arXiv subjects

Zhennan Zhou

Publications and source records attributed to Zhennan Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Fictitious Play for Mean Field Games with Optimal Stopping: Convergence and Computation

This paper studies mean field games with optimal stopping time (OSMFGs) where agents make optimal exit decisions. Such a model features a coupled obstacle problem and Fokker-Planck equation posing challenges on top of classic mean field games. The nonconvex nature of exit decisions renders the existence of a classic pure strategy equilibrium infeasible, necessitating the consideration of more complex mixed strategy equilibria. This paper proposes a generalized fictitious play algorithm that computes OSMFG mixed equilibria by iteratively solving pure strategy systems, i.e., approximating mixed strategies through averaging pure strategies according to a certain updating rule. The generalized fictitious play allows for a broad family of learning rates and the convergence to the mixed strategy equilibrium can be rigorously justified. The algorithm also incorporates efficient finite difference schemes of the pure strategy system. Numerical experiments demonstrate the effectiveness of the proposed method in robustly and efficiently computing mixed equilibria for OSMFGs.

math.NA↗

arXiv2026

Mean square error analysis of stochastic gradient and variance-reduced sampling algorithms

This paper considers mean square error (MSE) analysis for stochastic gradient sampling algorithms applied to underdamped Langevin dynamics under a global convexity assumption. A novel discrete Poisson equation framework is developed to bound the time-averaged sampling error. For the Stochastic Gradient UBU (SG-UBU) sampler, we derive an explicit MSE bound and establish that the numerical bias exhibits first-order convergence with respect to the step size $h$, with the leading error coefficient proportional to the variance of the stochastic gradient. The analysis is further extended to variance-reduced algorithms for finite-sum potentials, specifically the SVRG-UBU and SAGA-UBU methods. For these algorithms, we identify a phase transition phenomenon whereby the convergence rate of the numerical bias shifts from first to second order as the step size decreases below a critical threshold. Theoretical findings are validated by numerical experiments. In addition, we compare the computational cost that mini-batch SG-UBU and SVRG-UBU require to reach a prescribed accuracy, which indicates when variance reduction is worthwhile.

math.NA↗

arXiv2026

InverseScope: Scalable Activation Inversion for Interpreting Large Language Models

Understanding the internal representations of large language models (LLMs) is a central challenge in interpretability research. Existing feature interpretability methods often rely on strong structural assumptions--such as linearity or sparsity--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, InverseScope characterizes its encoded information by generating natural-language inputs that produce nearby activations, grounding abstract internal states in concrete language. To overcome the prohibitive cost of sampling in high-dimensional activation spaces, we propose a novel control-layer conditioning architecture that substantially improves sample efficiency compared to prior token-prepending approaches. We demonstrate that InverseScope reveals rich geometric structure in LLM representation spaces, including sentence-level linear analogies. The framework scales to state-of-the-art open-source models of up to 14B parameters and generalizes to out-of-distribution inputs, enabling systematic analysis of activation neighborhoods.

cs.LG↗

arXiv2026

Optimal Convergence Rate of Lie-Trotter Approximation for Quantum Thermal Averages

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.

quant-ph↗

arXiv2026

STOP: Structured On-Policy Pruning of Long-Form Reasoning in Low-Data Regimes

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.

cs.CL↗

arXiv2026

A Lagrangian Approach to Optimal Randomization

We develop an efficient method for solving non-convex constrained optimization problems that are pervasive in economics. The optimal solution to these problems often involves randomization. We employ a Lagrangian framework and prove that the value of the saddle point characterizing the optimal random solution equals the value of the deterministic dual problem. Our algorithm solves this dual via subgradient descent and recovers the optimal random solution directly from deterministic optima computed along the iterations. For many non-convex economic problems, our method is orders of magnitude faster than linear programming, making previously intractable lottery problems feasible. As an application, we solve for optimal Mirrleesian income taxation with multi-dimensional types. We show that heterogeneity in productivity and Frisch elasticity can make randomization welfare-improving over the optimal deterministic schedule.

econ.TH↗

arXiv2026

Resolving the Blow-Up: A Time-Dilated Numerical Framework for Multiple Firing Events in Mean-Field Neuronal Networks

