Zhennan Zhou

Jian-Guo Liu, Li Wang, Zhennan Zhou

We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is unconditionally stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we prove that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.

Jian-Guo Liu, Min Tang, Li Wang et al.

In this paper, we study the tumor growth equation along with various models for the nutrient component, including the \emph{in vitro} model and the \emph{in vivo} model. At the cell density level, the spatial availability of the tumor density $n$ is governed by the Darcy law via the pressure $p(n)=n^γ$. For finite $γ$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $γ\rightarrow \infty$, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

Jian-Guo Liu, Zheng Ma, Zhennan Zhou

In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally $\ell^1$ contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, $ (Δt)^α = O(Δx)$, where $α\in (0,1]$ is the fractional exponent in the derivative. When $α$ small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally $\ell^1$ contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.

Zhennan Zhou, Giovanni Russo

In this paper, we reformulate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Gaussian wave packets transform. With this approach, the highly oscillatory Schrödinger equation is equivalently transformed into another Schrödinger type wave equation, the $w$ equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the $w$ equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.

Jianfeng Lu, Zhennan Zhou

We develop a surface hopping algorithm based on frozen Gaussian approximation for semiclassical matrix Schrödinger equations, in the spirit of Tully's fewest switches surface hopping method. The algorithm is asymptotically derived from the Schrödinger equation with rigorous approximation error analysis. The resulting algorithm can be viewed as a path integral stochastic representation of the semiclassical matrix Schrödinger equations. Our results provide mathematical understanding to and shed new light on the important class of surface hopping methods in theoretical and computational chemistry.

Lei Li, Jian-Guo Liu, Zibu Liu et al.

There is a growing interest in investigating numerical approximations of the water wave equation in recent years, whereas the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In this work, we focus on a nonlocal hyperbolic model, which essentially inherits the features of the water wave equation, and is simplified from the latter. For the constant coefficient case, we carry out systematical stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and general Runge-Kutta method in time. In particular, we discover the optimal time step constraints, in the form of a modified CFL condition, when certain explicit Runge-Kutta method are applied. Besides, the convergence of the semi-discrete approximation of variable coefficient case is shown, which naturally connects to the water wave equation. Extensive numerical tests have been performed to verify the stability conditions and simulations of the simplified hyperbolic model in the high frequency regime and the water wave equation are also provided.

Chenjun Xu, Zhennan Zhou, Zhan Su et al.

Long chain-of-thought (Long CoT) reasoning improves performance on multi-step problems, but it also induces overthinking: models often generate low-yield reasoning that increases inference cost and latency. This inefficiency is especially problematic in low-data fine-tuning regimes, where real applications adapt reasoning models with limited supervision and cannot rely on large-scale teacher distillation or heavy test-time control. To address this, we propose STOP (Structured On-policy Pruning), an on-policy algorithm for analyzing and pruning long-form reasoning traces. STOP constructs self-distilled traces from the model. Then it maps each trace into a structured reasoning interface through node segmentation, taxonomy annotation, and reasoning-tree construction. On top of this interface, we introduce ECN (Earliest Correct Node), which retains the shortest prefix ending at the earliest node that both functions as an answering conclusion and yields the correct final answer, removing redundant post-solution reasoning while preserving semantic continuity. Experiments on DeepSeek-R1-Distill-Qwen-7B and DeepSeek-R1-Distill-LLaMA-3-8B across GSM8K, Math 500, and AIME 2024 show that STOP reduces generated tokens by 19.4-42.4% while largely preserving accuracy in low-data fine-tuning. Beyond efficiency, our analyses show that STOP induces much smaller distributional shift than teacher-guided pruning, improves the structural efficiency of generated reasoning, and reallocates reasoning effort away from redundant verification and backtracking toward more productive exploration.

Xuda Ye, Zhennan Zhou

The Lie--Trotter product formula is a foundational approximation for the quantum partition function, yet obtaining rigorous error bounds for the unbounded Hamiltonians common in physics remains a challenge. This paper provides a quantitative error analysis for this approximation across two systems. For a particle in a smooth, periodic potential, we establish an optimal convergence rate of $\mathcal O(1/N^2)$ for both the partition function and thermal averages, where $N$ is the number of imaginary time steps. We then extend this analysis to the more challenging case of a confining potential on $\mathbb R$, proving a nearly optimal rate of $\mathcal O((\log N+1)^{1.5}/N^2)$. The derived error bounds provide a firm mathematical foundation for the second-order accuracy of path integral simulations in quantum statistical mechanics.

