arXiv subjects

Lili Ju

Publications and source records attributed to Lili Ju.

At least 19 recordsLinked to original sources

arXiv2026

A Compact Residual-Screened Local Reduced-Order Model for Diffuse-Domain Shape Optimization

We generalize the compact residual-screened local reduced-order model (CL-ROM) from Brinkman-penalized flow to shape optimization with the diffuse domain method (DDM). As with Brinkman penalization, DDM extends evolving-domain problems to a fixed rectangular domain, allowing one grid for discretization and directly compatible snapshots from different designs. Unlike Brinkman penalization, DDM uses a phase field to weight operators and sources and treats Dirichlet, Neumann and Robin conditions through diffuse boundary terms. The CL-ROM mechanism carries over: recent full-order and corrected solutions determine local reduced spaces, while residual screening, correction with retained factors and objective verification decide when a full-order solve is needed. We derive reduced operators, residuals and sensitivities for each diffuse model, using Galerkin projection for the symmetric problems and least-squares Petrov--Galerkin projection for nonlinear flow. For the symmetric problems we prove a local approximation estimate that admits inexact, corrected snapshots. In numerical tests of shape identification, drag minimization in Navier--Stokes channel flow and compliance minimization of a cantilever, CL-ROM uses reduced solutions for nearly all design updates, reproduces the full-order objectives with relative differences below $5\times10^{-6}$, and is $3.85$ to $7.65$ times faster than a full-order baseline that reuses factorizations as preconditioners.

math.NA↗

arXiv2026

Quadrature Perturbations and High-Order Time Discretization for Prescribed-Feature Evolutionary GTransNet Galerkin Systems

In this paper we study quadrature perturbations and high-order time discretization for a prescribed-feature evolutionary generalized transferable neural network (Evo-GTransNet) Galerkin method for linear self-adjoint parabolic equations. Assembly quadrature, together with truncated singular value decomposition, is used to select a numerically resolved trial space and obtain coordinates with an identity mass matrix. Since this normalization depends on the assembly quadrature, the retained functions are also evaluated using a separate comparison quadrature, which is finer in the numerical experiments. For the two resulting semidiscrete systems, we derive an exact equation for the difference between their trajectories and an energy estimate that separates errors arising from the effective operator, load vector, and initial projection. The estimate is conditional on the trial space selected by the assembly quadrature. The retained rank, discarded mass, cross mass, and principal angles are used to assess this selection step. Refining the comparison quadrature yields stable retained spaces for the reported configurations, whereas deliberate underintegration leads to much larger perturbations. On the fixed retained systems, the temporal convergence of implicit midpoint, BDF, Radau IIA, and Gauss--Legendre methods is consistent with their classical orders over the resolved refinement ranges. The experiments distinguish the retained-coordinate error from the total solution error; the latter is measured against the PDE solution in the $L^2$ norm, with the error integrals evaluated by sufficiently fine quadrature. They also illustrate the importance of suitable starting values for BDF3 and show how the performance of IMEX-BDF schemes depends on the operator splitting and time step size.

math.NA↗

arXiv2026

Learning Optimal Robust-Pathwise Control of Elliptic Equations with Random Coefficients

Conventional control approaches to optimization problems under uncertainty entail a practical compromise: deterministic controls offer robustness at the expense of pathwise accuracy while stochastic controls can adapt to individual realizations but suffer from the curse of dimensionality. To achieve both robustness and adaptivity, a neural operator-based approach is proposed in this paper for distributed elliptic optimal control problems with high-dimensional random inputs. A data-driven training algorithm is first established to learn the mapping from coefficient realizations directly to their optimal pathwise controls. To bypass the burden of generating training data pairs, a physics-informed training algorithm is then proposed and recast as weakly supervised learning using pseudo label targets produced by standard adjoint-based control updates. Numerical experiments with rough coefficients are conducted to validate the effectiveness and efficiency of our robust-pathwise controls, which outperforms the benchmark robust deterministic control on unseen coefficient realizations owing to the generalization capability and fast inference of trained models.

math.NA↗

arXiv2026

AECSF: Adaptive Ensemble Conditional Score Filtering for High-Dimensional Nonlinear Data Assimilation

