Technical Summary: Physics-Informed Modeling for Wood Thermal Analysis and Prediction

Problem Statement

Wood materials possess complex, spatially varying thermal properties driven by anisotropic density and grain orientation, which challenge traditional architectural assumptions of material homogeneity. While data-driven deep learning approaches can map wood RGB images to thermal responses, they function as uninterpretable "black boxes." These models often prioritize statistical correlations over thermodynamic plausibility, potentially absorbing experimental noise rather than learning the underlying physical laws. Furthermore, the inherent intra-species diversity of wood (variations in grain, localized density, and unmodeled 3D subsurface structures) creates significant domain shifts that standard data-driven models struggle to generalize across, particularly in limited-data regimes.

Methodology

The authors propose integrating Partial Differential Equations (PDEs) into deep learning frameworks to predict pixel-level thermal responses of wood using RGB images and testbed temperature maps. The core of the approach is a normalized 2D steady-state heat transfer equation derived from fundamental 3D heat conduction principles, incorporating specific boundary conditions (Dirichlet, Robin, and Neumann) and infrared camera radiance formulations.

Two distinct architectures are presented to enforce this physical constraint:

1. Physics-Informed Convolutional Neural Networks (PICNNs):

   • Mechanism: Embeds the thermodynamic law as a soft penalty term in the loss function ($L = L_{data} + \lambda_{pde}L_{pde}$).
   • Operation: The network learns physical parameters ($\hat{r}_\alpha, \hat{r}_\beta, k^*, b^*$) from the RGB image. The PDE residual is calculated using a finite central difference method and minimized alongside the data reconstruction loss.
   • Constraint: Acts as a regularizer to guide the network toward physically plausible solutions without strictly hard-coding the solver.

2. Physics-Integrated Convolutional Neural Networks (PInteCNNs):

   • Mechanism: Hard-codes the thermodynamic law directly into the network architecture.
   • Operation: The network predicts spatially varying physical parameters ($\theta_\alpha, \theta_\beta, \theta_{bed}, \theta_s$) from the RGB image. These parameters are fed into an explicitly embedded, differentiable approximator-predictor-corrector solver executed within the forward pass.
   • Constraint: The solver iteratively refines the thermal response prediction ($\tilde{T}^*_{mea}$) based on the governing equation, enforcing physics as a hard structural bottleneck.

Data Strategy:
To address the scarcity of pixel-level thermal datasets and mitigate domain shifts, the authors introduce a physics-stratified data partitioning strategy. This involves:

• Calculating a mean transfer coefficient ($k^*$) and structural asymmetry ($\Delta k^*$) for each sample.
• Sorting samples by these physical metrics.
• Using a bivariate sorting algorithm with Kolmogorov-Smirnov (KS) statistics to distribute heterogeneous samples into training, validation, and test sets, ensuring statistical consistency across splits.

Key Contributions

1. Multimodal Data Acquisition and Analysis: The authors collected three real-world multimodal datasets (Poplar, Grandis Cross-Cut, and Grandis Radial-Cut). Cross-modal analysis revealed an empirical inverse correlation where visually darker regions correspond to higher thermal conductance. Deep feature analysis using Vision Transformers (DINOv3) demonstrated that intermediate network layers optimally capture the shared structural morphologies governing both visual appearance and thermal response.
2. Thermodynamic Model Formulation: A rigorous normalized 2D steady-state governing equation was derived, bridging the gap between testbed temperature maps and observed thermal responses. This formulation accounts for anisotropic diffusivity and boundary conditions.
3. Architecture Design: The paper introduces two frameworks (PICNN and PInteCNN) that enforce physical constraints via soft regularization and hard-coded solvers, respectively.
4. Quantitative Evaluation: A rigorous evaluation framework was established using a physics-stratified partitioning algorithm to ensure unbiased testing. The study characterizes the trade-offs between predictive accuracy, model interpretability, and intra-species diversity.

Results

The models were evaluated on three wood datasets using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and a strict pixel-wise accuracy metric ($\delta_{01}$).

• Performance: Both physics-informed approaches generally outperformed or matched the data-driven baseline, particularly in handling structural heterogeneity.
  • PICNN: Achieved the best overall performance on the Grandis-CC dataset (MAE: 0.2043) and showed consistent improvements on Poplar and Grandis-RC by tuning the PDE penalty weight ($\lambda_{pde}$).
  • PInteCNN: Demonstrated competitive results, particularly with a single corrector step. However, increasing solver iterations (e.g., to 2 steps) degraded performance due to the amplification of high-frequency noise by the Laplacian operator.
• Data Scarcity and Robustness: Sequential data integration ablation studies revealed that while data-driven models adapt quickly to limited data, they often hit performance plateaus as data volume increases. In contrast, physics-informed models (especially PICNN) exhibited more robust asymptotic scaling, bypassing statistical plateaus to achieve lower error limits as more diverse data was introduced.
• Interpretability: The models successfully extracted interpretable physical parameters. The stability of these parameters across training, validation, and test splits confirmed that the networks learned generalized physical representations rather than overfitting to specific data distributions.

Significance and Claims

The paper claims that embedding physical inductive biases into convolutional neural networks is a highly effective strategy for overcoming data limitations in the analysis of complex, heterogeneous materials like wood.

• Balancing Act: The study demonstrates that physics-informed modeling successfully balances predictive accuracy, physical interpretability, and the ability to handle intra-species diversity.
• Generalization: By rejecting non-physical noise through thermodynamic constraints, these frameworks enable models to generalize across different wood samples and extract underlying material properties (such as thermal anisotropy) that pure data-driven approaches obscure.
• Limitations: The authors modestly acknowledge that the current formulation relies on a 2D spatial simplification, approximating depth-wise heat transfer as a linear transformation. This approach ignores 3D internal structures (e.g., knots, localized density variations) which may affect surface thermal diffusion. Future work is suggested to explore 3D transient heat conduction solvers.

In conclusion, the proposed frameworks transform neural networks into interpretable tools capable of explicitly extracting physical parameters, offering a foundational predictive framework for responsive wood design and quality control.