arXiv subjects

Hyung Ju Hwang

Publications and source records attributed to Hyung Ju Hwang.

At least 19 recordsLinked to original sources

arXiv2026

FourierSpecNet: Neural Collision Operator Approximation Inspired by the Fourier Spectral Method for Solving the Boltzmann Equation

The Boltzmann equation, a fundamental model in kinetic theory, describes the evolution of particle distribution functions through a nonlinear, high-dimensional collision operator. However, its numerical solution remains computationally demanding, particularly for inelastic collisions and high-dimensional velocity domains. In this work, we propose the Fourier Neural Spectral Network (FourierSpecNet), a hybrid framework that integrates the Fourier spectral method with deep learning to approximate the collision operator in Fourier space efficiently. FourierSpecNet achieves resolution-invariant learning and supports zero-shot super-resolution, enabling accurate predictions at unseen resolutions without retraining. Beyond empirical validation, we establish a consistency result showing that the trained operator converges to the spectral solution as the discretization is refined. We evaluate our method on several benchmark cases, including Maxwellian and hard-sphere molecular models, as well as inelastic collision scenarios. The results demonstrate that FourierSpecNet offers competitive accuracy while significantly reducing computational cost compared to traditional spectral solvers. Our approach provides a robust and scalable alternative for solving the Boltzmann equation across both elastic and inelastic regimes.

cs.LG↗

arXiv2025

Learning from Imperfect Data: Robust Inference of Dynamic Systems using Simulation-based Generative Model

System inference for nonlinear dynamic models, represented by ordinary differential equations (ODEs), remains a significant challenge in many fields, particularly when the data are noisy, sparse, or partially observable. In this paper, we propose a Simulation-based Generative Model for Imperfect Data (SiGMoID) that enables precise and robust inference for dynamic systems. The proposed approach integrates two key methods: (1) physics-informed neural networks with hyper-networks that constructs an ODE solver, and (2) Wasserstein generative adversarial networks that estimates ODE parameters by effectively capturing noisy data distributions. We demonstrate that SiGMoID quantifies data noise, estimates system parameters, and infers unobserved system components. Its effectiveness is validated validated through realistic experimental examples, showcasing its broad applicability in various domains, from scientific research to engineered systems, and enabling the discovery of full system dynamics.

cs.LG↗

arXiv2024

Estimation of System Parameters Including Repeated Cross-Sectional Data through Emulator-Informed Deep Generative Model

Differential equations (DEs) are crucial for modeling the evolution of natural or engineered systems. Traditionally, the parameters in DEs are adjusted to fit data from system observations. However, in fields such as politics, economics, and biology, available data are often independently collected at distinct time points from different subjects (i.e., repeated cross-sectional (RCS) data). Conventional optimization techniques struggle to accurately estimate DE parameters when RCS data exhibit various heterogeneities, leading to a significant loss of information. To address this issue, we propose a new estimation method called the emulator-informed deep-generative model (EIDGM), designed to handle RCS data. Specifically, EIDGM integrates a physics-informed neural network-based emulator that immediately generates DE solutions and a Wasserstein generative adversarial network-based parameter generator that can effectively mimic the RCS data. We evaluated EIDGM on exponential growth, logistic population models, and the Lorenz system, demonstrating its superior ability to accurately capture parameter distributions. Additionally, we applied EIDGM to an experimental dataset of Amyloid beta 40 and beta 42, successfully capturing diverse parameter distribution shapes. This shows that EIDGM can be applied to model a wide range of systems and extended to uncover the operating principles of systems based on limited data.

cs.LG↗

arXiv2024

Estimating the Distribution of Parameters in Differential Equations with Repeated Cross-Sectional Data

