Shengbing Tang

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.

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.