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Shiheng Zhang

Publications and source records attributed to Shiheng Zhang.

At least 19 recordsLinked to original sources
arXiv2026

SAV Schemes with Decomposition-Induced Pullback Corrections for Gradient Flows

The scalar auxiliary variable (SAV) method constructs linear, unconditionally energy-stable time discretizations of gradient flows. Eliminating the auxiliary variable in a first-order SAV step shows that the state equation is a semi-implicit update augmented by a rank-one positive semidefinite correction from the previous nonlinear force. The multiple-SAV (MSAV) method produces this correction componentwise, with rank up to the number of energy components. This separates two mechanisms usually coupled in MSAV: the number of scalar variables tracking the nonlinear energy and the rank of the correction. We introduce a pullback-corrected SAV (PB-SAV) family that keeps a single scalar auxiliary variable but replaces the rank-one SAV correction by the pullback correction induced by an admissible component decomposition. The correction remains positive semidefinite, has rank at most the number of components, and may change from step to step without changing the scalar auxiliary variable. We prove modified-energy dissipation laws for fixed and step-dependent decompositions, establish first-order convergence after a fixed spatial discretization under standard smoothness and positivity assumptions, derive a refinement identity whose gain is an explicit weighted variance, and give a Sherman--Morrison--Woodbury implementation of the low-rank perturbation of the standard semi-implicit solve. We also show, in finite dimensions, that the pullback correction is the Gauss--Newton matrix of a least-squares representation of the nonlinear energy. Numerical experiments on finite-dimensional gradient flows, Allen--Cahn dynamics, a finite-rank nonlocal gradient flow, and a linear nonlocal Cahn--Hilliard model show regimes in which PB-SAV and SAV differ only in the first-order error constant and regimes in which PB-SAV reduces the trajectory error by a large factor.

math.NA↗
arXiv2026

A pullback-corrected scalar auxiliary variable optimizer with momentum and adaptive mobility

Objectives in scientific machine learning are often prescribed as a sum of several terms, such as the residual, boundary, initial, and data losses of a physics-informed neural network. In the pullback-corrected scalar auxiliary variable (PB--SAV) method, one scalar tracks the shifted objective while the component gradients build a positive semidefinite curvature correction of rank at most the number of components. We carry that correction into an optimizer with momentum and an adaptive mobility, applying it to the gradient and the stored momentum in a single implicit solve. A mobility that is nonincreasing in the Loewner order yields an exact modified energy law, covering Euclidean and AMSGrad-type choices; the corresponding identity for momentum appended after the solve carries a cross term of indefinite sign. For a fixed mobility we give a necessary and sufficient condition for local stability at a stationary point, depending on the Hessian minus twice the correction, and show that it also gives local geometric convergence for every scalar relaxation sequence. The implicit solve reduces to a dense system whose order is the number of components. In the forward Burgers comparison, four components reduce the mean tail objective by 64.7% and the final solution error by 50.2% relative to one component at the same learning rate and momentum settings.

stat.ML↗
arXiv2026

Robust High-Order Projector-Splitting Integrators

We develop a general framework for constructing robust high-order projector-splitting integrators for dynamical low-rank approximation. For a prescribed matrix increment, we show that the standard K-S-L projector-splitting step is equivalent to a reduced K-L step, thereby eliminating the explicit backward S-step. We then establish a central relaxed exactness property of the standard projector-splitting integrator: the rank-$r$ approximation inherits the accuracy of the numerical matrix increment, with an error bound independent of small singular values. The resulting schemes evolve fixed rank-$r$ factors and require neither basis augmentation nor rank truncation. As concrete examples, we combine the framework with selected second- and third-order Runge--Kutta methods to obtain robust high-order projector-splitting integrators. Numerical experiments confirm the predicted uniform convergence rates with respect to small singular values.

math.NA↗
arXiv2026

A Counterexample to Robust Second-Order Convergence of the Strang Projector-Splitting Integrator

