arXiv subjects

Young Ju Lee

Publications and source records attributed to Young Ju Lee.

17 recordsLinked to original sources

arXiv2026

ResiPhy-MDNF: A Residual-Based Physics-Aware Multilevel Discrete Neural Field Framework for PDE-Constrained Inverse Problems

Inverse problems governed by partial differential equations are difficult when observations are sparse and the unknown coefficient field contains both large- and small-scale structures. We introduce a residual-based, physics-aware multilevel discrete neural field framework, ResiPhy-MDNF, for such problems. The method couples a coarse-to-fine discrete neural field (DNF) optimizer with a residual-based graph neural network (GNN) transfer operator, called ResiPhy-GNN. At each level, the DNF directly optimizes trainable grid- or mesh-based state and coefficient arrays using the prescribed numerical model. Between levels, ResiPhy-GNN maps the coarse representation to the fine representation through a learned prolongation based on graph connectivity, spatial features, and residual information. The method requires neither surrogate models nor offline pretraining. We evaluate the framework on coefficient inversion in Darcy flow for subsurface modeling and on electrical impedance tomography (EIT). In the controlled Darcy test case, the $64^2\!\to128^2$ multilevel path uses $1.25\times$ the cumulative grid-work proxy of the direct single-level $128^2$ solve, while achieving $6.76\times$ lower permeability error and $10.4\times$ lower state error. On measured Kuopio Tomography Challenge 2023 EIT data, the framework improves the mean intersection-over-union after Otsu thresholding by roughly $3.4\%$ over the official linearized complete electrode model reconstruction and by $16.9\%$ over direct single-level discrete-field optimization. These results indicate that the same multilevel construction can be used across different coefficient structures, discretizations, and observation geometries.

math.NA↗

arXiv2026

Invariant Guided PINN for Fluid Flow Computation

Physics-informed neural networks (PINNs) often become difficult to optimize for incompressible flow problems with large spatial domains, multiscale stresses, or long-time invariant dynamics. We propose an invariant-guided PINN (IG-PINN) framework that uses partitioned training as a conservative preconditioning stage rather than as the final piecewise representation. A globally defined architecture is trained successively on spatial subdomains or temporal slabs; selected field traces, structural information, and conservative diagnostics are then transferred to a final global correction, yielding a single neural field on the full spatial or space-time domain. The framework is tested on two incompressible flow problems: steady Oldroyd--B flow past a confined cylinder and a rotational Newtonian flow with helicity diagnostics. In the Oldroyd--B case, IG-PINN transfers velocity, polymeric stress, and mass-flux information while avoiding pressure traces at artificial interfaces. In the helicity case, endpoint velocity is transferred through a hard temporal constraint and kinetic energy is controlled during slab training and residual global correction. The experiments demonstrate improved optimization robustness, reduced conservation errors for the cylinder wake, and controlled energy and helicity diagnostics for the transient rotational flow.

physics.flu-dyn↗

arXiv2026

Beyond binary scission: a generalized three-species cascade breakage model for wormlike micellar solutions

Wormlike micellar fluids exhibit complex rheological behavior driven by the continuous breakage and recombination of self-assembled micellar networks. Existing two-species models provide a coarse binary representation of the micellar population, limiting their ability to resolve intermediate structural states and broad relaxation spectra. To address this limitation, we develop a three-species cascade breakage model consisting of gel-network, long chains, and short chains. By introducing an intermediate micellar state, the model links the rapid relaxation of short fragments to the slow recovery of the gel-network within a unified kinetic framework. This additional structural pathway gives rise to a three-mode viscoelastic response, improves the high-frequency description of the dynamic moduli, and produces a non-monotone constitutive curve that evolves into a stress plateau with coexisting shear bands in Couette flow. This cascade mechanism also governs the transient response, including stress overshoot, hysteresis, and multistep relaxation after shear cessation. Overall, the proposed three-species model provides a physically interpretable framework for worm-like micellar shear banding, capturing the connection between cascade microstructural evolution, broad relaxation dynamics, and macroscopic flow localization.

cond-mat.soft↗

arXiv2026

fOGA: An Orthogonal Greedy Algorithm for Fractional Laplacian Problems

In this paper, we propose a numerical method for fractional Laplace equations that combines finite difference discretization with shallow neural network approximation. The fractional Laplace operator is discretized using a directional representation of Riemann--Liouville type, which leads to a finite difference approximation of the nonlocal operator. In two dimensions, the angular integral is approximated by a quadrature rule, and auxiliary points are introduced along each direction to facilitate the evaluation of the operator. Based on the resulting discrete system, the solution is then represented by a shallow neural network constructed through the orthogonal greedy algorithm (OGA).

