arXiv subjects

Qin Li

Publications and source records attributed to Qin Li.

At least 55 records · Page 3Linked to original sources

arXiv2026

Stochastic Modified Equations for Stochastic Gradient Descent in Infinite-Dimensional Hilbert Spaces

Inverse problems in scientific computing often require optimization over infinite-dimensional Hilbert spaces. A commonly used solver in such settings is stochastic gradient descent (SGD), where gradients are approximated using randomly sampled sub-objective functions. In this work we study the continuous-time limit of SGD in the small step-size regime. We show that the discrete dynamics can be approximated by a stochastic differential equation (SDE) driven by cylindrical Brownian motion. The analysis extends diffusion-approximation results previously established in Euclidean spaces to the infinite-dimensional setting. Two analytical difficulties arise in this extension. First, the cylindrical nature of the noise requires establishing well-posedness of the resulting stochastic evolution equation through appropriate structural conditions on the covariance operator. Second, since the randomness in SGD originates from discrete sampling while the limiting equation is driven by Gaussian noise, the comparison between the two dynamics must be carried out in a weak sense. We therefore introduce a suitable class of smooth functionals on the Hilbert space and prove that the discrepancy between SGD and the limiting SDE, when evaluated through these functionals, is of second order in the step size. Numerical experiments confirm the predicted convergence behavior.

math.OC↗

arXiv2026

Inference of interacting kernel in the mean-field regime

We study the problem of reconstructing interaction kernels in systems of interacting agents from macroscopic measurements when posed as an optimization problem. The reconstruction procedure depends on the formulation of the forward model, which may be given either by a finite-dimensional coupled ODE system tracking individual agent trajectories or by a mean-field PDE describing the evolution of the agent density. We investigate the similarities and differences between these two formulations in the mean-field regime. While the first variation derived from the particle system does not provide an unbiased estimator of the first variation associated with the limiting PDE, we prove that, under mild assumptions, the two are close in a weak sense with a convergence rate $\mathcal{O}(N^{-1/2})$. This rate is further confirmed by numerical evidences.

math.NA↗

arXiv2026

Random Coordinate Descent on the Wasserstein Space of Probability Measures

Optimization over the space of probability measures endowed with the Wasserstein-2 geometry is central to modern machine learning and mean-field modeling. However, traditional methods relying on full Wasserstein gradients often suffer from high computational overhead in high-dimensional or ill-conditioned settings. We propose a randomized coordinate descent framework specifically designed for the Wasserstein manifold, introducing both Random Wasserstein Coordinate Descent (RWCD) and Random Wasserstein Coordinate Proximal{-Gradient} (RWCP) for composite objectives. By exploiting coordinate-wise structures, our methods adapt to anisotropic objective landscapes where full-gradient approaches typically struggle. We provide a rigorous convergence analysis across various landscape geometries, establishing guarantees under non-convex, Polyak-Łojasiewicz, and geodesically convex conditions. Our theoretical results mirror the classic convergence properties found in Euclidean space, revealing a compelling symmetry between coordinate descent on vectors and on probability measures. The developed techniques are inherently adaptive to the Wasserstein geometry and offer a robust analytical template that can be extended to other optimization solvers within the space of measures. Numerical experiments on ill-conditioned energies demonstrate that our framework offers significant speedups over conventional full-gradient methods.

stat.ML↗

arXiv2026

Formal Deformation quantization as a Fréchet algebra

We define a Fréchet topology on the space $C^\infty(X)[[\hbar]]$ of formal smooth functions on a symplectic manifold $X$, by constructing a sequence of semi-norms on it. For any star product $\star$ on $C^\infty(X)[[\hbar]]$ making it a formal deformation quantization of $X$, we will show that the quantum product $\star$ is jointly continuous, and making it a Fréchet algebra. We will show a quantum Weierstrass theorem which says quantum polynomials are locally dense in all formal smooth functions. We will also show that the canonical trace of any formal deformation quantization is continuous under this Fréchet topology.

math.QA↗

arXiv2026

Correcting Auto-Differentiation in Neural-ODE Training

Does the use of auto-differentiation yield reasonable updates for deep neural networks (DNNs)? Specifically, when DNNs are designed to adhere to neural ODE architectures, can we trust the gradients provided by auto-differentiation? Through mathematical analysis and numerical evidence, we demonstrate that when neural networks employ high-order methods, such as Linear Multistep Methods (LMM) or Explicit Runge-Kutta Methods (ERK), to approximate the underlying ODE flows, brute-force auto-differentiation often introduces artificial oscillations in the gradients that prevent convergence. In the case of Leapfrog and 2-stage ERK, we propose simple post-processing techniques that effectively eliminates these oscillations, correct the gradient computation and thus returns the accurate updates.

cs.LG↗

arXiv2026

Cooperation in Public Goods Games over Uniform Random Hypergraphs with Game Transitions