Yifan Luo, Zhennan Zhou, Meitan Wang et al.

In this paper, we investigate the safety mechanisms of instruction fine-tuned large language models (LLMs). We discover that re-weighting MLP neurons can significantly compromise a model's safety, especially for MLPs in end-of-sentence inferences. We hypothesize that LLMs evaluate the harmfulness of prompts during end-of-sentence inferences, and MLP layers plays a critical role in this process. Based on this hypothesis, we develop 2 novel white-box jailbreak methods: a prompt-specific method and a prompt-general method. The prompt-specific method targets individual prompts and optimizes the attack on the fly, while the prompt-general method is pre-trained offline and can generalize to unseen harmful prompts. Our methods demonstrate robust performance across 7 popular open-source LLMs, size ranging from 2B to 72B. Furthermore, our study provides insights into vulnerabilities of instruction-tuned LLM's safety and deepens the understanding of the internal mechanisms of LLMs.

Yifan Luo, Yiming Tang, Chengfeng Shen et al.

Prompt Engineering (PE) has emerged as a critical technique for guiding Large Language Models (LLMs) in solving intricate tasks. Its importance is highlighted by its potential to significantly enhance the efficiency and effectiveness of human-machine interaction. As tasks grow increasingly complex, recent advanced PE methods have extended beyond the limitations of single-round interactions to embrace multi-round interactions, which allows for a deeper and more nuanced engagement with LLMs. In this paper, we propose an optimal control framework tailored for multi-round interactions with LLMs. This framework provides a unified mathematical structure that not only systematizes the existing PE methods but also sets the stage for rigorous analytical improvements. Furthermore, we extend this framework to include PE via ensemble methods and multi-agent collaboration, thereby enlarging the scope of applicability. By adopting an optimal control perspective, we offer fresh insights into existing PE methods and highlight theoretical challenges that warrant future research. Besides, our work lays a foundation for the development of more effective and interpretable PE methods.

Yifan Luo, Yang Zhan, Jiedong Jiang et al.

Sparse autoencoders (SAEs) have proven effective for extracting monosemantic features from large language models (LLMs), yet these features are typically identified in isolation. However, broad evidence suggests that LLMs capture the intrinsic structure of natural language, where the phenomenon of "feature splitting" in particular indicates that such structure is hierarchical. To capture this, we propose the Hierarchical Sparse Autoencoder (HSAE), which jointly learns a series of SAEs and the parent-child relationships between their features. HSAE strengthens the alignment between parent and child features through two novel mechanisms: a structural constraint loss and a random feature perturbation mechanism. Extensive experiments across various LLMs and layers demonstrate that HSAE consistently recovers semantically meaningful hierarchies, supported by both qualitative case studies and rigorous quantitative metrics. At the same time, HSAE preserves the reconstruction fidelity and interpretability of standard SAEs across different dictionary sizes. Our work provides a powerful, scalable tool for discovering and analyzing the multi-scale conceptual structures embedded in LLM representations.

Yifan Luo, Zhennan Zhou, Bin Dong

Understanding the internal representations of large language models (LLMs) is a central challenge in interpretability research. Existing feature interpretability methods often rely on strong assumptions about the structure of representations that may not hold in practice. In this work, we introduce InverseScope, an assumption-light and scalable framework for interpreting neural activations via input inversion. Given a target activation, we define a distribution over inputs that generate similar activations and analyze this distribution to infer the encoded information. To address the inefficiency of sampling in high-dimensional spaces, we propose a novel conditional generation architecture that significantly improves sample efficiency compared to previous method. We further introduce a quantitative evaluation protocol that tests interpretability hypotheses using the feature consistency rate computed over the sampled inputs. InverseScope scales inversion-based interpretability methods to larger models and practical tasks, enabling systematic and quantitative analysis of internal representations in real-world LLMs.