pith. sign in

arxiv: 2601.17437 · v2 · submitted 2026-01-24 · ⚛️ physics.comp-ph

Conformal Quantile Regression for Neural Probabilistic Constitutive Modeling

Bahador Bahmani This is my paper

Pith reviewed 2026-05-16 11:24 UTC · model grok-4.3

classification ⚛️ physics.comp-ph
keywords conformal quantile regressionprobabilistic constitutive modelingpolyconvex neural networksanisotropic soft materialsuncertainty quantificationdata-driven mechanicsthermodynamic consistency
0
0 comments X

The pith

Conformal quantile regression endows polyconvex neural models with probabilistic predictions for anisotropic soft tissue mechanics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework to model the mechanical behavior of anisotropic soft materials probabilistically by applying conformalized quantile regression to the tensor-valued outputs of neural networks built on strain invariants and polyconvexity. This produces uncertainty estimates around stress predictions without assuming any data distribution and without Monte Carlo sampling at inference time. The approach matters for patient-specific analysis because biological tissues show large inter-subject variability, so knowing the range of likely responses improves reliability in mechanical simulations. The method is constructed to attach directly to existing deterministic constitutive models while preserving thermodynamic consistency and extending robustly to data outside the training range.

Core claim

The central claim is that conformalized quantile regression applied to tensor-valued fields from a strain-invariant polyconvex neural constitutive model yields distribution-free probabilistic predictions for the mechanical response of anisotropic soft materials, with thermodynamic consistency maintained and computational efficiency preserved by avoiding sampling at inference.

What carries the argument

Conformalized quantile regression applied directly to the tensor-valued outputs of a strain-invariant, polyconvex neural constitutive model.

If this is right

  • Existing deterministic neural constitutive models can be upgraded to probabilistic versions through a plug-and-play application of conformal quantile regression.
  • Predictions maintain thermodynamic consistency because the underlying model remains polyconvex and strain-invariant.
  • No distributional assumptions on the data are needed and Monte Carlo sampling is avoided during inference.
  • The framework scales to large-parameter models and supports uncertainty propagation in large-scale finite element simulations.
  • Predictive performance holds in extrapolative regimes beyond the training data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Uncertainty bands could be propagated through patient-specific finite element models to quantify reliability of surgical or implant designs.
  • The distribution-free property may allow the method to capture stochasticity from heterogeneous tissue microstructures more robustly than parametric Bayesian alternatives.
  • Similar conformalization could be tested on other constrained tensor outputs in continuum mechanics where physical invariants must be respected.
  • Validation on real experimental datasets with measured inter-subject variability would reveal whether synthesized benchmarks understate practical uncertainty levels.

Load-bearing premise

Conformal quantile regression can be applied directly to the tensor-valued outputs of the polyconvex neural model without breaking thermodynamic consistency or requiring additional constraints on the quantile functions.

What would settle it

A concrete counter-example in which the quantile predictions from the conformalized model produce a stress tensor that violates polyconvexity, such as yielding negative strain energy density for some admissible deformation.

Figures

Figures reproduced from arXiv: 2601.17437 by Bahador Bahmani.

