Conformal Quantile Regression for Neural Probabilistic Constitutive Modeling

This paper proposes a computationally efficient, plug-and-play framework for probabilistic constitutive modeling of anisotropic soft tissues that leverages conformalized quantile regression on a thermodynamically consistent, polyconvex formulation to explicitly quantify predictive uncertainty without requiring distributional assumptions.

The Gist

Imagine you are a doctor trying to design a custom heart valve for a patient. To do this safely, you need to know exactly how that specific patient's soft tissue will stretch, squish, and snap back under pressure.

The problem is that human bodies are messy. No two people are exactly alike. One person's tissue might be slightly stiffer, another's slightly looser, and even the same tissue can behave differently depending on tiny variations in its microscopic structure.

For a long time, computer models used to predict this behavior were like rigid robots. They would say, "If you pull this tissue 10%, it will push back with exactly 5 Newtons of force." But in reality, the tissue might push back with 4.8, 5.2, or even 6 Newtons. Because the old models couldn't say, "It's probably 5, but it could be anywhere between 4 and 6," engineers were flying blind. If they guessed wrong, the design could fail.

This paper introduces a new way to build these models. Think of it as upgrading the robot from a rigid calculator to a wise, cautious weather forecaster.

Here is how the new system works, broken down into simple concepts:

1. The "Wise Weather Forecaster" (Probabilistic Modeling)

Instead of giving a single number (like "5 Newtons"), this new model gives a range of possibilities. It says, "I'm 95% sure the force will be between 4.5 and 5.5 Newtons."

• The Analogy: Imagine a weather app. An old app says, "It will rain at 2:00 PM." A smart app says, "There's a 90% chance of rain between 1:45 PM and 2:15 PM." The new model does this for tissue stress. It doesn't just guess the answer; it tells you how confident it is in that guess.

2. The "Guardrails" (Physics Constraints)

In the past, when scientists tried to make models that gave ranges, they often used complex math (Bayesian statistics) that was slow, expensive, and sometimes broke the laws of physics. The model might predict that a tissue could stretch infinitely or create energy out of nothing.

This paper uses a clever trick: Physics-Encoded Neural Networks.

• The Analogy: Imagine you are teaching a child to drive. Instead of letting them drive anywhere and hoping they don't crash, you put them in a car with guardrails and a speed limiter. No matter how they steer, the car physically cannot go off the road or break the speed limit.
• The authors built their AI model with "guardrails" built into its very structure. It is mathematically impossible for the model to predict something that violates the laws of thermodynamics or material stability. It learns the shape of the data while staying strictly within the rules of physics.

3. The "Fence Builder" (Conformal Quantile Regression)

Even with a good model, sometimes the AI gets overconfident. It might say, "I'm 99% sure the force is between 4.9 and 5.1," but the real answer is 5.5. That's dangerous.

To fix this, the authors use a method called Conformal Quantile Regression.

• The Analogy: Imagine you are building a fence around a garden to keep deer out.
  1. First, you build a fence based on your best guess of where the deer might jump (the Quantile part).
  2. Then, you test your fence with a few practice jumps. If the deer clears the fence, you realize your fence was too low.
  3. You measure how much the deer cleared the fence by, and you raise the entire fence by that amount (the Conformal part).
• This ensures that the final fence is guaranteed to be high enough to catch the deer, even if your initial guess was slightly off. In the paper, this guarantees that the "uncertainty range" is wide enough to actually catch the real answer most of the time.

4. Why This Matters (The "Plug-and-Play" Superpower)

The best part of this new method is that it's simple and fast.

• The Analogy: Think of existing computer models as a high-performance race car engine. Usually, to make that engine "smart" enough to predict uncertainty, you'd have to tear it apart and rebuild it with a massive, slow, complex transmission (like Bayesian methods).
• This new method is like a smart turbocharger you can bolt onto the existing engine. You don't have to rebuild the whole car. You just attach this new part, and suddenly, the engine not only runs fast but also tells you exactly how much fuel it might need under different conditions.

Summary

This paper solves a big problem in engineering: How do we trust computer models for unpredictable, squishy biological tissues?

They built a system that:

1. Predicts a range of outcomes instead of just one number.
2. Respects the laws of physics so it never makes impossible predictions.
3. Self-corrects to ensure its "safety net" is wide enough to catch real-world surprises.
4. Runs fast, making it practical for designing real medical devices and patient-specific treatments.

It turns a rigid, "one-size-fits-all" guess into a flexible, safety-conscious prediction tool that engineers can actually trust with human lives.

Technical Summary

1. Problem Statement

Biological soft tissues exhibit significant inter-subject variability and intrinsic stochasticity due to their complex microstructures. While data-driven constitutive modeling (using neural networks) has advanced, most existing approaches are deterministic. They fail to quantify predictive uncertainty, which limits their reliability for patient-specific analysis, design, and uncertainty propagation in large-scale mechanical simulations.

Key challenges in existing probabilistic approaches include:

• Bayesian Neural Networks (BNNs): Direct Bayesian treatment of high-dimensional network parameters is computationally prohibitive. Approximate inference (e.g., Variational Inference) often yields biased uncertainty estimates, and Markov Chain Monte Carlo (MCMC) methods are too slow for large-scale simulations.
• Thermodynamic Consistency: Enforcing physical constraints (e.g., polyconvexity, material stability) within a Bayesian framework is difficult, as priors must be restricted to admissible parameter spaces, complicating inference.
• Distributional Assumptions: Many methods assume specific parametric forms (e.g., Gaussian) for the conditional distribution, which is often invalid for highly nonlinear soft tissue behavior.

