Usually learning dynamical systems by data-driven methods requires large amount of training data, which may be time consuming and expensive. Active learning, which aims at choosing the most informative samples to make learning more efficient is a promising way to solve this issue. However, actively learning dynamical systems is difficult since it is not possible to arbitrarily sample the state-action space under the constraint of system dynamics. The state-of-the-art methods for actively learning dynamical systems iteratively search for an informative state-action pair by maximizing the differential entropy of the predictive distribution, or iteratively search for a long informative trajectory by maximizing the sum of predictive variances along the trajectory. These methods suffer from low efficiency or high computational complexity and memory demand. To solve these problems, this paper proposes novel and more sampleefficient methods which combine global and local explorations. As the global exploration, the agent searches for a relatively short informative trajectory in the whole state-action space of the dynamical system. Then, as the local exploration, an action sequence is optimized to drive the system’s state towards the initial state of the local informative trajectory found by the global exploration and the agent explores this local informative trajectory. Compared to the state-of-the-art methods, the proposed methods are capable of exploring the state-action space more efficiently, and have much lower computational complexity and memory demand. With the state-of-the-art methods as baselines, the advantages of the proposed methods are verified via various numerical examples.

Solving Algebra Problems with Geometry Diagrams (APGDs) is still a challenging problem because diagram processing is not studied as intensively as language processing. To work against this challenge, this paper proposes a hologram reasoning scheme and develops a high-performance method for solving APGDs by using this scheme. To reach this goal, it first defines a hologram, being a kind of graph, and proposes a hologram generator to convert a given APGD into a hologram, which represents the entire information of APGD and the relations for solving the problem can be acquired from it by a uniform way. Then HGR, a hologram reasoning method employs a pool of prepared graph models to derive algebraic equations, which is consistent with the geometric theorems. This method is able to be updated by adding new graph models into the pool. Lastly, it employs deep reinforcement learning to enhance the efficiency of model selection from the pool. The entire HGR not only ensures high solution accuracy with fewer reasoning steps but also significantly enhances the interpretability of the solution process by providing descriptions of all reasoning steps. Experimental results demonstrate the effectiveness of HGR in improving both accuracy and interpretability in solving APGDs.

Natural language to structured query language (Text2SQL) is a critical task in intelligent question answering and database interaction. A fundamental challenge lies in achieving semantic symmetry between natural language expressions and database schemas, as well as behavioral symmetry between query generation and error correction. Traditional approaches often struggle with comprehending domain-specific concepts and dynamically adapting to user’s feedback, breaking the desired symmetry between system output and user intent. Thispaper presents EvoAgent-SQL as a lightweight engineering framework for private domain data querying based on low-cost lightweight models. The framework comprises three core agents: (1) the Schema Grounding Agent (SGA) establishes a symmetric mapping from natural language concepts to database fields; (2) the Execution Agent (EA) generates SQL queries; and (3) the Reflection Agent (RA) mirrors the EA’s outputs by analyzing errors and proposing corrections, forming a reflective symmetry loop. When ambiguity arises, user feedback is incorporated as a symmetry-breaking signal, which the RA uses to restore alignment through iterative evolution with a memory mechanism. We evaluate the framework on an education-domain dataset and provide a reproducibility plan for releasing sanitized schemas, natural language questions, and reference SQL while withholding confidential institutional records. Experimental results demonstrate that EvoAgent-SQL enhances query execution accuracy, achieving a $13.7\%$ reduction in the relative error residual after one evolution cycle and a $17.6\%$ cumulative reduction after two cycles, suggesting practical adaptability in domain-specific Text2SQL tasks.