Symbolic regression with polynomial neural networks and polynomial neural ordinary differential equations (ODEs) are two recent and powerful approaches for equation recovery of many science and engineering problems. However, these methods provide point estimates for the model parameters and are currently unable to accommodate noisy data. We address this challenge by developing and validating the following Bayesian inference methods: the Laplace approximation, Markov Chain Monte Carlo (MCMC) sampling methods, and variational inference. We have found the Laplace approximation to be the best method for this class of problems. Our work can be easily extended to the broader class of symbolic neural networks to which the polynomial neural network belongs.

翻译：符号回归结合多项式神经网络与多项式神经常微分方程是近年来解决科学与工程问题中方程恢复的两类强大方法。然而，这些方法仅能提供模型参数的点估计，且目前无法处理含噪声数据。针对这一挑战，我们开发并验证了以下贝叶斯推理方法：拉普拉斯近似、马尔可夫链蒙特卡洛采样方法以及变分推理。研究发现，拉普拉斯近似是这类问题的最优方法。我们的工作可便捷扩展至多项式神经网络所属的更广泛符号神经网络范畴。