Figure 1
Figure 1. Figure 1: Synthetic example illustrating a smooth convex function (left) and its corresponding monotone derivative [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Predicted gradients obtained using convex energy parameterization (left) and gradient-based monotone [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Synthetic datasets generated using the Mooney–Rivlin constitutive model. (top) Twenty sets of material [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Conformal adjustments computed using two calibration strategies: (a) trajectory-wise calibration and (b) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Pinball loss for each loading case during training. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (top) Quantile regression uncertainty intervals. (bottom) Conformalized quantile regression. Dashed lines [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Anisotropic material model response with parameters adopted from [ [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Training, calibration, and test data. (left) Stress–strain responses for uniaxial tests along different directions. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Pinball loss training history for the arterial wall data. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Model predictions on the test data (black dots) and uncertainty estimates given by the quantile predictions [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Experimental data of porcine atrioventricular valve leaflets obtained under biaxial loading with varying [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Experimental data of porcine atrioventricular valve leaflets obtained under biaxial loading with varying [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Training loss evolution (left) and distribution of training times (right) over 10 randomly initialized neural [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Pinball loss training history for the porcine atrioventricular aalve leaflets data. [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a probabilistic constitutive modeling framework for anisotropic soft materials that applies conformalized quantile regression to the tensor-valued outputs of a strain-invariant, polyconvex neural network. It claims thermodynamic consistency, distribution-free uncertainty quantification, plug-and-play compatibility with existing deterministic models, computational efficiency without Monte Carlo sampling, and robust performance including in extrapolative regimes, with validation on several benchmark datasets from the literature.

Significance. If the central construction preserves polyconvexity and thermodynamic relations for the quantile functions, the framework would provide a practical route to uncertainty-aware, patient-specific biomechanical simulations. The distribution-free and non-sampling aspects would be genuine strengths for large-scale finite-element analyses.

major comments (1)
  1. [Abstract] Abstract: the claim of plug-and-play compatibility without extra constraints is load-bearing for the central result, yet the application of conformalized quantile regression directly to tensor components risks violating polyconvexity of the underlying strain-energy function and positive-definiteness of the tangent stiffness; no explicit mechanism is indicated for enforcing these properties on the quantile maps themselves.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The concern about preserving polyconvexity and thermodynamic consistency under conformal quantile regression is well taken, and we address it directly below while revising the manuscript for greater clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of plug-and-play compatibility without extra constraints is load-bearing for the central result, yet the application of conformalized quantile regression directly to tensor components risks violating polyconvexity of the underlying strain-energy function and positive-definiteness of the tangent stiffness; no explicit mechanism is indicated for enforcing these properties on the quantile maps themselves.

    Authors: We agree that an explicit mechanism for the quantile maps was not sufficiently detailed in the original submission. The core neural network employs a strain-invariant polyconvex architecture that enforces thermodynamic consistency and polyconvexity for the base strain-energy function; the stress tensor is obtained by automatic differentiation of this energy. Conformal quantile regression is applied post-hoc as a calibration step on the residuals of the derived stress components, without retraining or modifying the network weights. This preserves the plug-and-play property for any existing deterministic polyconvex model. To ensure the quantile bounds respect positive-definiteness of the tangent stiffness, the uncertainty intervals are constructed around the base-model stresses while the stiffness matrix itself is evaluated from the polyconvex energy at the mean prediction; numerical verification on the benchmark datasets confirms that the resulting tangent remains positive definite within the calibrated coverage. We have revised the abstract and added a new paragraph in Section 3.2 that explicitly describes this procedure, including the invariant-based residual calibration and the verification steps. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper presents conformalized quantile regression as a plug-and-play addition to existing strain-invariant polyconvex neural constitutive models, with claims of thermodynamic consistency and distribution-free uncertainty quantification resting on the base model's invariants and the conformal prediction framework. No load-bearing step equates a derived prediction or quantile output to the paper's own fitted parameters or self-citations by construction; the abstract and validation explicitly reference external benchmark datasets synthesized from literature. The central result therefore retains independent content beyond its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that a strain-invariant polyconvex formulation guarantees thermodynamic consistency when combined with conformal quantile regression on tensor fields.

axioms (1)
  • domain assumption Strain-invariant polyconvex formulation ensures thermodynamic consistency
    Explicitly stated as the foundation of the proposed framework in the abstract.

pith-pipeline@v0.9.0 · 5505 in / 1185 out tokens · 34686 ms · 2026-05-16T11:24:39.172806+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · 1 internal anchor

  1. [1]

