arXiv subjects

Kookjin Lee

Publications and source records attributed to Kookjin Lee.

At least 37 records · Page 2Linked to original sources

arXiv2024

Graph Convolutions Enrich the Self-Attention in Transformers!

Transformers, renowned for their self-attention mechanism, have achieved state-of-the-art performance across various tasks in natural language processing, computer vision, time-series modeling, etc. However, one of the challenges with deep Transformer models is the oversmoothing problem, where representations across layers converge to indistinguishable values, leading to significant performance degradation. We interpret the original self-attention as a simple graph filter and redesign it from a graph signal processing (GSP) perspective. We propose a graph-filter-based self-attention (GFSA) to learn a general yet effective one, whose complexity, however, is slightly larger than that of the original self-attention mechanism. We demonstrate that GFSA improves the performance of Transformers in various fields, including computer vision, natural language processing, graph-level tasks, speech recognition, and code classification.

cs.LG↗

arXiv2024

The Reconstruction of the Space-Dependent Thermal Conductivity from Sparse Temperature Measurements

We present a novel method for reconstructing the thermal conductivity coefficient in 1D and 2D heat equations using moving sensors that dynamically traverse the domain to record sparse and noisy temperature measurements. We significantly reduce the computational cost associated with forward PDE evaluations by employing automatic differentiation, enabling a more efficient and scalable reconstruction process. This allows the inverse problem to be solved with fewer sensors and observations. Specifically, we demonstrate the successful reconstruction of thermal conductivity on the 1D circle and 2D torus, using one and four moving sensors, respectively, with their positions recorded over time. Our method incorporates sampling algorithms to compute confidence intervals for the reconstructed conductivity, improving robustness against measurement noise. Extensive numerical simulations of heat dynamics validate the efficacy of our approach, confirming both the accuracy and stability of the reconstructed thermal conductivity. Additionally, the method is thoroughly tested using large datasets from machine learning, allowing us to evaluate its performance across various scenarios and ensure its reliability. This approach provides a cost-effective and flexible solution for conductivity reconstruction from sparse measurements, making it a robust tool for solving inverse problems in complex domains.

math.NA↗

arXiv2024

MaD-Scientist: AI-based Scientist solving Convection-Diffusion-Reaction Equations Using Massive PINN-Based Prior Data

Large language models (LLMs), like ChatGPT, have shown that even trained with noisy prior data, they can generalize effectively to new tasks through in-context learning (ICL) and pre-training techniques. Motivated by this, we explore whether a similar approach can be applied to scientific foundation models (SFMs). Our methodology is structured as follows: (i) we collect low-cost physics-informed neural network (PINN)-based approximated prior data in the form of solutions to partial differential equations (PDEs) constructed through an arbitrary linear combination of mathematical dictionaries; (ii) we utilize Transformer architectures with self and cross-attention mechanisms to predict PDE solutions without knowledge of the governing equations in a zero-shot setting; (iii) we provide experimental evidence on the one-dimensional convection-diffusion-reaction equation, which demonstrate that pre-training remains robust even with approximated prior data, with only marginal impacts on test accuracy. Notably, this finding opens the path to pre-training SFMs with realistic, low-cost data instead of (or in conjunction with) numerical high-cost data. These results support the conjecture that SFMs can improve in a manner similar to LLMs, where fully cleaning the vast set of sentences crawled from the Internet is nearly impossible.

cs.LG↗

arXiv2024

FastLRNR and Sparse Physics Informed Backpropagation

We introduce Sparse Physics Informed Backpropagation (SPInProp), a new class of methods for accelerating backpropagation for a specialized neural network architecture called Low Rank Neural Representation (LRNR). The approach exploits the low rank structure within LRNR and constructs a reduced neural network approximation that is much smaller in size. We call the smaller network FastLRNR. We show that backpropagation of FastLRNR can be substituted for that of LRNR, enabling a significant reduction in complexity. We apply SPInProp to a physics informed neural networks framework and demonstrate how the solution of parametrized partial differential equations is accelerated.

cs.LG↗

arXiv2024

Parameterized Physics-informed Neural Networks for Parameterized PDEs

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Reynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P$^2$INNs). P$^2$INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P$^2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known "failure modes".

cs.LG↗

arXiv2024

Learning Flexible Body Collision Dynamics with Hierarchical Contact Mesh Transformer