The classical Strang projector-splitting integrator is widely observed to converge with order two, whereas the error analysis that remains uniform as the smallest singular value retained in the low-rank approximation tends to zero proves only order one. We show that this gap is intrinsic under the standard assumptions. We construct $3\times3$ matrix differential equations that are $C^2$ in time and smooth in the matrix variable, with rank-two initial data that satisfy uniform boundedness, Lipschitz, tangency-defect, and regularity bounds. Nevertheless, the exact-subflow Strang method has a nonzero $h^2$ term in the local error over one periodic forcing cycle consisting of four Strang steps. Repetition of that cycle rules out a global second-order bound whose constant and stepsize threshold are independent of the retained singular values. The mechanism is a rapid rotation of the factor directions associated with the small singular value: first-order consistency is preserved, but the changing projection spaces prevent the cancellation normally expected from a symmetric Strang composition. Hence the robust first-order result cannot, under these assumptions alone, be upgraded to robust second order for the classical projector-splitting method.

math.NA↗
arXiv2026

Online Learning in Stackelberg Security Games with Adaptive Attacker Sequences and Time-Varying Attack Intensities

This work studies no-regret online learning in Repeated Stackelberg Security Games with time-varying attack intensities. We formulate an extended security game in which an attacker may select multiple targets and derive an exact mixed-integer linear programming oracle under a optimistic tie-breaking rule. Under full-information feedback, the oracle is integrated with Follow-the-Perturbed-Leader and yields expected $\mathcal{O}(\sqrt{T})$ regret against non-anticipating sequences with time-varying follower numbers, attack intensities, and attacker types. Under bandit feedback, we consider multiple followers sharing a fixed attacker type and use a barycentric-spanner construction to reconstruct utility estimates from aggregate attack observations, obtaining expected $\mathcal{O}(T^{2/3})$ regret. Extensive simulations demonstrate the robustness and effectiveness of our approach under full and partial information feedback.

cs.GT↗
arXiv2026

Low-Rank Evolutionary Deep Neural Networks via Adaptive Tangent-Space Reduction

Evolutionary deep neural networks (EDNNs) solve time-dependent partial differential equations by evolving the neural-network parameters sequentially in time through a local least-squares problem. Their main computational bottleneck is that each time step requires the solution of a dense linear system whose dimension equals the total number of trainable parameters. We propose a low-rank evolutionary deep neural network (LR-EDNN) method that reduces this cost through adaptive tangent-space projection. This construction replaces direct bilinear low-rank factor evolution by a linear reduced problem while preserving the sequential-in-time structure of EDNN. We construct the reduced Jacobian directly through layerwise Jacobian-vector products, without forming the full Jacobian. We further establish a finite-time comparison estimate: the deviation of the LR-EDNN trajectory from full EDNN is bounded by a discrete Grönwall accumulation of the local tangent-space projection defects, with amplification governed by the assumed Lipschitz and directional-coercivity constants. Numerical experiments on a porous-medium equation with drift, one- and two-dimensional Allen-Cahn equations, and two-dimensional viscous Burgers' equations demonstrate that LR-EDNN substantially reduces computational cost while maintaining the accuracy and qualitative fidelity of the full EDNN solver when the rank is chosen adequately.

stat.ML↗
arXiv2026

Asymptotic Preservation and Uniform Accuracy of Diffusion and Flow-Matching Samplers

