论文标题
一种用于求解差分方程的神经符号方法
A Neuro-Symbolic Method for Solving Differential and Functional Equations
论文作者
论文摘要
当使用神经网络求解微分方程时，它们通常以黑框函数的形式产生解决方案，这些函数在数学上无法直接解释。我们引入了一种生成符号表达式的方法，以求解微分方程，同时利用深度学习训练方法。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，从而使其可扩展，紧凑且易于适应各种任务和配置。作为该方法的一部分，我们提出了一种新型的神经体系结构，用于学习数学表达式以优化可自定义的目标。该系统旨在始终返回有效的符号公式，当找不到或找不到微分方程的精确分析解决方案时，就会生成有用的近似值。我们通过示例说明如何将我们的方法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概括为对其他数学任务（包括集成和功能方程）的符号解决方案。
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.
