# Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks

## TL;DR

This paper introduces a neural network-based framework for deriving constitutive relations in mechanical systems directly from observational data, enabling improved predictive modeling and uncertainty quantification.

## Contribution

The paper proposes a novel neural network approach for learning constitutive relations from data, outperforming traditional methods and integrating uncertainty quantification.

## Key findings

- Neural networks outperform piecewise linear and radial basis functions in certain cases.
- The framework successfully quantifies uncertainty through confidence intervals.
- Numerical examples demonstrate the effectiveness of the proposed method.

## 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.