Diffusion and Gaussian-interpolant flow-matching samplers approach data through a terminal noise floor $\varepsilon$, a singular limit for manifold-supported or rank-deficient data. We study two properties of a complete sampler specification, comprising its update rule, time grid, and terminal rule. Asymptotic preservation (AP) means a stable and consistent zero-noise discretization with a step count bounded independently of $\varepsilon$. Uniform accuracy (UA) of order $p$ means that, at numerical resolution $h$, the endpoint $W_2$ error is $O(h^p)$ with a floor-independent constant. Bounded log-noise stepping fails AP because its step count diverges. Stopping a stable base solver at a positive switching scale $a$ and appending one map fitted to the analytic normal mode restores AP. On smooth compact boundaryless manifolds, the standard map has exact-input error $O(a^2-\varepsilon^2)$ and sharp zero-floor error $Θ(a^2)$. A base solver with a floor-uniform order-$p$ estimate on the resolved interval retains that order when $a=O(h^{p/2})$, provided the terminal transfer factor remains bounded. Along exact trajectories, the posterior-mean identity $D(x(σ),σ)=x(σ)-σx'(σ)$ cancels the linear terminal defect and enables higher-order fitted maps. A three-evaluation Hermite construction is uniformly third order for exact switching-scale input over $0\le\varepsilon\le a$, and a seven-evaluation construction is fourth order at zero. We classify representative diffusion and flow-matching specifications by AP and UA. On EDM and Rectified Flow checkpoints, a paired decomposition separates base-integration from terminal-completion error and predicts held-out same-seed endpoint errors.

cs.LG↗
arXiv2026

Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance

Classifier-free guidance (CFG) is the standard way to strengthen class-conditioning in diffusion and flow-matching samplers, yet at large guidance it oversaturates and destabilizes, symptoms practitioners suppress with more steps or limited-interval schedules. We analyze CFG through an asymptotic-preserving, numerical-analysis lens. Building on a recent result that the deterministic DDIM step is the unique fitted operator for the unguided terminal layer, exact on the final small-sigma stretch of sampling, we show that guidance re-stiffens exactly the discriminative subspace to an anomalous exponent 1+w. DDIM is therefore no longer fitted there, and on coarse meshes its guided residual diverges as sigma_min goes to zero. We prove a guided clock barrier with three ordered step-size thresholds, and read one-step oversaturation as its endpoint: a solver artifact on the calibration model rather than the continuous guided law. The same analysis yields a one-coefficient, zero-extra-NFE repair: replace CFG's w(r-1) by r^(1+w)-r on the guidance direction. On the calibration model's discriminative crossover, this removes CFG's sigma_min-divergent blow-up and is first-order accurate against the exact guided flow as sigma_min goes to zero. On learned CIFAR-10 checkpoints, and as a cross-domain smoke test on Stable Diffusion 1.5 DDIM, it acts as a high-guidance stabilizer at no extra cost rather than a universal quality knob: it cuts residual amplification and saturation, gives 9/9 point-FID wins over CFG on the tested grid, and preserves classifier-proxy target accuracy in the hard-cell blocks. We report the limits alongside: it is not a universal image-quality win, and against a dense vanilla-CFG reference it is not a uniformly better integrator of that field.

cs.LG↗
arXiv2026

A separable and asymptotic-preserving dynamical low-rank method for the Vlasov-Poisson-Fokker-Planck system

We present a dynamical low-rank (DLR) method for the Vlasov-Poisson-Fokker-Planck (VPFP) system. Our main contributions are two-fold: (i) a conservative spatial discretization of the Fokker-Planck operator that factors into velocity-only and space-only components, enabling efficient low-rank projection, and (ii) a time discretization within the DLR framework that properly handles stiff collisions. We propose both first-order and second-order low-rank IMEX schemes. For the first-order scheme, we prove an asymptotic-preserving (AP) property when the field fluctuation is small. Numerical experiments demonstrate accuracy, robustness, and AP property at modest ranks.