    Akinjide R Akintunde, Kristin S Miller, and Daniele E Schiavazzi. Bayesian inference of constitutive model parameters from uncertain uniaxial experiments on murine tendons.Journal of the mechanical behavior of biomedical materials, 96:285–300, 2019

    work page 2019

  2. [2]

    The influence of structural variations on the constitutive response and strain variations in thin fibrous materials.Acta Materialia, 203: 116460, 2021

    Mossab Alzweighi, Rami Mansour, Jussi Lahti, Ulrich Hirn, and Artem Kulachenko. The influence of structural variations on the constitutive response and strain variations in thin fibrous materials.Acta Materialia, 203: 116460, 2021

    work page 2021

  3. [3]

    Input convex neural networks

    Brandon Amos, Lei Xu, and J Zico Kolter. Input convex neural networks. InInternational conference on machine learning, pages 146–155. PMLR, 2017. 19 CQR for Anisotropic Soft MaterialsA PREPRINT 101 103 105 Epoch 100 101 Loss Quantile Figure 14: Pinball loss training history for the porcine atrioventricular aalve leaflets data

    work page 2017

  4. [4]

    A review of the characterizations of soft tissues used in human body modeling: scope, limitations, and the path forward.Journal of Tissue Viability, 32(2):286–304, 2023

    Nicole Arnold, Justin Scott, and Tamara Reid Bush. A review of the characterizations of soft tissues used in human body modeling: scope, limitations, and the path forward.Journal of Tissue Viability, 32(2):286–304, 2023

    work page 2023

  5. [5]

    A mechanics-informed artificial neural network approach in data-driven constitutive modeling.International Journal for Numerical Methods in Engineering, 123(12):2738– 2759, 2022

    Faisal As’ ad, Philip Avery, and Charbel Farhat. A mechanics-informed artificial neural network approach in data-driven constitutive modeling.International Journal for Numerical Methods in Engineering, 123(12):2738– 2759, 2022

    work page 2022

  6. [6]

    Physics-constrained symbolic model discovery for polyconvex incom- pressible hyperelastic materials.International Journal for Numerical Methods in Engineering, 125(15):e7473, 2024

    Bahador Bahmani and WaiChing Sun. Physics-constrained symbolic model discovery for polyconvex incom- pressible hyperelastic materials.International Journal for Numerical Methods in Engineering, 125(15):e7473, 2024

    work page 2024

  7. [7]

    Convexity conditions and existence theorems in nonlinear elasticity.Archive for rational mechanics and Analysis, 63(4):337–403, 1976

    John M Ball. Convexity conditions and existence theorems in nonlinear elasticity.Archive for rational mechanics and Analysis, 63(4):337–403, 1976

    work page 1976

  8. [8]

    Variational inference: A review for statisticians.Journal of the American statistical Association, 112(518):859–877, 2017

    David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians.Journal of the American statistical Association, 112(518):859–877, 2017

    work page 2017

  9. [9]

    Use of deep neural networks for uncertain stress functions with extensions to impact mechanics

    Garrett Blum, Ryan Doris, Diego Klabjan, Horacio Espinosa, and Ron Szalkowski. Use of deep neural networks for uncertain stress functions with extensions to impact mechanics. In2024 9th International Conference on Big Data Analytics (ICBDA), pages 248–257. IEEE, 2024

    work page 2024

  10. [10]

    Weight uncertainty in neural net- work

    Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural net- work. InInternational conference on machine learning, pages 1613–1622. PMLR, 2015

    work page 2015

  11. [11]

    Uncertainty quantification for constitutive model calibration of brain tissue.Journal of the mechanical behavior of biomedical materials, 85:237–255, 2018

    Patrick T Brewick and Kirubel Teferra. Uncertainty quantification for constitutive model calibration of brain tissue.Journal of the mechanical behavior of biomedical materials, 85:237–255, 2018

    work page 2018

  12. [12]

    Fifty shades of brain: a review on the mechanical testing and modeling of brain tissue.Archives of Computational Methods in Engineering, 2019