Differential equations are pivotal in modeling and understanding the dynamics of various systems, offering insights into their future states through parameter estimation fitted to time series data. In fields such as economy, politics, and biology, the observation data points in the time series are often independently obtained (i.e., Repeated Cross-Sectional (RCS) data). With RCS data, we found that traditional methods for parameter estimation in differential equations, such as using mean values of time trajectories or Gaussian Process-based trajectory generation, have limitations in estimating the shape of parameter distributions, often leading to a significant loss of data information. To address this issue, we introduce a novel method, Estimation of Parameter Distribution (EPD), providing accurate distribution of parameters without loss of data information. EPD operates in three main steps: generating synthetic time trajectories by randomly selecting observed values at each time point, estimating parameters of a differential equation that minimize the discrepancy between these trajectories and the true solution of the equation, and selecting the parameters depending on the scale of discrepancy. We then evaluated the performance of EPD across several models, including exponential growth, logistic population models, and target cell-limited models with delayed virus production, demonstrating its superiority in capturing the shape of parameter distributions. Furthermore, we applied EPD to real-world datasets, capturing various shapes of parameter distributions rather than a normal distribution. These results effectively address the heterogeneity within systems, marking a substantial progression in accurately modeling systems using RCS data.

stat.ML↗

arXiv2024

Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations

Learning the mapping between two function spaces has garnered considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural operator (FNO) was recently proposed to learn solution operators, and it achieved an excellent performance. In this study, we propose a novel \textit{pseudo-differential integral operator} (PDIO) to analyze and generalize the Fourier integral operator in FNO. PDIO is inspired by a pseudo-differential operator, which is a generalized differential operator characterized by a certain symbol. We parameterize this symbol using a neural network and demonstrate that the neural network-based symbol is contained in a smooth symbol class. Subsequently, we verify that the PDIO is a bounded linear operator, and thus is continuous in the Sobolev space. We combine the PDIO with the neural operator to develop a \textit{pseudo-differential neural operator} (PDNO) and learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by utilizing Darcy flow and the Navier-Stokes equation. The obtained results indicate that the proposed PDNO outperforms the existing neural operator approaches in most experiments.

cs.LG↗

arXiv2024

Sobolev Training for Operator Learning

This study investigates the impact of Sobolev Training on operator learning frameworks for improving model performance. Our research reveals that integrating derivative information into the loss function enhances the training process, and we propose a novel framework to approximate derivatives on irregular meshes in operator learning. Our findings are supported by both experimental evidence and theoretical analysis. This demonstrates the effectiveness of Sobolev Training in approximating the solution operators between infinite-dimensional spaces.

cs.LG↗

arXiv2024

Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids

Scientific computing using deep learning has seen significant advancements in recent years. There has been growing interest in models that learn the operator from the parameters of a partial differential equation (PDE) to the corresponding solutions. Deep Operator Network (DeepONet) and Fourier Neural operator, among other models, have been designed with structures suitable for handling functions as inputs and outputs, enabling real-time predictions as surrogate models for solution operators. There has also been significant progress in the research on surrogate models based on graph neural networks (GNNs), specifically targeting the dynamics in time-dependent PDEs. In this paper, we propose GraphDeepONet, an autoregressive model based on GNNs, to effectively adapt DeepONet, which is well-known for successful operator learning. GraphDeepONet exhibits robust accuracy in predicting solutions compared to existing GNN-based PDE solver models. It maintains consistent performance even on irregular grids, leveraging the advantages inherited from DeepONet and enabling predictions on arbitrary grids. Additionally, unlike traditional DeepONet and its variants, GraphDeepONet enables time extrapolation for time-dependent PDE solutions. We also provide theoretical analysis of the universal approximation capability of GraphDeepONet in approximating continuous operators across arbitrary time intervals.

cs.LG↗

arXiv2023

HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork

Fast and accurate predictions for complex physical dynamics are a significant challenge across various applications. Real-time prediction on resource-constrained hardware is even more crucial in real-world problems. The deep operator network (DeepONet) has recently been proposed as a framework for learning nonlinear mappings between function spaces. However, the DeepONet requires many parameters and has a high computational cost when learning operators, particularly those with complex (discontinuous or non-smooth) target functions. This study proposes HyperDeepONet, which uses the expressive power of the hypernetwork to enable the learning of a complex operator with a smaller set of parameters. The DeepONet and its variant models can be thought of as a method of injecting the input function information into the target function. From this perspective, these models can be viewed as a particular case of HyperDeepONet. We analyze the complexity of DeepONet and conclude that HyperDeepONet needs relatively lower complexity to obtain the desired accuracy for operator learning. HyperDeepONet successfully learned various operators with fewer computational resources compared to other benchmarks.

