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Zhiwei Gao

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math.NA Numerical Analysis cond-mat.mtrl-sci Machine Learning cond-mat.soft Information Retrieval Robotics Social and Information Networks

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preprint2026arXiv

Nonlinear parametrization solver for fractional Burgers equations

Fractional Burgers equations pose substantial challenges for classical numerical methods due to the combined effects of nonlocality and shock-forming nonlinear dynamics. In particular, linear approximation frameworks-such as spectral, finite-difference, or discontinuous Galerkin methods-often suffer from Gibbs-type oscillations or require carefully tuned stabilization mechanisms, whose effectiveness degrades in transport-dominated and long-time integration regimes. In this work, we introduce a sequential-in-time nonlinear parametrization (STNP) for solving fractional Burgers equations, including models with a fractional Laplacian or with nonlocal nonlinear fluxes. The solution is represented by a nonlinear parametric ansatz, and the parameter evolution is obtained by projecting the governing dynamics onto the tangent space of the parameter manifold through a regularized least-squares formulation at each time step. This yields a well-posed and stable time-marching scheme that preserves causality and avoids global-in-time optimization. We provide a theoretical analysis of the resulting projected dynamics, including a stability estimate and an a posteriori error bound that explicitly decomposes the total error into contributions from initial condition fitting, projection residuals, and discretization of fractional operators. Our analysis clarifies the stabilizing role of regularization and quantifies its interaction with the nonlocal discretization error. Numerical experiments for both fractional Burgers models demonstrate that STNP achieves oscillation-free shock resolution and accurately captures long-time dynamics. The method consistently outperforms high-order spectral schemes augmented with spectral vanishing viscosity, while requiring significantly fewer degrees of freedom and avoiding ad hoc stabilization.

preprint2026arXiv

Proposal-Guided Greedy Surrogate Refinement for PDE-Driven High-Dimensional Rare-Event Estimation

Accurate surrogate construction for PDE-driven high-dimensional rare-event simulation is challenging when performance evaluations are expensive. Since a globally accurate surrogate may require many high-fidelity evaluations, adaptive importance sampling provides a natural localization tool: its evolving proposal distribution progressively identifies the failure-relevant region. Motivated by this observation, we propose a surrogate-assisted adaptive importance sampling framework that refines the surrogate locally along the evolving proposal, rather than over the entire input space. The surrogate combines an encoder with a neural network, providing a low-dimensional latent representation for both prediction and sample selection. At each adaptive iteration, candidates drawn from the current proposal are selected by a greedy latent-space rule balancing proximity to the estimated failure boundary and sample diversity. The selected samples are evaluated by the high-fidelity model and used to refine the surrogate, which then guides the subsequent cross-entropy-type adaptive proposal update. We establish one-step proposal stability bounds under local surrogate errors, together with surrogate-induced misclassification and finite-sample estimation error bounds. Numerical experiments on multimodal benchmarks and PDE-driven rare-event problems up to 100 dimensions show that the proposed method achieves accuracy comparable to true-model adaptive importance sampling while requiring substantially fewer high-fidelity evaluations.

preprint2023arXiv

Failure-informed adaptive sampling for PINNs

Physics-informed neural networks (PINNs) have emerged as an effective technique for solving PDEs in a wide range of domains. It is noticed, however, the performance of PINNs can vary dramatically with different sampling procedures. For instance, a fixed set of (prior chosen) training points may fail to capture the effective solution region (especially for problems with singularities). To overcome this issue, we present in this work an adaptive strategy, termed the failure-informed PINNs (FI-PINNs), which is inspired by the viewpoint of reliability analysis. The key idea is to define an effective failure probability based on the residual, and then, with the aim of placing more samples in the failure region, the FI-PINNs employs a failure-informed enrichment technique to adaptively add new collocation points to the training set, such that the numerical accuracy is dramatically improved. In short, similar as adaptive finite element methods, the proposed FI-PINNs adopts the failure probability as the posterior error indicator to generate new training points. We prove rigorous error bounds of FI-PINNs and illustrate its performance through several problems.

preprint2022arXiv

A Sufficient Condition for Convex Hull Property in General Convex Spatio-Temporal Corridors

Motion planning is one of the key modules in autonomous driving systems to generate trajectories for self-driving vehicles to follow. A common motion planning approach is to generate trajectories within semantic safe corridors. The trajectories are generated by optimizing parametric curves (e.g. Bezier curves) according to an objective function. To guarantee safety, the curves are required to satisfy the convex hull property, and be contained within the safety corridors. The convex hull property however does not necessary hold for time-dependent corridors, and depends on the shape of corridors. The existing approaches only support simple shape corridors, which is restrictive in real-world, complex scenarios. In this paper, we provide a sufficient condition for general convex, spatio-temporal corridors with theoretical proof of guaranteed convex hull property. The theorem allows for using more complicated shapes to generate spatio-temporal corridors and minimizing the uncovered search space to $O(\frac{1}{n^2})$ compared to $O(1)$ of trapezoidal corridors, which can improve the optimality of the solution. Simulation results show that using general convex corridors yields less harsh brakes, hence improving the overall smoothness of the resulting trajectories.

preprint2021arXiv

A Variational Integrator for the Discrete Element Method

A novel implicit integration scheme for the Discrete Element Method (DEM) based on the variational integrator approach is presented. The numerical solver provides a fully dynamical description that, notably, reduces to an energy minimisation scheme in the quasi-static limit. A detailed derivation of the numerical method is presented for the Hookean contact model and tested against an established open source DEM package that uses the velocity-Verlet integration scheme. These tests compare results for a single collision, long-term stability and statistical quantities of ensembles of particles. Numerically, the proposed integration method demonstrates equivalent accuracy to the velocity-Verlet method.

preprint2020arXiv

NAIST COVID: Multilingual COVID-19 Twitter and Weibo Dataset

Since the outbreak of coronavirus disease 2019 (COVID-19) in the late 2019, it has affected over 200 countries and billions of people worldwide. This has affected the social life of people owing to enforcements, such as "social distancing" and "stay at home." This has resulted in an increasing interaction through social media. Given that social media can bring us valuable information about COVID-19 at a global scale, it is important to share the data and encourage social media studies against COVID-19 or other infectious diseases. Therefore, we have released a multilingual dataset of social media posts related to COVID-19, consisting of microblogs in English and Japanese from Twitter and those in Chinese from Weibo. The data cover microblogs from January 20, 2020, to March 24, 2020. This paper also provides a quantitative as well as qualitative analysis of these datasets by creating daily word clouds as an example of text-mining analysis. The dataset is now available on Github. This dataset can be analyzed in a multitude of ways and is expected to help in efficient communication of precautions related to COVID-19.

preprint2016arXiv

Resonant TMR inversion in LiF/EuS based spin-filter tunnel junctions

Resonant tunneling can lead to inverse tunnel magnetoresistance when impurity levels rather than direct tunneling dominate the transport process. We fabricated hybrid magnetic tunnel junctions of CoFe/LiF/EuS/Ti, with an epitaxial LiF energy barrier joined with a polycrystalline EuS spin-filter bar-rier. Due to the water solubility of LiF, the devices were fully packaged in situ. The devices showed sizeable positive TMR up to 16% at low bias voltages but clearly inverted TMR at higher bias voltages. The TMR inversion depends sensitively on the thickness of LiF, and the tendency of inversion disap-pears when LiF gets thick enough and recovers its intrinsic properties.