    Silvia Budday, Timothy C Ovaert, Gerhard A Holzapfel, Paul Steinmann, and Ellen Kuhl. Fifty shades of brain: a review on the mechanical testing and modeling of brain tissue.Archives of Computational Methods in Engineering, 2019

    work page 2019

  13. [13]

    Polyconvex neural networks for hyperelastic constitutive models: A rectifi- cation approach.Mechanics Research Communications, 125:103993, 2022

    Peiyi Chen and Johann Guilleminot. Polyconvex neural networks for hyperelastic constitutive models: A rectifi- cation approach.Mechanics Research Communications, 125:103993, 2022

    work page 2022

  14. [14]

    Uncertainty-aware digital twins: Robust model predictive control using time-series deep quantile learning.Journal of Mechanical Design, 148(2):021702, 2026

    Yi-Ping Chen, Ying-Kuan Tsai, Vispi Karkaria, and Wei Chen. Uncertainty-aware digital twins: Robust model predictive control using time-series deep quantile learning.Journal of Mechanical Design, 148(2):021702, 2026

    work page 2026

  15. [15]

    Shape-constrained regression using sum of squares polynomials.Operations Research, 73(1):543–559, 2025

    Mihaela Curmei and Georgina Hall. Shape-constrained regression using sum of squares polynomials.Operations Research, 73(1):543–559, 2025

    work page 2025

  16. [16]

    A Bayesian approach to accounting for variability in mechanical properties in biomaterials

    Srikrishna Doraiswamy and Arun R Srinivasa. A bayesian approach to accounting for variability in mechanical properties in biomaterials.arXiv preprint arXiv:1312.2856, 2013

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  17. [17]

    Tensor basis gaussian process models of hyperelastic materials

    Ari L Frankel, Reese E Jones, and Laura P Swiler. Tensor basis gaussian process models of hyperelastic materials. Journal of Machine Learning for Modeling and Computing, 1(1), 2020

    work page 2020

  18. [18]

    Monotone piecewise cubic interpolation.SIAM Journal on Numerical Analysis, 17(2):238–246, 1980

    Frederick N Fritsch and Ralph E Carlson. Monotone piecewise cubic interpolation.SIAM Journal on Numerical Analysis, 17(2):238–246, 1980. 20 CQR for Anisotropic Soft MaterialsA PREPRINT

    work page 1980

  19. [19]

    Jan N Fuhg and Nikolaos Bouklas. On physics-informed data-driven isotropic and anisotropic constitutive mod- els through probabilistic machine learning and space-filling sampling.Computer Methods in Applied Mechanics and Engineering, 394:114915, 2022

    work page 2022

  20. [20]

    Learning hyperelastic anisotropy from data via a tensor basis neural network.Journal of the Mechanics and Physics of Solids, 168:105022, 2022

    Jan N Fuhg, Nikolaos Bouklas, and Reese E Jones. Learning hyperelastic anisotropy from data via a tensor basis neural network.Journal of the Mechanics and Physics of Solids, 168:105022, 2022

    work page 2022

  21. [21]

    A review on data-driven constitutive laws for solids

    Jan Niklas Fuhg, Govinda Anantha Padmanabha, Nikolaos Bouklas, Bahador Bahmani, WaiChing Sun, Niko- laos N Vlassis, Moritz Flaschel, Pietro Carrara, and Laura De Lorenzis. A review on data-driven constitutive laws for solids.arXiv preprint arXiv:2405.03658, 2024

    work page arXiv 2024

  22. [22]

    Nonlinear solid mechanics: a continuum approach for engineering science, 2002

    Gerhard A Holzapfel. Nonlinear solid mechanics: a continuum approach for engineering science, 2002

    work page 2002

  23. [23]

    Mod- elling non-symmetric collagen fibre dispersion in arterial walls.Journal of the royal society interface, 12(106): 20150188, 2015

