arXiv subjects

Jiwei Jia

Publications and source records attributed to Jiwei Jia.

At least 19 recordsLinked to original sources

arXiv2026

Calibrated Pressure-Observable Born and Hessian Actions for Quantum-Assisted Waveform Inversion

We construct a pressure-consistent operator-and-readout interface for Born, adjoint, and Gauss--Newton actions in constant-density acoustic full-waveform inversion (FWI) using Schrödingerised propagation. The energy variables $π=c^{-1}\partial_t u$ and $q=\nabla u$ yield an auxiliary-space Hamiltonian, while physical pressure $p=cπ$ depends explicitly on wavespeed. Its derivative $D(cπ)[c_0](δc)=c_0δπ+δc\,π_0$ combines propagated wavefield sensitivity with a direct receiver-calibration term. Duhamel and receiver-row differentiation retain both contributions in the Born map, its adjoint, and the Gauss--Newton normal action. We prove a conditional consistency estimate with a periodic second-order finite-difference specialization and give a resource model for state preparation, normalization, quadrature, and selected-output measurement. A compiled nine-qubit instance realizes structured preparation, product-formula propagation, a derivative-LCU block, and calibrated pressure-overlap measurements. Bernoulli samples from ideal-circuit probabilities drive a four-parameter hybrid inversion. A two-qubit VQLS circuit represents the normalized update direction, while normal-system assembly, line search, and model refresh remain classical. Finite differences, tangent and reverse-adjoint recurrences, autodiff JVP/VJP evaluations, and explicit Jacobians verify the discrete Born, adjoint, and normal actions. Smooth periodic refinement confirms second-order convergence, whereas omitting receiver calibration leaves an order-one Born error and substantially changes the regularized Gauss--Newton direction. All ten predeclared finite-shot runs reduce the initial model error. These results specify the physical-pressure derivative and selected-output measurements needed to connect Schrödingerised propagation to a local FWI update.

quant-ph↗

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

McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation

Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy improves large-domain scalability by adding new coarse levels while finetuning only the newly introduced parameters. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.

math.NA↗

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

Second-Order Area/Volume-Preserving PFEMs for Surface Diffusion via Simpson--Boole Geometric Identities

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 is based on exact geometric variation identities along a quadratic temporal interpolation path. The induced area variation in 2D is evaluated exactly by Simpson's rule, while the induced volume variation in 3D is evaluated exactly by Boole's rule. The resulting fully discrete schemes preserve the enclosed area or volume exactly, without introducing an auxiliary Lagrange multiplier for the geometric constraint. They can be assembled on BGN-predicted auxiliary geometries and are therefore compatible with existing second-order BGN-type implementations. Numerical experiments demonstrate the expected second-order behavior, area/volume conservation, and good mesh quality for both curve and surface evolutions.

math.NA↗

arXiv2026

Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems

Operator learning is an emerging interdisciplinary field that integrates machine learning with scientific computing. By mapping infinite-dimensional function spaces, this approach provides an efficient surrogate modeling framework for high-dimensional partial differential equations (PDEs). Compared to traditional numerical solvers, it achieves a superior trade-off between computational complexity and approximation accuracy, demonstrating significant advantages in many-query tasks such as real-time prediction and parameter sweeps. Given the stringent accuracy requirements of both forward simulation and inverse inference, as well as the precision bottlenecks of existing operator learning methods in handling complex boundaries or long-term evolution, we propose the Starter-Iterator Neural Operator (SINO). Our framework reinterprets the initialization strategies and iterative formats of traditional iterative methods through neural networks, establishing an efficient approach for spectral-spatiotemporal collaborative modeling. Specifically, the frequency-domain initialization module captures globally stable low-frequency features, while the time-domain learning module focuses on optimizing local solution residuals, thereby effectively overcoming the inherent limitations of conventional single-domain modeling approaches. Extensive experiments on typical dynamical systems such as the Navier-Stokes equations and acoustic wave equations, as well as practical applications including super-resolution imaging and weather forecasting, demonstrate that SINO achieves outstanding performance in numerical accuracy, generalization capability, and robustness.

math.NA↗

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

Neural Preconditioned Born Series: A Metric-Matched Framework for Learning-based Preconditioners

High-frequency Helmholtz problems in heterogeneous media remain challenging for both classical iterative methods and end-to-end neural PDE solvers. We propose Neural Preconditioned Born Series (NPBS), a learned iterative preconditioning framework that operates in preconditioned residual coordinates induced by the Convergent Born Series (CBS). Existing learned Born-series methods primarily use Born-style unrolling for forward wavefield prediction, while learned Helmholtz preconditioners are usually formulated in physical residual coordinates. NPBS fills this gap by recasting Born-series iteration as shifted-Laplacian left preconditioning, and replacing the CBS preconditioner with a learned residual-to-correction map in the Born-preconditioned coordinates. The left preconditioner further induces a residual metric, which yields a metric-matched training objective that aligns optimization with the preconditioned geometry used at inference. On heterogeneous Helmholtz benchmarks, metric-matched NPBS reduces iteration counts by up to $1.9\times$ over direct residual learning, with gains increasing from $1.2\times$ to $1.9\times$ as the wavenumber rises. Compared to classical CBS, learned NPBS reduces stationary iteration counts by over $20\times$; when used as a preconditioner for FGMRES, it further achieves the lowest wall-clock time among all evaluated methods. The same metric-matched formulation also improves convergence on convection--diffusion--reaction systems and Newton linear systems for nonlinear PDEs, indicating that residual-metric matching is a general design principle for neural preconditioners.

math.NA↗

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

The pressure-robust weak Galerkin finite element method for Stokes-Darcy problem

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 the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme. We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient. Moreover, the errors of the velocity function and the pressure function reach the optimal convergence orders under the energy norm, as validated by both theoretical analysis and numerical results.

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↗

arXiv2020

The impact of multilateral imported cases of COVID-19 on the epidemic control in China

Nowadays, the epidemic of COVID-19 in China is under control. However, the epidemic are developing rapidly around the world. Due to the normal migration of population, China is facing high risk from imported cases. The potential specific medicine and vaccine is still in the process of clinical trials. Currently, controlling the impact of imported cases is the key to prevent new outbreak of COVID-19 in China. In this paper, we propose two impulsive systems to describe the impact of multilateral imported cases of COVID-19. Based on the published data, we simulate and discussed the epidemic trends under different control strategies. We compare four different scenarios and show the corresponding medical burden. The results help to design appropriate control strategy for imported cases in practice.

q-bio.PE↗