Research

My research concerns singularly perturbed partial differential equations, mainly in fluid mechanics. A small parameter ε — viscosity or diffusivity — creates thin boundary and interior layers, and asymptotic analysis tells us a great deal about them in advance: where the layers sit, how thick they are, the limit problem as ε → 0, and the explicit form of the boundary-layer correctors. Conservation laws and weak formulations add further structure.

Much of my earlier work concerns the analysis of these layers. The question that guides my recent work is computational: where should this structure be encoded in a learning-based method — in the trial space, in the discrete residual, or in the learned solution operator? For each choice, I study the effect on accuracy, on robustness as ε → 0, and on computational cost. Classical numerical methods face the same question: layer-adapted meshes put this information in the mesh, asymptotic-preserving schemes in the scheme, and enriched methods in the approximation space.

Research program: building the mathematical structure of a PDE — asymptotic, variational, conservation, interface — into neural approximations, neural discretizations, and neural operators.

Recent Results in Numbers

• Corrector in the trial space. With the same network, optimizer, and collocation points, changing only the trial function reduces the relative L² error for a reaction–diffusion problem at ε = 10⁻⁸ from 3.1 × 10⁻¹ to 1.2 × 10⁻⁴. A Shishkin mesh with 50 points, which uses the same information in the mesh, gives 4.6 × 10⁻⁴.

• Flux form in the loss. For a nonlinear convection–diffusion problem, the singular-layer finite volume network keeps the error between 1.5 × 10⁻⁴ and 1.6 × 10⁻⁴ for ε from 10⁻⁴ down to 10⁻¹², while the error of the plain finite volume network grows from 4.8 to 9.2 × 10⁴. For Burgers’ equation at ε = 10⁻⁸, training time drops from 78.9 s to 0.15 s.

• Weak form by construction. In a staggered discontinuous Galerkin neural network, the trace conditions hold exactly, and the training loss equals the squared distance between the projection of the network and the finite element solution.

Singular Perturbations and Boundary Layers 

I study boundary, corner, and interior layers arising in viscous and diffusive limits of partial differential equations. This includes problems involving curved or non-smooth domains, ill-prepared initial or boundary data, and multiple interacting scales. My work aims to clarify how such structures influence stability, convergence, and accuracy in both analysis and computation of solutions.

This analysis is also where the corrector used in computation comes from: corner correctors from nested expansions, explicit tanh profiles for interior layers in Burgers’ equation, boundary-fitted correctors for curved and non-smooth domains, and vorticity-layer correctors for plane-parallel and Navier–Stokes flows. For problems without a closed-form corrector, such as the Prandtl layer under the no-slip condition, I am developing explicit, computable corrector hierarchies that can be used in the same way.

Structure in the Trial Space: Singular-Layer Neural Networks

A recent focus of my research is singular-layer neural networks, hybrid methods that combine asymptotic analysis with learning-based and conservative discretizations. The guiding observation is that, in a singularly perturbed problem, the hardest part of the solution — a thin boundary or interior layer — is often a single function that asymptotic analysis already provides explicitly.

Singular-layer physics-informed neural networks (sl-PINN) and finite volume neural networks (sl-FVNN) build this analytic corrector directly into the trial space, so the network only needs to learn the smooth remainder. The resulting methods stay accurate for extremely small parameters, such as ε = 10⁻⁸, where standard PINNs fail to resolve the layer. The same idea has been carried from one-dimensional boundary layers to interior layers in Burgers’ equation, corner layers in rectangular domains, curved boundaries, and periodic pipe flows.

The key idea in one picture: for the reaction–diffusion problem −εu″ + u = f, the solution uε splits into a smooth limit u0 and a boundary-layer corrector Θε that carries all of the stiffness on a width √ε. Singular-layer methods supply Θε analytically and let the network learn only the smooth part.

Structure in the Discretization

A second direction uses the network as part of a finite volume or discontinuous Galerkin method, so that local conservation, numerical fluxes, and interface conditions hold by construction. In finite volume neural networks (FVNN, sl-FVNN), the loss is a cell-by-cell flux balance rather than a pointwise residual; in staggered discontinuous Galerkin neural networks (SDGNN), the trace conditions are built into the architecture. In the singularly perturbed setting, uniformity in ε is an additional requirement, and since the loss is a discrete distance, it can also be used for a posteriori error control.

Structure in the Solution Operator

A third, newer direction is to learn the map (μ, ε) ↦ u(μ, ε), where μ denotes coefficients, data, or geometry. Standard operator learning loses accuracy as ε → 0. My approach is to include the corrector in the architecture and to train with discrete weak residuals instead of large sets of high-fidelity solutions. The central question is whether a single learned operator can remain accurate across the singular limit; this work is at an early stage.

Applications: Wall-Bounded and Geophysical Flows

The main application is incompressible flow at small viscosity: wall-bounded flows, where the Prandtl layer is the central difficulty, and rotating and stratified geophysical flows — including simplified forms of the primitive equations and atmospheric boundary layers — where additional scales and anisotropies lead to complex layer interactions. These problems require repeated computations over many parameters, and because finite volume and discontinuous Galerkin residuals are local, the methods parallelize well; the next step is to move from one- and two-dimensional examples to three-dimensional problems.

Current Directions

Ongoing work extends singular-layer neural network methods to vorticity layers of the Navier–Stokes equations and to staggered discontinuous Galerkin frameworks, and develops explicit corrector hierarchies that make Prandtl-type viscous boundary layers computable. Papers in preparation:

• Singular layer PINN and FVNN methods for vorticity layers of the Navier–Stokes equations, with T.-Y. Chang, Y. Hong, and C.-Y. Jung.

• Explicit corrector hierarchies for slowly varying viscous boundary layers (single-authored).

• Staggered discontinuous Galerkin neural network methods for scalar elliptic problems, with C.-Y. Jung, B. Kwon, and K. Kwon.

Recent Invited Talks

2026 lecture series during the Fulbright appointment in Korea: Analysis-Informed Neural Network Methods for Singularly Perturbed Problems.

• Oct 2026 — PDE & Applied Math Seminar, KAIST

• Oct 2026 — Math Seminar, Yonsei University

• Sep 2026 — MINDS Seminar, POSTECH

• Sep 2026 — Mathematics Department Seminar, Korea University

• Sep 2026 — PDE & Applied Math Seminar, Seoul National University

• Jul 2026 — Applied Math Seminar, Sangmyung University

• Jul 2026 — PDE and Machine Learning Seminar, Kyungpook National University

• Jun 2026 — Machine Learning Seminar, UNIST

• Nov 2025 — SIAM Conference on Analysis of Partial Differential Equations, Pittsburgh (minisymposium co-organizer)

• May 2025 — Workshop for Research and Workforce Development in Fluid Mechanics, University of Nebraska–Lincoln

• Apr 2025 — Department Colloquium, South Dakota State University

• Mar 2025 — PDE Seminar, University of Nebraska–Lincoln

• Jan 2025 — POSTECH International Conference on Navier–Stokes Equations and Numerical Analysis

• Jan 2025 — PDE Seminar, Inha University

Publications Related to This Research

Selected publications that illustrate aspects of this work are listed on the Publications page. A complete list of publications is available in my curriculum vitae.