数学前沿

We build structure-preserving reduced-order models for both challenge problems of the AePW-4 High-Speed Working Group. They are a clamped thin panel at Mach 1.92 (RC-19) and a cantilever plate at Mach 5.85 (HyMAX). The published data for each case hold no time-resolved response and no repeated runs. The physics must therefore be carried by the form of the model. One construction gives two models per problem. A Ritz-Galerkin reference model describes the plate. A learned model keeps the reference model's exact potential, fits the remaining generalized forces, and trains only a dissipation that cannot add energy. Its energy balance holds for any weights by construction. On the trained model the worst relative residual is 2.2e-15. On RC-19, shock relations predict a pressure ratio of 1.521 across the panel. Pressure-sensitive paint measured the same value. Thirty runs of the learned model place the mean dominant frequency within 1.0% of the measured 236 Hz. At one periodic and one chaotic condition, the learned model reproduces the response regime of a published reduced-order model in all sixteen runs. It does so without that model's 113 Hz spectral peak, which the measurements do not show. Measured displacements can also correct the chaotic response through one more force that respects the energy balance. We tested this on synthetic data at the experiment's 5 kHz sampling rate. The state error is 1.25% of the panel thickness without noise and 3.5% with 10 micrometers of noise. On HyMAX the learned model follows the reference model's trailing-edge displacement for 65 ms with a relative error of 3.7%. These results show that a model whose structure carries the physics can match measured flutter behavior when the available data cannot train it.

Policy gradient methods are among the most successful approaches for solving challenging reinforcement learning problems. Despite their empirical success, many state-of-the-art policy gradient algorithms for discounted problems estimate update directions using a sampling distribution that differs from the discounted state distribution required by the policy gradient theorem. In this work, we analyze the impact of this distribution mismatch on policy gradient methods. Specifically, {for tabular direct and softmax parameterizations, we establish global convergence of objective values under biased updates}. {For tabular softmax policies, we further derive the exact leading-order coefficients of the asymptotic objective errors under biased and normalized exact updates. Their ratio provides an explicit criterion for determining when distribution mismatch accelerates or slows asymptotic convergence.} We then extend the analysis to more general policy parameterizations by deriving explicit bounds on both the state distribution mismatch and the induced normalized policy-gradient mismatch in episodic and continuing MDPs. These mismatches vanish at least linearly as the discount factor approaches one. Under an additional smoothness assumption, we further show that the limit inferior of the squared norm of the normalized exact policy-gradient direction along the biased iterates is bounded by $O((1-\gamma)^2)$ in both settings. Our results provide theoretical insights into the robustness of policy gradient methods to distribution mismatch and shed light on the gap between their theoretical formulations and practical implementations.

The assessment of planktonic standing stocks and microorganism structures is critical for understanding upper ocean biological processes. Currently, autonomous underwater vehicles (AUVs) equipped with in-situ optical imaging and artificial intelligence (AI) methods offer a promising solution for persistent surveillance, mapping and monitoring of planktonic life. However, current AI methods often lack robustness in dynamic, unstructured environments, where environmental noise and non-biological artifacts lead to frequent misclassifications. Standard convolutional neural network (CNN) classifiers often struggle with such conditions, leading to misclassifications that require time-consuming manual validation by marine biologists. To address this issue, we propose a novel robustness verification framework for in-situ plankton classifiers based on reachability analysis. We also introduce a continuous-time neural ordinary differential equation (neural ODE) classification model leveraging the high-resolution imaging capabilities of the SilCam particle imager. In this paper, we demonstrate the effectiveness of the proposed framework by formally verifying the robustness of the neural ODE model against environmental perturbations. We demonstrate that our verification framework acts as an automated filter providing formal guarantees of model stability against ambiguous data, thereby improving the reliability of autonomous sampling and reducing the post-processing workload.

