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Doyoon Kim

Publications and source records attributed to Doyoon Kim.

At least 19 recordsLinked to original sources
arXiv2026

Failure of zero extension in parabolic Sobolev spaces

We show that spatial zero extension across the boundary may fail in parabolic Sobolev spaces $\mathring{\mathcal{H}}^1_p((0,T) \times \Omega)$, which can also be characterized as $$ L_p(0,T;\mathring{W}^1_p(\Omega))\cap W^1_p(0,T; W^{-1}_{p}(\Omega)). $$ More precisely, for any $p\in [1, \infty)$, we construct a function $u\in \mathring{\mathcal{H}}^1_p((0,T)\times \mathbb{R}^d_+)$ whose zero extension does not belong to $\mathcal{H}^1_p((0,T)\times \mathbb{R}^d)$. The obstruction occurs even for a flat boundary and is caused by a self-similar boundary layer concentrated at the initial-boundary corner, which produces a boundary supported normal flux defect after zero extension. We also discuss the suitability of various Sobolev-type spaces as solution spaces for parabolic equations in divergence form.

math.AP↗
arXiv2025

$L_p$-estimates of the conormal derivative problem for parabolic equations with time measurable coefficients and $A_p$-weights

This paper investigates weighted mixed-norm estimates for divergence-type parabolic equations on Reifenberg-flat domains with the conormal derivative boundary condition. The leading coefficients are assumed to be merely measurable in the time variable and to have small mean oscillations in the spatial variables. In deriving the boundary estimates, we overcome a regularity issue by employing half-time derivative estimates.

math.AP↗
arXiv2025

Trace Regularity PINNs: Enforcing $\mathrm{H}^{\frac{1}{2}}(\partial \Omega)$ for Boundary Data

We propose an enhanced physics-informed neural network (PINN), the Trace Regularity Physics-Informed Neural Network (TRPINN), which enforces the boundary loss in the Sobolev-Slobodeckij norm $H^{1/2}(\partial \Omega)$, the correct trace space associated with $H^1(\Omega)$. We reduce computational cost by computing only the theoretically essential portion of the semi-norm and enhance convergence stability by avoiding denominator evaluations in the discretization. By incorporating the exact $H^{1/2}(\partial \Omega)$ norm, we show that the approximation converges to the true solution in the $H^{1}(\Omega)$ sense, and, through Neural Tangent Kernel (NTK) analysis, we demonstrate that TRPINN can converge faster than standard PINNs. Numerical experiments on the Laplace equation with highly oscillatory Dirichlet boundary conditions exhibit cases where TRPINN succeeds even when standard PINNs fail, and show performance improvements of one to three decimal digits.

cs.LG↗
arXiv2025

Causal-Paced Deep Reinforcement Learning

Designing effective task sequences is crucial for curriculum reinforcement learning (CRL), where agents must gradually acquire skills by training on intermediate tasks. A key challenge in CRL is to identify tasks that promote exploration, yet are similar enough to support effective transfer. While recent approach suggests comparing tasks via their Structural Causal Models (SCMs), the method requires access to ground-truth causal structures, an unrealistic assumption in most RL settings. In this work, we propose Causal-Paced Deep Reinforcement Learning (CP-DRL), a curriculum learning framework aware of SCM differences between tasks based on interaction data approximation. This signal captures task novelty, which we combine with the agent's learnability, measured by reward gain, to form a unified objective. Empirically, CP-DRL outperforms existing curriculum methods on the Point Mass benchmark, achieving faster convergence and higher returns. CP-DRL demonstrates reduced variance with comparable final returns in the Bipedal Walker-Trivial setting, and achieves the highest average performance in the Infeasible variant. These results indicate that leveraging causal relationships between tasks can improve the structure-awareness and sample efficiency of curriculum reinforcement learning. We provide the full implementation of CP-DRL to facilitate the reproduction of our main results at https://github.com/Cho-Geonwoo/CP-DRL.

cs.LG↗
arXiv2024

$L_p$-estimates for parabolic equations in divergence form with a half-time derivative

We establish the unique solvability of solutions in Sobolev spaces to linear parabolic equations in a more general form than those in the literature. A distinguishing feature of our equations is the inclusion of a half-order time derivative term on their right-hand side. We anticipate that such equations will prove useful in various problems involving time evolution terms. Notably, the coefficients of the equations exhibit significant irregularity, being merely measurable with respect to the temporal variable or one spatial variable.

math.AP↗
arXiv2023

Sobolev spaces and trace theorems for time-fractional evolution equations

We establish trace and extension theorems for evolutionary equations with the Caputo fractional derivatives in (weighted) $L_p$ spaces. To achieve this, we identify weighted Sobolev and Besov spaces with mixed norms that accommodate solution spaces and their initial values well-suited for equations involving time-fractional derivatives. Our analysis encompasses both time-fractional sub-diffusion and super-diffusion equations. We also provide observations on the initial behavior of solutions to time-fractional equations.