math.NA↗
arXiv2026

Planner-Admissible Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning

Sparse goal-conditioned planning with few cost-to-go labels can be viewed as a graph-PDE Dirichlet extension problem: extend sparse labels on a goal-dependent boundary to unlabelled graph vertices so that greedy rollouts reach the goal. We study which graph value extensions are planner-admissible under the operational argmin-Q planner. Our main result is a local action-gap certificate: if the surrogate value error along the rollout stays below half the true action gap, then the greedy rollout reaches the goal. Absolutely Minimal Lipschitz Extension (AMLE), the p=infinity endpoint of the graph p-Laplacian family, instantiates this certificate through a comparison-principle fill-distance bound. Harmonic extension, by contrast, can mis-rank local actions because its values reflect boundary hitting probabilities rather than shortest-path greedy order. On 120 AntMaze layout-derived graph configurations, harmonic extension achieves 0.584 aggregate rollout success, while AMLE reaches 0.970. Finite high-p methods also enter a high-success regime, with success 0.903 for p=4, 0.973 for p=8, and 0.982 for a fixed-budget p=16 solver, though the p=16 row is not used as a converged endpoint ranking due to incomplete solver certification. Mechanism audits show that many rollout decisions occur in AMLE-compatible but harmonic-incompatible local geometry, and that AMLE corrects most harmonic inversions on the rollout-weighted decision scope.

cs.LG↗
arXiv2026

On the stability of the low-rank projector-splitting integrators for hyperbolic and parabolic equations

We study the stability of a class of dynamical low-rank methods--the projector-splitting integrator (PSI)--applied to linear hyperbolic and parabolic equations. Using a von Neumann-type analysis, we investigate the stability of such low-rank time integrator coupled with standard spatial discretizations, including upwind and central finite difference schemes, under two commonly used formulations: discretize-then-project (DtP) and project-then-discretize (PtD). For hyperbolic equations, we show that the stability conditions for DtP and PtD are the same under Lie-Trotter splitting, and that the stability region can be significantly enlarged by using Strang splitting. For parabolic equations, despite the presence of a negative S-step, unconditional stability can still be achieved by employing Crank-Nicolson or a hybrid forward-backward Euler scheme in time stepping. While our analysis focuses on simplified model problems, it offers insight into the stability behavior of PSI for more complex systems, such as those arising in kinetic theory.

math.NA↗
arXiv2025

Dual-level Progressive Hardness-Aware Reweighting for Cross-View Geo-Localization

Cross-view geo-localization (CVGL) between drone and satellite imagery remains challenging due to severe viewpoint gaps and the presence of hard negatives, which are visually similar but geographically mismatched samples. Existing mining or reweighting strategies often use static weighting, which is sensitive to distribution shifts and prone to overemphasizing difficult samples too early, leading to noisy gradients and unstable convergence. In this paper, we present a Dual-level Progressive Hardness-aware Reweighting (DPHR) strategy. At the sample level, a Ratio-based Difficulty-Aware (RDA) module evaluates relative difficulty and assigns fine-grained weights to negatives. At the batch level, a Progressive Adaptive Loss Weighting (PALW) mechanism exploits a training-progress signal to attenuate noisy gradients during early optimization and progressively enhance hard-negative mining as training matures. Experiments on the University-1652 and SUES-200 benchmarks demonstrate the effectiveness and robustness of the proposed DPHR, achieving consistent improvements over state-of-the-art methods.

cs.CV↗
arXiv2025

Asymptotic-Preserving Dynamical Low-Rank Method for the Stiff Nonlinear Boltzmann Equation

In kinetic theory, numerically solving the full Boltzmann equation is extremely expensive. This is because the Boltzmann collision operator involves a high-dimensional, nonlinear integral that must be evaluated at each spatial grid point and every time step. The challenge becomes even more pronounced in the fluid (strong collisionality) regime, where the collision operator exhibits strong stiffness, causing explicit time integrators to impose severe stability restrictions. In this paper, we propose addressing this problem through a dynamical low-rank (DLR) approximation. The resulting algorithm requires evaluating the Boltzmann collision operator only $r^2$ times, where $r$, the rank of the approximation, is much smaller than the number of spatial grid points. We propose a novel DLR integrator, called the XL integrator, which reduces the number of steps compared to the available alternatives (such as the projector splitting or basis update & Galerkin (BUG) integrator). For a class of problems including the Boltzmann collision operator which enjoys a separation property between physical and velocity space, we further propose a specialized version of the XL integrator, called the sXL integrator. This version requires solving only one differential equation to update the low-rank factors. Furthermore, the proposed low-rank schemes are asymptotic-preserving, meaning they can capture the asymptotic fluid limit in the case of strong collisionality. Our numerical experiments demonstrate the efficiency and accuracy of the proposed methods across a wide range of regimes, from non-stiff (kinetic) to stiff (fluid).