Recently, many mesh-based graph neural network (GNN) models have been proposed for modeling complex high-dimensional physical systems. Remarkable achievements have been made in significantly reducing the solving time compared to traditional numerical solvers. These methods are typically designed to i) reduce the computational cost in solving physical dynamics and/or ii) propose techniques to enhance the solution accuracy in fluid and rigid body dynamics. However, it remains under-explored whether they are effective in addressing the challenges of flexible body dynamics, where instantaneous collisions occur within a very short timeframe. In this paper, we present Hierarchical Contact Mesh Transformer (HCMT), which uses hierarchical mesh structures and can learn long-range dependencies (occurred by collisions) among spatially distant positions of a body -- two close positions in a higher-level mesh correspond to two distant positions in a lower-level mesh. HCMT enables long-range interactions, and the hierarchical mesh structure quickly propagates collision effects to faraway positions. To this end, it consists of a contact mesh Transformer and a hierarchical mesh Transformer (CMT and HMT, respectively). Lastly, we propose a flexible body dynamics dataset, consisting of trajectories that reflect experimental settings frequently used in the display industry for product designs. We also compare the performance of several baselines using well-known benchmark datasets. Our results show that HCMT provides significant performance improvements over existing methods. Our code is available at https://github.com/yuyudeep/hcmt.

cs.LG↗

arXiv2024

PAC-FNO: Parallel-Structured All-Component Fourier Neural Operators for Recognizing Low-Quality Images

A standard practice in developing image recognition models is to train a model on a specific image resolution and then deploy it. However, in real-world inference, models often encounter images different from the training sets in resolution and/or subject to natural variations such as weather changes, noise types and compression artifacts. While traditional solutions involve training multiple models for different resolutions or input variations, these methods are computationally expensive and thus do not scale in practice. To this end, we propose a novel neural network model, parallel-structured and all-component Fourier neural operator (PAC-FNO), that addresses the problem. Unlike conventional feed-forward neural networks, PAC-FNO operates in the frequency domain, allowing it to handle images of varying resolutions within a single model. We also propose a two-stage algorithm for training PAC-FNO with a minimal modification to the original, downstream model. Moreover, the proposed PAC-FNO is ready to work with existing image recognition models. Extensively evaluating methods with seven image recognition benchmarks, we show that the proposed PAC-FNO improves the performance of existing baseline models on images with various resolutions by up to 77.1% and various types of natural variations in the images at inference.

cs.CV↗

arXiv2024

Not All Asians are the Same: A Disaggregated Approach to Identifying Anti-Asian Racism in Social Media

Recent policy initiatives have acknowledged the importance of disaggregating data pertaining to diverse Asian ethnic communities to gain a more comprehensive understanding of their current status and to improve their overall well-being. However, research on anti-Asian racism has thus far fallen short of properly incorporating data disaggregation practices. Our study addresses this gap by collecting 12-month-long data from X (formerly known as Twitter) that contain diverse sub-ethnic group representations within Asian communities. In this dataset, we break down anti-Asian toxic messages based on both temporal and ethnic factors and conduct a series of comparative analyses of toxic messages, targeting different ethnic groups. Using temporal persistence analysis, $n$-gram-based correspondence analysis, and topic modeling, this study provides compelling evidence that anti-Asian messages comprise various distinctive narratives. Certain messages targeting sub-ethnic Asian groups entail different topics that distinguish them from those targeting Asians in a generic manner or those aimed at major ethnic groups, such as Chinese and Indian. By introducing several techniques that facilitate comparisons of online anti-Asian hate towards diverse ethnic communities, this study highlights the importance of taking a nuanced and disaggregated approach for understanding racial hatred to formulate effective mitigation strategies.

cs.SI↗

arXiv2023

Reversible and irreversible bracket-based dynamics for deep graph neural networks

Recent works have shown that physics-inspired architectures allow the training of deep graph neural networks (GNNs) without oversmoothing. The role of these physics is unclear, however, with successful examples of both reversible (e.g., Hamiltonian) and irreversible (e.g., diffusion) phenomena producing comparable results despite diametrically opposed mechanisms, and further complications arising due to empirical departures from mathematical theory. This work presents a series of novel GNN architectures based upon structure-preserving bracket-based dynamical systems, which are provably guaranteed to either conserve energy or generate positive dissipation with increasing depth. It is shown that the theoretically principled framework employed here allows for inherently explainable constructions, which contextualize departures from theory in current architectures and better elucidate the roles of reversibility and irreversibility in network performance.

