Learning constitutive relations from indirect observations using deep neural networks

Abstract

We present a new approach for predictive modeling and its uncertainty quantification for mechanical systems, where coarse-grained models such as constitutive relations are derived directly from observation data. We explore the use of a neural network to represent the unknown constitutive relations, compare the neural networks with piecewise linear functions, radial basis functions, and radial basis function networks, and show that the neural network outperforms the others in certain cases. We analyze the approximation error of the neural networks using a scaling argument. The training and predicting processes in our framework combine the finite element method, automatic differentiation, and neural networks (or other function approximators). Our framework also allows uncertainty quantification in the form of confidence intervals. Numerical examples on a multiscale fiber-reinforced plate problem and a nonlinear rubbery membrane problem from solid mechanics demonstrate the effectiveness of our framework.

A computationally tractable framework for nonlinear dynamic multiscale modeling of membrane woven fabrics

“…In all cases, the line search routine provided in Reference 30 is used: it attempts to enforce the Wolfe conditions 29 using a sequence of polynomial interpolations. Note that the BFGS algorithm is appropriate in this case because the datasets are relatively small; 31 for larger datasets, the stochastic gradient descent method is suggested for training.…”

Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

“…In fluid mechanics, a potential application of this discovery can lead to a better understanding of turbulence dominated by multiple spatiotemporal scales. Moreover, we can learn constitutive relations from indirect observations using deep neural networks (e.g., see Reference [147]). Alternatively, to address the key limitation of the black‐box learning methods, we can exploit to use of SR as a principle for identifying relations and operators that are related to the underlying system dynamics.…”

Deep learning or interpolation for inverse modelling of heat and fluid flow problems?

“…See also He et al (2020) for the approximation properties of finite element functions by DNNs. For completeness, we also refer to Raissi et al (2019) for physics-informed neural networks (PINNs), and also Huang et al (2020) for a DNN-based approach to learn constitutive relations from observations.…”