Lin Chensen, Li Zhen, Lu Lu, Cai Shengze, Maxey Martin, Karniadakis George Em
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA.
Department of Mechanical Engineering, Clemson University, Clemson, South Carolina 29634, USA.
J Chem Phys. 2021 Mar 14;154(10):104118. doi: 10.1063/5.0041203.
Simulating and predicting multiscale problems that couple multiple physics and dynamics across many orders of spatiotemporal scales is a great challenge that has not been investigated systematically by deep neural networks (DNNs). Herein, we develop a framework based on operator regression, the so-called deep operator network (DeepONet), with the long-term objective to simplify multiscale modeling by avoiding the fragile and time-consuming "hand-shaking" interface algorithms for stitching together heterogeneous descriptions of multiscale phenomena. To this end, as a first step, we investigate if a DeepONet can learn the dynamics of different scale regimes, one at the deterministic macroscale and the other at the stochastic microscale regime with inherent thermal fluctuations. Specifically, we test the effectiveness and accuracy of the DeepONet in predicting multirate bubble growth dynamics, which is described by a Rayleigh-Plesset (R-P) equation at the macroscale and modeled as a stochastic nucleation and cavitation process at the microscale by dissipative particle dynamics (DPD). First, we generate data using the R-P equation for multirate bubble growth dynamics caused by randomly time-varying liquid pressures drawn from Gaussian random fields (GRFs). Our results show that properly trained DeepONets can accurately predict the macroscale bubble growth dynamics and can outperform long short-term memory networks. We also demonstrate that the DeepONet can extrapolate accurately outside the input distribution using only very few new measurements. Subsequently, we train the DeepONet with DPD data corresponding to stochastic bubble growth dynamics. Although the DPD data are noisy and we only collect sparse data points on the trajectories, the trained DeepONet model is able to predict accurately the mean bubble dynamics for time-varying GRF pressures. Taken together, our findings demonstrate that DeepONets can be employed to unify the macroscale and microscale models of the multirate bubble growth problem, hence providing new insight into the role of operator regression via DNNs in tackling realistic multiscale problems and in simplifying modeling with heterogeneous descriptions.
模拟和预测跨多个时空尺度耦合多种物理和动力学的多尺度问题是一个巨大的挑战，深度神经网络（DNN）尚未对其进行系统研究。在此，我们开发了一个基于算子回归的框架，即所谓的深度算子网络（DeepONet），其长期目标是通过避免用于拼接多尺度现象的异构描述的脆弱且耗时的“握手”接口算法来简化多尺度建模。为此，作为第一步，我们研究DeepONet是否能够学习不同尺度 regime 的动力学，一个是确定性宏观尺度，另一个是具有固有热涨落的随机微观尺度 regime。具体而言，我们测试了DeepONet在预测多速率气泡生长动力学方面的有效性和准确性，该动力学在宏观尺度上由瑞利 - 普莱斯方程（R - P）描述，在微观尺度上通过耗散粒子动力学（DPD）建模为随机成核和空化过程。首先，我们使用R - P方程生成由从高斯随机场（GRF）中随机抽取的时变液体压力引起的多速率气泡生长动力学数据。我们的结果表明，经过适当训练的DeepONets能够准确预测宏观尺度的气泡生长动力学，并且可以优于长短期记忆网络。我们还证明，DeepONet仅使用很少的新测量值就能在输入分布之外进行准确的外推。随后，我们使用与随机气泡生长动力学对应的DPD数据训练DeepONet。尽管DPD数据有噪声，并且我们仅在轨迹上收集了稀疏的数据点，但训练后的DeepONet模型能够准确预测时变GRF压力下的平均气泡动力学。综上所述，我们的研究结果表明，DeepONets可用于统一多速率气泡生长问题的宏观尺度和微观尺度模型，从而为通过DNN进行算子回归在解决实际多尺度问题和简化异构描述建模中的作用提供了新的见解。
