In nature, the behaviors of many complex systems can be described by parsimonious math equations. Automatically distilling these equations from limited data is cast as a symbolic regression process which hitherto remains a grand challenge. Keen efforts in recent years have been placed on tackling this issue and demonstrated success in symbolic regression. However, there still exist bottlenecks that current methods struggle to break when the discrete search space tends toward infinity and especially when the underlying math formula is intricate. To this end, we propose a novel Reinforcement Symbolic Regression Machine (RSRM) that masters the capability of uncovering complex math equations from only scarce data. The RSRM model is composed of three key modules: (1) a Monte Carlo tree search (MCTS) agent that explores optimal math expression trees consisting of pre-defined math operators and variables, (2) a Double Q-learning block that helps reduce the feasible search space of MCTS via properly understanding the distribution of reward, and (3) a modulated sub-tree discovery block that heuristically learns and defines new math operators to improve representation ability of math expression trees. Biding of these modules yields the state-of-the-art performance of RSRM in symbolic regression as demonstrated by multiple sets of benchmark examples. The RSRM model shows clear superiority over several representative baseline models.

翻译：自然界中，许多复杂系统的行为可通过简洁的数学方程加以描述。从有限数据中自动提取这些方程的过程被称为符号回归，而这一领域至今仍是一项重大挑战。近年来，研究者在解决该问题上投入了大量努力，并在符号回归中取得了成功。然而，当离散搜索空间趋于无穷大，尤其是底层数学公式极其复杂时，现有方法仍面临难以突破的瓶颈。为此，我们提出了一种新颖的强化符号回归机器（RSRM），它能够仅从稀疏数据中掌握发现复杂数学方程的能力。RSRM模型由三个关键模块组成：（1）蒙特卡洛树搜索（MCTS）智能体，用于探索由预定义数学运算符和变量构成的最优数学表达式树；（2）双Q学习模块，通过合理理解奖励分布来帮助缩小MCTS的可行搜索空间；（3）调制子树发现模块，通过启发式学习并定义新的数学运算符，提升数学表达式树的表示能力。这三个模块的协同作用使RSRM在符号回归中展现出最先进的性能，这在多个基准测试中得到了验证。RSRM模型在多个代表性基线模型上表现出明显的优越性。