Jiwei Jia

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 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.

Ye Lin, Wentao Liu, Young Ju Lee et al.

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.

Juntao Wang, Xinliang Liu, Jiwei Jia

Loss functions for learning-based PDE preconditioners implicitly choose a \emph{metric} in which residuals are matched, yet most approaches still optimize an unpreconditioned Euclidean residual norm. For indefinite operators such as the high-frequency Helmholtz equation, this default metric can make both learning and iterative correction overly sensitive to near-resonant spectral components, while classical preconditioning succeeds precisely by reshaping the residual geometry. We show that the Born Series and shifted-Laplacian left preconditioning are linked by the identity $ I-G_ηV_η= G_ηA = L_η^{-1}A, $ which turns the reference Green operator $G_η$ into a natural Riesz-map residual metric $ R_η= G_η^\ast G_η$ and suggests measuring the physical residual via $ \|r\|_{R_η}=\|G_ηr\|_2. $ Building on this viewpoint, we propose a \emph{Neural Preconditioned Born Series} (NPBS) iteration that replaces the scalar CBS relaxation with a residual-driven neural operator, together with a metric-matched Born-series-inspired loss $\mathcal{L}_{\mathrm{bs}}^{R_η}$. The framework is architecture-agnostic and supports fast $\mathcal{O}(N\log N)$ evaluation via FFT/DST/DCT. Numerical experiments on heterogeneous Helmholtz problems demonstrate the effectiveness of our method, and its advantage becomes more pronounced as the systems grow more ill-conditioned; we then extend the framework to other PDE classes, including convection--diffusion--reaction equations and linearized Newton systems for nonlinear PDEs, where it also yields substantial iteration reductions.

Ye Lin, Jiwei Jia, Young Ju Lee et al.

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.

Zheng Lu, Young Ju Lee, Jiwei Jia et al.

This paper develops a helicity-aware physics-informed neural network framework for incompressible non-Newtonian flow in rotational form. In addition to the energy law and the incompressibility constraint, helicity is a fundamental geometric quantity that characterizes the topology of vortex lines and plays an important role in the physical fidelity of long-time flow simulations. While helicity-preserving discretizations have been studied extensively in finite difference, finite element, and other structure-preserving settings, their realization within neural network solvers remains largely unexplored. Motivated by this gap, we propose a neural formulation in which vorticity is computed directly from the neural velocity field by automatic differentiation rather than learned as an independent output, thereby avoiding compatibility errors that pollute the helicity balance. To improve robustness and scalability, we combine two algorithmic ingredients: an overlapping spatial domain decomposition inspired by finite-basis physics-informed neural networks (FBPINNs), and a causal slab-wise temporal continuation strategy for long-time transient simulations. The local subnetworks are blended by explicitly normalized super-Gaussian window functions, which yield a smooth partition of unity, while the temporal evolution is advanced sequentially across time slabs by transferring the converged solution on one slab to the next. The resulting spatiotemporal framework provides a stable and physically meaningful approach for helicity-aware simulation of incompressible non-Newtonian flows.