When neural networks are used to solve differential equations, they usually produce solutions in the form of black-box functions that are not directly mathematically interpretable. We introduce a method for generating symbolic expressions to solve differential equations while leveraging deep learning training methods. Unlike existing methods, our system does not require learning a language model over symbolic mathematics, making it scalable, compact, and easily adaptable for a variety of tasks and configurations. As part of the method, we propose a novel neural architecture for learning mathematical expressions to optimize a customizable objective. The system is designed to always return a valid symbolic formula, generating a useful approximation when an exact analytic solution to a differential equation is not or cannot be found. We demonstrate through examples how our method can be applied on a number of differential equations, often obtaining symbolic approximations that are useful or insightful. Furthermore, we show how the system can be effortlessly generalized to find symbolic solutions to other mathematical tasks, including integration and functional equations.

中文翻译：

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一种求解微分方程和泛函方程的神经符号方法

当神经网络用于求解微分方程时，它们通常以不能直接用数学解释的黑盒函数的形式产生解。我们介绍了一种在利用深度学习训练方法的同时生成符号表达式来求解微分方程的方法。与现有方法不同，我们的系统不需要通过符号数学学习语言模型，使其具有可扩展性、紧凑性并且易于适应各种任务和配置。作为该方法的一部分，我们提出了一种新颖的神经架构，用于学习数学表达式以优化可定制的目标。该系统旨在始终返回有效的符号公式，从而在未找到或无法找到微分方程的精确解析解时生成有用的近似值。我们通过示例展示了我们的方法如何应用于许多微分方程，通常会获得有用或有见地的符号近似值。此外，我们展示了如何毫不费力地将系统泛化为其他数学任务的符号解，包括积分和函数方程。