Neural network based data-driven operator learning schemes have shown tremendous potential in computational mechanics. DeepONet is one such neural network architecture which has gained widespread appreciation owing to its excellent prediction capabilities. Having said that, being set in a deterministic framework exposes DeepONet architecture to the risk of overfitting, poor generalization and in its unaltered form, it is incapable of quantifying the uncertainties associated with its predictions. We propose in this paper, a Variational Bayes DeepONet (VB-DeepONet) for operator learning, which can alleviate these limitations of DeepONet architecture to a great extent and give user additional information regarding the associated uncertainty at the prediction stage. The key idea behind neural networks set in Bayesian framework is that, the weights and bias of the neural network are treated as probability distributions instead of point estimates and, Bayesian inference is used to update their prior distribution. Now, to manage the computational cost associated with approximating the posterior distribution, the proposed VB-DeepONet uses \textit{variational inference}. Unlike Markov Chain Monte Carlo schemes, variational inference has the capacity to take into account high dimensional posterior distributions while keeping the associated computational cost low. Different examples covering mechanics problems like diffusion reaction, gravity pendulum, advection diffusion have been shown to illustrate the performance of the proposed VB-DeepONet and comparisons have also been drawn against DeepONet set in deterministic framework.
翻译:以数据驱动的网络驱动的操作员学习计划在计算力方面显示出巨大的潜力。DeepONet是这种神经网络结构之一,由于它的预测能力极佳,这种神经网络结构得到了广泛的理解。在确定性框架中设置的DeepONet结构面临过大、不全面化的风险,并且以其未改变的形式,它无法量化其预测的不确定性。我们在本文件中提议,一个Variational Bayes DeepONet(VB-DeepONet)用于操作员学习,这样可以大大减轻DeepONet结构的这些局限性,并给用户更多关于预测阶段相关不确定性的信息。在Bayesian框架中设置的神经网络背后的关键思想是,将神经网络的重量和偏差视为概率分布,而不是点估计,而Bayesincience被用来更新其先前的分布。现在,管理与近距离比较分布相联的计算成本成本成本,拟议的VB-DeepONet(Deepent)利用了DeepONet结构的这些限制,给用户更多关于预测阶段相关不确定性的信息。在Dreal conditional developal developal developal restial developtional developal develoption sliverstion sliverstion sliverstional resmisstional resmissational exptional prestitional ex ex expal prestitional disal distional distional exptional exptional dismpltional sabliverstional dismpltional sapertional dismtional sableglegleglegleglex sablex sabild sabild sabild sabild sabal sabild sabild sapeal sabal sabal sapeal sapeutdal sapeal sapeal sapeal sapeal sapeal sapeal sapeal saild saild sabledal sapeal lical lical lical lical saildal