Bayesian state estimation for high-dimensional nonlinear dynamical systems entails a fundamental tension between statistical fidelity and computational tractability, as particle weights can collapse, while Gaussian ensemble updates can miss non-Gaussian posterior structure. Score-based diffusion filters offer a sampling-based alternative, but existing training-free score filters often rely on heuristic likelihood corrections, which can compromise posterior accuracy by neglecting uncertainty about the system state associated with each noisy reverse particle. To address these issues, we propose AECSF, a training-free adaptive ensemble conditional score filter. AECSF constructs an analytically tractable score estimator from the conditional Tweedie identity, which recasts noisy posterior score estimation as estimating the conditional mean of the system state given a noisy reverse particle and the observation. To estimate these conditional means efficiently, AECSF employs a shared adaptive weighted proposal ensemble, while particle-specific conditional weights yield an estimate for each noisy reverse particle without separate proposal sampling. The proposal ensemble is updated using reverse-particle information within the same reverse-diffusion run to improve conditional-mean estimation. Theoretically, we characterize when a fixed weighted proposal measure yields the exact noisy posterior score. Under stated assumptions, we establish a bound relating conditional-mean estimation errors to reverse-sampling endpoint error. Numerical experiments demonstrate that AECSF improves the accuracy of posterior sampling and nonlinear filtering in high-dimensional problems with limited forecast ensembles.

stat.ML↗

arXiv2026

Efficient and Accurate Lift-and-Decoupling Generalized Transferable Neural Networks for Elliptic and Moving Interface Problems

In this paper, we propose a highly efficient and accurate method, called ``lift-and-decoupling generalized transferable neural network" (LD-GTransNet), for numerical solution of elliptic and moving interface problems. Our method makes use of the multi-layer GTransNet approach and incorporates a novel {lift-and-decoupling} strategy, in which the hidden-layer neuron parameters are preset and the physical coordinates and the added auxiliary variable are processed by separate subnetworks. Such design enables a unified global approximation of piecewise smooth solutions without explicit domain decomposition, while preserving network expressivity and avoiding the redundancy inherent in conventional {lift} approaches. The resulting formulation reduces to a linear least-squares problem, thereby eliminating the need of nonlinear training and ensuring the high computational efficiency and accuracy. Extensive numerical experiments on benchmark interface problems in two and three dimensions demonstrate that, compared with existing neural network-based and traditional numerical methods, our proposed LD-GTransNet consistently delivers superior performance, particularly for problems with high-contrast coefficients and complex interface geometries.

math.NA↗

arXiv2026

FREESIA: Covariance-Aware Posterior Transport for Expressive and Scalable Data Assimilation

Data assimilation aims to infer the state of complex dynamical systems based on observational data. However, accurate inference of the multimodal posteriors induced by nonlinear or non-injective observation operators remains a key challenge under high-dimensional and sparse observation conditions. Ensemble filters scale to high dimensions but are confined by restrictive distributional assumptions, while training-free generative filters (e.g., EnSF, EnFF) alleviate this limitation but may introduce structural errors and hinder information propagation under sparse observations. To address these issues, we propose a training-free, asymptotically exact posterior transport method. Firstly, a covariance-aware posterior transport scheme is designed, which embeds the forecast cross-covariance into flow-based transport and accurately recovers unobserved states while preserving the non-Gaussian posterior structure. Furthermore, the method combines a tractable observation-adaptive proposal with posterior correction, ensuring accurate approximation of the nonlinear posterior distribution. Finally, we establish the corresponding posterior flow theory, from which the asymptotic exactness of the proposed method relative to finite-ensemble surrogates and the Wasserstein error bound are derived. Experiments on Double-Well, Lorenz-96, and Kolmogorov flow show that the proposed method captures complex posterior structure and remains accurate under sparse, nonlinear, and non-injective observations. In the sparse non-injective setting, it reduces RMSE by 56% relative to the best baseline.

stat.ML↗

arXiv2026

Structure-preserving generalized transferable neural networks for the Cahn-Hilliard equation

