Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modeling

Developing reliable constitutive models for hyperelastic materials is often hindered by sparse, noisy, or heterogeneous experimental data. Traditional machine learning approaches often provide deterministic point predictions that fail to account for aleatoric uncertainty (innate data noise), which is critical for safe engineering simulations. This paper addresses this by proposing a framework to learn uncertainty-aware constitutive laws that satisfy fundamental physical constraints.

The authors propose two architectures: Interval Physics-Augmented Neural Networks (iPANNs) and Fuzzy Physics-Augmented Neural Networks (fPANNs).

• iPANNs learn three distinct branches of a free energy density function—a mean, a lower bound, and an upper bound—using automatic differentiation to ensure the resulting stress predictions enclose noisy observations.
• fPANNs extend this by using alpha-cut interpolation to create a nested family of admissible responses, allowing for a graded, membership-based description of uncertainty.

Both models incorporate physical priors such as objectivity, consistency, and polyconvexity, and utilize smoothed L0 regularization to ensure the learned energy representations remain interpretable. The training process employs a two-stage transfer-learning strategy, where a mean response is learned first, followed by fine-tuning the bounds.

The framework successfully produces compact, physics-consistent models that effectively enclose noisy stress data across various synthetic test cases, including scenarios with heteroscedastic noise and shifted noise means. The authors demonstrate that these learned bounds generalize well to unseen test data and can be integrated into finite element simulations to propagate uncertainty from the material model to the structural response.

This work provides a robust, non-probabilistic alternative to Bayesian neural networks for uncertainty quantification in mechanics. By avoiding the need for complex probability distribution assumptions, the method is particularly well-suited for engineering applications where data is limited and worst-case scenario bounds are more valuable than probabilistic distributions.