    Gerhard A Holzapfel, Justyna A Niestrawska, Ray W Ogden, Andreas J Reinisch, and Andreas J Schriefl. Mod- elling non-symmetric collagen fibre dispersion in arterial walls.Journal of the royal society interface, 12(106): 20150188, 2015

    work page 2015

  24. [24]

    What are bayesian neural network posteriors really like? InInternational conference on machine learning, pages 4629–4640

    Pavel Izmailov, Sharad Vikram, Matthew D Hoffman, and Andrew Gordon Gordon Wilson. What are bayesian neural network posteriors really like? InInternational conference on machine learning, pages 4629–4640. PMLR, 2021

    work page 2021

  25. [25]

    Intra-and inter-individual variability in the mechanical properties of the human skin from in vivo measurements on 20 volunteers.Skin Research and Technology, 23(4):491–499, 2017

    Emmanuelle Jacquet, Jérôme Chambert, Julien Pauchot, and Patrick Sandoz. Intra-and inter-individual variability in the mechanical properties of the human skin from in vivo measurements on 20 volunteers.Skin Research and Technology, 23(4):491–499, 2017

    work page 2017

  26. [26]

    Biaxial mechanical data of porcine atrioventricular valve leaflets.Data in brief, 21: 358–363, 2018

    Samuel Jett, Devin Laurence, Robert Kunkel, Anju R Babu, Katherine Kramer, Ryan Baumwart, Rheal Towner, Yi Wu, and Chung-Hao Lee. Biaxial mechanical data of porcine atrioventricular valve leaflets.Data in brief, 21: 358–363, 2018

    work page 2018

  27. [27]

    Bayesian-euclid: Discovering hyperelastic material laws with uncertainties.Computer Methods in Applied Mechanics and Engineering, 398:115225, 2022

    Akshay Joshi, Prakash Thakolkaran, Yiwen Zheng, Maxime Escande, Moritz Flaschel, Laura De Lorenzis, and Siddhant Kumar. Bayesian-euclid: Discovering hyperelastic material laws with uncertainties.Computer Methods in Applied Mechanics and Engineering, 398:115225, 2022

    work page 2022

  28. [28]

    Karl A Kalina, Jörg Brummund, WaiChing Sun, and Markus Kästner. Neural networks meet anisotropic hypere- lasticity: A framework based on generalized structure tensors and isotropic tensor functions.Computer Methods in Applied Mechanics and Engineering, 437:117725, 2025

    work page 2025

  29. [29]

    Polyconvex anisotropic hyperelasticity with neural networks.Journal of the Mechanics and Physics of Solids, 159:104703, 2022

    Dominik K Klein, Mauricio Fernández, Robert J Martin, Patrizio Neff, and Oliver Weeger. Polyconvex anisotropic hyperelasticity with neural networks.Journal of the Mechanics and Physics of Solids, 159:104703, 2022

    work page 2022

  30. [30]

    Quantile regression.Journal of economic perspectives, 15(4):143–156, 2001

    Roger Koenker and Kevin F Hallock. Quantile regression.Journal of economic perspectives, 15(4):143–156, 2001

    work page 2001

  31. [31]

    Neural networks meet hyperelasticity: A guide to enforcing physics.Journal of the Mechanics and Physics of Solids, 179:105363, 2023

    Lennart Linden, Dominik K Klein, Karl A Kalina, Jörg Brummund, Oliver Weeger, and Markus Kästner. Neural networks meet hyperelasticity: A guide to enforcing physics.Journal of the Mechanics and Physics of Solids, 179:105363, 2023

    work page 2023

  32. [32]

    A new family of constitutive artificial neural networks towards automated model discovery.Computer Methods in Applied Mechanics and Engineering, 403:115731, 2023

    Kevin Linka and Ellen Kuhl. A new family of constitutive artificial neural networks towards automated model discovery.Computer Methods in Applied Mechanics and Engineering, 403:115731, 2023

    work page 2023

  33. [33]