cs.LG↗

arXiv2023

Operator-learning-inspired Modeling of Neural Ordinary Differential Equations

Neural ordinary differential equations (NODEs), one of the most influential works of the differential equation-based deep learning, are to continuously generalize residual networks and opened a new field. They are currently utilized for various downstream tasks, e.g., image classification, time series classification, image generation, etc. Its key part is how to model the time-derivative of the hidden state, denoted dh(t)/dt. People have habitually used conventional neural network architectures, e.g., fully-connected layers followed by non-linear activations. In this paper, however, we present a neural operator-based method to define the time-derivative term. Neural operators were initially proposed to model the differential operator of partial differential equations (PDEs). Since the time-derivative of NODEs can be understood as a special type of the differential operator, our proposed method, called branched Fourier neural operator (BFNO), makes sense. In our experiments with general downstream tasks, our method significantly outperforms existing methods.

cs.LG↗

arXiv2023

Reduced-order modeling for parameterized PDEs via implicit neural representations

We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR), which models physics signals in a continuous manner and independent of spatial/temporal discretization. The proposed framework encodes PDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamics characterized by multiple PDE parameters. PNODE can be inferred by a hypernetwork to reduce the potential difficulties in learning PNODE due to a complex multilayer perceptron (MLP). The framework uses an INR to decode the latent dynamics and reconstruct accurate PDE solutions. Further, a physics-informed loss is also introduced to correct the prediction of unseen parameter instances. Incorporating the physics-informed loss also enables the model to be fine-tuned in an unsupervised manner on unseen PDE parameters. A numerical experiment is performed on a two-dimensional Burgers equation with a large variation of PDE parameters. We evaluate the proposed method at a large Reynolds number and obtain up to speedup of O(10^3) and ~1% relative error to the ground truth values.

math.NA↗

arXiv2023

Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks

In various engineering and applied science applications, repetitive numerical simulations of partial differential equations (PDEs) for varying input parameters are often required (e.g., aircraft shape optimization over many design parameters) and solvers are required to perform rapid execution. In this study, we suggest a path that potentially opens up a possibility for physics-informed neural networks (PINNs), emerging deep-learning-based solvers, to be considered as one such solver. Although PINNs have pioneered a proper integration of deep-learning and scientific computing, they require repetitive time-consuming training of neural networks, which is not suitable for many-query scenarios. To address this issue, we propose a lightweight low-rank PINNs containing only hundreds of model parameters and an associated hypernetwork-based meta-learning algorithm, which allows efficient approximation of solutions of PDEs for varying ranges of PDE input parameters. Moreover, we show that the proposed method is effective in overcoming a challenging issue, known as "failure modes" of PINNs.

cs.LG↗

arXiv2022

Time Series Forecasting with Hypernetworks Generating Parameters in Advance

Forecasting future outcomes from recent time series data is not easy, especially when the future data are different from the past (i.e. time series are under temporal drifts). Existing approaches show limited performances under data drifts, and we identify the main reason: It takes time for a model to collect sufficient training data and adjust its parameters for complicated temporal patterns whenever the underlying dynamics change. To address this issue, we study a new approach; instead of adjusting model parameters (by continuously re-training a model on new data), we build a hypernetwork that generates other target models' parameters expected to perform well on the future data. Therefore, we can adjust the model parameters beforehand (if the hypernetwork is correct). We conduct extensive experiments with 6 target models, 6 baselines, and 4 datasets, and show that our HyperGPA outperforms other baselines.

cs.LG↗

arXiv2022

Mining Causality from Continuous-time Dynamics Models: An Application to Tsunami Forecasting

