Predictive Modeling with Learned Constitutive Relations from Indirect Observations

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 neural networks to represent the unknowns functions (e.g., constitutive relations). Its counterparts, like piecewise linear functions and radial basis functions, are compared, and the strength of neural networks is explored. The training and predictive processes in this framework seamlessly combine the finite element method, automatic differentiation, and neural networks (or its counterparts). Under mild assumptions, we establish convergence guarantees. This framework also allows uncertainty quantification analysis in the form of intervals of confidence. Numerical examples on a multiscale fiber-reinforced plate problem and a nonlinear rubbery membrane problem from solid mechanics demonstrate the effectiveness of the framework.