math.NA↗
arXiv2024

Spectral-Temporal Fusion Representation for Person-in-Bed Detection

This study is based on the ICASSP 2025 Signal Processing Grand Challenge's Accelerometer-Based Person-in-Bed Detection Challenge, which aims to determine bed occupancy using accelerometer signals. The task is divided into two tracks: "in bed" and "not in bed" segmented detection, and streaming detection, facing challenges such as individual differences, posture variations, and external disturbances. We propose a spectral-temporal fusion-based feature representation method with mixup data augmentation, and adopt Intersection over Union (IoU) loss to optimize detection accuracy. In the two tracks, our method achieved outstanding results of 100.00% and 95.55% in detection scores, securing first place and third place, respectively.

eess.SP↗
arXiv2024

Independent Feature Enhanced Crossmodal Fusion for Match-Mismatch Classification of Speech Stimulus and EEG Response

It is crucial for auditory attention decoding to classify matched and mismatched speech stimuli with corresponding EEG responses by exploring their relationship. However, existing methods often adopt two independent networks to encode speech stimulus and EEG response, which neglect the relationship between these signals from the two modalities. In this paper, we propose an independent feature enhanced crossmodal fusion model (IFE-CF) for match-mismatch classification, which leverages the fusion feature of the speech stimulus and the EEG response to achieve auditory EEG decoding. Specifically, our IFE-CF contains a crossmodal encoder to encode the speech stimulus and the EEG response with a two-branch structure connected via crossmodal attention mechanism in the encoding process, a multi-channel fusion module to fuse features of two modalities by aggregating the interaction feature obtained from the crossmodal encoder and the independent feature obtained from the speech stimulus and EEG response, and a predictor to give the matching result. In addition, the causal mask is introduced to consider the time delay of the speech-EEG pair in the crossmodal encoder, which further enhances the feature representation for match-mismatch classification. Experiments demonstrate our method's effectiveness with better classification accuracy, as compared with the baseline of the Auditory EEG Decoding Challenge 2023.

eess.AS↗
arXiv2024

SAV-based entropy-dissipative schemes for a class of kinetic equations

We introduce novel entropy-dissipative numerical schemes for a class of kinetic equations, leveraging the recently introduced scalar auxiliary variable (SAV) approach. Both first and second order schemes are constructed. Since the positivity of the solution is closely related to entropy, we also propose positivity-preserving versions of these schemes to ensure robustness, which include a scheme specially designed for the Boltzmann equation and a more general scheme using Lagrange multipliers. The accuracy and provable entropy-dissipation properties of the proposed schemes are validated for both the Boltzmann equation and the Landau equation through extensive numerical examples.

math.NA↗
arXiv2024

Structure preserving schemes for a class of Wasserstein gradient flows

We introduce in this paper two time discretization schemes tailored for a range of Wasserstein gradient flows. These schemes are designed to preserve mass, positivity and to be uniquely solvable. In addition, they also ensure energy dissipation in many typical scenarios. Through extensive numerical experiments, we demonstrate the schemes' robustness, accuracy and efficiency.

math.NA↗
arXiv2023

An Element-wise RSAV Algorithm for Unconstrained Optimization Problems

We present a novel optimization algorithm, element-wise relaxed scalar auxiliary variable (E-RSAV), that satisfies an unconditional energy dissipation law and exhibits improved alignment between the modified and the original energy. Our algorithm features rigorous proofs of linear convergence in the convex setting. Furthermore, we present a simple accelerated algorithm that improves the linear convergence rate to super-linear in the univariate case. We also propose an adaptive version of E-RSAV with Steffensen step size. We validate the robustness and fast convergence of our algorithm through ample numerical experiments.

math.OC↗