arXiv subjects

Thanh Tran

Publications and source records attributed to Thanh Tran.

At least 19 recordsLinked to original sources

arXiv2026

Error Estimates for a Linear Fully-Discrete Finite Element Method for the Ferromagnetic Magnetohydrodynamical Model

The ferromagnetic magnetohydrodynamic equations are a non-linear PDE system that models the flow of an electrically conducting fluid that also has intrinsic magnetisation. In this paper, we propose a fully-discrete linear finite element scheme based on Euler's method to approximate the solutions to these equations and perform an error analysis for the scheme. Numerical experiments have been included to corroborate our theoretical results.

math.NA↗

arXiv2026

Error analysis of a divergence-preserving mixed finite element scheme for the incompressible Hall--magnetohydrodynamic equations

The incompressible Hall-magnetohydrodynamics (Hall--MHD) system presents substantial analytical and computational challenges due to its stiff, highly nonlinear Hall term and the strict requirement that the magnetic field remains solenoidal. In this paper, we study a Voigt-regularised Hall--MHD system, which is of independent analytical interest and provides a physically consistent, well-posed regularisation of the original model. We propose, analyse, and implement a structure-preserving, linear, fully discrete finite element method for this regularised problem. Using finite element exterior calculus and a mixed formulation, the spatial discretisation enforces the divergence-free condition on the magnetic field exactly, while a skew-symmetric, linearly implicit time discretisation yields unconditional energy stability. We establish optimal convergence rates for the Voigt-regularised problem and, additionally, derive error estimates for the unregularised Hall--MHD system, with the Voigt regularisation playing a crucial role in the non-resistive regime. Finally, numerical simulations in both 2.5D and 3D corroborate the theoretical results and demonstrate the physical fidelity of the scheme.

math.NA↗

arXiv2025

Retardance of lab grown diamond substrates as a function of thickness: momentum-drift random walk model

This work studies the correlation between mean retardance and thickness of diamond substrates grown homoepitaxially via microwave plasma-enhanced chemical vapor deposition (MPCVD). We measure the retardance of a diamond substrate in two orientations: perpendicular and parallel to the growth direction. Our experimental results demonstrate that the correlation between mean retardance and thickness differs for these orientations. When measured perpendicular to the growth direction, the mean retardance is approximately proportional to the square root of the substrate thickness. In contrast, when measured parallel to the growth direction, we observe a generally higher mean retardance and an approximately linear correlation with thickness. This anisotropy arises not from differences in stress magnitude but from differences in the interlayer correlation of the principal stress axes, as evidenced by correlation coefficients between the azimuth angles of consecutive layers in the diamond crystal. To simulate the integrated retardance of diamond wafers, we propose a two-dimensional random walk model with momentum drift, which captures the diamond crystal tendency to preserve the azimuth angle across the samples. By optimizing the momentum factor, we show that the model can closely match experimental data. The momentum factor is found higher along the growth direction, which is consistent with the calculated correlation coefficients. Furthermore, both the model and experiments indicate that retardance-to-thickness ratios of thin samples converge toward similar base retardances in both orientations. These findings establish a quantitative framework for interpreting birefringence in diamond substrates, with implications for material selection and development in thermal management, quantum sensing, high-power electronics, and optical applications.

cond-mat.mtrl-sci↗

arXiv2025

Unmasking Fake Careers: Detecting Machine-Generated Career Trajectories via Multi-layer Heterogeneous Graphs

The rapid advancement of Large Language Models (LLMs) has enabled the generation of highly realistic synthetic data. We identify a new vulnerability, LLMs generating convincing career trajectories in fake resumes and explore effective detection methods. To address this challenge, we construct a dataset of machine-generated career trajectories using LLMs and various methods, and demonstrate that conventional text-based detectors perform poorly on structured career data. We propose CareerScape, a novel heterogeneous, hierarchical multi-layer graph framework that models career entities and their relations in a unified global graph built from genuine resumes. Unlike conventional classifiers that treat each instance independently, CareerScape employs a structure-aware framework that augments user-specific subgraphs with trusted neighborhood information from a global graph, enabling the model to capture both global structural patterns and local inconsistencies indicative of synthetic career paths. Experimental results show that CareerScape outperforms state-of-the-art baselines by 5.8-85.0% relatively, highlighting the importance of structure-aware detection for machine-generated content.

cs.CR↗

arXiv2025

Strong pathwise solutions for a class of stochastic thermo-magneto-hydrodynamic-type systems with multiplicative noise