The evolution of cooperation is a central enigma in evolutionary game theory. Traditionally, the combination of pairwise networks and repeated Public Goods Games with a single state fails to adequately describe realistic group interaction scenarios. On the one hand, pairwise networks lack clear group definitions. On the other hand, a participant's decision affects not only competitors' fitness but also the state of the surrounding environment. To address this problem, we propose a Public Goods Game with game transition mechanisms based on Uniform Random Hypergraphs. In our model, game groups formed by hyperedges transition between two types of games, one with abundant public resources and the other with scarce public resources. The transition probability is closely related to the strategies of players within the hyperedges. By developing a Monte Carlo simulation framework that incorporates payoff accumulation, strategy imitation, and game state transitions, we aim to reveal the coevolutionary patterns of strategies and game states in group interactions. Our study highlights a nonlinear relationship between defection sensitivity and cooperation frequency under game transitions, as well as the asymmetric effects of the two sensitivities in state-dependent transitions. These observations open new directions for how to approach social dilemmas.

physics.soc-ph↗

arXiv2026

From Natural Language to Executable Properties for Property-based Testing of Mobile Apps

Property-based testing (PBT) is a popular software testing methodology and is effective in validating the functionality of mobile applications (apps for short). However, its adoption in practice remains limited, largely due to the manual effort and technical expertise required to specify executable properties. In this experience paper, we propose a novel structured property synthesis approach that automatically translates property descriptions in natural language into executable properties, and implement it in a tool named iPBT. Our approach decomposes the problem into UI semantic grounding and executable property synthesis. It first builds an enriched widget context via multimodal LLMs to align visual elements with their functional semantics, and then uses an LLM with in-context learning to generate framework-specific executable properties. We evaluate iPBT with a closed-source LLM (GPT-4o) and an open-source LLM (DeepSeek-V3) on 124 diverse property descriptions derived from an existing benchmark dataset. iPBT achieves 95.2% (118/124) accuracy on both LLMs. Notably, an ablation study reveals that the enriched widget context contributes to an absolute improvement of up to 20.2% (from 75.0% to 95.2%). A user study with 10 participants demonstrates that iPBT reduces the time required to write executable properties by 56%, suggesting substantially lower manual effort. Furthermore, evaluations on 1,180 linguistically diverse variations demonstrate iPBT's robustness (87.6% accuracy), indicating its capability to handle varied expressions.

cs.SE↗

arXiv2026

Control of a Uniformly Magnetized Plasma with External Electric Fields

Stabilizing plasma dynamics through externally applied electric and magnetic fields is a fundamental control problem. We study this question for a plasma evolving under a uniform external magnetic field. Although the governing dynamics are nonlinear, a linear analysis based on the Laplace-Fourier transform yields actionable insight. In particular, by controlling the location of the roots of the dispersion relation, we propose a general control strategy that restores stability, with the free-streaming solution recovered as a special case. Numerical experiments for Gaussian equilibria and for the Dory-Guest-Harris instability show that the proposed control suppresses the unstable modes and stabilizes the dynamics, in agreement with our theoretical predictions.

physics.plasm-ph↗

arXiv2026

Safe and Stylized Trajectory Planning for Autonomous Driving via Diffusion Model

Achieving safe and stylized trajectory planning in complex real-world scenarios remains a critical challenge for autonomous driving systems. This paper proposes the SDD Planner, a diffusion-based framework designed to effectively reconcile safety constraints with driving styles in real time. The framework integrates two core modules: a Multi-Source Style-Aware Encoder, which employs distance-sensitive attention to fuse dynamic agent data and environmental contexts for heterogeneous safety-style perception; and a Style-Guided Dynamic Trajectory Generator, which adaptively modulates priority weights within the diffusion denoising process to generate user-preferred yet safe trajectories. Extensive experiments demonstrate that SDD Planner achieves state-of-the-art performance. On the StyleDrive benchmark, it improves the SM-PDMS metric by 3.9% over WoTE, the strongest baseline. Furthermore, on the NuPlan Test14 and Test14-hard benchmarks, SDD Planner ranks first with overall scores of 91.76 and 80.32, respectively, outperforming leading methods such as PLUTO. Real-vehicle closed-loop tests further confirm that SDD Planner maintains high safety standards while aligning with preset driving styles, validating its practical applicability for real-world deployment.

cs.RO↗

arXiv2026

Supervised tax compliance and evasion from a spatial evolutionary game perspective

