Many equations arising in science currently cannot be solved by available analytical techniques and are therefore solved numerically, without yielding explicit symbolic expressions. Existing symbolic regression approaches can recover symbolic expressions, but require training data obtained from the underlying process, rather than the governing equation alone. We propose the Symbolic Equation Solver (SES), a framework that formulates equation solving as an optimization problem over differentiable symbolic models. SES constructs its objective from the equation together with initial or boundary conditions, eliminating the need for paired input-output data. The learned model is expressed in explicit symbolic form, enabling further analysis. We evaluate SES on representative algebraic and differential equations, including a system of algebraic equations, an equation with transcendental terms, an ordinary differential equation, and partial differential equations with different initial or boundary conditions. Across these settings, SES recovers compact symbolic expressions that match the corresponding analytical solutions.

翻译：科学领域中的许多方程目前无法通过现有解析技术求解，因此需采用数值方法求解，而无法得到显式符号表达式。现有符号回归方法虽可恢复符号表达式，但需依赖底层过程生成的训练数据，而非仅基于控制方程。我们提出符号方程求解器（SES），该框架将方程求解表述为可微符号模型上的优化问题。SES基于方程及初始/边界条件构建目标函数，无需成对输入-输出数据。学习到的模型以显式符号形式表示，便于进一步分析。我们在代表性代数方程与微分方程上评估SES，包括代数方程组、含超越项的方程、常微分方程以及具有不同初始/边界条件的偏微分方程。在这些场景下，SES均能恢复与对应解析解匹配的紧凑符号表达式。