This paper is concerned with a structure-preserving neural network-based framework for the Cahn-Hilliard equation in mixed form. We employ generalized transferable neural networks (GTransNet) for spatial approximation and stabilized backward differentiation formulas (BDF) for temporal discretization. The resulting first- and second-order in time GTransNet-BDF schemes are shown to conserve mass and satisfy energy stability at the time-discrete level. The schemes are implemented by a collocation-based method, in which a least-squares system with constant coefficient matrix needs to be solved at each time step. The solution of this system, which determines the output-layer weights of the network, violates mass conservation due to the expected nonzero least-squares residual. To overcome this issue, we introduce a novel post-processing mass-conserving projection that enforces the mass constraint through a minimization problem, whose solution can be computed at negligible computational cost. A key advantage of the proposed method lies in its predetermined hidden layers and mesh-free nature, making the method applicable to complex domains, variable mobility, and long-time simulations. Extensive numerical experiments in two and three dimensions verify convergence, mass conservation, and energy dissipation as well as demonstrate the accuracy and robustness of the proposed GTransNet-BDF schemes.

math.NA↗

arXiv2026

Evo-GTransNet for Parabolic PDEs: A Fixed-Feature Galerkin Method of Lines with Quadrature-Mass Orthonormalization

In this paper, we develop an evolutionary generalized transferable neural network (Evo-GTransNet) solver for parabolic partial differential equations, formulated as a retained-space fixed-feature Galerkin method of lines. A GTransNet provides the prescribed spatial dictionary, while only the retained output coefficients evolve, thereby avoiding nonlinear training during time integration. To address severe mass-matrix ill-conditioning, we apply a quadrature-weighted truncated singular value decomposition (SVD) to select the numerically resolved trial space, followed by a separate rescaling that makes its basis orthonormal with respect to the assembly-quadrature mass inner product. Rank truncation modifies the approximation space, whereas the subsequent orthonormalization changes only its coordinate representation and preserves the retained discrete functions in exact arithmetic. The resulting semidiscrete coefficient system has an identity mass matrix, and we establish a semidiscrete energy law for symmetric linear parabolic problems. With the implicit midpoint scheme for time discretization, we further prove the contractivity of the method and derive a conditional fully discrete error estimate in which the error is controlled by the retained-space approximation error and the consistency defects. Numerical experiments demonstrate second-order temporal convergence using repeated feature samples together with separate assembly and validation quadratures, and quantify the accuracy of the retained space in the presence of severe raw-mass ill-conditioning. For the high-frequency and multiscale benchmark problems considered here, GTransNet achieves the smallest mean validation errors among the tested fixed-feature dictionaries at the same nominal output dimension.

math.NA↗

arXiv2026

Fully discrete parameter-robust error analysis of a grad-div stabilized Crank-Nicolson artificial compressibility method for the Navier-Stokes equations

The artificial compressibility method (ACM) relaxes the incompressibility constraint in the Navier-Stokes equations by introducing a perturbation term proportional to the time derivative of the pressure, scaled by a small positive parameter $\varepsilon$. Consequently, the ACM system inherently involves two small parameters: the fluid viscosity $ν$ and the artificial compressibility parameter $\varepsilon$. While existing temporal analyses of ACM provide insights into its behavior, rigorous fully discrete error estimates that are robust with respect to both parameters remain a significant gap in the literature. In this paper, we propose a second-order Crank-Nicolson fully discrete ACM scheme and establish its parameter-robust optimal error estimates. To enhance stability and ensure robustness, we incorporate two distinct grad-div stabilization strategies: one facilitates the decoupling of velocity and pressure computations, while the other guarantees robustness with respect to both $ν$ and $\varepsilon$. For spatial discretization, we employ the Scott-Vogelius finite element pair, which is crucial for the decoupling and error analysis. The resulting parameter-uniform bounds are crucial for ensuring the long-time accuracy of the scheme, circumventing the exponential dependence on the Reynolds number typically introduced by Grönwall's lemma. Numerical experiments are provided to validate the theoretical findings and demonstrate the efficiency of the proposed methods.

math.NA↗

arXiv2026

Generalized Transferable Neural Networks for Steady-State Partial Differential Equations