Taxation constitutes a fundamental component of modern national economic systems, exerting profound impacts on both societal functioning and governmental operations. In this paper, we employ an interdependent network approach to model the coevolution between citizens and regulators within a taxation system that fundamentally constitutes a public goods game framework with complex interactive dynamics. In a game layer, citizens engage in public goods games, facing the social dilemma of tax compliance (cooperation) versus evasion (defection). Tax compliance supports the sustainability of public finances while tax evasion presents markedly stronger short-term incentives. In a regulatory layer, fair regulators punish tax evaders, while corrupt regulators keep silent due to bribes. Governmental regulatory interventions introduce critical institutional constraints that alter the traditional equilibrium of the game. Importantly, there exists a strategy update not only among citizens but also among regulators. Our results indicate that strengthening penalties can effectively curb tax evasion, and the influence of bribery on both tax compliance rates and the proportion of fair regulators is nonlinear. Additionally, increasing regulators' salaries and intensifying the crackdown on corrupt regulators can foster the emergence of fair regulators, thereby reducing tax evasion among citizens. The results offer practical policy implications, suggesting that balanced deterrence and institutional fairness are essential to sustaining compliance, and point to the need for future empirical validation and model extensions.

physics.soc-ph↗

arXiv2026

Optimal design for linear models via gradient flow

Optimal experimental design (OED) aims to choose the observations in an experiment to be as informative as possible, according to certain statistical criteria. In the linear case (when the observations depend linearly on the unknown parameters), it seeks the optimal weights over rows of the design matrix $\mA$ under certain criteria. Classical OED assumes a discrete design space and thus a design matrix with finite dimensions. In many practical situations, however, the design space is continuous-valued, so that the OED problem is one of optimizing over a continuous-valued design space. The objective becomes a functional over the probability measure, instead of a function of a finite dimensional vector. This change of perspective requires a new set of techniques to optimize over probability measures, and Wasserstein gradient flow becomes a natural candidate. Both the first-order criticality and the convexity properties of the OED objective are presented. Computationally, the Monte Carlo particle method is used to translate the gradient flow equation formulation into a numerical algorithm. This algorithm is applied to two elliptic inverse problems.

math.NA↗

arXiv2026

Effects of Stochastic Games on Evolutionary Dynamics in Structured Populations

Continuously changing environments have a paramount role in the evolution of cooperative behavior. Previous works have shown that the transitions among different games, as the feedback between behaviors and environments, can promote cooperative behavior significantly. Quantitative analysis, however, is limited to homogeneous populations, while realistic populations in nature are often more complex and highly heterogeneous. We hereby provide an analytical treatment of when the evolution of cooperation can be supported in stochastic games, applying to arbitrary spatial heterogeneity and payoff structure. We highlight that the rule and frequency of game changes can have surprisingly diverse effects on evolutionary outcomes, depending on the governing social dilemmas. While stochastic games favor the evolution of cooperation in donation games, this is not the case for public goods games and snowdrift games. Hence, our framework and model results offer a more subtle insight into the long-standing enigma.

physics.soc-ph↗

arXiv2026

Comparative study of equilibrium and non-equilibrium predictions by different models for a hypersonic cone at high-altitude

Targeting a cone with the half-angle as 10-deg at M = 27 and H = 72 km, simulations were conducted comparatively to analyze the predictions by different equilibrium and non-equilibrium gas models. Following validation and grid studies, systematic comparisons on aerodynamic performance, flow structures, and characteristic distributions were performed. The key findings are: (1) While the overall flow structures are broadly similar, discrepancies exist in the features at the base locations, e.g., the diverse high-temperature distributions. Notably, the vibrational temperatures distribute differently under slip and non-slip boundary conditions near the wall; (2) The equilibrium gas model predicts higher drag coefficient, wall heat flux, and skin friction than those of non-equilibrium models. Predictions also vary among the non-equilibrium models themselves. Specifically, compared to the three-temperature model, the one- and two-temperature models predict larger drag coefficients with the relative difference exceeding 5%. Nevertheless, the results from the three-temperature model with and without slip conditions are largely consistent; (3) The disparities between equilibrium and non-equilibrium characteristics are primarily manifested in the shock layer and wake regions. Within these regions, the overall temperature for the equilibrium gas is lower than that for the non-equilibrium cases, while in the latter specific non-equilibrium features are distinctly exhibited, e.g., the translational-rotational temperature is generally higher than that from the one-temperature model, and the vibrational-electronic temperature shows the opposite trend. Notably, in the slip flow within the three-temperature model, the translational-rotational temperature is higher and, particularly, the vibrational temperature is even larger than counterparts of the non-slip flows near the wall and base center line.

physics.flu-dyn↗

arXiv2026

Data selection: at the interface of PDE-based inverse problem and randomized linear algebra

All inverse problems rely on data to recover unknown parameters, yet not all data are equally informative. This raises the central question of data selection. A distinctive challenge in PDE-based inverse problems is their inherently infinite-dimensional nature: both the parameter space and the design space are infinite, which greatly complicates the selection process. Somewhat unexpectedly, randomized numerical linear algebra (RNLA), originally developed in very different contexts, has provided powerful tools for addressing this challenge. These methods are inherently probabilistic, with guarantees typically stating that information is preserved with probability at least 1-p when using N randomly selected, weighted samples. Here, the notion of "information" can take different mathematical forms depending on the setting. In this review, we survey the problem of data selection in PDE-based inverse problems, emphasize its unique infinite-dimensional aspects, and highlight how RNLA strategies have been adapted and applied in this context.

