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Researcher profile

Jae Yong Lee

Jae Yong Lee contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine Learningmath.NANumerical AnalysisArtificial IntelligenceComputer Visionmath.OCNeural and Evolutionary Computingphysics.comp-phSymbolic Computation

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Published work

6 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

Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation

Discovering governing differential equations from observational data is a fundamental challenge in scientific machine learning. Existing symbolic regression approaches rely primarily on quantitative metrics; however, real-world differential equation modeling also requires incorporating domain knowledge to ensure physical plausibility. To address this gap, we propose DoLQ, a method for discovering ordinary differential equations with LLM-based qualitative and quantitative evaluation. DoLQ employs a multi-agent architecture: a Sampler Agent proposes dynamic system candidates, a Parameter Optimizer refines equations for accuracy, and a Scientist Agent leverages an LLM to conduct both qualitative and quantitative evaluations and synthesize their results to iteratively guide the search. Experiments on multi-dimensional ordinary differential equation benchmarks demonstrate that DoLQ achieves superior performance compared to existing methods, not only attaining higher success rates but also more accurately recovering the correct symbolic terms of ground truth equations. Our code is available at https://github.com/Bon99yun/DoLQ.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Unbiased and Second-Order-Free Training for High-Dimensional PDEs

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

DIVeR: Real-time and Accurate Neural Radiance Fields with Deterministic Integration for Volume Rendering

DIVeR builds on the key ideas of NeRF and its variants -- density models and volume rendering -- to learn 3D object models that can be rendered realistically from small numbers of images. In contrast to all previous NeRF methods, DIVeR uses deterministic rather than stochastic estimates of the volume rendering integral. DIVeR's representation is a voxel based field of features. To compute the volume rendering integral, a ray is broken into intervals, one per voxel; components of the volume rendering integral are estimated from the features for each interval using an MLP, and the components are aggregated. As a result, DIVeR can render thin translucent structures that are missed by other integrators. Furthermore, DIVeR's representation has semantics that is relatively exposed compared to other such methods -- moving feature vectors around in the voxel space results in natural edits. Extensive qualitative and quantitative comparisons to current state-of-the-art methods show that DIVeR produces models that (1) render at or above state-of-the-art quality, (2) are very small without being baked, (3) render very fast without being baked, and (4) can be edited in natural ways.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach

The issue of the relaxation to equilibrium has been at the core of the kinetic theory of rarefied gas dynamics. In the paper, we introduce the Deep Neural Network (DNN) approximated solutions to the kinetic Fokker-Planck equation in a bounded interval and study the large-time asymptotic behavior of the solutions and other physically relevant macroscopic quantities. We impose the varied types of boundary conditions including the inflow-type and the reflection-type boundaries as well as the varied diffusion and friction coefficients and study the boundary effects on the asymptotic behaviors. These include the predictions on the large-time behaviors of the pointwise values of the particle distribution and the macroscopic physical quantities including the total kinetic energy, the entropy, and the free energy. We also provide the theoretical supports for the pointwise convergence of the neural network solutions to the \textit{a priori} analytic solutions. We use the library \textit{PyTorch}, the activation function \textit{tanh} between layers, and the \textit{Adam} optimizer for the Deep Learning algorithm.

0 citations0 reviewsNo code linkedOpen access