cs.LG↗

arXiv2023

Solving PDE-constrained Control Problems Using Operator Learning

The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.

math.OC↗

arXiv2023

The Deep Minimizing Movement Scheme

Solutions of certain partial differential equations (PDEs) are often represented by the steepest descent curves of corresponding functionals. Minimizing movement scheme was developed in order to study such curves in metric spaces. Especially, Jordan-Kinderlehrer-Otto studied the Fokker-Planck equation in this way with respect to the Wasserstein metric space. In this paper, we propose a deep learning-based minimizing movement scheme for approximating the solutions of PDEs. The proposed method is highly scalable for high-dimensional problems as it is free of mesh generation. We demonstrate through various kinds of numerical examples that the proposed method accurately approximates the solutions of PDEs by finding the steepest descent direction of a functional even in high dimensions.

math.NA↗

arXiv2023

Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)

Physics-Informed Neural Networks (PINNs) have become a prominent application of deep learning in scientific computation, as they are powerful approximators of solutions to nonlinear partial differential equations (PDEs). There have been numerous attempts to facilitate the training process of PINNs by adjusting the weight of each component of the loss function, called adaptive loss-balancing algorithms. In this paper, we propose an Augmented Lagrangian relaxation method for PINNs (AL-PINNs). We treat the initial and boundary conditions as constraints for the optimization problem of the PDE residual. By employing Augmented Lagrangian relaxation, the constrained optimization problem becomes a sequential max-min problem so that the learnable parameters $λ$ adaptively balance each loss component. Our theoretical analysis reveals that the sequence of minimizers of the proposed loss functions converges to an actual solution for the Helmholtz, viscous Burgers, and Klein--Gordon equations. We demonstrate through various numerical experiments that AL-PINNs yield a much smaller relative error compared with that of state-of-the-art adaptive loss-balancing algorithms.

cs.LG↗

arXiv2022

opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation

We propose a hybrid framework opPINN: physics-informed neural network (PINN) with operator learning for approximating the solution to the Fokker-Planck-Landau (FPL) equation. The opPINN framework is divided into two steps: Step 1 and Step 2. After the operator surrogate models are trained during Step 1, PINN can effectively approximate the solution to the FPL equation during Step 2 by using the pre-trained surrogate models. The operator surrogate models greatly reduce the computational cost and boost PINN by approximating the complex Landau collision integral in the FPL equation. The operator surrogate models can also be combined with the traditional numerical schemes. It provides a high efficiency in computational time when the number of velocity modes becomes larger. Using the opPINN framework, we provide the neural network solutions for the FPL equation under the various types of initial conditions, and interaction models in two and three dimensions. Furthermore, based on the theoretical properties of the FPL equation, we show that the approximated neural network solution converges to the a priori classical solution of the FPL equation as the pre-defined loss function is reduced.

math.NA↗

arXiv2022

Machine Learning-Based COVID-19 Patients Triage Algorithm using Patient-Generated Health Data from Nationwide Multicenter Database

A prompt severity assessment model of patients confirmed for having infectious diseases could enable efficient diagnosis while alleviating burden on the medical system. This study aims to develop a SARS-CoV-2 severity assessment model and establish a medical system that allows patients to check the severity of their cases and informs them to visit the appropriate clinic center based on past treatment data of other patients with similar severity levels. This paper provides the development processes of a severity assessment model using machine learning techniques and its application on SARS-CoV-2 patients. The proposed model is trained on a nationwide dataset provided by a Korean government agency and only requires patients' basic personal data, allowing them to judge the severity of their own cases. After modeling, the boosting-based decision tree model was selected as the classifier while mortality rate was interpreted as the probability score. The dataset was collected from all Korean citizens who were confirmed with COVID-19 between February, 2020 and July, 2021. The experiments achieved high model performance with an approximate precision of $0{\cdot}923$ and AUROC score of $0{\cdot}950$ [$95$% Tolerance Interval $0{\cdot}940$-$0{\cdot}958$, $95$% Confidence Interval $0{\cdot}949$-$0{\cdot}950$]. Moreover, our experiments identified the most important variables affecting the severity in the model via sensitivity analysis. The prompt severity assessment model for managing infectious people has been attained through using a nationwide dataset. It has demonstrated its superior performance by surpassing that of conventional risk assessments. With the model's high performance and easily accessible features, the triage algorithm is expected to be particularly useful when patients monitor their health status by themselves through smartphone applications.