math.NA↗

arXiv2026

Alternatives to the Laplacian for Scalable Spectral Clustering with Group Fairness Constraints

Recent research has focused on mitigating algorithmic bias in clustering by incorporating fairness constraints into algorithmic design. Notions such as disparate impact, community cohesion, and cost per population have been implemented to enforce equitable outcomes. Among these, group fairness (balance) ensures that each protected group is proportionally represented within every cluster. However, incorporating balance as a metric of fairness into spectral clustering algorithms has led to computational times that can be improved. This study aims to enhance the efficiency of spectral clustering algorithms by reformulating the constrained optimization problem using a new formulation derived from the Lagrangian method and the Sherman-Morrison-Woodbury (SMW) identity, resulting in the Fair-SMW algorithm. Fair-SMW employs three alternatives to the Laplacian matrix with different spectral gaps to generate multiple variations of Fair-SMW, achieving clustering solutions with comparable balance to existing algorithms while offering improved runtime performance. We present the results of Fair-SMW, evaluated using the Stochastic Block Model (SBM) to measure both runtime efficiency and balance across real-world network datasets, including LastFM, FacebookNet, Deezer, and German. We achieve an improvement in computation time that is twice as fast as the state-of-the-art, and also flexible enough to achieve twice as much balance.

cs.LG↗

arXiv2026

Boundary neuron method for solving partial differential equations

We propose a boundary neuron method with random features (BNM-RF) for solving partial differential equations. The method approximates the unknown boundary function by a shallow network within the boundary integral formulation. With randomly sampled and fixed hidden parameters, the computation reduces to a linear least squares problem for the output coefficients, which avoids gradient based nonconvex optimization. This construction retains the dimensionality reduction of boundary integral equations and the linear solution structure of the random feature method. For elliptic problems, we establish convergence analysis by combining kernel-based method with random feature approximation, and obtain error bounds on both the boundary and the interior solution. Numerical experiments on Laplace and Helmholtz problems, including interior and exterior cases, show that the proposed method achieves competitive accuracy relative to the boundary element method and favorable performance relative to boundary integral neural networks in the tested settings with only few neurons. Overall, the proposed method provides a practical framework for combining boundary integral equations with neural network for problems on complex geometries and unbounded domains.

math.NA↗

arXiv2026

R-PINN: Recovery-type a-posteriori estimator enhanced adaptive PINN

In recent years, with the advancements in machine learning and neural networks, algorithms using physics-informed neural networks (PINNs) to solve PDEs have gained widespread applications. While these algorithms are well-suited for a wide range of equations, they often exhibit suboptimal performance when applied to equations with large local gradients, resulting in substantial localized errors. To address this issue, this paper proposes an adaptive PINN algorithm designed to improve accuracy in such cases. The core idea of the algorithm is to adaptively adjust the distribution of collocation points based on the recovery-type a-posterior error of the current numerical solution, enabling a better approximation of the true solution. This approach is inspired by the adaptive finite element method. By combining the recovery-type a-posteriori estimator, a gradient-recovery estimator commonly used in the adaptive finite element method (FEM) with PINNs, we introduce the Recovery-type a-posteriori estimator enhanced adaptive PINN (R-PINN) and compare its performance with a typical adaptive PINN algorithm, FI-PINN. Our results demonstrate that R-PINN achieves faster convergence with fewer adaptive points and significantly outperforms in the cases with multiple regions of large errors than FI-PINN. Notably, our method is a hybrid numerical approach for solving partial differential equations, integrating adaptive FEM with PINNs.

math.NA↗

arXiv2025

Orthogonal greedy algorithm for linear operator learning with shallow neural network

Greedy algorithms, particularly the orthogonal greedy algorithm (OGA), have proven effective in training shallow neural networks for fitting functions and solving partial differential equations (PDEs). In this paper, we extend the application of OGA to the tasks of linear operator learning, which is equivalent to learning the kernel function through integral transforms. Firstly, a novel greedy algorithm is developed for kernel estimation rate in a new semi-inner product, which can be utilized to approximate the Green's function of linear PDEs from data. Secondly, we introduce the OGA for point-wise kernel estimation to further improve the approximation rate, achieving orders of accuracy improvement across various tasks and baseline models. In addition, we provide a theoretical analysis on the kernel estimation problem and the optimal approximation rates for both algorithms, establishing their efficacy and potential for future applications in PDEs and operator learning tasks.

