Training-free guided generation is a widely used and powerful technique that allows the end user to exert further control over the generative process of flow/diffusion models. Generally speaking, two families of techniques have emerged for solving this problem for gradient-based guidance: namely, posterior guidance (i.e., guidance by projecting the current sample to the target distribution via the target prediction model) and end-to-end guidance (i.e., guidance by performing backpropagation throughout the entire ODE solve). In this work, we show that these two seemingly separate families can actually be unified by looking at the posterior guidance as a greedy strategy of end-to-end guidance. We explore the theoretical connections between these two families and provide an in-depth theoretical understanding of these two techniques relative to the continuous ideal gradients. Motivated by this analysis, we then show a method for interpolating between these two families enabling a trade-off between compute and accuracy of the guidance gradients.

The constant $\large \pi$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections among formulas remain unknown, missing unifying theories that could unveil deeper understanding. The absence of a unifying theory reflects a broader challenge across math and science: knowledge is typically accumulated through isolated discoveries, while deeper connections often remain hidden. In this work, we present an automated framework for the unification of mathematical formulas. Our system combines large language models (LLMs) for systematic formula harvesting, an LLM-code feedback loop for validation, and a novel symbolic algorithm for clustering and eventual unification. We demonstrate this methodology on the hallmark case of $\large \pi$, an ideal testing ground for symbolic unification. Applying this approach to 455,050 arXiv papers, we validate 385 distinct formulas for $\large \pi$ and prove relations between 360 (94\%) of them, of which 166 (43\%) can be derived from a single mathematical object--linking canonical formulas by Euler, Gauss, Brouncker, and newer ones from algorithmic discoveries by the Ramanujan Machine. Our method generalizes to other constants, including $e$, $\zeta(3)$, and Catalan's constant, demonstrating the potential of AI-assisted mathematics to uncover hidden structures and unify knowledge across domains.

We introduce training-free looped transformers, in which a lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning, continued training, or architectural changes. Unlike prior looped transformer methods that train with the looped structure end-to-end, we retrofit recurrence onto pretrained models at test time. We show that naive block reapplication usually degrades performance, highlighting the importance of the loop application strategy. Motivated by viewing a pre-norm transformer block as a forward Euler step on an ODE, we instead treat looping as a refinement of the same approximation, replacing one large update with smaller damped sub-steps. Across seven dense, sparse MoE, and MLA+MoE model families, our method improves Qwen3-4B-Instruct by +2.64 pp on MMLU-Pro, Qwen3-30B-A3B-Instruct by +1.14 pp on CommonsenseQA, and Moonlight-16B-A3B-Instruct by +1.20 pp on OpenBookQA.

Protein-ligand binding prediction is a fundamental problem in AI-driven drug discovery. Previous work focused on supervised learning methods for small molecules where binding affinity data is abundant, but it is hard to apply the same strategy to other ligand classes like antibodies where labelled data is limited. In this paper, we explore unsupervised approaches and reformulate binding energy prediction as a generative modeling task. Specifically, we train an energy-based model on a set of unlabelled protein-ligand complexes using SE(3) denoising score matching (DSM) and interpret its log-likelihood as binding affinity. Our key contribution is a new equivariant rotation prediction network for SE(3) DSM called Neural Euler's Rotation Equations (NERE). It predicts a rotation by modeling the force and torque between protein and ligand atoms, where the force is defined as the gradient of an energy function with respect to atom coordinates. Using two protein-ligand and antibody-antigen binding affinity prediction benchmarks, we show that NERE outperforms all unsupervised baselines (physics-based potentials and protein language models) in both cases and surpasses supervised baselines in the antibody case.

Consequently, recent methods have aimed to tailor the models and training procedures to stabilise the discrete updates.

We study first-order optimization algorithms obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: symplectic Euler (S), explicit Euler (E) and implicit Euler (I) schemes. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves the accelerated rate for minimizing both strongly convex function and convex function. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.

We present the results of a large-scale computational analysis of mathematical papers from the ArXiv repository, demonstrating a comprehensive system that not only detects mathematical errors but provides complete referee reports with journal tier recommendations. Our automated analysis system processed over 37,000 papers across multiple mathematical categories, revealing significant error rates and quality distributions. Remarkably, the system identified errors in papers spanning three centuries of mathematics, including seven works by Leonhard Euler (1707-1783) in just 403 papers analyzed from the History category, as well as errors by Peter Gustav Lejeune Dirichlet (1805-1859) and contemporary Fields medalists. In Dynamical Systems (math.DS), we observed the highest error rate of 11.4% (2,347 errors in 20,666 papers), while Numerical Analysis (math.NA) showed 9.6% (2,271 errors in 23,761 papers). History and Overview (math.HO) exhibited 13.6% errors in preliminary analysis, including seven papers by Euler. In contrast, Geometric Topology (math.GT) showed 3.6% and Category Theory (math.CT) exhibited the lowest rate at 6.1% (228 errors in 3,720 papers). Beyond error detection, the system evaluated papers for journal suitability, recommending 0.4% for top generalist journals, 15.5% for top field-specific journals, and categorizing the remainder across specialist venues. These findings demonstrate both the universality of mathematical error across all eras and the feasibility of automated comprehensive mathematical peer review at scale. This work demonstrates that the methodology, while applied here to mathematics, is discipline-agnostic and could be readily extended to physics, computer science, and other fields represented in the ArXiv repository.

Visual-inertial fusion is crucial for a large amount of intelligent and autonomous applications, such as robot navigation and augmented reality. To bootstrap and achieve optimal state estimation, the spatial-temporal displacements between IMU and cameras must be calibrated in advance. Most existing calibration methods adopt continuous-time state representation, more specifically the B-spline. Despite these methods achieve precise spatial-temporal calibration, they suffer from high computational cost caused by continuous-time state representation. To this end, we propose a novel and extremely efficient calibration method that unleashes the power of discrete-time state representation. Moreover, the weakness of discrete-time state representation in temporal calibration is tackled in this paper. With the increasing production of drones, cellphones and other visual-inertial platforms, if one million devices need calibration around the world, saving one minute for the calibration of each device means saving 2083 work days in total. To benefit both the research and industry communities, our code will be open-source.