Lei Xu

Junjie Zhang, Jiayu Liu, Wenbin Liu et al.

Existing LLM-based theorem provers have achieved impressive results on formal mathematics benchmarks, yet they remain confined to acting as autonomous agents that prove a stated proposition. In this paper, we propose MathCoPilot, a human-in-the-loop system that embodies a new human--AI symbiotic paradigm for mathematical research, in which the mathematician steers the high-level mathematical direction while AI agents carry out the detailed formalization and proof work under continuous human guidance. MathCoPilot unifies three core capabilities: (1) an interactive workbench where the mathematician and AI agents collaborate through a living proof blueprint that decomposes a proof into navigable steps the human can directly inspect, direct, and refine; (2) automated proving skill orchestration with adaptive knowledge base search and Lean-integrated iterative verification; and (3) topic-driven paper retrieval and automated formalization into a verified Lean knowledge base. Using MathCoPilot, we systematically compare four state-of-the-art LLMs, including Gemini~3.1~Pro, GPT-5.4, and Claude~Opus~4.7, on a FormalMATH subset and on two real PDE theorems requiring deep domain expertise, evaluating their ability to produce verified Lean~4 proofs and to identify errors in deliberately incorrect proofs. Our results show that while current models can handle undergraduate-level problems with high success rates under favorable autoformalization conditions, substantial challenges remain for domain-specific theorems requiring genuine mathematical understanding.

Rabimba Karanjai, Yang Lu, Hemanth Hegadehalli Madhavarao et al.

Large Language Models are increasingly deployed in Security Operations Centers for log analysis tasks including summarization, alert triage, and threat investigation. These systems ingest logs from external-facing services and process network logs as natural language contexts to generate security insights. We demonstrate that this architectural pattern introduces a critical vulnerability: adversaries can embed prompt injection payloads in log-generating fields that persist in storage and are executed when analysts query the LLM, achieving what we term passive prompt injection. We present LogInject, a systematic framework for evaluating these threats. Using LogInject-1.0, a benchmark of 12,847 log entries including 2,569 adversarial samples, we evaluate three production LLMs across four attack objectives: activity concealment, false positive generation, information exfiltration, and output hijacking. Our findings reveal an up to 88.2% attack success rate (83.4% average across models) under the baseline conditions. We introduce Context Stitching, a novel technique that fragments payloads across multiple log entries to evade stateless filters while exploiting LLM long-context reasoning, achieving a 76.4% success rate. As mitigation, we evaluate layered defenses by combining input filtering, prompt hardening, and output validation, demonstrating a 90.4% attack reduction, although 8.4% residual vulnerability persists. Our results establish that LLM-based log analysis creates an inherent confused deputy vulnerability where untrusted data and trusted instructions compete indistinguishably for model attention, requiring defense in-depth architectures and continued human oversight for security-critical decisions.

Rabimba Karanjai, Hemanth Madhavarao, Lei Xu et al.

The interconnected nature of global financial systems makes them vulnerable to systemic risks, where the failure of a few institutions can trigger catastrophic cascading defaults. Traditional risk models often fail to capture the complex, non-linear dynamics of these networks. While Graph Neural Networks (GNNs) have shown promise in modeling relational data, they primarily learn correlative patterns and function as black boxes, offering little insight into the causal mechanisms of shock propagation. This limitation is critical for regulators who require explainable models to perform stress tests and devise effective interventions. We introduce CausalGraphX, a novel framework that integrates GNNs with counterfactual reasoning to provide explainable assessments of systemic risk. CausalGraphX employs a Graph Attention mechanism to learn representations of institutional vulnerability and uses an adversarial regularization technique to ensure these representations capture causal drivers rather than spurious correlations. Furthermore, we propose an optimization-based approach to generate counterfactual explanations, answering questions such as, "What minimum capital injection would have prevented Bank A's default under a specific stress scenario?" We validate CausalGraphX on large-scale synthetic financial networks. Our results demonstrate that CausalGraphX significantly outperforms traditional and deep learning baselines in predicting cascading defaults while providing sparse, plausible, and actionable counterfactual explanations.

Peng Yang, Wenyu Lei, Liwei Xu

We develop a complete flux finite volume method for convection-diffusion equations that works on arbitrary meshes in two and three dimensions and for discrete spaces of any polynomial degree. Unlike standard finite volume discretizations, where the numerical flux is directly approximated from the flux definition, we derive the exact normal flux across each control volume edge/face from the underlying PDE. This exact flux splits naturally into a homogeneous part (the classical Scharfetter-Gummel flux) and an inhomogeneous part expressed via a Green's function that incorporates the tangential flux and the source term. The resulting formulation is exactly equivalent to the continuous equation and, once the discrete space is chosen, yields high-order schemes without any correction or stabilization. For piecewise linear spaces, the scheme achieves optimal second-order accuracy in convection-dominated regimes and can preserve positivity on moderately coarse meshes. For quadratic spaces, standard finite volume methods, based on the Lagrange elements or B-splines, fail to attain optimal $L^2$ convergence unless the control volume mesh is specially designed. The proposed complete flux scheme, however, always achieves optimal $L^2$ convergence independently of the control volume mesh. Numerical experiments in two and three dimensions confirm the robustness and optimal accuracy of the approach.

Maojun Li, Jiancheng Wang, Zeyu Xia et al.

In this paper, we derive a new model to simulate the incompressible resistive magnetohydrodynamic (MHD) free surface flow. A thermodynamically consistent diffuse interface method is adopted to characterize the moving interface in the modeling process. The formal convergence of the proposed MHD free surface flow model to the sharp interface model is established via a matched asymptotic argument, and the model can be solved without the need for sophisticated free surface capturing schemes. We design a fully decoupled linear finite element scheme that preserves the divergence-free constraint of the magnetic field at a discrete level. The reliability and robustness of the proposed model and algorithm are validated through numerical investigations of the magnetic damping effect on bubble dynamics. In particular, we provide a quantitative numerical comparison of the present results with those obtained from an inductionless MHD model and a sharp interface arbitrary Lagrangian--Eulerian model.