math.NA↗

arXiv2024

Greedy Algorithm for Neural Networks for Indefinite Elliptic Problems

The paper presents a priori error analysis of the shallow neural network approximation to the solution to the indefinite elliptic equation and and cutting-edge implementation of the Orthogonal Greedy Algorithm (OGA) tailored to overcome the challenges of indefinite elliptic problems, which is a domain where conventional approaches often struggle due to nontraditional difficulties due to the lack of coerciveness. A rigorous a priori error analysis that shows the neural networks ability to approximate indefinite problems is confirmed numerically by OGA methods. We also present a discretization error analysis of the relevant numerical quadrature. In particular, massive numerical implementations are conducted to justify the theory, some of which showcase the OGAs superior performance in comparison to the traditional finite element method. This advancement illustrates the potential of neural networks enhanced by OGA to solve intricate computational problems more efficiently, thereby marking a significant leap forward in the application of machine learning techniques to mathematical problem-solving.

math.NA↗

arXiv2024

Green Multigrid Network

GreenLearning networks (GL) directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential equations (PDEs). For many PDEs, the corresponding Green's function exhibits asymptotic smoothness. In this paper, we propose a framework named Green Multigrid networks (GreenMGNet), an operator learning algorithm designed for a class of asymptotically smooth Green's functions. Compared with the pioneering GL, the new framework presents itself with better accuracy and efficiency, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, Green's function is modeled as a piecewise function to take into account its singular behavior in some parts of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output(AugNN) so that it can capture singularity accurately. Second, the asymptotic smoothness property of Green's function is used to leverage the Multi-Level Multi-Integration (MLMI) algorithm for both the training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectiveness of the proposed method. On average, GreenMGNet achieves $3.8\%$ to $39.15\%$ accuracy improvement. To match the accuracy level of GL, GreenMGNet requires only about $10\%$ of the full grid data, resulting in a $55.9\%$ and $92.5\%$ reduction in training time and GPU memory cost for one-dimensional test problems, and a $37.7\%$ and $62.5\%$ reduction for two-dimensional test problems.

math.NA↗

arXiv2024

An Unconstrained Formulation of Some Constrained Partial Differential Equations and its Application to Finite Neuron Methods

In this paper, we present a new framework how a PDE with constraints can be formulated into a sequence of PDEs with no constraints, whose solutions are convergent to the solution of the PDE with constraints. This framework is then used to build a novel finite neuron method to solve the 2nd order elliptic equations with the Dirichlet boundary condition. Our algorithm is the first algorithm, proven to lead to shallow neural network solutions with an optimal H1 norm error. We show that a widely used penalized PDE, which imposes the Dirichlet boundary condition weakly can be interpreted as the first element of the sequence of PDEs within our framework. Furthermore, numerically, we show that it may not lead to the solution with the optimal H1 norm error bound in general. On the other hand, we theoretically demonstrate that the second and later elements of a sequence of PDEs can lead to an adequate solution with the optimal H1 norm error bound. A number of sample tests are performed to confirm the effectiveness of the proposed algorithm and the relevant theory.

math.NA↗

arXiv2024

Helicity-conservative Physics-informed Neural Network Model for Navier-Stokes Equations

We design the helicity-conservative physics-informed neural network model for the Navier-Stokes equation in the ideal case. The key is to provide an appropriate PDE model as loss function so that its neural network solutions produce helicity conservation. Physics-informed neural network model is based on the strong form of PDE. We compare the proposed Physics-informed neural network model and a relevant helicity-conservative finite element method. We arrive at the conclusion that the strong form PDE is better suited for conservation issues. We also present theoretical justifications for helicity conservation as well as supporting numerical calculations.

physics.comp-ph↗

arXiv2023

Finite flocking time of the nonlinear Cucker--Smale model with Rayleigh friction type using the discrete $p$-Laplacian

The study of collective behavior in multi-agent systems has attracted the attention of many researchers due to its wide range of applications. Among them, the Cucker-Smale model was developed to study the phenomenon of flocking, and various types of extended models have been actively proposed and studied in recent decades. In this study, we address open questions of the Cucker--Smale model with norm-type Rayleigh friction: {\bf (i)} The positivity of the communication weight, {\bf (ii)} The convergence of the norm of the velocities of agents, {\bf (iii)} The direction of the velocities of agents. For problems (i) and (ii), we present the nonlinear Cucker--Smale model with norm-type Rayleigh friction, where the nonlinear Cucker--Smale model is generalized to a nonlinear model by applying a discrete $p$-Laplacian operator. For this model, we present conditions that guarantee that the norm for velocities of agents converges to 0 or a positive value, and we also show that the regular communication weight satisfies the conditions given in this study. In particular, we present a condition for the initial configuration to obtain that the norm of agent velocities converges to only some positive value. By contrast, problem (iii) is not solved by the norm-type nonlinear model. Thus, we propose a nonlinear Cucker--Smale model with a vector-type Rayleigh friction for problem (iii). In parallel to the first model, we show that the direction of the agents' velocities can be controlled by parameters in the nonlinear Cucker--Smale model with the vector-type Rayleigh friction.

