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arxiv: 2606.23402 · v1 · pith:P4IF2TJFnew · submitted 2026-06-22 · 💻 cs.LG

Physics-Informed Modeling for Wood Thermal Analysis and Prediction

Jingren Xie , Alex John Buckthal , Ryan Anthony O'Connor , Isak Worre Foged , Dim P. Papadopoulos This is my paper

Pith reviewed 2026-06-26 08:46 UTC · model grok-4.3

classification 💻 cs.LG
keywords physics-informed neural networkswood thermal propertiesheat transfer equationconvolutional neural networksmaterial heterogeneitythermal predictiondeep learning
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The pith

Physics-informed convolutional networks that embed the steady-state heat equation outperform data-driven models on predicting pixel-level thermal responses of heterogeneous wood from RGB images.

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

The paper develops two physics-informed neural network architectures that incorporate a normalized 2D steady-state heat transfer equation to predict how wood materials of different species respond to temperature changes at the pixel level. Traditional data-driven methods treat the mapping from visual images to thermal maps as a black box and can overfit to noise, but these approaches enforce thermodynamic consistency either as a loss penalty or by building a solver into the network. By testing on real datasets of poplar and eucalyptus wood samples, the work shows that adding physical constraints improves both prediction quality and the ability to extract meaningful parameters like thermal conductivity variations within a species. This matters because wood's natural variability challenges assumptions of uniform material properties in applications like building design and energy efficiency.

Core claim

Embedding physical inductive biases successfully balances predictive accuracy, physical interpretability, and intra-species diversity, outperforming data-driven approaches in handling complex wood material heterogeneity and enabling the extraction of interpretable physical parameters.

What carries the argument

PICNNs that embed physics as a soft penalty term in the loss function and PInteCNNs that hard-code an analytical approximator-predictor-corrector solver for the heat equation directly into the CNN.

If this is right

  • Improved prediction of thermal responses while respecting physical laws for heterogeneous materials.
  • Ability to extract interpretable physical parameters such as thermal conductivity variations.
  • Handling of intra-species diversity in wood without assuming material homogeneity.
  • Outperformance over pure data-driven CNNs on real-world multimodal wood datasets.

Where Pith is reading between the lines

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

  • This method may generalize to other materials with spatially varying properties where PDEs are known.
  • Hard-coded physics integration could enable more efficient inference compared to penalty-based methods.
  • Future work might extend the steady-state assumption to time-dependent heat transfer for dynamic scenarios.

Load-bearing premise

The normalized 2D steady-state heat transfer equation derived from the general heat equation is an adequate model for the thermal behavior of the tested wood samples under the experimental conditions.

What would settle it

A held-out set of wood samples where a standard convolutional neural network produces lower prediction error on thermal maps than either the penalty-based or hard-coded physics models, or where extracted parameters fail to correlate with separate lab measurements of thermal conductivity.

Figures

Figures reproduced from arXiv: 2606.23402 by Alex John Buckthal, Dim P. Papadopoulos, Isak Worre Foged, Jingren Xie, Ryan Anthony O'Connor.

Figure 1
Figure 1. Figure 1: Overview of three thermal prediction frameworks. Top Left: The data-driven [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Selected data samples in the Poplar (top three rows), Grandis-CC (middle three [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The experimental setup for thermal measurement. The experimental setup consists [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sample-wise distributions of mean and standard deviation values for grayscale [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sample-wise distributions of the thermal-to-testbed mapping coefficients [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative results of the best-performing test samples (minimum MAE) across [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative results of the worst-performing test samples (maximum MAE) across [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Overall sample distributions of grayscale image intensities (left), thermal re [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Pixel-to-pixel binned distributions of mean and standard deviation values for [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Layer-wise RGB image-thermal feature similarity for DINOv3 models [ [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Probability distributions of physical metrics ( [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the Main Path for Data-Driven, PICNN, and PInteCNN across [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
read the original abstract