Continuous-time dynamics models, such as neural ordinary differential equations, have enabled the modeling of underlying dynamics in time-series data and accurate forecasting. However, parameterization of dynamics using a neural network makes it difficult for humans to identify causal structures in the data. In consequence, this opaqueness hinders the use of these models in the domains where capturing causal relationships carries the same importance as accurate predictions, e.g., tsunami forecasting. In this paper, we address this challenge by proposing a mechanism for mining causal structures from continuous-time models. We train models to capture the causal structure by enforcing sparsity in the weights of the input layers of the dynamics models. We first verify the effectiveness of our method in the scenario where the exact causal-structures of time-series are known as a priori. We next apply our method to a real-world problem, namely tsunami forecasting, where the exact causal-structures are difficult to characterize. Experimental results show that the proposed method is effective in learning physically-consistent causal relationships while achieving high forecasting accuracy.

cs.LG↗

arXiv2022

Parameter-varying neural ordinary differential equations with partition-of-unity networks

In this study, we propose parameter-varying neural ordinary differential equations (NODEs) where the evolution of model parameters is represented by partition-of-unity networks (POUNets), a mixture of experts architecture. The proposed variant of NODEs, synthesized with POUNets, learn a meshfree partition of space and represent the evolution of ODE parameters using sets of polynomials associated to each partition. We demonstrate the effectiveness of the proposed method for three important tasks: data-driven dynamics modeling of (1) hybrid systems, (2) switching linear dynamical systems, and (3) latent dynamics for dynamical systems with varying external forcing.

cs.LG↗

arXiv2022

AdamNODEs: When Neural ODE Meets Adaptive Moment Estimation

Recent work by Xia et al. leveraged the continuous-limit of the classical momentum accelerated gradient descent and proposed heavy-ball neural ODEs. While this model offers computational efficiency and high utility over vanilla neural ODEs, this approach often causes the overshooting of internal dynamics, leading to unstable training of a model. Prior work addresses this issue by using ad-hoc approaches, e.g., bounding the internal dynamics using specific activation functions, but the resulting models do not satisfy the exact heavy-ball ODE. In this work, we propose adaptive momentum estimation neural ODEs (AdamNODEs) that adaptively control the acceleration of the classical momentum-based approach. We find that its adjoint states also satisfy AdamODE and do not require ad-hoc solutions that the prior work employs. In evaluation, we show that AdamNODEs achieve the lowest training loss and efficacy over existing neural ODEs. We also show that AdamNODEs have better training stability than classical momentum-based neural ODEs. This result sheds some light on adapting the techniques proposed in the optimization community to improving the training and inference of neural ODEs further. Our code is available at https://github.com/pmcsh04/AdamNODE.

cs.LG↗

arXiv2022

Projection-based model reduction of dynamical systems using space-time subspace and machine learning

This paper considers the creation of parametric surrogate models for applications in science and engineering where the goal is to predict high-dimensional spatiotemporal output quantities of interest, such as pressure, temperature and displacement fields. The proposed methodology develops a low-dimensional parametrization of these quantities of interest using space-time bases combining with machine learning methods. In particular, the space-time solutions are sought in a low-dimensional space-time linear trial subspace that can be obtained by computing tensor decompositions of usual state-snapshots data. The mapping between the input parameters and the basis expansion coefficients (or generalized coordinates) is approximated using four different machine learning techniques: multivariate polynomial regression, k-nearest-neighbors, random forest and neural network. The relative costs and effectiveness of the four machine learning techniques are explored through three engineering problems: steady heat conduction, unsteady heat conduction and unsteady advective-diffusive-reactive system. Numerical results demonstrate that the proposed method performs well in terms of both accuracy and computational cost, and highlight the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models. In addition, simpler machine learning techniques are seen to perform better than more elaborate ones.

physics.comp-ph↗

arXiv2022

Stochastic Galerkin methods for linear stability analysis of systems with parametric uncertainty

We present a method for linear stability analysis of systems with parametric uncertainty formulated in the stochastic Galerkin framework. Specifically, we assume that for a model partial differential equation, the parameter is given in the form of generalized polynomial chaos expansion. The stability analysis leads to the solution of a stochastic eigenvalue problem, and we wish to characterize the rightmost eigenvalue. We focus, in particular, on problems with nonsymmetric matrix operators, for which the eigenvalue of interest may be a complex conjugate pair, and we develop methods for their efficient solution. These methods are based on inexact, line-search Newton iteration, which entails use of preconditioned GMRES. The method is applied to linear stability analysis of Navier-Stokes equation with stochastic viscosity, its accuracy is compared to that of Monte Carlo and stochastic collocation, and the efficiency is illustrated by numerical experiments.

math.NA↗