Conformal Quantile Regression for Probabilistic Constitutive Modeling of Anisotropic Soft Materials

TL;DR

This work tackles the challenge of modeling anisotropic soft tissues with substantial inter-subject variability by introducing a probabilistic constitutive framework that yields calibrated uncertainty without distributional assumptions. It leverages a physics-encoded, polyconvex backbone and models tensor-valued stresses via tensorial quantile regression, augmented with conformal calibration to achieve distribution-free marginal coverage. A key innovation is the use of monotone univariate networks to parameterize invariant-gradient fields, ensuring thermodynamic consistency and improved trainability, together with a conformal adjustment step that guarantees finite-sample coverage. Across synthetic and experimental datasets, the approach demonstrates accurate quantile predictions, robust extrapolation, and practical uncertainty quantification suitable for large-scale mechanical simulations.

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.

Figures (14)

• Figure 1: Synthetic example illustrating a smooth convex function (left) and its corresponding monotone derivative (right), used to compare performances of convex-function and gradient-based monotone parameterizations.
• Figure 2: Predicted gradients obtained using convex energy parameterization (left) and gradient-based monotone parameterization (right). The monotone approach more accurately captures the underlying stepwise gradient structure in this example.
• Figure 3: Synthetic datasets generated using the Mooney--Rivlin constitutive model. (top) Twenty sets of material parameters are randomly sampled. (bottom) Mean and standard deviation computed over 500 parameter realizations.
• Figure 4: Conformal adjustments computed using two calibration strategies: (a) trajectory-wise calibration and (b) pooled calibration across all trajectories. The horizontal axis denotes the number of calibration trajectories randomly selected from a batch of 256 trajectories. For each setting, the statistics of the conformal adjustments are computed over 10 independent random subsamples drawn from the calibration batch.
• Figure 5: Pinball loss for each loading case during training.

Theorems & Definitions (3)

• Remark 2.1: Other choices beyond neural networks
• Remark 2.2
• Remark 3.1