Deep learning has emerged as a compelling framework for scientific and engineering computing, motivating growing interest in neural network-based solvers for partial differential equations (PDEs). Within this landscape, network architectures with deterministic feature construction have become an appealing approach, offering both high accuracy and computational efficiency in practice. Among them, the transferable neural network (TransNet) is a special class of shallow neural networks (i.e., single-hidden-layer architectures), whose hidden-layer parameters are predetermined according to the principle of uniformly distributed partition hyperplanes. Although TransNet has demonstrated strong performance in solving PDEs with relatively smooth solutions, its accuracy and stability may deteriorate in the presence of highly oscillatory solution structures, where activation saturation and system conditioning issues become limiting factors. In this paper, we propose a generalized transferable neural network (GTransNet) for solving steady-state PDEs, which augments the original TransNet design with additional hidden layers while preserving its interpretable feature-generation mechanism. In particular, the first hidden layer of GTransNet retains TransNet's parameter sampling strategy but incorporates an additional symmetry constraint on the neuron biases, while the subsequent hidden layers omit bias terms and employ a variance-controlled sampling strategy for selecting neuron weights.

math.NA↗

arXiv2026

Matched Asymptotic Expansions-Based Transferable Neural Networks for Singular Perturbation Problems

In this paper, by utilizing the theory of matched asymptotic expansions, an efficient and accurate neural network method, named as "MAE-TransNet", is developed for solving singular perturbation problems in general dimensions, whose solutions usually change drastically in some narrow boundary layers. The TransNet is a two-layer neural network with specially pre-trained hidden-layer neurons. In the proposed MAE-TransNet, the inner and outer solutions produced from the matched asymptotic expansions are first approximated by a TransNet with nonuniform hidden-layer neurons and a TransNet with uniform hidden-layer neurons, respectively. Then, these two solutions are combined with a matching term to obtain the composite solution, which approximates the asymptotic expansion solution of the singular perturbation problem. This process enables the MAE-TransNet method to retain the precision of the matched asymptotic expansions while maintaining the efficiency and accuracy of TransNet. Meanwhile, the rescaling of the sharp region allows the same pre-trained network parameters to be applied to boundary layers with various thicknesses, thereby improving the transferability of the method. Notably, for coupled boundary layer problems, a computational framework based on MAE-TransNet is also constructed to effectively address issues resulting from the lack of relevant matched asymptotic expansion theory in such problems. Our MAE-TransNet is compared with TransNet, PINN, and Boundary-Layer PINN on various benchmark problems including 1D linear and nonlinear problems with boundary layers, the 2D Couette flow problem, a 2D coupled boundary layer problem, and the 3D Burgers vortex problem. Numerical results demonstrate that MAE-TransNet significantly outperforms other neural network methods in capturing the characteristics of boundary layers, improving the accuracy, and reducing the computational cost.

physics.comp-ph↗

arXiv2025

Parallel-in-Time Solution of Allen-Cahn Equations by Integrating Operator Learning into the Parareal Method

While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to improve the accuracy of deep learning-based solutions for time-dependent PDEs is to use the learned solution as the coarse propagator in the Parareal method and a traditional numerical method as the fine solver. However, successful integration of deep learning into the Parareal method requires consistency between the coarse and fine solvers, particularly for PDEs exhibiting rapid changes such as sharp transitions. To ensure such consistency, we propose to use the convolutional neural networks (CNNs) to learn the fully discrete time-stepping operator defined by the traditional numerical scheme used as the fine solver. We demonstrate the effectiveness of the proposed method in solving the classical and mass-conservative Allen-Cahn (AC) equations. Through iterative updates in the Parareal algorithm, our approach achieves a significant computational speedup compared to traditional fine solvers while converging to high-accuracy solutions. Our results highlight that the proposed Parareal algorithm effectively accelerates simulations, particularly when implemented on multiple GPUs, and converges to the desired accuracy in only a few iterations. Another advantage of our method is that the CNNs model is trained on trajectories-based on random initial conditions, such that the trained model can be used to solve the AC equations with various initial conditions without re-training. This work demonstrates the potential of integrating neural network methods into the parallel-in-time frameworks for efficient and accurate simulations of time-dependent PDEs.

math.NA↗

arXiv2025

Optimal error estimates of the diffuse domain method for parabolic equations

In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM employs a phase-field function to extend the original parabolic problem to a similar but slightly modified problem defined over a larger rectangular domain that contains the target physical domain. Based on the weighted Sobolev spaces, we rigorously establish the convergence of the diffuse domain solution to the original solution as the interface thickness parameter goes to zero, together with the corresponding optimal error estimates under the weighted $L^2$ and $H^1$ norms. Numerical experiments are also presented to validate the theoretical results.