In this paper, we develop a rigorous optimal control-theoretic approach to Transformer training that respects key structural constraints such as (i) realized-input-independence during execution, (ii) the ensemble control nature of the problem, and (iii) positional dependence. We model the Transformer architecture as a discrete-time controlled particle system with shared actions, exhibiting noise-free controlled McKean--Vlasov dynamics, and we show that lifting the dynamics to probability measures produces a fully-observed Markov decision process (MDP). Positional encodings are incorporated into the state space to preserve the sequence order under lifting. Using the dynamic programming principle, we establish the existence of globally optimal policies under mild assumptions of compactness. We further prove that closed-loop policies in the lifted problem is equivalent to an initial-distribution dependent open-loop policy, which are realized-input-independent and compatible with standard Transformer training. To train a Transformer, we propose a triply quantized training procedure for the lifted MDP by quantizing the state space, the space of probability measures, and the action space, and show that any optimal policy for the triply quantized model is near-optimal for the original training problem. Finally, we establish stability and empirical consistency properties of the lifted model by showing that the value function is continuous with respect to the perturbations of the initial empirical measures and convergence of policies as the data size increases. This approach provides a globally optimal and robust theoretical alternative to gradient-based training without requiring smoothness or convexity.

Sharpness-Aware Minimization (SAM) has established itself as a widely used method for training deep neural networks. Despite its popularity, existing convergence guarantees for SAM-type methods require smoothness, a restrictive assumption. Generalized smoothness conditions have recently been proposed in the analysis of gradient methods but remain underexplored for SAM. To analyze the effect of SAM's perturbation under generalized smoothness, we use relative gradient distortion to quantify how the sharpness-aware direction departs from the original gradient. Under $(L_0,L_1)$-smoothness, we show that this distortion can become unbounded near stationarity for SAM and at large gradients for its unnormalized variant (USAM), with examples where SAM cycles and USAM diverges. We then establish a necessary condition for uniform distortion control among perturbations along the gradient whose nonnegative multipliers depend only on its norm. Motivated by this condition, we propose Curvature-Scaled SAM (CS-SAM), which scales both the perturbation and the descent step by suitable curvature upper bounds, and prove that it uniformly bounds the distortion. Under both $(L_0,L_1)$- and $(H_0,H_1)$-smoothness, we establish deterministic $\mathcal{O}(1/T)$ rates for the best squared gradient norm on nonconvex objectives, last-iterate $\mathcal{O}(1/T)$ rates for the function gap on convex objectives with a minimizer, and linear convergence under the Polyak-Lojasiewicz condition. With zero perturbation, our guarantees recover the rates for gradient descent with $(L_0,L_1)$- or $(H_0,H_1)$-based step-sizes. Minibatch experiments on computer vision tasks demonstrate competitive performance for both variants, with the $(H_0,H_1)$ variant achieving the highest mean best-test accuracy in all four settings against tuned SAM and USAM with constant and cosine annealing schedules.

Safe robotic task planning requires planners that can reliably achieve task goals while satisfying safety constraints. Classical planners search over explicit symbolic task representations and can provide formal guarantees with respect to encoded safety constraints, but require rigid input formats. Their computational cost also increases rapidly with the search space. In contrast, large language models (LLMs) offer flexible task interfaces and strong generalization capabilities, but they are large, power-hungry, and pre-trained with little robot-specific safety data and knowledge. Thus, they can be difficult to deploy locally on robotic systems with limited resources. To address these limitations, we present SLoSa, a neuro-symbolic compression and fine-tuning framework for training foundation-model-based robotic task planners with Small size, Low power demand, and Safety awareness. The framework first constructs a safety-aware dataset using classical planners, an advanced teacher LLM, and formal verifiers, followed by two training stages: supervised fine-tuning and reinforcement learning (RL) fine-tuning guided by a verification-based reward machine to enforce safety alignment. We evaluate SLoSa across four Planning Domain Definition Language (PDDL) domains, two household simulation environments, local edge platforms, and a desktop robotic arm. Across the PDDL tasks, verifier-guided post-training raises the success rate of compact 4B-14B planners to 70.5-76.0%, while INT4 deployment on a Jetson Orin requires 17.85-26.41 W of average power. Results in household simulators and on the physical robot further demonstrate the applicability of SLoSa across diverse robotic task settings.

``What are the irreducible conditions that are sufficient to produce an outcome?'' is one of the most common questions that recur across computation and science. Its answers, the minimal sufficient witnesses, are what we mean by explanations, mechanisms and reasons. These problems usually ask for multiple minimal witnesses, yet standard RL methods may reveal only one solution or redundant ones. We formalize this problem as minimal-witness identification and introduce Minimal-Witness Reinforcement Learning (MWRL). MWRL takes the union of the sets certified by successful proposals sampled from the policy and credits each proposal for the coverage the group union would lose without that proposal. This credit assignment, derived directly from the problem definition, unifies the demands for minimality and recovery of alternatives from a single black-box verifier bit. Under this principle, we derive a value iteration planner that recovers the entire family of witnesses and a policy gradient method that can scale to large language models. Across different experimental settings, MWRL recovers most minimal witnesses, while other methods return redundant supersets or a single witness. By making witness families learnable from verifier feedback, MWRL expands the scope of reinforcement learning beyond single-solution optimization. Our code is available at https://github.com/TSUITUENYUE/MWRL.