    Kevin Linka, Markus Hillgärtner, Kian P Abdolazizi, Roland C Aydin, Mikhail Itskov, and Christian J Cyron. Constitutive artificial neural networks: A fast and general approach to predictive data-driven constitutive model- ing by deep learning.Journal of Computational Physics, 429:110010, 2021

    work page 2021

  34. [34]

    Discovering uncertainty: Bayesian constitutive artificial neural networks.Computer Methods in Applied Mechanics and Engineering, 433:117517, 2025

    Kevin Linka, Gerhard A Holzapfel, and Ellen Kuhl. Discovering uncertainty: Bayesian constitutive artificial neural networks.Computer Methods in Applied Mechanics and Engineering, 433:117517, 2025

    work page 2025

  35. [35]

    Bayesian calibration of hyperelastic constitutive models of soft tissue.Journal of the mechanical behavior of biomedical materials, 59:108–127, 2016

    Sandeep Madireddy, Bhargava Sista, and Kumar Vemaganti. Bayesian calibration of hyperelastic constitutive models of soft tissue.Journal of the mechanical behavior of biomedical materials, 59:108–127, 2016

    work page 2016

  36. [36]

    Courier Corporation, 1994

    Jerrold E Marsden and Thomas JR Hughes.Mathematical foundations of elasticity. Courier Corporation, 1994

    work page 1994

  37. [37]

    Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materials.Journal of the Mechanics and Physics of Solids, page 106352, 2025

    Denisa Martonová, Alain Goriely, and Ellen Kuhl. Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materials.Journal of the Mechanics and Physics of Solids, page 106352, 2025

    work page 2025

  38. [38]

    Discovering uncertainty: Gaussian constitutive neural networks with correlated weights.Computational Mechanics, pages 1–13, 2025

    Jeremy A McCulloch and Ellen Kuhl. Discovering uncertainty: Gaussian constitutive neural networks with correlated weights.Computational Mechanics, pages 1–13, 2025

    work page 2025

  39. [39]

    A variable selection approach to monotonic regression with bernstein polynomials.Journal of Applied Statistics, 38(5):961–976, 2011

    S McKay Curtis and Sujit K Ghosh. A variable selection approach to monotonic regression with bernstein polynomials.Journal of Applied Statistics, 38(5):961–976, 2011

    work page 2011

  40. [40]

    Inference using shape-restricted regression splines

    Mary C Meyer. Inference using shape-restricted regression splines. 2008. 21 CQR for Anisotropic Soft MaterialsA PREPRINT

    work page 2008

  41. [41]

    L Angela Mihai, Thomas E Woolley, and Alain Goriely. Stochastic isotropic hyperelastic materials: constitutive calibration and model selection.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2211):20170858, 2018

    work page 2018

  42. [42]

    Deep learning predicts path-dependent plasticity.Proceedings of the National Academy of Sciences, 116(52):26414–26420, 2019

    Mojtaba Mozaffar, Ramin Bostanabad, W Chen, K Ehmann, Jian Cao, and MA Bessa. Deep learning predicts path-dependent plasticity.Proceedings of the National Academy of Sciences, 116(52):26414–26420, 2019

    work page 2019

  43. [43]

    Fast and flexible methods for monotone polynomial fitting.Journal of Statistical Computation and Simulation, 86(15):2946–2966, 2016

    Kevin Murray, S Müller, and BA Turlach. Fast and flexible methods for monotone polynomial fitting.Journal of Statistical Computation and Simulation, 86(15):2946–2966, 2016

    work page 2016

  44. [44]

    Springer Science & Business Media, 2012

    Radford M Neal.Bayesian learning for neural networks, volume 118. Springer Science & Business Media, 2012

    work page 2012

  45. [45]

    Courier Corporation, 1997

    Raymond W Ogden.Non-linear elastic deformations. Courier Corporation, 1997

    work page 1997

  46. [46]

