This work explores using the physics-inspired AI Feynman symbolic regression algorithm to automatically rediscover a fundamental equation in astronomy -- the Equation of the Centre. Through the introduction of observational and inductive biases corresponding to the physical nature of the system through data preprocessing and search space restriction, AI Feynman was successful in recovering the first-order analytical form of this equation from lunar ephemerides data. However, this manual approach highlights a key limitation in its reliance on expert-driven coordinate system selection. We therefore propose an automated preprocessing extension to find the canonical coordinate system. Results demonstrate that targeted domain knowledge embedding enables symbolic regression to rediscover physical laws, but also highlight further challenges in constraining symbolic regression to derive physics equations when leveraging domain knowledge through tailored biases.

Refusal refers to the functional behavior enabling safety-aligned language models to reject harmful or unethical prompts. Following the growing scientific interest in mechanistic interpretability, recent work encoded refusal behavior as a single direction in the model's latent space; e.g., computed as the difference between the centroids of harmful and harmless prompt representations. However, emerging evidence suggests that concepts in LLMs often appear to be encoded as a low-dimensional manifold embedded in the high-dimensional latent space. Motivated by these findings, we propose a novel method leveraging Self-Organizing Maps (SOMs) to extract multiple refusal directions. To this end, we first prove that SOMs generalize the prior work's difference-in-means technique. We then train SOMs on harmful prompt representations to identify multiple neurons. By subtracting the centroid of harmless representations from each neuron, we derive a set of multiple directions expressing the refusal concept. We validate our method on an extensive experimental setup, demonstrating that ablating multiple directions from models' internals outperforms not only the single-direction baseline but also specialized jailbreak algorithms, leading to an effective suppression of refusal. Finally, we conclude by analyzing the mechanistic implications of our approach.

We study GCS-TSP, a new variant of the Traveling Salesman Problem (TSP) defined over a Graph of Convex Sets (GCS) -- a powerful representation for trajectory planning that decomposes the configuration space into convex regions connected by a sparse graph. In this setting, edge costs are not fixed but depend on the specific trajectory selected through each convex region, making classical TSP methods inapplicable. We introduce GHOST, a hierarchical framework that optimally solves the GCS-TSP by combining combinatorial tour search with convex trajectory optimization. GHOST systematically explores tours on a complete graph induced by the GCS, using a novel abstract-path-unfolding algorithm to compute admissible lower bounds that guide best-first search at both the high level (over tours) and the low level (over feasible GCS paths realizing the tour). These bounds provide strong pruning power, enabling efficient search while avoiding unnecessary convex optimization calls. We prove that GHOST guarantees optimality and present a bounded-suboptimal variant for time-critical scenarios. Experiments show that GHOST is orders-of-magnitude faster than unified mixed-integer convex programming baselines for simple cases and uniquely handles complex trajectory planning problems involving high-order continuity constraints and an incomplete GCS.

Alpha factor mining is pivotal in quantitative investment for identifying predictive signals from complex financial data. While traditional formulaic alpha mining relies on human expertise, contemporary automated methods, such as those based on genetic programming or reinforcement learning, often struggle with search inefficiency or yield alpha factors that are difficult to interpret. This paper introduces a novel framework that integrates Large Language Models (LLMs) with Monte Carlo Tree Search (MCTS) to overcome these limitations. Our framework leverages the LLM's instruction-following and reasoning capability to iteratively generate and refine symbolic alpha formulas within an MCTS-driven exploration. A key innovation is the guidance of MCTS exploration by rich, quantitative feedback from financial backtesting of each candidate factor, enabling efficient navigation of the vast search space. Furthermore, a frequent subtree avoidance mechanism is introduced to enhance search diversity and prevent formulaic homogenization, further improving performance. Experimental results on real-world stock market data demonstrate that our LLM-based framework outperforms existing methods by mining alphas with superior predictive accuracy and trading performance. The resulting formulas are also more amenable to human interpretation, establishing a more effective and efficient paradigm for formulaic alpha mining.

Greedy search methods like Greedy Best-First Search (GBFS) and Enforced Hill-Climbing (EHC) often struggle when faced with Uninformed Heuristic Regions (UHRs) like heuristic local minima or plateaus. In this work, we theoretically and empirically compare two popular methods for escaping UHRs in breadth-first search (BrFS) and restarting random walks (RRWs). We first derive the expected runtime of escaping a UHR using BrFS and RRWs, based on properties of the UHR and the random walk procedure, and then use these results to identify when RRWs will be faster in expectation than BrFS. We then evaluate these methods for escaping UHRs by comparing standard EHC, which uses BrFS to escape UHRs, to variants of EHC called EHC-RRW, which use RRWs for that purpose. EHC-RRW is shown to have strong expected runtime guarantees in cases where EHC has previously been shown to be effective. We also run experiments with these approaches on PDDL planning benchmarks to better understand their relative effectiveness for escaping UHRs.

