Symbolic regression is a machine learning technique that can learn the governing formulas of data and thus has the potential to transform scientific discovery. However, symbolic regression is still limited in the complexity and dimensionality of the systems that it can analyze. Deep learning on the other hand has transformed machine learning in its ability to analyze extremely complex and high-dimensional datasets. We propose a neural network architecture to extend symbolic regression to parametric systems where some coefficient may vary but the structure of the underlying governing equation remains constant. We demonstrate our method on various analytic expressions, ODEs, and PDEs with varying coefficients and show that it extrapolates well outside of the training domain. The neural network-based architecture can also integrate with other deep learning architectures so that it can analyze high-dimensional data while being trained end-to-end. To this end we integrate our architecture with convolutional neural networks to analyze 1D images of varying spring systems.

翻译：符号回归是一种机器学习技术，能够学习数据中的支配性公式，因此具有变革科学发现的潜力。然而，符号回归在可分析系统的复杂性和维度上仍存在局限性。另一方面，深度学习的出现深刻改变了机器学习领域，使其能够分析极其复杂和高维度的数据集。我们提出了一种神经网络架构，将符号回归扩展到参数化系统中——此类系统中部分系数可能变化，但底层支配方程的结构保持不变。我们在各种解析表达式、常微分方程和偏微分方程上验证了该方法，并展示了其在训练域外具备良好的外推能力。该神经网络架构还可与其他深度学习架构集成，从而在端到端训练的同时分析高维数据。为此，我们将该架构与卷积神经网络集成，分析了变化弹簧系统的一维图像。