In large-scale excitatory neuronal networks, rapid synchronization manifests as {multiple firing events (MFEs)}, mathematically characterized by a finite-time blow-up of the neuronal firing rate in the mean-field Fokker-Planck equation. Standard numerical methods struggle to resolve this singularity due to the divergent boundary flux and the instantaneous nature of the population voltage reset. In this work, we propose a robust {multiscale numerical framework based on time dilation}. By transforming the governing equation into a dilated timescale proportional to the firing activity, we desingularize the blow-up, effectively stretching the instantaneous synchronization event into a resolved mesoscopic process. This approach is shown to be physically consistent with the {microscopic cascade mechanism} underlying MFEs and the system's inherent fragility. To implement this numerically, we develop a hybrid scheme that utilizes a {mesh-independent flux criterion} to switch between timescales and a semi-analytical ``moving Gaussian'' method to accurately evolve the post-blowup Dirac mass. Numerical benchmarks demonstrate that our solver not only captures steady states with high accuracy but also efficiently reproduces periodic MFEs, matching Monte Carlo simulations without the severe time-step restrictions associated with particle cascades.

math.NA↗

arXiv2026

A Stabilized Numerical Framework for Necrotic Tumor Growth via Coupled Boundary Integral and Obstacle Solvers

We present a robust computational framework for Hele-Shaw tumor growth with necrotic cores, a problem identified as the incompressible limit of the Porous Media Equation. Simulating this system presents a fundamental challenge: while the outer boundary evolves via advection, the inner necrotic interface is defined by an obstacle problem and lacks an explicit advection structure, causing standard schemes to fail. To address this, we introduce a stabilized predictor-corrector strategy that iteratively resolves the bidirectional coupling between the nutrient-pressure fields and the domain geometry, ensuring robust time-stepping for both the advection-driven outer surface and the obstacle-defined necrotic core. We establish rigorous convergence theory for the single-interface case and demonstrate the method's robustness in capturing the topological transition of necrotic core nucleation and complex geometric evolution.

math.NA↗

arXiv2026

From Atoms to Trees: Building a Structured Feature Forest with Hierarchical Sparse Autoencoders

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.

cs.AI↗

arXiv2026

Projection-Based Memory Kernel Coupling Theory for Quantum Dynamics: A Stable Framework for Non-Markovian Simulations

We present a projection-based, stability-preserving methodology for computing time correlation functions in open quantum systems governed by generalized quantum master equations with non-Markovian effects. Building upon the memory kernel coupling theory framework, our approach transforms the memory kernel hierarchy into a system of coupled linear differential equations through Mori-Zwanzig projection, followed by spectral projection onto stable eigenmodes to ensure numerical stability. By systematically eliminating unstable modes while preserving the physically relevant dynamics, our method guaranties long-time convergence without introducing artificial damping or ad hoc modifications. The theoretical framework maintains mathematical rigor through orthogonal projection operators and spectral decomposition. Benchmark calculations on the spin-boson model show excellent agreement with exact hierarchical equations of motion results while achieving significant computational efficiency. This approach provides a versatile and reliable framework for simulating non-Markovian dynamics in complex systems.

quant-ph↗

arXiv2025

Frozen Gaussian sampling algorithms for simulating Markovian open quantum systems in the semiclassical regime

Simulating Markovian open quantum systems in the semiclassical regime poses a grand challenge for computational physics, as the highly oscillatory nature of the dynamics imposes prohibitive resolution requirements on traditional grid-based methods. To overcome this barrier, this paper introduces an efficient Frozen Gaussian Sampling (FGS) algorithm based on the Wigner-Fokker-Planck phase-space formulation. The proposed algorithm exhibits two transformative advantages. First, for the computation of physical observables, its sampling error is independent of the semiclassical parameter $\varepsilon$, thus fundamentally breaking the prohibitive computational scaling faced by grid methods in the semiclassical limit. Second, its mesh-free nature entirely eliminates the boundary-induced instabilities that constrain long-time grid-based simulations. Leveraging these capabilities, the FGS algorithm serves as a powerful investigatory tool for exploring the long-time behavior of open quantum systems. Specifically, we provide compelling numerical evidence for the existence of steady states in strongly non-harmonic potentials-a regime where rigorous analytical results are currently lacking.