cs.LG↗

arXiv2021

NEAR: Neighborhood Edge AggregatoR for Graph Classification

Learning graph-structured data with graph neural networks (GNNs) has been recently emerging as an important field because of its wide applicability in bioinformatics, chemoinformatics, social network analysis and data mining. Recent GNN algorithms are based on neural message passing, which enables GNNs to integrate local structures and node features recursively. However, past GNN algorithms based on 1-hop neighborhood neural message passing are exposed to a risk of loss of information on local structures and relationships. In this paper, we propose Neighborhood Edge AggregatoR (NEAR), a framework that aggregates relations between the nodes in the neighborhood via edges. NEAR, which can be orthogonally combined with Graph Isomorphism Network (GIN), gives integrated information that describes which nodes in the neighborhood are connected. Therefore, NEAR can reflect additional information of a local structure of each node beyond the nodes themselves in 1-hop neighborhood. Experimental results on multiple graph classification tasks show that our algorithm makes a good improvement over other existing 1-hop based GNN-based algorithms.

cs.LG↗

arXiv2021

Prior Preference Learning from Experts:Designing a Reward with Active Inference

Active inference may be defined as Bayesian modeling of a brain with a biologically plausible model of the agent. Its primary idea relies on the free energy principle and the prior preference of the agent. An agent will choose an action that leads to its prior preference for a future observation. In this paper, we claim that active inference can be interpreted using reinforcement learning (RL) algorithms and find a theoretical connection between them. We extend the concept of expected free energy (EFE), which is a core quantity in active inference, and claim that EFE can be treated as a negative value function. Motivated by the concept of prior preference and a theoretical connection, we propose a simple but novel method for learning a prior preference from experts. This illustrates that the problem with inverse RL can be approached with a new perspective of active inference. Experimental results of prior preference learning show the possibility of active inference with EFE-based rewards and its application to an inverse RL problem.

cs.LG↗

arXiv2021

Sobolev Training for Physics Informed Neural Networks

Physics Informed Neural Networks (PINNs) is a promising application of deep learning. The smooth architecture of a fully connected neural network is appropriate for finding the solutions of PDEs; the corresponding loss function can also be intuitively designed and guarantees the convergence for various kinds of PDEs. However, the rate of convergence has been considered as a weakness of this approach. This paper proposes Sobolev-PINNs, a novel loss function for the training of PINNs, making the training substantially efficient. Inspired by the recent studies that incorporate derivative information for the training of neural networks, we develop a loss function that guides a neural network to reduce the error in the corresponding Sobolev space. Surprisingly, a simple modification of the loss function can make the training process similar to \textit{Sobolev Training} although PINNs is not a fully supervised learning task. We provide several theoretical justifications that the proposed loss functions upper bound the error in the corresponding Sobolev spaces for the viscous Burgers equation and the kinetic Fokker--Planck equation. We also present several simulation results, which show that compared with the traditional $L^2$ loss function, the proposed loss function guides the neural network to a significantly faster convergence. Moreover, we provide the empirical evidence that shows that the proposed loss function, together with the iterative sampling techniques, performs better in solving high dimensional PDEs.

math.NA↗

arXiv2021

Traveling Wave Solutions of Partial Differential Equations via Neural Networks

This paper focuses on how to approximate traveling wave solutions for various kinds of partial differential equations via artificial neural networks. A traveling wave solution is hard to obtain with traditional numerical methods when the corresponding wave speed is unknown in advance. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter. We proved that under a mild assumption, the neural network solution converges to the analytic solution and the free parameter accurately approximates the wave speed as the corresponding loss tends to zero for the Keller-Segel equation. We also demonstrate in the experiments that reducing loss through training assures an accurate approximation of the traveling wave solution and the wave speed for the Keller-Segel equation, the Allen-Cahn model with relaxation, and the Lotka-Volterra competition model.

math.NA↗

arXiv2021

Lagrangian dual framework for conservative neural network solutions of kinetic equations

In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.

math.NA↗