We establish the existence and uniqueness of strong solutions, in both the PDE and probabilistic sense, for a broad class of nonlinear stochastic partial differential equations (SPDEs) on a bounded domain $\mathscr{O}\subset \mathbb{R}^d$, $d\in \{1,2,3\}$, driven by multiplicative Gaussian noise. The solutions are global in time for $d\in \{1,2\}$. This theory simultaneously covers several physically relevant systems, including a class of stochastic generalised Burgers equations, a class of stochastic damped Navier--Stokes equations, stochastic magnetohydrodynamics (MHD), stochastic B\'enard convection in porous media, stochastic convective dynamo system, stochastic thermo-magneto-micropolar fluids, and stochastic diffusive tropical climate model, for which previous results only provide analytically weak martingale or pathwise solutions. The proof relies on Galerkin approximation and compactness argument. Up to a suitable stopping time, we derive strong moment bounds and verify a Cauchy property for the approximate solutions, in the absence of any inherent cancellation structure. By employing a localisation argument and a Gronwall-type lemma for stochastic processes, we establish the existence and uniqueness of maximal strong pathwise solutions, which are global when $d\in \{1,2\}$.

math.AP↗

arXiv2025

Error analysis of a fully discrete structure-preserving finite element scheme for a diffuse-interface model of tumour growth

We develop a linear fully discrete structure-preserving finite element method for a diffuse-interface model of tumour growth. The system couples a Cahn--Hilliard type equation with a nonlinear reaction-diffusion equation for nutrient concentration and admits a dissipative energy law at the continuous level. For the discretisation, we employ a scalar auxiliary variable (SAV) formulation together with a mixed finite element method for the Cahn--Hilliard part and standard conforming finite elements for the reaction-diffusion equation in space, combined with a first-order Euler time-stepping scheme. The resulting method is unconditionally energy-stable, mass-preserving, and inherits a discrete energy dissipation law associated with the SAV-based approximate energy functional, while requiring the solution of only linear systems at each time step. Under suitable regularity assumptions on the exact solution, we derive rigorous error estimates in $L^2$, $H^1$, and $L^\infty$ norms, establishing first-order accuracy in time and optimal-order accuracy in space. A key step in this analysis is the proof of boundedness of the numerical solutions in $L^\infty$. Numerical experiments validate the theoretical convergence rates and demonstrate the robustness of the method in capturing characteristic phenomena such as aggregation and chemotactic tumour growth.

math.NA↗

arXiv2025

Customize Multi-modal RAI Guardrails with Precedent-based predictions

A multi-modal guardrail must effectively filter image content based on user-defined policies, identifying material that may be hateful, reinforce harmful stereotypes, contain explicit material, or spread misinformation. Deploying such guardrails in real-world applications, however, poses significant challenges. Users often require varied and highly customizable policies and typically cannot provide abundant examples for each custom policy. Consequently, an ideal guardrail should be scalable to the multiple policies and adaptable to evolving user standards with minimal retraining. Existing fine-tuning methods typically condition predictions on pre-defined policies, restricting their generalizability to new policies or necessitating extensive retraining to adapt. Conversely, training-free methods struggle with limited context lengths, making it difficult to incorporate all the policies comprehensively. To overcome these limitations, we propose to condition model's judgment on "precedents", which are the reasoning processes of prior data points similar to the given input. By leveraging precedents instead of fixed policies, our approach greatly enhances the flexibility and adaptability of the guardrail. In this paper, we introduce a critique-revise mechanism for collecting high-quality precedents and two strategies that utilize precedents for robust prediction. Experimental results demonstrate that our approach outperforms previous methods across both few-shot and full-dataset scenarios and exhibits superior generalization to novel policies.

cs.LG↗

arXiv2025

AS400-DET: Detection using Deep Learning Model for IBM i (AS/400)

This paper proposes a method for automatic GUI component detection for the IBM i system (formerly and still more commonly known as AS/400). We introduce a human-annotated dataset consisting of 1,050 system screen images, in which 381 images are screenshots of IBM i system screens in Japanese. Each image contains multiple components, including text labels, text boxes, options, tables, instructions, keyboards, and command lines. We then develop a detection system based on state-of-the-art deep learning models and evaluate different approaches using our dataset. The experimental results demonstrate the effectiveness of our dataset in constructing a system for component detection from GUI screens. By automatically detecting GUI components from the screen, AS400-DET has the potential to perform automated testing on systems that operate via GUI screens.

cs.CV↗

arXiv2025

A mixed finite element method for a class of fourth-order stochastic evolution equations with multiplicative noise