math.DS↗

arXiv2023

Companion-Based Multi-Level Finite Element Method for Computing Multiple Solutions of Nonlinear Differential Equations

The use of nonlinear PDEs has led to significant advancements in various fields, such as physics, biology, ecology, and quantum mechanics. However, finding multiple solutions for nonlinear PDEs can be a challenging task, especially when suitable initial guesses are difficult to obtain. In this paper, we introduce a novel approach called the Companion-Based Multilevel finite element method (CBMFEM), which can efficiently and accurately generate multiple initial guesses for solving nonlinear elliptic semi-linear equations with polynomial nonlinear terms using finite element methods with conforming elements. We provide a theoretical analysis of the error estimate of finite element methods using an appropriate notion of isolated solutions, for the nonlinear elliptic equation with multiple solutions and present numerical results obtained using CBMFEM which are consistent with the theoretical analysis.

math.NA↗

arXiv2021

Existence and Stability of Global Solutions to a regularized Oldroyd-B Model in its Vorticity Formulation

We present a new regularized Oldroyd-B model in three dimensions which satisfies an energy estimate analogous to that of the standard model, and maintains the positive semi-definiteness of the conformation tensor. This results in the unique existence and stability of global solutions in a periodic domain. To be precise, given an initial velocity $u_0$ and initial conformation tensor $σ_0$, both with components in $H^2$, we obtain a velocity $u$ and conformation tensor $σ$ both with components in $C([0, T]; H^2)$ for all $T>0$. Assuming better regularity for the initial data allows us to obtain better regularity for the solutions. We treat both the diffusive and non-diffusive cases of the model. Notably, the regularization in the equation for the conformation tensor in our new model has been applied only to the velocity, rather than to the conformation tensor, unlike other available regularization techniques \cite{barrett;suli2009}. This is desired since the stress, and thus the conformation tensor, is typically less regular than the velocity for the creeping flow of non-Newtonian fluids. In \cite{constantinnote} the existence and regularity of solutions to the non-regularized two dimensional diffusive Oldroyd-B model was established. However, the proof cannot be generalized to three dimensions nor to the non-diffusive case. The proposed regularization overcomes these obstacles. Moreover, we show that the solutions in the diffusive case are stable in the $H^2$ norm. In the non-diffusive case, we are able to establish that the solutions are stable in the $L^2$ norm. Furthermore, we show that as the diffusivity parameter goes to zero, our solutions converge in the $L^2$ norm to the non-diffusive solution.

math.AP↗

arXiv2021

Locally conservative immersed finite element method for elliptic interface problems

In this paper, we introduce the locally conservative enriched immersed finite element method (EIFEM) to tackle the elliptic problem with interface. The immersed finite element is useful for handling interface with mesh unfit with the interface. However, all the currently available method under IFEM framework may not be designed to consider the flux conservation. We provide an efficient and effective remedy for this issue by introducing a local piecewise constant enrichment, which provides the locally conservative flux. We have also constructed and analyzed an auxiliary space preconditioner for the resulting system based on the application of algebraic multigrid method. The new observation in this work is that by imposing strong Dirichlet boundary condition for the standard IFEM part of EIFEM, we are able to remove the zero eigen-mode of the EIFEM system while still imposing the Dirichlet boundary condition weakly assigned to the piecewise constant enrichment part of EIFEM. A couple of issues relevant to the piecewise constant enrichment given for the mesh unfit to the interface has been discussed and clarified as well. Numerical tests are provided to confirm the theoretical development.

math.NA↗

arXiv2012

Axisymmetric Stokes equations in polygonal domains: regularity and finite element approximations

We study the regularity and finite element approximation of the axisymmetric Stokes problem on a polygonal domain $Ω$. In particular, taking into account the singular coefficients in the equation and non-smoothness of the domain, we establish the well-posedness and full regularity of the solution in new weighted Sobolev spaces $\maK^m_{μ, 1}(Ω)$. Using our a priori results, we give a specific construction of graded meshes on which the Taylor-Hood mixed method approximates singular solutions at the optimal convergence rate. Numerical tests are presented to confirm the theoretical results in the paper.

math.NA↗