Youngjoon Hong

Sanghyeon Kim, Hyunmo Yang, Younghyun Kim et al.

The recent surge in large-scale foundation models has spurred the development of efficient methods for adapting these models to various downstream tasks. Low-rank adaptation methods, such as LoRA, have gained significant attention due to their outstanding parameter efficiency and no additional inference latency. This paper investigates a more general form of adapter module based on the analysis that parallel and sequential adaptation branches learn novel and general features during fine-tuning, respectively. The proposed method, named Hydra, due to its multi-head computational branches, combines parallel and sequential branch to integrate capabilities, which is more expressive than existing single branch methods and enables the exploration of a broader range of optimal points in the fine-tuning process. In addition, the proposed adaptation method explicitly leverages the pre-trained weights by performing a linear combination of the pre-trained features. It allows the learned features to have better generalization performance across diverse downstream tasks. Furthermore, we perform a comprehensive analysis of the characteristics of each adaptation branch with empirical evidence. Through an extensive range of experiments, encompassing comparisons and ablation studies, we substantiate the efficiency and demonstrate the superior performance of Hydra. This comprehensive evaluation underscores the potential impact and effectiveness of Hydra in a variety of applications. Our code is available on \url{https://github.com/extremebird/Hydra}

Youngjoon Hong

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.

Namgyu Kang, Jaemin Oh, Youngjoon Hong et al.

The numerical approximation of partial differential equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and nonlinear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive bias of MLPs. However, they usually require high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting. In addition, the fixed positions of the mesh parameters restrict their flexibility, making accurate approximation of complex PDEs challenging. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/

Youngjoon Hong, Seungchan Ko, Jaeyong Lee

In this paper, we provide a theoretical analysis of a type of operator learning method without data reliance based on the classical finite element approximation, which is called the finite element operator network (FEONet). We first establish the convergence of this method for general second-order linear elliptic PDEs with respect to the parameters for neural network approximation. In this regard, we address the role of the condition number of the finite element matrix in the convergence of the method. Secondly, we derive an explicit error estimate for the self-adjoint case. For this, we investigate some regularity properties of the solution in certain function classes for a neural network approximation, verifying the sufficient condition for the solution to have the desired regularity. Finally, we will also conduct some numerical experiments that support the theoretical findings, confirming the role of the condition number of the finite element matrix in the overall convergence.

Jaemin Oh, Jinsil Lee, Youngjoon Hong

Nudging is an empirical data assimilation technique that incorporates an observation-driven control term into the model dynamics. The trajectory of the nudged system approaches the true system trajectory over time, even when the initial conditions differ. For linear state space models, such control terms can be derived under mild assumptions. However, designing effective nudging terms becomes significantly more challenging in the nonlinear setting. In this work, we propose neural network nudging, a data-driven method for learning nudging terms in nonlinear state space models. We establish a theoretical existence result based on the Kazantzis--Kravaris--Luenberger observer theory. The proposed approach is evaluated on three benchmark problems that exhibit chaotic behavior: the Lorenz 96 model, the Kuramoto--Sivashinsky equation, and the Kolmogorov flow.

Bryce Chudomelka, Youngjoon Hong, Hyunwoo Kim et al.

Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is associated with solving these systems. We explore the performance and accuracy of various neural architectures on both linear and nonlinear differential equations by creating accurate training sets with the spectral element method. Next, we implement a novel Legendre-Galerkin Deep Neural Network (LGNet) algorithm to predict solutions to differential equations. By constructing a set of a linear combination of the Legendre basis, we predict the corresponding coefficients, $α_i$ which successfully approximate the solution as a sum of smooth basis functions $u \simeq \sum_{i=0}^{N} α_i \varphi_i$. As a computational example, linear and nonlinear models with Dirichlet or Neumann boundary conditions are considered.

Byungjoo Kim, Bryce Chudomelka, Jinyoung Park et al.

Deep neural networks have achieved state-of-the-art performance in a variety of fields. Recent works observe that a class of widely used neural networks can be viewed as the Euler method of numerical discretization. From the numerical discretization perspective, Strong Stability Preserving (SSP) methods are more advanced techniques than the explicit Euler method that produce both accurate and stable solutions. Motivated by the SSP property and a generalized Runge-Kutta method, we propose Strong Stability Preserving networks (SSP networks) which improve robustness against adversarial attacks. We empirically demonstrate that the proposed networks improve the robustness against adversarial examples without any defensive methods. Further, the SSP networks are complementary with a state-of-the-art adversarial training scheme. Lastly, our experiments show that SSP networks suppress the blow-up of adversarial perturbations. Our results open up a way to study robust architectures of neural networks leveraging rich knowledge from numerical discretization literature.

Youngjoon Hong, Bongsuk Kwon, Byung-Jun Yoon

We consider the optimal experimental design problem for an uncertain Kuramoto model, which consists of N interacting oscillators described by coupled ordinary differential equations. The objective is to design experiments that can effectively reduce the uncertainty present in the coupling strengths between the oscillators, thereby minimizing the cost of robust control of the uncertain Kuramoto model. We demonstrate the importance of quantifying the operational impact of the potential experiments in designing optimal experiments.

Youngjoon Hong, Djoko Wirosoetisno

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.