math.AP↗
arXiv2023

Painsight: An Extendable Opinion Mining Framework for Detecting Pain Points Based on Online Customer Reviews

As the e-commerce market continues to expand and online transactions proliferate, customer reviews have emerged as a critical element in shaping the purchasing decisions of prospective buyers. Previous studies have endeavored to identify key aspects of customer reviews through the development of sentiment analysis models and topic models. However, extracting specific dissatisfaction factors remains a challenging task. In this study, we delineate the pain point detection problem and propose Painsight, an unsupervised framework for automatically extracting distinct dissatisfaction factors from customer reviews without relying on ground truth labels. Painsight employs pre-trained language models to construct sentiment analysis and topic models, leveraging attribution scores derived from model gradients to extract dissatisfaction factors. Upon application of the proposed methodology to customer review data spanning five product categories, we successfully identified and categorized dissatisfaction factors within each group, as well as isolated factors for each type. Notably, Painsight outperformed benchmark methods, achieving substantial performance enhancements and exceptional results in human evaluations.

cs.AI↗
arXiv2023

Time fractional parabolic equations with partially SMO coefficients

We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in $(t,x_1)$ for $a^{ij}$, $1 \leq i,j \leq d$, $(i,j) \neq (1,1)$. The coefficient $a^{11}$ is merely measurable locally either in $t$ or $x_1$. As functions of the remaining variables, the coefficients have small mean oscillations. We consider mixed norm Sobolev spaces with Muckenhoupt weights. Our results generalize previous work on parabolic equations with time fractional derivatives to a much larger class of coefficients and solution spaces.

math.AP↗
arXiv2022

Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms

We prove the Lp,q-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.

math.AP↗
arXiv2022

Parabolic Systems with measurable coefficients in weighted Sobolev spaces

In this paper we present a weighted $L_p$-theory of parabolic systems on a half space. The leading coefficients are assumed to be only measurable in $t$ and have small bounded mean oscillations (BMO) with respect to $x$, and the lower order coefficients are allowed to blow up near the boundary.

math.AP↗
arXiv2022

Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients

We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm $L_{s,q}$-estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with $s=q=2$, the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

math.AP↗
arXiv2021

Boundedness of non-local operators with spatially dependent coefficients and $L_p$-estimates for non-local equations

We prove the boundedness of the non-local operator \[ \mathcal{L}^a u(x)=\int_{\mathbb{R}^d} \left(u(x+y)-u(x)-χ_α(y)\big(\nabla u(x),y\big)\right) a(x,y)\frac{dy}{|y|^{d+α}} \] from $H_{p,w}^α(\mathbb{R}^d)$ to $L_{p,w}(\mathbb{R}^d)$ for the whole range of $p \in (1,\infty)$, where $w$ is a Muckenhoupt weight. The coefficient $a(x,y)$ is bounded, merely measurable in $y$, and Hölder continuous in $x$ with an arbitrarily small exponent. We extend the previous results by removing the largeness assumption on $p$ as well as considering weighted spaces with Muckenhoupt weights. Using the boundedness result, we prove the unique solvability in $L_p$ spaces of the corresponding parabolic and elliptic non-local equations.

math.AP↗
arXiv2021

Time fractional parabolic equations with measurable coefficients and embeddings for fractional parabolic Sobolev spaces

We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.

math.AP↗
arXiv2020

Deep Transfer Learning for Automated Diagnosis of Skin Lesions from Photographs

Melanoma is not the most common form of skin cancer, but it is the most deadly. Currently, the disease is diagnosed by expert dermatologists, which is costly and requires timely access to medical treatment. Recent advances in deep learning have the potential to improve diagnostic performance, expedite urgent referrals and reduce burden on clinicians. Through smart phones, the technology could reach people who would not normally have access to such healthcare services, e.g. in remote parts of the world, due to financial constraints or in 2020, COVID-19 cancellations. To this end, we have investigated various transfer learning approaches by leveraging model parameters pre-trained on ImageNet with finetuning on melanoma detection. We compare EfficientNet, MnasNet, MobileNet, DenseNet, SqueezeNet, ShuffleNet, GoogleNet, ResNet, ResNeXt, VGG and a simple CNN with and without transfer learning. We find the mobile network, EfficientNet (with transfer learning) achieves the best mean performance with an area under the receiver operating characteristic curve (AUROC) of 0.931$\pm$0.005 and an area under the precision recall curve (AUPRC) of 0.840$\pm$0.010. This is significantly better than general practitioners (0.83$\pm$0.03 AUROC) and dermatologists (0.91$\pm$0.02 AUROC).

cs.CV↗
arXiv2020

An approach for weighted mixed-norm estimates for parabolic equations with local and non-local time derivatives

We give a unified approach to weighted mixed-norm estimates and solvability for both the usual and time fractional parabolic equations in nondivergence form when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend the previous result in [6] for unmixed $L_p$-estimates without weights.

math.AP↗