Neural ordinary differential equations (neural ODE) gained attention in safety critical settings such as continuous-time controllers for cyber-physical systems and classifiers integrated into automated decision pipelines, raising the question whether their behavior can be formally verified. Existing tools dedicated to neural ODE provide only a single reachability call without iterative input-set refinement, limiting the precision of their verdicts to whatever one reachability call can deliver. We present TNODEV, the first formal verifier for neural ODE that integrates a falsification checker, a fast interval-based reachability backend based on continuous-time mixed monotonicity, a verification and refinement loop with three input-set splitting heuristics, and a parallel scheduler in a single end-to-end pipeline. TNODEV supports safe-set inclusion verification on pure neural ODE, neural ODE in closed loop with a neural network controller and general neural ODE (GNODE), with the safe set specified either as an interval or as the half-space intersection induced by a target classification label. We evaluate TNODEV on a range of benchmarks across safe-set inclusion and classification-robustness properties, including a direct reachability comparison against NNV 2.0 and CORA and a verification comparison against NNV 2.0 on MNIST general neural ODE classifiers.

In this paper, we study continuous-time reinforcement learning, where the state follows an unknown stochastic differential equation but only a fixed off-policy batch of discrete-time transitions is available. Standard discrete-time Q-learning becomes problematic as the sampling interval shrinks, because differences between action values vanish. Continuous-time alternatives, such as Deep Advantage Updating and martingale q-learning, preserve the action signal but jointly learn a value function and an action-dependent critic. Building on the PhiBE-Q formulation (arXiv:2606.21925), we develop a model-free PhiBE policy iteration (PhiBE-PI) framework that uses a single state-action critic while preserving the continuous-time action-ranking scale. At each iteration, the framework learns the critic directly from sampled increments and uses it to improve the policy. The same batch is reused throughout, without explicitly estimating the drift or diffusion. We further develop an implementation under linear function approximation and establish a local end-to-end suboptimality guarantee for the learned policy. Our analysis quantifies how errors from time discretization, function approximation, and finite-sample policy evaluation propagate through policy iteration. It accounts for the dependence of successive policies on the reused data, so the guarantee does not require a fresh batch at each iteration. Moreover, we identify a bias-variance trade-off: smaller sampling intervals improve the approximation to the continuous-time problem but can increase statistical error at a fixed sample size. Thus, finer time discretization does not necessarily lead to better policy performance.

Equipping Large Language Models (LLMs) to execute reliable multi-step workflows has become a central challenge in artificial intelligence. Despite recent advances in LLMs' agentic capabilities, most agent systems still lack formal methods for specifying, verifying, and debugging their workflow and execution trajectories. This challenge mirrors a long-standing problem in mathematics, where the ambiguity of natural languages (NLs) motivates the development of formal languages (FLs). Inspired by this paradigm, we propose **Lean4Agent**, to the best of our knowledge, the first framework that uses Lean4, a dependent-type FL to model and verify agent behavior. **Lean4Agent** launches **FormalAgentLib**, an extensible Lean4 library for formally modeling and verifying agent workflows' semantic consistency under explicit assumptions, and enabling localization of execution-time failures revealed by trajectories. Building on **FormalAgentLib**, we further develop **LeanEvolve**, which applies results in **FormalAgentLib** to revise workflows to enhance its capability. Extensive experiments on a hard problem subset of SWE-Bench-Verified and a subset of ELAIP-Bench across 5 leading LLMs indicate that the verification-passing workflows outperform the failing ones by an average of **11.94%**, and **LeanEvolve** further improves SWE performance by **7.47%** on average. Furthermore, **Lean4Agent** establishes a foundation for a new field of using expressive dependent-type FL to formally model and verify agent behavior.

In this paper, we propose a geometric approach to deriving attention mechanisms and exploring potential new transformer architectures. We begin by formalizing the notions of knowledge worlds and attention pairings, then demonstrate how this framework naturally leads to the architectures of standard attention, causal attention, linear attention, tensor attention, and recurrent neural networks. Finally, we introduce several new geometric spaces that may inspire novel neural network architectures.

