#01Nov 5, 2026
math.AP
Rigorously Certified Neural-Network Computation of Nonlinear Water Waves at Near-Machine Precision
Michal Shavit, Yongji Wang, Tristan Buckmaster
We present a computational framework that combines machine-learning-based discovery with rigorous computer-assisted numerical certification and formal verification for strongly nonlinear partial differential equations. We demonstrate the approach for periodic gravity-capillary traveling water waves. Physics-informed neural networks are used as a search mechanism, with the wave profile, wave speed and, in continuation experiments, surface tension learned directly from the governing equations without training data. Through a staged training and refinement procedure, the neural networks achieve near-machine-precision residuals of the governing equations, even in strongly nonlinear regimes. A single formulation recovers high-amplitude Stokes waves, Crapper waves and Wilton ripples. Continuation from a high-amplitude Wilton ripple in surface tension reveals new high-amplitude candidate solutions. We freeze one candidate as an explicit finite representation and rigorously certify a uniform bound on its governing-equation residual using interval arithmetic, accounting for floating-point roundoff, Fourier truncation, aliasing and grid-to-continuum errors. This independently validates the neural-network accuracy. We formalize the certification argument in Lean, which machine-checks the deductions from certified numerical bounds to the residual theorem. Both the interval certification and formalization were developed with substantial assistance from a large language model, while the validity of the result remains independent of the language model: the numerical bounds are rigorously enclosed by interval arithmetic and the subsequent mathematical deductions are checked by the Lean kernel. Together, these steps provide a pipeline from neural discovery to rigorous numerical certification and machine-checked verification with auditable mathematical guarantees.
#02Nov 5, 2026
math.OC
From What Is Used to What Is Missing: Rethinking Dynamic Rank in LoRA
Peng Xue, Guodong Fan, Guang-Yong Chen and 1 more
Parameter-efficient fine-tuning with LoRA is governed by rank, which controls parameter budget and locally accessible task updates. Existing dynamic-rank methods mainly focus on capacity allocation, leaving unresolved whether the current adaptation space is sufficient for the task. We characterize the first-order information boundary of standard factor gradients, showing that their unobservable subspace coincides with the normal space of the low-rank manifold. Thus, factor gradients alone cannot distinguish under-optimization within the current space from task-relevant directions beyond its first-order reach. This motivates a geometric view of rank evolution, where tangent information optimizes existing degrees of freedom and the normal spectrum quantifies the value of new ones. Under a local quadratic model, we show that leading normal singular directions yield the locally optimal rank expansion, with spectral energy determining its gain. Building on this characterization, we propose GEAR, which turns the tangent--normal decomposition into rank evolution through tangent-space optimization, matrix-free normal-spectrum estimation for rank expansion, and task-induced geometric initialization. Across language and vision benchmarks, GEAR achieves the best average performance in all evaluated settings with substantially fewer trainable parameters. On LLaMA3-8B, it raises the average commonsense score from 86.61 to 88.45 while reducing trainable parameters from 42.0M to 14.4M.
#03Nov 5, 2026
cs.IR
CacheGraft: Enhancing Retrieval-Augmented Generation via Latent-Space Document Enrichment
Vihan Lakshman, Peter Baile Chen, Zhuohan Gu and 1 more
Recent work in machine learning has demonstrated that transformer-based language model agents can communicate state not only through text, but also directly in latent space via key-value (KV) caches. In this paper, we argue that such cache-to-cache translation can provide utility beyond the original motivating application of real-time agent communication and can instead be utilized for retrieval-augmented generation (RAG). Building on the Cache-to-Cache (C2C) fusion framework, our proposed system CacheGraft encodes each corpus passage offline into a fused KV cache representation between a larger sharer model (4B parameters in our experiments) and a smaller receiver model (0.6B parameters) such that this cache can be injected into the receiver at query time with no need for the larger model. Across five QA benchmarks and two independent locally-run judges, CacheGraft improves answer quality over the text-only receiver baseline in 29 out of 30 comparisons, of which 17 of the 30 are statistically significant. Notably, these gains are zero-shot with respect to the fuser, which was never trained on retrieval data. These initial results suggest that latent-space document enrichment may be a promising avenue for enhancing the performance of RAG systems, particularly in compute-constrained settings where it is only feasible to deploy a small model at inference time.
#04Nov 5, 2026
cs.LG
Spectral symmetry fingerprints in equivariant representations
Jiacheng Cen, Anyi Li, Ning Lin and 3 more
Symmetry governs the organization of physical structures, while symmetry breaking reveals their transformations. In geometric deep learning, symmetry imposes rank constraints on equivariant representations, limiting the expressive capacity of equivariant graph neural networks (GNNs). Here we show that these constraints can instead be exploited: the singular-value spectra of equivariant representations provide a unified fingerprint of structural symmetry. We introduce spectral symmetry fingerprints (SymFP), which enable discrete point-group identification through rank-template matching and continuous symmetry analysis by quantifying spectral energy beyond target rank constraints. SymFP is orientation invariant and architecture agnostic, enabling a parameter-free GNN implementation. Its spectra can be reused across multiple target groups, and the resulting leakage scores satisfy subgroup monotonicity. Across synthetic structures, molecules, coordination complexes and large biological assemblies, SymFP accurately identifies point groups, tracks controlled symmetry breaking and reveals local and degree-resolved symmetry changes that can be obscured by global geometric measures. These results establish the spectral structure of equivariant representations as a general framework for characterizing symmetry across physical systems.
#05Nov 5, 2026
cs.CV
WeLM-Image: A Unified Image Generation Model towards Visual Intelligence
Canmiao Fu, Chen Li, Guohui Zhang and 13 more
We present \textbf{WeLM-Image}, a unified image foundation model that integrates multimodal understanding into text-to-image generation, instruction-guided editing, and multi-reference synthesis. Designed for single-GPU deployment in WeChat's large-scale consumer applications, the model processes text, reference images, and target latents as one interleaved sequence. A Mixture-of-Transformers (MoT) backbone combines shared attention with independent understanding and generation parameters to limit optimization interference. Equipping each path with Mixture-of-Experts (MoE) modules provides sparse capacity, with 31.7B stored parameters and approximately 5.4B generation parameters activated per target latent token. A generation-oriented data engine combines a knowledge-grounded visual vocabulary with large-scale visual clustering to balance semantic coverage and editing tasks, while training-time evaluation guides iterative data refinement. Bilingual annotations and verified editing supervision support progressive training up to native 1.5K resolution. We optimize generation and editing separately with rubric- and checklist-based rewards, then consolidate the specialized policies through noise-aware on-policy distillation. In the reported overall comparisons, the 1536-resolution model ranks first among the evaluated open-source models on PRISM-Bench, Qwen-Image-Bench, and UnicBench. MeanFlow-based few-step distillation yields WeLM-Image-Turbo, which generates a $1024\times1024$ image in 4.98\,s on a single NVIDIA H20 GPU using eight guidance-free steps.