Cure State Estimation in Frontal Polymerization from Temperature Fields via Cascaded Inverse-Forward Deep Learning with Uncertainty Quantification

Technical Summary: Cure State Estimation in Frontal Polymerization via Cascaded Inverse-Forward Deep Learning

Problem Statement
Frontal polymerization (FP) is an energy-efficient manufacturing strategy for thermosets and composites, relying on self-propagating exothermic reaction fronts. A critical challenge in FP is that the degree of cure ($\alpha$), which dictates the final mechanical properties, cannot be measured directly during processing. While infrared (IR) thermography provides real-time temperature fields ($T$), these signals are primarily used to track front kinematics rather than the underlying cure state. Predicting the spatiotemporal cure field from temperature observations is complicated by significant parametric uncertainty: key process parameters, specifically the convective heat-transfer coefficient ($h$) and the initial degree of cure ($\alpha_0$), are unknown and vary based on fixture conditions and resin history. Conventional finite-element (FE) simulations are accurate but computationally prohibitive (minutes to hours per case) for real-time deployment, and existing deep learning surrogates for FP typically require these parameters as known inputs, failing to address the coupled inverse-forward nature of the problem.

Methodology
The authors propose a simulation-validated, two-stage cascaded deep learning pipeline designed for real-time digital twin deployment. The framework operates on high-fidelity MOOSE finite-element simulation data of dicyclopentadiene (DCPD) frontal ring-opening metathesis polymerization (FROMP).

1. Stage 1: Inverse Parameter Estimation

   • Input: An early-time temperature history (11 frames, $\approx$5 seconds).
   • Task: Estimate the unknown process parameters $\theta = (h, \alpha_0)$.
   • Architecture: A hybrid Conv2D-BiLSTM encoder. Spatial features are extracted via four 2D convolutional layers with skip connections, followed by a bidirectional LSTM to capture temporal dynamics. A multi-layer perceptron (MLP) head outputs bounded estimates for $h$ and $\alpha_0$ using sigmoid activation.
   • Training: Supervised learning with range-normalized mean squared error (MSE) loss.

2. Stage 2: Causal Forward Cure Prediction

   • Input: Streaming temperature fields and the Stage 1 parameter estimates ($\hat{h}, \hat{\alpha}_0$).
   • Task: Predict the evolving cure field $\hat{\alpha}(x, y, t)$ in real-time.
   • Architecture: A SpatialEncoder-ConvLSTM model. A 2D spatial encoder processes each temperature frame, which is then integrated into a recurrent hidden state via a ConvLSTM cell. Crucially, the model employs Feature-wise Linear Modulation (FiLM) to condition the hidden states on the estimated parameters, allowing a single model to adapt to varying operating conditions without retraining.
   • Constraint: The architecture is strictly causal, using only past and current temperature data ($t \leq \tau$) to predict the cure at time $t$.

3. Uncertainty Quantification (UQ) Framework
   A four-phase workflow decomposes total predictive variance into three components:

   • Parametric Uncertainty: Propagation of Stage 1 estimation errors through Stage 2, quantified via the linearized delta method.
   • Aleatoric Uncertainty: Input-dependent data uncertainty captured by a heteroscedastic deep ensemble (predicting mean and variance).
   • Epistemic Uncertainty: Model uncertainty estimated via ensemble variance and MC dropout.
   • Calibration: Split conformal prediction is used to convert raw predictive intervals into calibrated prediction intervals with guaranteed marginal coverage on held-out data.

Key Contributions

1. First End-to-End Inverse-Forward Pipeline for FP: Unlike prior surrogates that assume known process parameters, this pipeline directly maps observable temperature fields to cure fields by first estimating latent parameters.
2. Causal Parameter-Conditioned Architecture: The introduction of FiLM conditioning in a recurrent ConvLSTM structure enables robust generalization across operating conditions (varying $h$ and $\alpha_0$) within a single model.
3. Multi-Source Variance Decomposition: The framework explicitly separates and quantifies aleatoric, epistemic, and parametric uncertainties, identifying that parametric errors (specifically from $\alpha_0$) dominate downstream cure uncertainty.
4. Calibrated Prediction Intervals: The application of split conformal prediction yields prediction intervals with 91.4% empirical coverage on a disjoint test set, providing a rigorous basis for process monitoring.

Results

• Accuracy: On an interpolation test split, the end-to-end pipeline achieves a cure degree Root-Mean-Square Error (RMSE) of 0.0035 ($R^2 = 0.9999$). This represents a 1.5$\times$ degradation compared to an "oracle" Stage 2 model using ground-truth parameters, indicating high fidelity.
• Speed: End-to-end inference runs in approximately 104 ms per 41-frame trajectory on a single NVIDIA L40S GPU, comfortably meeting the $\approx$500 ms inter-frame budget for real-time monitoring.
• Error Propagation: Analysis reveals that errors in estimating $\alpha_0$ account for >90% of the propagated cure variance, while errors in $h$ contribute minimally. This identifies $\alpha_0$ estimation as the primary bottleneck for accuracy.
• Robustness: The FiLM-conditioned model significantly outperforms non-conditioned variants (5.2$\times$ lower RMSE on interpolation, 6.7$\times$ on extrapolation), demonstrating its necessity for handling varying process conditions.
• Uncertainty: The total predictive standard deviation is 0.0073. The variance is dominated by aleatoric uncertainty (84.8%), followed by parametric uncertainty (14.4%) and epistemic uncertainty (0.9%). Conformal intervals achieved 91.4% marginal coverage on a held-out test set, though a localized coverage trough was observed near $\alpha_0 \approx 0.15$.

Significance and Claims
The paper positions this work as a simulation-validated prototype for a real-time, IR-driven digital twin for frontal polymerization. The authors claim that this framework bridges the gap between high-fidelity physics simulations and real-time manufacturing constraints by:

• Eliminating the need for direct cure measurement or known process parameters.
• Providing calibrated uncertainty bounds essential for decision support (e.g., quality gating).
• Offering a modular architecture where inverse and forward modules can be improved independently.

The authors explicitly state that experimental validation against measured thermal imaging data and sim-to-real transfer remain future work. The current results are strictly limited to the DCPD formulation and 2D geometry used in the MOOSE simulations. The framework is presented as a foundational step toward closed-loop control, with the control arm reported separately in companion work.