for applications requiring stable learned dynamics and memory-efficient model construction.

Date and time: 2026-11-12, 2:30 pm to 4:30 pm

Location: MK317

Committee:
Dr. Elizabeth Qian (advisor), School of Aerospace Engineering
Dr. Keegan Moore, School of Aerospace Engineering
Dr. Joseph Oefelein, School of Aerospace Engineering
Dr. Benjamin Peherstorfer, École Polytechnique Fédérale de Lausanne
Dr. Christopher G. Baker, Nvidia Corporation