Wood materials exhibit complex, spatially varying thermal properties that challenge traditional architectural assumptions of material homogeneity. Although data-driven approaches can directly map wood RGB images to their corresponding thermal responses, they operate as uninterpretable black boxes that prioritize statistical correlation and may absorb experimental noise rather than thermodynamic plausibility. To address these limitations, we present physics-informed deep learning frameworks that integrate partial differential equations (PDEs) to predict pixel-level thermal responses of spatially heterogeneous wood materials using wood RGB images and testbed temperature maps. Specifically, we investigate two distinct approaches to enforcing a normalized 2D steady-state heat transfer equation derived from the general heat transfer equation: Physics-Informed Convolutional Neural Networks (PICNNs), which embed physics as a soft penalty term in the loss function, and Physics-Integrated Convolutional Neural Networks (PInteCNNs), which hard-code an analytical approximator-predictor-corrector solver directly into convolutional neural networks. To validate our proposed approaches, we collect three real-world multimodal datasets of Poplar, Grandis Cross-Cut (Grandis-CC), and Grandis Radial-Cut (Grandis-RC) wood samples. We further demonstrate that embedding physical inductive biases successfully balances predictive accuracy, physical interpretability, and intra-species diversity, outperforming data-driven approaches in handling complex wood material heterogeneity and enabling the extraction of interpretable physical parameters. Project: https://zekifayes.github.io/pim

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

2 major / 0 minor

Summary. The manuscript proposes two physics-informed frameworks, PICNNs (soft penalty on the PDE) and PInteCNNs (hard-coded analytical solver), that embed a normalized 2D steady-state heat transfer equation derived from the general heat equation to predict pixel-level thermal responses of spatially heterogeneous wood from RGB images and temperature maps. It introduces three real-world multimodal datasets (Poplar, Grandis-CC, Grandis-RC) and claims that the physics-informed approaches outperform purely data-driven baselines in accuracy, interpretability, and extraction of physical parameters while capturing intra-species diversity.

Significance. If the quantitative results demonstrate clear gains over baselines with proper validation and if the chosen PDE is shown to be an adequate model, the work would illustrate a practical way to inject thermodynamic constraints into image-to-field prediction tasks, offering a template for interpretable modeling of heterogeneous materials.

major comments (2)
  1. [Abstract] Abstract: the central claims of outperformance and interpretable parameter extraction are asserted without any quantitative metrics, error bars, baseline comparisons, or validation protocol details, rendering it impossible to evaluate whether the data support the claims.
  2. [PDE derivation and model section] PDE model (derivation and experimental justification): the normalized 2D steady-state heat transfer equation is adopted as the inductive bias, yet the manuscript provides no analysis or evidence showing that transient effects, moisture-dependent conductivity, anisotropy, or out-of-plane heat flow remain negligible for the Poplar, Grandis-CC, and Grandis-RC samples under the reported testbed conditions. If any of these are material, the soft or hard enforcement of the PDE would impose an incorrect constraint.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims of outperformance and interpretable parameter extraction are asserted without any quantitative metrics, error bars, baseline comparisons, or validation protocol details, rendering it impossible to evaluate whether the data support the claims.

    Authors: We agree that the abstract should include quantitative support. In the revised manuscript we will add specific metrics (e.g., mean squared error reductions with standard deviations across the three datasets), explicit baseline comparisons, and a concise description of the validation protocol. revision: yes

  2. Referee: [PDE derivation and model section] PDE model (derivation and experimental justification): the normalized 2D steady-state heat transfer equation is adopted as the inductive bias, yet the manuscript provides no analysis or evidence showing that transient effects, moisture-dependent conductivity, anisotropy, or out-of-plane heat flow remain negligible for the Poplar, Grandis-CC, and Grandis-RC samples under the reported testbed conditions. If any of these are material, the soft or hard enforcement of the PDE would impose an incorrect constraint.

    Authors: We acknowledge the need to justify the modeling assumptions. The testbed protocol used equilibration periods and controlled moisture to approximate steady-state conditions; we will add a dedicated subsection with literature references on wood thermal behavior and any sensitivity checks feasible from the existing datasets to support that the neglected effects remain small. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives a normalized 2D steady-state heat transfer equation from the general heat equation (standard reduction, no self-reference), then embeds it via soft penalty in PICNNs or hard-coded solver in PInteCNNs. Training and validation use independently collected multimodal datasets from Poplar, Grandis-CC, and Grandis-RC samples, with explicit comparison to data-driven baselines. No load-bearing self-citations, fitted inputs renamed as predictions, or self-definitional steps appear; claims rest on empirical performance rather than reducing to inputs by construction. The chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the normalized 2D heat equation is presented as derived from the general equation without further detail.

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