We develop a fully discrete, semi-implicit mixed finite element method for approximating solutions to a class of fourth-order stochastic partial differential equations (SPDEs) with non-globally Lipschitz and non-monotone nonlinearities, perturbed by spatially smooth multiplicative Gaussian noise. The proposed scheme is applicable to a range of physically relevant nonlinear models, including the stochastic Landau--Lifshitz--Baryakhtar (sLLBar) equation, the stochastic convective Cahn--Hilliard equation with mass source, and the stochastic regularised Landau--Lifshitz--Bloch (sLLB) equation, among others. To overcome the difficulties posed by the interplay between the nonlinearities and the stochastic forcing, we adopt a `truncate-then-discretise' strategy: the nonlinear term is first truncated before discretising the resulting modified problem. We show that the strong solution to the truncated system converges in probability to that of the original problem. A fully discrete numerical scheme is then proposed for the truncated problem. Assuming initial data in $\mathbb{H}^2$, we utilise parabolic smoothing estimates and the temporal H\"older continuity of the solution to establish both convergence in probability and strong convergence (with quantitative rates) for the two fields used in the mixed formulation. Numerical simulations are provided to support the theoretical results.

math.NA↗

arXiv2025

Tree-Sliced Wasserstein Distance with Nonlinear Projection

Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures. This approach enhances the ability to capture the topological structures of integration domains in Sliced Optimal Transport while maintaining low computational costs. Building on this foundation, we propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, substituting the linear projections in earlier versions with general projections, while ensuring the injectivity of the associated Radon Transform and preserving the well-definedness of the resulting metric. By designing appropriate projections, we construct efficient metrics for measures on both Euclidean spaces and spheres. Finally, we validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets. Applications include gradient flows, self-supervised learning, and generative models, where our methods demonstrate significant improvements over recent SW and TSW variants.

cs.LG↗

arXiv2024

Well-Posedness for a Magnetohydrodynamical Model with Intrinsic Magnetisation

Ferromagnetic magnetohydrodynamics concerns the study of conducting fluids with intrinsic magnetisation under the influence of a magnetic field. It is a generalisation of the magnetohydrodynamical equations and takes into account the dynamics of the magnetisation of a fluid. First proposed by Lingam (Lingam, `Dissipative effects in magnetohydrodynamical models with intrinsic magnetisation', Communications in Nonlinear Science and Numerical Simulation Vol 28, pp 223-231, 2015), the usual equations of magnetohydrodynamics, namely the Navier-Stokes equation and the induction equation, are coupled with the Landau-Lifshitz-Gilbert equation. In this paper, the local existence, uniqueness and regularity of weak solutions to this system are discussed.

math.AP↗

arXiv2024

Global attractor and robust exponential attractors for some classes of fourth-order nonlinear evolution equations

We study the long-time behaviour of solutions to some classes of fourth-order nonlinear PDEs with non-monotone nonlinearities, which include the Landau--Lifshitz--Baryakhtar (LLBar) equation (with all relevant fields and spin torques) and the convective Cahn--Hilliard/Allen--Cahn (CH-AC) equation with a proliferation term, in dimensions $d=1,2,3$. Firstly, we show the global well-posedness, as well as the existence of global and exponential attractors with finite fractal dimensions for these problems. In the case of the exchange-dominated LLBar equation and the CH-AC equation without convection, an estimate for the rate of convergence of the solution to the corresponding stationary state is given. Finally, we show the existence of a robust family of exponential attractors when $d\leq 2$. As a corollary, exponential attractor of the LLBar equation is shown to converge to that of the Landau--Lifshitz--Bloch equation in the limit of vanishing exchange damping, while exponential attractor of the convective CH-AC equation is shown to converge to that of the convective Allen--Cahn equation in the limit of vanishing diffusion coefficient.

math.AP↗

arXiv2024

Equivariant Neural Functional Networks for Transformers

This paper systematically explores neural functional networks (NFN) for transformer architectures. NFN are specialized neural networks that treat the weights, gradients, or sparsity patterns of a deep neural network (DNN) as input data and have proven valuable for tasks such as learnable optimizers, implicit data representations, and weight editing. While NFN have been extensively developed for MLP and CNN, no prior work has addressed their design for transformers, despite the importance of transformers in modern deep learning. This paper aims to address this gap by providing a systematic study of NFN for transformers. We first determine the maximal symmetric group of the weights in a multi-head attention module as well as a necessary and sufficient condition under which two sets of hyperparameters of the multi-head attention module define the same function. We then define the weight space of transformer architectures and its associated group action, which leads to the design principles for NFN in transformers. Based on these, we introduce Transformer-NFN, an NFN that is equivariant under this group action. Additionally, we release a dataset of more than 125,000 Transformers model checkpoints trained on two datasets with two different tasks, providing a benchmark for evaluating Transformer-NFN and encouraging further research on transformer training and performance.

