Walk-on-Spheres (WoS) enables mesh-free solutions of elliptic PDEs, but conventional methods repeatedly discard sampled trajectories, limiting their efficiency in interactive applications. We introduce reusable transport skeletons, a persistent representation that organizes stochastic walks into a shared computational graph. Our framework enables efficient reuse across spatial queries and boundary conditions, while supporting progressive refinement, structure-preserving compression, and local adaptation to dynamic geometry. It further enables interactive editing of transport between selected regions. Experiments demonstrate substantial computational savings while preserving solution accuracy. By transforming transient stochastic trajectories into reusable transport structures, our method provides a flexible foundation for interactive Monte Carlo PDE solving.
Existing LLM logical-reasoning benchmarks largely focus on object-level inference under fixed predicates and rules, leaving higher-order reasoning over rules and policies underexplored. We introduce HOLogicBench, a 1,379-instance benchmark that systematically evaluates five representative higher-order reasoning operations, with Isabelle/HOL formalizations and proof-verified answers. Beyond answer accuracy, HOLogicBench provides solver-verified reference reasoning paths for each instance, enabling fine-grained evaluation of model-generated reasoning against valid derivations. Across 16 model configurations, average accuracy is 53.86\%, with the best configuration reaching 65.99\%. Further analyses reveal that LLMs exhibit limited alignment with valid derivations, remain vulnerable to minimal counterfactual perturbations, and display distinct failure patterns across higher-order reasoning types.
Yucheng Wu, Chengyi Xie, Chenpeng Wang, Liangming Pan
Activation steering offers training-free and dynamic control of target behaviors in large language models, but existing methods can impair general capabilities and require domain-specific intervention designs. We introduce Capability-Preserving Neuron Steering (CANS), a unified framework that balances target-behavior control with capability preservation. To make target-behavior improvement and general-capability preservation tractable, we introduce two proxies: log-probabilities of target tokens in model generated responses, and output-distribution KL divergence on general-purpose inputs. Combining the local approximation of the two proxies yields a closed-form solution, enabling lightweight steering without iterative model training or additional inference-time computation. The resulting neuron-wise scaling coefficients achieve effective steering while limiting general-capability loss. A shared data-construction and intervention procedure supports adaptation across domains using sampled responses with target-behavior filtering and target-span annotations. Across safety, toxicity, and hallucination control on Qwen and Llama models, CANS achieves comparable or better steering while better preserving MMLU, BBH, and IFEval performance than SFT, CAA, and neuron-level baselines. On Qwen3-4B, CANS reduces HarmBench attack success from $15.50\%$ to $1.75\%$ with a $1.11$-point decrease in average capability. These results support CANS as a practical approach to behavioral control across domains.