math.NA↗

arXiv2026

Modeling Epidemic Dynamics of Mutant Strains with Evolutionary Game-based Vaccination Behavior

The outbreak of mutant strains and vaccination behaviors have been the focus of recent epidemiological research, but most existing epidemic models failed to simultaneously capture viral mutation and consider the complexity and behavioral dynamics of vaccination. To address this gap, we develop an extended SIRS model that distinguishes infections with the original strain and a mutant strain, and explicitly introduces a vaccinated compartment state. At the behavioral level, we employ evolutionary game theory to model individual vaccination decisions, where strategies are determined by both neighbors' choices and the current epidemiological situation. This process corresponds to the time-varying vaccination rate of susceptible individuals transitioning to vaccinated individuals at the epidemic spreading level. We then couple the epidemic and vaccination behavioral processes through the microscopic Markov chain approach (MMCA) and finally investigate the evolutionary dynamics via numerical simulations. The results show that our framework can effectively mitigate outbreaks across different disease scenarios. Sensitivity analysis further reveals that vaccination uptake is most strongly influenced by vaccine cost, efficacy, and perceived risk of side effects. Overall, this behavior-aware modeling framework captures the co-evolution of viral mutation and vaccination behavior, providing quantitative and theoretical support for designing effective public health vaccination policies.

q-bio.PE↗

arXiv2025

DEVAL: A Framework for Evaluating and Improving the Derivation Capability of Large Language Models

Assessing the reasoning ability of Large Language Models (LLMs) over data remains an open and pressing research question. Compared with LLMs, human reasoning can derive corresponding modifications to the output based on certain kinds of changes to the input. This reasoning pattern, which relies on abstract rules that govern relationships between changes of data, has not been comprehensively described or evaluated in LLMs. In this paper, we formally define this reasoning pattern as the Derivation Relation (DR) and introduce the concept of Derivation Capability (DC), i.e. applying DR by making the corresponding modification to the output whenever the input takes certain changes. To assess DC, a systematically constructed evaluation framework named DEVAL is proposed and used to evaluate five popular LLMs and one Large Reasoning Model in seven mainstream tasks. The evaluation results show that mainstream LLMs, such as GPT-4o and Claude3.5, exhibit moderate DR recognition capabilities but reveal significant drop-offs on applying DR effectively in problem-solving scenarios. To improve this, we propose a novel prompt engineering approach called Derivation Prompting (DP). It achieves an average improvement of 15.2% in DC for all tested LLMs, outperforming commonly used prompt engineering techniques.

cs.LG↗

arXiv2025

Reputation assimilation mechanism for sustaining cooperation

Keeping a high reputation, by contributing to common efforts, plays a key role in explaining the evolution of collective cooperation among unrelated agents in a complex society. Nevertheless, it is not necessarily an individual feature, but may also reflect the general state of a local community. Consequently, a person with a high reputation becomes attractive not just because we can expect cooperative acts with higher probability, but also because such a person is involved in a more efficient group venture. These observations highlight the cumulative and socially transmissible nature of reputation. Interestingly, these aspects were completely ignored by previous works. To reveal the possible consequences, we introduce a spatial public goods game in which we use an assimilated reputation simultaneously characterizing the individual and its neighbors' reputation. Furthermore, a reputation-dependent synergy factor is used to capture the high (or low) quality of a specific group. Through extensive numerical simulations, we investigate how cooperation and extended reputation co-evolve, thereby revealing the dynamic influence of the assimilated reputation mechanism on the emergence and persistence of cooperation. By fostering social learning from high-reputation individuals and granting payoff advantages to high-reputation groups via an adaptive multiplier, the assimilated reputation mechanism promotes cooperation, ultimately to the systemic level.

physics.soc-ph↗

arXiv2025

Forward Euler for Wasserstein Gradient Flows: Breakdown and Regularization

Wasserstein gradient flows have become a central tool for optimization problems over probability measures. A natural numerical approach is forward-Euler time discretization. We show, however, that even in the simple case where the energy functional is the Kullback-Leibler (KL) divergence against a smooth target density, forward-Euler can fail dramatically: the scheme does not converge to the gradient flow, despite the fact that the first variation $\nabla\frac{δF}{δρ}$ remains formally well defined at every step. We identify the root cause as a loss of regularity induced by the discretization, and prove that a suitable regularization of the functional restores the necessary smoothness, making forward-Euler a viable solver that converges in discrete time to the global minimizer.

math.NA↗