quant-ph↗

arXiv2025

Recursive contracts in non-convex environments

In this paper we examine non-convex dynamic optimization problems with forward looking constraints. We prove that the recursive multiplier formulation in \cite{marcet2019recursive} gives the optimal value if one assumes that the planner has access to a public randomization device and forward looking constraints only have to hold in expectations. Whether one formulates the functional equation as a sup-inf problem or as an inf-sup problem is essential for the timing of the optimal lottery and for determining which constraints have to hold in expectations. We discuss for which economic problems the use of lotteries can be considered a reasonable assumption. We provide a general method to recover the optimal policy from a solution of the functional equation. As an application of our results, we consider the Ramsey problem of optimal government policy and give examples where lotteries are essential for the optimal solution.

econ.TH↗

arXiv2025

Meta-DSP: A Meta-Learning Approach for Data-Driven Nonlinear Compensation in High-Speed Optical Fiber Systems

Nonlinear effects in high-speed optical fiber systems fundamentally limit channel capacity. While traditional Digital Backward Propagation (DBP) with adaptive filters addresses these effects, its computational complexity remains impractical. Data-driven solutions like Filtered DBP (FDBP) reduce complexity but critically lack inherent generalization: Their nonlinear compensation capability cannot be naturally extended to new transmission rates or WDM channel counts without retraining on newly collected data. We propose Meta-DSP, a novel signal processing pipeline combining: (1) Meta-DBP, a meta-learning-based DBP model that generalizes across transmission parameters without retraining, and (2) XPM-ADF, a carefully engineered adaptive filter designed to address multi-channel nonlinear distortions. The system demonstrates strong generalization, learning from 40 Gbaud single-channel data and successfully applying this knowledge to higher rates (80/160 Gbaud) and multi-channel configurations (up to 21 channels). Experimental results show Meta-DSP improves Q-factor by 0.55 dB over CDC in challenging scenarios while reducing computational complexity 10$\times$ versus DBP. This work provides a scalable solution for nonlinear compensation in dynamic optical networks, balancing performance with practical computational constraints.

eess.SP↗

arXiv2025

Nonsymmetric traveling wave solution to a Hele-Shaw type tumor growth model

We consider a Hele-Shaw model that describes tumor growth subject to nutrient supply. The model is derived by taking the incompressible limit of porous medium type equations, and the boundary instability of this model was recently studied in \cite{feng2022tumor} using asymptotic analysis. In this paper, we further prove the existence of nonsymmetric traveling wave solutions to the model in a two dimensional tube-like domain, which reflect intrinsic boundary instability in tumor growth dynamics.

math.AP↗

arXiv2025

Crossover from ballistic transport to normal diffusion: a kinetic view

The crossover between dispersion patterns has been frequently observed in various systems. Inspired by the pathway-based kinetic model for E. coli chemotaxis that accounts for the intracellular adaptation process and noise, we propose a kinetic model that can exhibit a crossover from ballistic transport to normal diffusion at the population level. At the particle level, this framework aligns with a stochastic individual-based model. Using numerical simulations and rigorous asymptotic analysis, we demonstrate this crossover both analytically and computationally. Notably, under suitable scaling, the model reveals two distinct limits in which the macroscopic densities exhibit either ballistic transport or normal diffusion.

math.AP↗

arXiv2024

Weak formulation and spectral approximation of a Fokker-Planck equation for neural ensembles

In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method.

math.NA↗

arXiv2024

Jailbreak Instruction-Tuned LLMs via end-of-sentence MLP Re-weighting

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.

cs.CL↗

arXiv2024

Noisy integrate-and-fire equation: continuation after blow-up

The integrate and fire equation is a classical model for neural assemblies which can exhibit finite time blow-up. A major open problem is to understand how to continue solutions after blow-up. Here we study an approach based on random discharge models and a change of time which generates a classical global solution to the expense of a strong absorption rate 1/$ε$. We prove that in the limit $ε$ $\rightarrow$ 0 + , a global solution is recovered where the integrate and fire equation is reformulated with a singular measure. This describes the dynamics after blow-up and also gives information on the blow-up phenomena itself.The major difficulty is to handle nonlinear terms. To circumvent it, we establish two new estimates, a kind of equi-integrability of the discharge measure and a L 2 estimate of the density. The use of the new timescale turns out to be fundamental for those estimates.

math.AP↗