This paper charts the evolution of the Prolog textbook Simply Logical -- Intelligent Reasoning by Example from print with runnable programmes on a 3.5-inch diskette published in 1994, via various intermediate online versions, to the fully interactive online edition available today. Our aim is to investigate -- from both the writer's and the reader's perspectives -- the potential impact of technology on educational materials. The three authors of this paper present three distinct and complementary points of view, which come together to shape an interactive online book that offers an immersive learning experience. Peter describes the philosophy behind the original book and experiences teaching from it. Jan demonstrates how contemporary web programming has enabled a fully interactive realisation of the book's philosophy. Kacper reports how different technologies can be adopted to build a versatile authoring toolkit that underpins the interactive online edition of the book. Collating these distinct yet coherent perspectives allows us to reflect on future opportunities for interactive Prolog teaching materials arising from the continuous development of web technologies. The insights we offer should be of interest to teachers of Prolog as well as those wanting to find out about the latest educational technologies.

Large language models increasingly automate optimization modeling from natural language problems, producing written mathematics, solver code and a returned report. Verifying one optimization problem across these artifacts requires semantic comparison: unequal models can attain the same objective, equivalent models can use different auxiliary variables, and mutually consistent artifacts can omit the same source requirement. ORSMT provides a typed semantic interface and exact SMT relation queries. It aligns observable decisions, projects auxiliary variables, proves or refutes feasible region and objective relations, localizes refutations to implicated rows or domains and links source texts when available. It then reconstructs one artifact from another, and each adopted candidate passes the required relation checks. Across 3,034 problems and six generator sizes, ORSMT improves the model-code match rate in all 72 benchmark and generator combinations before selection. Protected selection raises the aggregate rate from 43.2% to 88.2% and the share that are both correct and matched from 25.2% to 40.2%. On the 1,905 refuted outputs, certified voting under the shared fallback makes 50.9% of outputs correct and matched, compared with 44.8% for ReLoop and 49.4% for Opt-Verifier. A separate one-round source revision followed by exact reconstruction restores a model-code match for 84.4% of the refuted pairs.

Cutting planes (cuts) remove fractional solutions from mixed-integer linear programs (MILPs) without excluding any feasible integer solutions, making them a key ingredient in modern MILP solvers. Current state-of-the-art solvers rely on fine-tuned handcrafted cut strategies, which, despite their effectiveness, remain inherently static and can be suboptimal. More recently, learning-based approaches have been proposed to design adaptive cut strategies; however, most of these methods are based on neural networks, offering limited interpretability and providing no guarantees of convergence. In this paper, we propose \emph{Genetic Programming and Statistics-based Cut Selection} (GPSCS), the first learning-based framework that explicitly generalizes handcrafted cut selection heuristics into a fully interpretable policy. First, a genetic programming model evolves a symbolic scoring function that ranks candidate cuts using features derived from the cuts and the solving context. Second, a lightweight statistics-based mechanism adaptively determines the number of top-ranked cuts to add at each separation round by detecting significant score dropouts. We evaluate GPSCS within the SCIP solver on seven classes of synthetic and real-world benchmark instances, with a strong emphasis on generalization: evaluation instances are on average 41 times harder than those used for training. Experimental results show that GPSCS improves average solving time by $6\%$ compared to SCIP's default cut selection strategy and achieves a two-fold speedup over both a recent neural-network-based approach and a no-cut baseline, demonstrating strong robustness and generalization across heterogeneous MILP families.

We study causal self-attention dynamics -- a toy model for decoder Transformers -- which we interpret as a non-exchangeable interacting particle system. Adapting cumulant expansions to the triangular causal dependency structure of the model, and appealing to non-hierarchical methods to estimate correlations using Glauber calculus, we prove a quantitative mean-field limit result and a next-order characterization of correlations. For iid uniformly distributed tokens, the limiting correlation equation can be solved in closed form and we obtain a rigorous explanation of the empirically observed \emph{lost-in-the-middle} phenomenon: the token retrieval profile, as a function of the source position in the prompt, is $\mathsf{U}$-shaped, with primacy, recency, and a unique interior minimum under an explicit smallness condition.

