13 papers · 1 filter
ResiPhy-MDNF: A Residual-Based Physics-Aware Multilevel Discrete Neural Field Framework for PDE-Constrained Inverse Problems
Zheng Lu, Jiwei Jia, Young Ju Lee
Inverse problems governed by partial differential equations are difficult when observations are sparse and the unknown coefficient field contains both large- and small-scale struct…
McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation
Jiwei Jia, Xinliang Liu, Juntao Wang +1
Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coar…
Second-Order Area/Volume-Preserving PFEMs for Surface Diffusion via Simpson--Boole Geometric Identities
Zhiqing Pan, Jiwei Jia, Lian Zhang
We propose second-order-in-time parametric finite element methods for surface diffusion of closed curves in two dimensions and closed surfaces in three dimensions. The construction…
Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems
Kuilin Qin, Lianfang Wang, Xu Sun +4
Operator learning is an emerging field at the intersection of machine learning and scientific computing. By learning mappings between function spaces, neural operators provide data…
Boundary neuron method for solving partial differential equations
Ye Lin, Wentao Liu, Young Ju Lee +1
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 net…
Born-Preconditioned Learning for Iterative Neural PDE Solvers
Juntao Wang, Jiwei Jia, Xinliang Liu
Simulating high-frequency waves in spatially varying media remains challenging for both classical iterative methods and neural PDE solvers. We propose Neural Preconditioned Born Se…
R-PINN: Recovery-type a-posteriori estimator enhanced adaptive PINN
Rongxin Lu, Jiwei Jia, Young Ju Lee +2
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 appli…
Orthogonal greedy algorithm for linear operator learning with shallow neural network
Ye Lin, Jiwei Jia, Young Ju Lee +1
Greedy algorithms, particularly the orthogonal greedy algorithm (OGA), have proven effective in training shallow neural networks for fitting functions and solving partial different…
Greedy Algorithm for Neural Networks for Indefinite Elliptic Problems
Qingguo Hong, Jiwei Jia, Young Ju Lee +1
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 th…
fOGA: An Orthogonal Greedy Algorithm for Fractional Laplacian Problems
Ruitong Shan, Young Ju Lee, Jiwei Jia
In this paper, we propose a numerical method for fractional Laplace equations that combines finite difference discretization with shallow neural network approximation. The fraction…
The pressure-robust weak Galerkin finite element method for Stokes-Darcy problem
Jiwei Jia, Lin Yang, Qilong Zhai
In this paper, we propose a pressure-robust weak Galerkin (WG) finite element scheme to solve the Stokes-Darcy problem. To construct the pressure-robust numerical scheme, we use th…
Green Multigrid Network
Ye Lin, Young Ju Lee, Jiwei Jia
GreenLearning networks (GL) directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential e…
An Unconstrained Formulation of Some Constrained Partial Differential Equations and its Application to Finite Neuron Methods
Jiwei Jia, Young Ju Lee, Ruitong Shan
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 o…