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Source author record

Youngjoon Hong

Youngjoon Hong appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
math.NAMachine LearningNumerical AnalysisArtificial Intelligencemath.APComputational Engineering, Finance, and Sciencemath.DSmath.PRphysics.ao-phphysics.app-phphysics.comp-ph

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Published work

8 published item(s)

preprint2026arXiv

Data-Free Asymptotics-Informed Operator Networks for Singularly Perturbed PDEs

Recent advances in machine learning (ML) have opened new possibilities for solving partial differential equations (PDEs), yet robust performance in challenging regimes remains limited. In particular, singularly perturbed differential equations exhibit sharp boundary or interior layers with rapid transitions, where standard ML surrogates often fail without extensive resolution. Generating training data for such problems is also costly, as accurate reference solutions typically require massive adaptive mesh refinement. In this work, we propose eFEONet, an enriched Finite Element Operator Network tailored to singularly perturbed problems. Guided by classical singular perturbation theory, eFEONet augments the operator-learning framework with specialized enrichment basis functions that encode the asymptotic structure of layer solutions. This design enables accurate approximation of sharp transitions without relying on large datasets, and can operate with minimal supervision-or even in a data-free manner under appropriate settings. We further provide a rigorous convergence analysis of the proposed method and demonstrate its effectiveness through extensive experiments on representative problems featuring both boundary and interior layers.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Discontinuous Galerkin finite element operator network for solving non-smooth PDEs

We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework that combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial differential equations (PDEs) with discontinuous coefficients and non-smooth solutions. Unlike traditional operator learning models such as DeepONet and Fourier Neural Operator, which require large paired datasets and often struggle near sharp features, our approach minimizes the residual of a DG-based weak formulation using the Symmetric Interior Penalty Galerkin (SIPG) scheme. DG-FEONet predicts element-wise solution coefficients via a neural network, enabling data-free training without the need for precomputed input-output pairs. We provide theoretical justification through convergence analysis and validate the model's performance on a series of one- and two-dimensional PDE problems, demonstrating accurate recovery of discontinuities, strong generalization across parameter space, and reliable convergence rates. Our results highlight the potential of combining local discretization schemes with machine learning to achieve robust, singularity-aware operator approximation in challenging PDE settings.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Semi-Supervised Neural Super-Resolution for Mesh-Based Simulations

Mesh-based simulations provide high-fidelity solutions to partial differential equations (PDEs), but achieving such accuracy typically requires fine meshes, leading to substantial computational overhead. Super-resolution techniques aim to mitigate this cost by reconstructing high-resolution (HR), high-fidelity solutions from low-cost, low-resolution (LR) counterparts. However, training neural networks for super-resolution often demands large amounts of expensive HR supervision data. To address this challenge, we propose SuperMeshNet, an HR data-efficient super-resolution framework for mesh-based simulations aided by message passing neural networks (MPNNs). At its core, SuperMeshNet introduces complementary learning, a semi-supervised approach that effectively leverages both 1) a small amount of paired LR-HR data and 2) abundant unpaired LR data via two jointly trained, complementary MPNN-based models. Additionally, our model is enriched by inductive biases, which are empirically shown to further improve super-resolution performance. Extensive experiments demonstrate that SuperMeshNet requires 90% less HR data to achieve even lower root mean square error (RMSE) than that of the fully supervised benchmark without the inductive biases. The source code and datasets are available at https://github.com/jykim-git/SuperMeshNet.git.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Sobolev Approximation of Deep ReLU Networks in Log-Barron Space

Universal approximation theorems show that neural networks can approximate any continuous function; however, the number of parameters may grow exponentially with the ambient dimension, so these results do not fully explain the practical success of deep models on high-dimensional data. Barron space theory addresses this: if a target function belongs to a Barron space, a two-layer network with $n$ parameters achieves an $O(n^{-1/2})$ approximation error in $L^2$. Yet classical Barron spaces $\mathscr{B}^{s+1}$ still require stronger regularity than Sobolev spaces $H^s$, and existing depth-sensitive results often assume constraints such as $sL \le 1/2$. In this paper, we introduce a log-weighted Barron space $\mathscr{B}^{\log}$, which requires a strictly weaker assumption than $\mathscr{B}^s$ for any $s>0$. For this new function space, we first study embedding properties and carry out a statistical analysis via the Rademacher complexity. Then we prove that functions in $\mathscr{B}^{\log}$ can be approximated by deep ReLU networks with explicit depth dependence. We then define a family $\mathscr{B}^{s,\log}$, establish approximation bounds in the $H^1$ norm, and identify maximal depth scales under which these rates are preserved. Our results clarify how depth reduces regularity requirements for efficient representation, offering a more precise explanation for the performance of deep architectures beyond the classical Barron setting, and for their stable use in high-dimensional problems used today.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Semi-analytic PINN methods for singularly perturbed boundary value problems

We propose a new semi-analytic physics informed neural network (PINN) to solve singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that offers a promising perspective for finding numerical solutions to partial differential equations. The PINNs have shown impressive performance in solving various differential equations including time-dependent and multi-dimensional equations involved in a complex geometry of the domain. However, when considering stiff differential equations, neural networks in general fail to capture the sharp transition of solutions, due to the spectral bias. To resolve this issue, here we develop the semi-analytic PINN methods, enriched by using the so-called corrector functions obtained from the boundary layer analysis. Our new enriched PINNs accurately predict numerical solutions to the singular perturbation problems. Numerical experiments include various types of singularly perturbed linear and nonlinear differential equations.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Numerical approximation of the singularly perturbed heat equation in a circle

In this article we study the two dimensional singularly perturbed heat equation in a circular domain. The aim is to develop a numerical method with a uniform mesh, avoiding mesh refinement at the boundary thanks to the use of a relatively simple representation of the boundary layer. We provide the asymptotic expansion of the solution at first order and derive the boundary layer element resulting from the boundary layer analysis. We then perform the convergence analysis introducing the boundary layer element in the finite element space thus obtaining what is called an "enriched Galerkin space". Finally we present and comment on numerical simulations using a quasi-uniform grid and the modified finite element method.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Numerical weather prediction in two dimensions with topography, using a finite volume method

We aim to study a finite volume scheme to solve the two dimensional inviscid primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions to the system of equations. In that respect, a version of a projection method is introduced to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. The resulting scheme allows for a significant reduction of the errors near the topography when compared to more standard finite volume schemes. In the numerical simulations, we first present the associated good convergence results that are satisfied by the solutions simulated by our scheme when compared to particular analytic solutions. We then report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated. The numerical results show that such a forcing is responsible for recurrent large-scale patterns to emerge in the temperature and velocity fields.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Timestepping schemes for the 3d Navier-Stokes equations: small solutions and short times

It is well known that the solution of the 3d Navier--Stokes equations remains bounded if the initial data and the forcing are sufficiently small relative to the viscosity, and for a finite time given any bounded initial data. In this article, we consider two temporal discretisations (semi-implicit and fully implicit) of the 3d Navier--Stokes equations in a periodic domain and prove that their solutions remain bounded in $H^1$ subject to essentially the same smallness conditions (on initial data, forcing or time) as the continuous system and to suitable timestep restrictions.

0 citations0 reviewsNo code linkedOpen access