Using symbolic regression to discover physical laws from observed data is an emerging field. In previous work, we combined genetic algorithm (GA) and machine learning to present a data-driven method for discovering a wave equation. Although it managed to utilize the data to discover the two-dimensional (x, z) acoustic constant-density wave equation \(u_{tt}=v^2(u_{xx}+u_{zz})\) (subscripts of the wavefield, u, are second derivatives in time and space) in a homogeneous medium, it did not provide the complete equation form, where the velocity term is represented by a coefficient rather than directly given by \(v^2\). In this work, we redesign the framework, encoding both velocity information and candidate functional terms simultaneously. Thus, we use GA to simultaneously evolve the candidate functional and coefficient terms in the library. Also, we consider here the physics rationality and interpretability in the randomly generated potential wave equations, by ensuring that both-hand sides of the equation maintain balance in their physical units. We demonstrate this redesigned framework using the acoustic wave equation as an example, showing its ability to produce physically reasonable expressions of wave equations from noisy and sparsely observed data in both homogeneous and inhomogeneous media. Also, we demonstrate that our method can effectively discover wave equations from a more realistic observation scenario.

使用符号回归从观测数据中发现物理定律是一个新兴领域。在之前的工作中，我们结合遗传算法（GA）和机器学习提出了一种数据驱动的方法来发现波动方程。尽管它设法利用数据发现了二维 ( x , z ) 声学常密度波动方程\(u_{tt}=v^2(u_{xx}+u_{zz})\) （下标波场u是均匀介质中时间和空间的二阶导数），它没有提供完整的方程形式，其中速度项由系数表示，而不是直接由\(v^2\)给出。在这项工作中，我们重新设计了框架，同时编码速度信息和候选功能项。因此，我们使用 GA 同时演化库中的候选函数项和系数项。此外，我们在这里通过确保方程两边在其物理单位中保持平衡来考虑随机生成的势波动方程的物理合理性和可解释性。我们以声波方程为例演示了这个重新设计的框架，展示了它从均匀和非均匀介质中的噪声和稀疏观测数据生成物理上合理的波动方程表达式的能力。此外，我们还证明了我们的方法可以从更真实的观测场景中有效地发现波动方程。