cs.LG↗

arXiv2024

Equivariant Polynomial Functional Networks

Neural Functional Networks (NFNs) have gained increasing interest due to their wide range of applications, including extracting information from implicit representations of data, editing network weights, and evaluating policies. A key design principle of NFNs is their adherence to the permutation and scaling symmetries inherent in the connectionist structure of the input neural networks. Recent NFNs have been proposed with permutation and scaling equivariance based on either graph-based message-passing mechanisms or parameter-sharing mechanisms. However, graph-based equivariant NFNs suffer from high memory consumption and long running times. On the other hand, parameter-sharing-based NFNs built upon equivariant linear layers exhibit lower memory consumption and faster running time, yet their expressivity is limited due to the large size of the symmetric group of the input neural networks. The challenge of designing a permutation and scaling equivariant NFN that maintains low memory consumption and running time while preserving expressivity remains unresolved. In this paper, we propose a novel solution with the development of MAGEP-NFN (Monomial mAtrix Group Equivariant Polynomial NFN). Our approach follows the parameter-sharing mechanism but differs from previous works by constructing a nonlinear equivariant layer represented as a polynomial in the input weights. This polynomial formulation enables us to incorporate additional relationships between weights from different input hidden layers, enhancing the model's expressivity while keeping memory consumption and running time low, thereby addressing the aforementioned challenge. We provide empirical evidence demonstrating that MAGEP-NFN achieves competitive performance and efficiency compared to existing baselines.

cs.LG↗

arXiv2024

Well-Posedness and Finite Element Approximation for the Landau-Lifshitz-Gilbert Equation with Spin-Torques

Spin currents act on ferromagnets by exerting a torque on the magnetisation. This torque is modelled by appending additional terms to the Landau-Lifshitz-Gilbert equation motivating the study of the non-homogeneous Landau-Lifshitz-Gilbert equation. We first prove the existence and uniqueness of high regularity local solutions to this equation using the Faedo-Galerkin method. Then we construct a numerical method for the problem and prove that it converges to a global weak solution of the PDE. Numerical simulations of the problem are also included.

math.AP↗

arXiv2024

The Landau--Lifshitz--Bloch equation on polytopal domains: Unique existence and finite element approximation

The Landau--Lifshitz--Bloch equation (LLBE) describes the evolution of the magnetic spin field in ferromagnets at high temperatures. In this paper, we study the numerical approximation of the LLBE on bounded polytopal domains in $\mathbb{R}^d$, where $d\le 3$. We first establish the existence and uniqueness of strong solutions to the LLBE and propose a linear, fully discrete, conforming finite element scheme for its approximation. While this scheme is shown to converge, the obtained rate is suboptimal. To address this shortcoming, we introduce a viscous (pseudo-parabolic) regularisation of the LLBE, which we call the $\epsilon$-LLBE. For this regularised problem, we prove the unique existence of strong solutions and establish a rate of convergence of the solution $\boldsymbol{u}^\epsilon$ of the $\epsilon$-LLBE to the solution $\boldsymbol{u}$ of the LLBE as $\epsilon\to 0^+$. Furthermore, we propose a linear, fully discrete, conforming finite element scheme to approximate the solution of the $\epsilon$-LLBE. Given sufficiently smooth initial data, error analysis is performed to show stability and uniform-in-time convergence of the scheme. Finally, several numerical simulations are presented to corroborate our theoretical results.

math.NA↗

arXiv2024

The stochastic Landau--Lifshitz--Baryakhtar equation: Global solution and invariant measure

The Landau--Lifshitz--Baryakhtar (LLBar) equation perturbed by both additive and multiplicative noises is a system of fourth order stochastic PDEs which models the evolution of magnetic spin fields in ferromagnetic materials at elevated temperatures, taking into account longitudinal damping, long-range interactions, spin current, and noise-induced phenomena at high temperatures. In this paper, we show the existence of a unique pathwise solution (which is analytically strong) to the stochastic LLBar equation posed in a bounded domain $\mathscr{D}\subset \mathbb{R}^d$, where $d=1,2,3$. We also prove the convergence of this pathwise solution to that of the stochastic Landau--Lifshitz--Bloch (LLB) equation in the limit of vanishing exchange relaxation parameter. Finally, we show the Feller property of the Markov semigroup associated with the strong solution, and prove the existence of nontrivial invariant measures. For temperatures above the Curie temperature, exponential stability of the solution and uniqueness of invariant measure are obtained under certain dissipativity conditions.

math.PR↗

arXiv2024

Analytically controlling laser-induced electron phase in sub-cycle motion

Precise control of the electron phase accumulated during its sub-cycle motion within intense laser fields is essential in strong-field physics, yet remains mostly indirect and complicated so far. In this Letter, we develop a novel approach to control this sub-cycle electron phase by tuning a low-frequency electric field applied on a centrosymmetric gaseous target during its interaction with a few-cycle infrared laser pulse. Our method is based on a universal analytical relation between the low-frequency electric field and its induced harmonic frequency shift, derived by the strong-field approximation. This simple relation and its universality are confirmed numerically by directly solving the time-dependent Schr\"odinger equation. Moreover, we discuss the benefits of the discovered relation in \textit{in situ} applications, including continuously and precisely tuning XUV waves and developing a new method of comprehensively sampling THz pulse.

physics.optics↗