We prove that for every open $U \subseteq \mathbb R^n$ the spherical projection of every gradient trajectory $\theta(t)$, $t \geq 0$, of a definable $C^1$-potential $F \colon U \to \mathbb R$ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable $C^1$-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field. Notably, the gradient trajectory in this result is an arbitrary solution of the gradient flow differential equation (whose solutions may not be unique) with no further assumptions: The gradient trajectory may be unbounded and diverge to infinity or it may be convergent. The established theorem therefore provides affirmative answers to both the gradient conjecture at infinity for definable potentials and the definable Thom gradient conjecture. We apply our theorem to the training of deep neural networks (DNNs) to conclude that, in the case of definable and continuously differentiable activation and loss functions, the spherical projection of the DNN training flow must always converge (directional convergence).

We determine the minimum number of directional commands needed to solve alevel of Order of the Sinking Star, modeled as a crystal-exchangepuzzle on an 11*11 board. A command swapsthe player with the nearest crystal in the selected row or column, regardlessof distance; only an empty ray permits an ordinary unit walk. Two gates admitthe player to the exit exactly while all sixteen crystals occupy their targetcells. We give an explicit 93-command solution and exclude every shortersolution by an exact, cost-bounded breadth-first search. The search representsa position by its seventeen occupied cells and one active cell. Its admissibleheuristic partitions command costs between a compressed core abstraction andresidual motion, strengthened by a necessary activation detour and a cornervisit condition. The completed computation retained 605,349,215 occupied setsand 1,863,925,136 states. We prove the reductions and pruning conditions,describe the exact storage scheme, and provide source code, execution logs,and a reproduction driver. The result is a computer-assisted proof for thespecified instance, with an explicit implementation trust boundary.

Large retail chains must tailor assortment and pricing decisions to local customer preferences, yet even retailers with extensive sales records often have limited data for estimating demand at the store or local market level, particularly in product categories with short selling windows. Local estimation (fitting a demand model for each store on its own) can therefore be unreliable, while pooling data across stores improves estimation precision at the cost of ignoring local demand heterogeneity. We study how retailers can balance these competing effects through transfer learning under a multinomial logit (MNL) demand model. We propose a shrinkage estimator that combines store-specific demand estimates with information pooled from other stores. Unlike the linear or quadratic settings underlying much of the existing theoretical literature on information transfer in operations, the nonlinear MNL likelihood does not admit a straightforward bias-variance decomposition. By exploiting its self-concordant-like properties, we establish finite-sample performance guarantees that characterize when information transfer improves demand estimation and how the appropriate degree of shrinkage depends on local sample size, feature dimensionality, and cross-store heterogeneity. We further derive regret bounds for downstream assortment and pricing decisions. Numerical experiments demonstrate the operational benefits of shrinkage across different data environments. An empirical study using data from several hundred stores of a European apparel retailer shows that shrinkage improves out-of-sample product prioritization and revenue capture relative to both local estimation and full pooling. Our findings demonstrate how retailers can leverage information across their store networks to improve revenue management decisions while preserving local demand differences.

We investigate the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces. We show that the covariance operator of a weak second-order Radon probability measure on such a space is uniquely determined by a symmetric positive-definite function. In the Gaussian case, we characterize those functions that arise from covariance operators in terms of $\gamma$-radonifying operators. Building on these results, we extend Driscoll's theorem to the reproducing kernel Banach space setting.

A learner's present predictions do not fully determine how it will learn next. We study the dynamics of hidden learning structure through two related operations. Fiber Regulation Theory (FRT) describes protected repair of existing function and the learning responses left available after repair. Fiber Acquisition Theory (FAT) describes the formation and renewal of trainable structure, distinguishing neutral installation, access to a learning signal, and additional functional directions. The two operations also constrain one another. A repair obligation can force a change in a learning channel, while acquired features can make new protected repairs feasible or reduce the minimum coefficient cost of existing ones. We characterize joint admissibility and response reachability, and give a finite supervised-learning construction in which the two operations together produce a favorable response absent from either operation alone. We implement protected head repair and fixed-capacity LoRA renewal in Qwen2.5-32B with AdamW, under replay and output distillation. Across two training trajectories differing in seed and within-task example order, the coordinated policy achieves higher teacher-forced Exact than either separate policy in all four comparisons. It also has the lowest macro NLL under both learning rules on the second trajectory. The implementations realize structure-conditioned self-regulation within specified information and action constraints: accessible state readouts and arrived evidence guide repairs and structural renewals, and ordinary learning continues from the modified state.