Conformal Quantile Regression for Neural Probabilistic Constitutive Modeling

TL;DR

This paper introduces a probabilistic modeling framework for soft tissue materials that quantifies uncertainty using conformalized quantile regression, ensuring thermodynamic consistency and computational efficiency.

Contribution

It presents a simple, plug-and-play, distribution-free approach to add uncertainty quantification to existing deterministic constitutive models for soft tissues.

Findings

The method provides reliable probabilistic predictions for anisotropic soft materials.

It is scalable, easy to train, and avoids Monte Carlo sampling during inference.

Validated on multiple benchmark datasets from literature.

Abstract

Biological soft tissues exhibit substantial inter-subject variability, making the automation of constitutive material modeling essential for patient-specific analysis and design. Such materials are not only highly nonlinear but also display intrinsic stochasticity arising from their complex and heterogeneous microstructure. Despite recent advances in data-driven constitutive modeling, most existing approaches remain deterministic and fail to quantify predictive uncertainty, thereby limiting their reliability in downstream mechanical analyses. In this work, we propose a probabilistic, data-driven constitutive modeling framework for anisotropic soft materials that explicitly accounts for uncertainty through conformalized quantile regression applied to tensor-valued fields. The proposed framework is built upon a strain-invariant, polyconvex formulation that ensures thermodynamic consistency and promotes robust predictive performance, including in extrapolative regimes. A key advantage of the proposed approach is its simplicity: it can be applied in a plug-and-play manner to endow existing deterministic models with probabilistic predictions, while remaining distribution-free and requiring no assumptions on the underlying data distribution. Moreover, the method is straightforward to train, scalable to models with a large number of parameters, and avoids Monte Carlo sampling at inference, making it computationally efficient and well suited for uncertainty propagation in large-scale mechanical simulations. The proposed method is validated using several benchmark datasets synthesized and collected from the literature.