math.NA↗

arXiv2025

Learning Singularity-Encoded Green's Functions with Application to Iterative Methods

Green's function provides an inherent connection between theoretical analysis and numerical methods for elliptic partial differential equations, and general absence of its closed-form expression necessitates surrogate modeling to guide the design of effective solvers. Unfortunately, numerical computation of Green's function remains challenging due to its doubled dimensionality and intrinsic singularity. In this paper, we present a novel singularity-encoded learning approach to resolve these problems in an unsupervised fashion. Our method embeds the Green's function within a one-order higher-dimensional space by encoding its prior estimate as an augmented variable, followed by a neural network parametrization to manage the increased dimensionality. By projecting the trained neural network solution back onto the original domain, our deep surrogate model exploits its spectral bias to accelerate conventional iterative schemes, serving either as a preconditioner or as part of a hybrid solver. The effectiveness of our proposed method is empirically verified through numerical experiments with two and four dimensional Green's functions, achieving satisfactory resolution of singularities and acceleration of iterative solvers.

math.NA↗

arXiv2025

Convergence analysis of the dynamically regularized Lagrange multiplier method for the incompressible Navier-Stokes equations

This paper is concerned with temporal convergence analysis of the recently introduced Dynamically Regularized Lagrange Multiplier (DRLM) method for the incompressible Navier-Stokes equations. A key feature of the DRLM approach is the incorporation of the kinetic energy evolution through a quadratic dynamic equation involving a time-dependent Lagrange multiplier and a regularization parameter. We apply the backward Euler method with an explicit treatment of the nonlinear convection term and show the unique solvability of the resulting first-order DRLM scheme. Optimal error estimates for the velocity and pressure are established based on a uniform bound on the Lagrange multiplier and mathematical induction. Numerical results confirm the theoretical convergence rates and error bounds that decay with respect to the regularization parameter.

math.NA↗

arXiv2025

A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions

We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.

stat.CO↗

arXiv2025

A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks

Solving partial differential equations (PDEs) with neural networks (NNs) has shown great potential in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDE formulas in the strong or weak sense, without explicitly considering some intrinsic physical properties, such as mass and momentum conservation, or energy dissipation. This limitation may result in nonphysical or unstable numerical solutions, particularly in long-term simulations. To address this issue, we propose ``Sidecar'', a novel framework that enhances the physical consistency of existing NN solvers for solving parabolic PDEs. Inspired by the time-dependent spectral renormalization approach, our Sidecar framework introduces a small network as a copilot, guiding the primary function-learning NN solver to respect the structure-preserving properties. Our framework is highly flexible, allowing the preservation of various physical quantities for different PDEs to be incorporated into a wide range of NN solvers. Experimental results on some benchmark problems demonstrate significant improvements brought by the proposed framework to both accuracy and structure preservation of existing NN solvers.

cs.LG↗

arXiv2025

Data-Driven Reduced-Order Models for Port-Hamiltonian Systems with Operator Inference

Hamiltonian operator inference has been developed in [Sharma, H., Wang, Z., Kramer, B., Physica D: Nonlinear Phenomena, 431, p.133122, 2022] to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. The method constructs a low-dimensional model using only data and knowledge of the functional form of the Hamiltonian. The resulting ROMs preserve the intrinsic structure of the system, ensuring that the mechanical and physical properties of the system are maintained. In this work, we extend this approach to port-Hamiltonian systems, which generalize Hamiltonian systems by including energy dissipation, external input, and output. Based on snapshots of the system's state and output, together with the information about the functional form of the Hamiltonian, reduced operators are inferred through optimization and are then used to construct data-driven ROMs. To further alleviate the complexity of evaluating nonlinear terms in the ROMs, a hyper-reduction method via discrete empirical interpolation is applied. Accordingly, we derive error estimates for the ROM approximations of the state and output. Finally, we demonstrate the structure preservation, as well as the accuracy of the proposed port-Hamiltonian operator inference framework, through numerical experiments on a linear mass-spring-damper problem and a nonlinear Toda lattice problem.

math.NA↗