2. Methodology

The authors propose a frequentist, data-driven framework that combines physics-constrained neural networks with Conformalized Quantile Regression (CQR). The methodology consists of three main pillars:

A. Physics-Constrained Deterministic Backbone

The model is built upon a strain-invariant, polyconvex formulation for anisotropic hyperelastic materials (based on the framework of [18]).

• Strain Energy Density ($\psi$): Represented as a sum of univariate functions of isotropic and anisotropic strain invariants ($I_k$). This additive structure ensures polyconvexity, guaranteeing thermodynamic consistency and mechanical stability.
• Parameterization Strategy: Instead of parameterizing the strain energy $\psi$ directly (which requires convex neural networks), the model parameterizes the derivatives of the energy functions ($\tilde{g}_k = d\tilde{f}_k/dI_k$).
  • These derivatives are modeled using Monotone Neural Networks (MNNs).
  • Advantage: By enforcing non-negative weights and non-decreasing activation functions, the network guarantees that the derivative is non-negative and monotone by construction. This approach is computationally more efficient and easier to train than parameterizing the convex energy function directly.
  • Stress Calculation: The First Piola-Kirchhoff stress ($P$) is computed directly from these learned gradients, avoiding the need for automatic differentiation during inference.

B. Tensorial Quantile Regression

To handle uncertainty, the deterministic unknown functions are replaced by quantile functions.

• Componentwise Approach: The stress tensor $P$ is treated as a random variable. The model learns conditional quantiles for each stress component ($P_{iJ}$) at specific levels (e.g., lower $\alpha_{lo}$ and upper $\alpha_{hi}$).
• Additive Structure: The quantile stress is modeled as an additive sum of the learned monotone functions:
  $$q_\alpha(F) = \sum \frac{\partial \tilde{I}_k}{\partial F} \tilde{g}_{\alpha k}(\tilde{I}_k)$$
• Loss Function: The model is trained using the pinball loss (quantile loss) to minimize the error between predicted and observed stress quantiles. A non-crossing penalty is added to ensure the lower quantile does not exceed the upper quantile.

C. Conformalized Quantile Regression (CQR) for Calibration

Raw quantile regression often suffers from miscalibration in finite-sample settings. The authors apply Conformal Prediction to guarantee coverage.

• Nonconformity Scores: A calibration set is used to compute scores measuring how far observed stresses fall outside the predicted quantile intervals.
• Trajectory-Level Calibration: Recognizing that experimental data points within a single loading path are correlated, the authors propose a trajectory-level calibration strategy. The maximum nonconformity score along an entire loading trajectory is used, rather than pointwise scores.
• Calibrated Intervals: The final predictive interval is constructed by expanding the raw quantile bounds by a conformal adjustment factor derived from the calibration set. This provides finite-sample, distribution-free marginal coverage guarantees without assuming a specific data distribution.

3. Key Contributions

1. Probabilistic Constitutive Modeling: A novel framework that extends deterministic, physics-constrained neural constitutive models to a probabilistic setting without resorting to Bayesian inference.
2. Gradient-Based Parameterization: A demonstration that parameterizing stress-response gradients via monotone networks is more efficient and accurate than parameterizing the energy function directly via convex networks.
3. Tensorial CQR: An extension of Conformalized Quantile Regression to tensor-valued fields (stress tensors), enabling calibrated uncertainty quantification for anisotropic materials.
4. Plug-and-Play Simplicity: The method requires no assumptions on the underlying data distribution and can be applied to existing deterministic models to add probabilistic outputs.
5. Computational Efficiency: The method avoids Monte Carlo sampling at inference, making it as fast as a deterministic model and suitable for large-scale uncertainty propagation.

4. Results

The method was validated on three numerical examples:

1. Synthetic Mooney-Rivlin Data (Isotropic):
   • Demonstrated the ability to capture heteroscedasticity (non-constant variance) in stress-strain responses.
   • Showed that trajectory-level calibration yields more conservative and reliable uncertainty bounds compared to pooled pointwise calibration.
2. Synthetic Arterial Wall Data (Anisotropic, Extrapolation):
   • Tested on data generated from a Holzapfel-Gasser-Ogden model with added noise.
   • The model successfully extrapolated to strain levels beyond the training range, maintaining physical admissibility.
   • Uncertainty estimates were meaningful, though they narrowed at extreme extrapolation, highlighting the need for adaptive sampling in those regimes.
3. Experimental Porcine Valve Data:
   • Applied to real biaxial experimental data of porcine atrioventricular valve leaflets.
   • The model captured the complex anisotropic behavior and provided uncertainty intervals that covered the majority of unseen test data (different loading ratios), even in regions with high data scatter.

5. Significance and Limitations

Significance:

• Reliability: Provides rigorous, distribution-free uncertainty quantification essential for safety-critical medical device design and patient-specific simulations.
• Scalability: By avoiding MCMC and Bayesian priors, the method scales to large neural networks and large datasets.
• Physical Admissibility: Ensures that all probabilistic predictions (quantiles) strictly satisfy thermodynamic laws (polyconvexity, stability), a feature often lost in standard probabilistic ML approaches.

Limitations:

• Aleatoric vs. Epistemic: The framework primarily captures aleatoric uncertainty (data variability). It does not explicitly model epistemic uncertainty (model inadequacy or lack of data), which may lead to overconfidence in extrapolation regimes.
• Additive Assumption: The current formulation assumes an additive separation of strain invariants, excluding explicit cross-invariant coupling terms.
• Symmetry Classes: While capable of handling transverse isotropy and lower symmetries, the current structural tensor formulation may require extension (higher-order tensors) to fully capture trigonal or hexagonal symmetries.

In conclusion, this work offers a robust, efficient, and theoretically grounded solution for uncertainty quantification in soft tissue mechanics, bridging the gap between data-driven flexibility and physical rigor.