Jonghyeon Lee - Research

Broadly speaking, my research is about using kernel methods (Gaussian processes) to make differential equations more interpretable, both in terms of solving them and revealing key features of the behavior of solutions.

Reduced Order Models

A reduced order model (ROM) is a mathematical approximation of a computationally complex problem, in this case a nonlinear PDE, which lowers the degrees of freedom required to solve it. ROMs have applications to high-dimensional problems in fluids and biomechanics. Our proposed kernel-based ROM (KROM) overcomes challenges in existing ROMs and kernel methods by a particular constructing a kernel composed of other solutions to the same PDE and effectively reducing the number of collocation points required to solve the problem by exploiting a sparse Cholesky approximation of the precision matrix. This kernel, which we call the empirical kernel, outperforms a Matérn kernel in accuracy in our examples - we have shown the 2D Navier-Stokes equation at time t=5 as an illustrative example:

Koopman Operators

The Koopman operator is a linear operator which maps a nonlinear dynamical system to a dynamical system which is linear in infinite dimensions. While many researchers in this field attempt to approximate the Koopman operator directly, we instead study it by simply solving the linear PDE for the eigenfunctions with kernel regression. These eigenfunctions allow us to identify the domain of attraction and the stable and unstable manifolds of an equilibrium point. Kernels provide robust convergence results and error bounds in terms of the dimension of the domain and the number of collocation points, making them ideal for this kind of problem. Our paper has additional applications to constructing Lyapunov functions of dynamical systems and quasi-potential functions of stochastic differential equations.

Normal Form Transformations

Chaotic dynamical systems can be mapped locally around an equilibrium to another system which captures key features of the flow; this is known as the Poincaré normal form. I am working on learning normal forms for the Moore-Greitzer PDE system which is encountered in engine dynamics, connecting solutions of Moore-Greitzer and their different failure modes (surge and stall) to normal forms and learning the transformation to the normal form. The paper should be out very soon, stay tuned!