Symbolic Regression is the study of algorithms that automate the search for analytic expressions that fit data. While recent advances in deep learning have generated renewed interest in such approaches, the development of symbolic regression methods has not been focused on physics, where we have important additional constraints due to the units associated with our data. Here we present $\Phi$-SO, a Physical Symbolic Optimization framework for recovering analytical symbolic expressions from physics data using deep reinforcement learning techniques by learning units constraints. Our system is built, from the ground up, to propose solutions where the physical units are consistent by construction. This is useful not only in eliminating physically impossible solutions, but because the "grammatical" rules of dimensional analysis restrict enormously the freedom of the equation generator, thus vastly improving performance. The algorithm can be used to fit noiseless data, which can be useful for instance when attempting to derive an analytical property of a physical model, and it can also be used to obtain analytical approximations to noisy data. We test our machinery on a standard benchmark of equations from the Feynman Lectures on Physics and other physics textbooks, achieving state-of-the-art performance in the presence of noise (exceeding 0.1%) and show that it is robust even in the presence of substantial (10%) noise. We showcase its abilities on a panel of examples from astrophysics.

翻译：符号回归是研究自动化搜索数据拟合解析表达式的算法。虽然深度学习的近期进展重新激发了对此类方法的兴趣，但符号回归方法的发展并未聚焦于物理学领域——该领域因数据关联的单位而存在重要额外约束。本文提出Φ-SO（物理符号优化框架），通过深度强化学习技术学习单位约束，从物理数据中恢复解析符号表达式。我们的系统从底层设计之初就致力于生成物理单位天然一致的解。这不仅有助于排除物理不可能的解，还因为量纲分析的"语法"规则极大限制了方程生成器的自由度，从而显著提升性能。该算法既可拟合无噪声数据（例如在推导物理模型解析性质时），也可用于获取含噪数据的解析近似。我们使用费曼物理学讲义及其他物理教科书中的标准方程基准进行测试，在存在噪声（超过0.1%）情况下达到业界领先水平，并证明即使在10%显著噪声下仍保持鲁棒性。我们通过天体物理学案例展示了该框架的能力。