Four-Period Detailed Design Period 1: Topic Selection and Initial Exploration Period 2: Principle Analysis and Model Design Period 3: Model Construction and Refinement Period 4: "Historical Technology Expo" with presentations [Includes detailed student reflection prompts, extension activities, and troubleshooting guidance...] Base Model: Generic Outline Interdisciplinary Lesson Plan Design Learning Objectives: Help students understand how physics influences historical progress... Cultivate ability to analyze social factors behind technological development... Class Schedule: Four periods covering physics review, historical technologies, case study, and modern applications. Assessment: Class participation, group reports, reflection journals [Subsequent periods contain only high-level bullet points without actionable details...] 12 Qualitative Analysis This comparison reveals dramatic capability differences for complex generation tasks. The Base Model produces only a generic outline with vague bullet points--entirely insufficient for classroom use. Both AutoSynth and Expert-Designed models generate outstanding, comprehensive lesson plans with detailed objectives, granular activities, and sophisticated assessment schemes. The subtle differences reflect their optimization processes: AutoSynth emphasizes systematic difficulty coverage (likely from iterative refinement), while Expert-Designed showcases deep assessment design expertise. Both represent quantum leaps over baseline, validating that specialized workflows-- automated or manual--are essential for professional-grade content. This supports our quantitative findings (Table 1): while Au-toSynth achieves lower human preference (51 percent vs 96 percent), it produces genuinely high-quality outputs far superior to baseline capabilities.

We consider the problem of certified robustness for sequence classification against edit distance perturbations. Naturally occurring inputs of varying lengths (e.g., sentences in natural language processing tasks) present a challenge to current methods that employ fixed-rate deletion mechanisms and lead to suboptimal performance. To this end, we introduce AdaptDel methods with adaptable deletion rates that dynamically adjust based on input properties. We extend the theoretical framework of randomized smoothing to variable-rate deletion, ensuring sound certification with respect to edit distance. We achieve strong empirical results in natural language tasks, observing up to 30 orders of magnitude improvement to median cardinality of the certified region, over state-of-the-art certifications.

In recent years, graph neural networks (GNNs) have been widely applied in tackling combinatorial optimization problems. However, existing methods still suffer from limited accuracy when addressing that on complex graphs and exhibit poor scalability, since full training requires loading the whole adjacent matrix and all embeddings at a time, the it may results in out of memory of a single machine. This limitation significantly restricts their applicability to large-scale scenarios. To address these challenges, we propose a distributed GNN-based training framework for combinatorial optimization. In details, firstly, large graph is partition into several small subgraphs. Then the individual subgraphs are full trained, providing a foundation for efficient local optimization. Finally, reinforcement learning (RL) are employed to take actions according to GNN output, to make sure the restrictions between cross nodes can be learned. Extensive experiments are conducted on both real large-scale social network datasets (e.g., Facebook, Youtube) and synthetically generated high-complexity graphs, which demonstrate that our framework outperforms state-of-the-art approaches in both solution quality and computational efficiency. Moreover, the experiments on large graph instances also validate the scalability of the model.

Mixed-Integer Linear Programming (MILP) is a cornerstone of combinatorial optimization, yet solving large-scale instances remains a significant computational challenge. Recently, Graph Neural Networks (GNNs) have shown promise in accelerating MILP solvers by predicting high-quality solutions. However, we identify that existing methods misalign with the intrinsic structure of MILP problems at two levels. At the leaning objective level, the Binary Cross-Entropy (BCE) loss treats variables independently, neglecting their relative priority and yielding plausible logits. At the model architecture level, standard GNN message passing inherently smooths the representations across variables, missing the natural competitive relationships within constraints. To address these challenges, we propose CoCo-MILP, which explicitly models inter-variable Contrast and intra-constraint Competition for advanced MILP solution prediction. At the objective level, CoCo-MILP introduces the Inter-Variable Contrastive Loss (VCL), which explicitly maximizes the embedding margin between variables assigned one versus zero. At the architectural level, we design an Intra-Constraint Competitive GNN layer that, instead of homogenizing features, learns to differentiate representations of competing variables within a constraint, capturing their exclusionary nature. Experimental results on standard benchmarks demonstrate that CoCo-MILP significantly outperforms existing learning-based approaches, reducing the solution gap by up to 68.12% compared to traditional solvers. Our code is available at https://github.com/happypu326/CoCo-MILP.