Zhang Michael, Kim Samuel, Lu Peter Y, Soljacic Marin
IEEE Trans Neural Netw Learn Syst. 2024 Nov;35(11):16775-16787. doi: 10.1109/TNNLS.2023.3297978. Epub 2024 Oct 29.
Symbolic regression is a machine learning technique that can learn the equations governing 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 and partial differential equations (PDEs) with varying coefficients and show that it extrapolates well outside of the training domain. The proposed neural-network-based architecture can also be enhanced by integrating with other deep learning architectures such that it can analyze high-dimensional data while being trained end-to-end. To this end, we demonstrate the scalability of our architecture by incorporating a convolutional encoder to analyze 1-D images of varying spring systems.
符号回归是一种机器学习技术，它可以学习支配数据的方程，因此有潜力改变科学发现。然而，符号回归在其能够分析的系统的复杂性和维度方面仍然存在局限性。另一方面，深度学习在分析极其复杂和高维数据集的能力方面改变了机器学习。我们提出了一种神经网络架构，将符号回归扩展到参数系统，其中一些系数可能会变化，但基础支配方程的结构保持不变。我们在具有不同系数的各种解析表达式和偏微分方程（PDE）上展示了我们的方法，并表明它在训练域之外能很好地外推。所提出的基于神经网络的架构还可以通过与其他深度学习架构集成来增强，以便它能够在端到端训练的同时分析高维数据。为此，我们通过合并卷积编码器来分析不同弹簧系统的一维图像，展示了我们架构的可扩展性。
