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.

发展和验证了拉普拉斯逼近、马尔可夫链蒙特卡罗采样方法和变分推断等贝叶斯推断方法，发现拉普拉斯逼近是解决这类问题的最佳方法。我们的工作可以轻松扩展到多项式神经网络等符号神经网络的更广泛类别。