    Monotone regression splines in action.Statistical science, pages 425–441, 1988

    James O Ramsay. Monotone regression splines in action.Statistical science, pages 425–441, 1988

    work page 1988

  47. [47]

    Bayesian model selection of hyperelastic models for simple and pure shear at large deformations.Computers & Structures, 156:101–109, 2015

    TG Ritto and LCS Nunes. Bayesian model selection of hyperelastic models for simple and pure shear at large deformations.Computers & Structures, 156:101–109, 2015

    work page 2015

  48. [48]

    Conformalized quantile regression.Advances in neural information processing systems, 32, 2019

    Yaniv Romano, Evan Patterson, and Emmanuel Candes. Conformalized quantile regression.Advances in neural information processing systems, 32, 2019

    work page 2019

  49. [49]

    Integrating uncertainty awareness into conformal- ized quantile regression

    Raphael Rossellini, Rina Foygel Barber, and Rebecca Willett. Integrating uncertainty awareness into conformal- ized quantile regression. InInternational Conference on Artificial Intelligence and Statistics, pages 1540–1548. PMLR, 2024

    work page 2024

  50. [50]

    Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors.Journal of the Mechanics and Physics of Solids, 56(12):3486–3506, 2008

    Jörg Schröder, Patrizio Neff, and Vera Ebbing. Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors.Journal of the Mechanics and Physics of Solids, 56(12):3486–3506, 2008

    work page 2008

  51. [51]

    Sajjad Seyedsalehi, Liangliang Zhang, Jongeun Choi, and Seungik Baek. Prior distributions of material param- eters for bayesian calibration of growth and remodeling computational model of abdominal aortic wall.Journal of biomechanical engineering, 137(10):101001, 2015

    work page 2015

  52. [52]

    Monotone approximation.Pacific Journal of Mathematics, 15(2):667–671, 1965

    Oved Shisha. Monotone approximation.Pacific Journal of Mathematics, 15(2):667–671, 1965

    work page 1965

  53. [53]

    Monotonic networks.Advances in neural information processing systems, 10, 1997

    Joseph Sill. Monotonic networks.Advances in neural information processing systems, 10, 1997

    work page 1997

  54. [54]

    Data-driven tissue mechanics with polyconvex neural ordinary differential equations.Computer Methods in Applied Mechanics and Engineering, 398:115248, 2022

    Vahidullah Tac, Francisco Sahli Costabal, and Adrian B Tepole. Data-driven tissue mechanics with polyconvex neural ordinary differential equations.Computer Methods in Applied Mechanics and Engineering, 398:115248, 2022

    work page 2022

  55. [55]

    Gregory H Teichert, Anirudh R Natarajan, Anton Van der Ven, and Krishna Garikipati. Machine learning materi- als physics: Integrable deep neural networks enable scale bridging by learning free energy functions.Computer Methods in Applied Mechanics and Engineering, 353:201–216, 2019

    work page 2019

  56. [56]

    Nn-euclid: Deep-learning hyperelasticity without stress data.Journal of the Mechanics and Physics of Solids, 169:105076, 2022

    Prakash Thakolkaran, Akshay Joshi, Yiwen Zheng, Moritz Flaschel, Laura De Lorenzis, and Siddhant Kumar. Nn-euclid: Deep-learning hyperelasticity without stress data.Journal of the Mechanics and Physics of Solids, 169:105076, 2022

    work page 2022

  57. [57]

    Geometric deep learning for computational mechanics part i: Anisotropic hyperelasticity.Computer Methods in Applied Mechanics and Engineering, 371:113299, 2020

    Nikolaos N Vlassis, Ran Ma, and WaiChing Sun. Geometric deep learning for computational mechanics part i: Anisotropic hyperelasticity.Computer Methods in Applied Mechanics and Engineering, 371:113299, 2020. 22

    work page 2020