Neural Field Thermal Tomography: A Differentiable Physics Framework for Non-Destructive Evaluation

Tao Zhong; Yixun Hu; Dongzhe Zheng; Aditya Sood; Christine Allen-Blanchette

cs.LGMar 11, 2026

Neural Field Thermal Tomography: A Differentiable Physics Framework for Non-Destructive Evaluation

Authors: Tao Zhong, Yixun Hu, Dongzhe Zheng, Aditya Sood, Christine Allen-Blanchette

Organizations: Princeton University

Abstract

Inverse problems for stiff parabolic partial differential equations (PDEs), such as the inverse heat conduction problem (IHCP), are severely ill-posed: the forward map rapidly damps high-frequency interior structure before it reaches the boundary. Soft-constrained physics-informed neural networks (PINNs), which embed the PDE as a residual penalty, suffer from gradient pathology in this regime and tend to fit boundary measurements while leaving the interior field essentially untouched. We propose Neural Field Thermal Tomography (NeFTY), a hard-constrained neural field framework for label-free three-dimensional inverse heat conduction. NeFTY represents the unknown diffusivity as a continuous coordinate-based neural network, and at every optimization step passes the candidate field through a differentiable implicit-Euler heat solver with harmonic-mean interface flux, so that the governing PDE holds exactly on the discretization rather than as a soft penalty. Adjoint gradients propagate the surface reconstruction error back to the network weights at solver-level memory cost, making test-time inversion tractable on a single GPU. Across synthetic 3D benchmarks, NeFTY substantially outperforms soft-constrained PINN variants and a voxel-grid baseline on label-free volumetric recovery, and it transfers to real thermography data, surpassing classical signal-processing baselines in both defect segmentation and depth estimation. Additional details at https://cab-lab-princeton.github.io/nefty/

Figures & tables

Figure 1: IHCP setup. A high-speed camera measures the transient surface temperature following pulsed laser heating; NeFTY reconstructs the 3D subsurface diffusivity field from these measurements.

Figure 2: Empirical ill-posedness. Distinct interiors yield nearly indistinguishable surface profiles, illustrating Prop. 2 .

Figure 3: NeFTY pipeline. A coordinate MLP $f_{\theta}$ maps each grid node $\mathbf{x}$ through a frequency-annealed positional encoding $\gamma(\cdot)$ into a bounded diffusivity $\alpha_{\theta}(\mathbf{x})$ (§ 4.1 ). A differentiable implicit-Euler solver with harmonic-mean coefficients evolves the temperature $\mathbf{T}^{n}$ on a Cartesian grid (§ 4.2 ). Surface frames are compared to measurements; the adjoint state $\boldsymbol{\mu}^{n}$ is propagated backward through the same sparse system to assemble the gradient on $\theta$ at constant memory in time (§ 4.3 ).

	Homogeneous				Layered
Method	MSE ( $10^{-4}$ ) $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	IoU $\uparrow$	MSE ( $10^{-4}$ ) $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	IoU $\uparrow$
Data-driven (zero-shot)
U-Net (Sound-Only) [ 45 ]	$14.73\pm 3.83$	$14.83\pm 0.71$	$0.83\pm 0.02$	$0.00\pm 0.00$	$10.17\pm 2.21$	$15.42\pm 0.98$	$0.88\pm 0.02$	$0.00\pm 0.00$
Label-free
PINN [ 44 , 7 ]	$208.4\pm 26.2$	$-0.24\pm 0.06$	$0.04\pm 0.00$	$0.015\pm 0.003$	$200.4\pm 17.8$	$1.42\pm 0.27$	$0.04\pm 0.00$	$0.018\pm 0.004$
SPINN [ 8 ]	$32.71\pm 10.02$	$9.12\pm 0.93$	$0.45\pm 0.06$	$0.001\pm 0.001$	$33.84\pm 8.57$	$9.71\pm 0.77$	$0.53\pm 0.07$	$0.000\pm 0.000$

Table 1: Label-free 3D thermal diffusivity reconstruction on synthetic

1{,}000

-sample benchmark (mean

\pm

95% CI).

\uparrow

higher is better,

\downarrow

lower is better. Best label-free entry per metric in bold . PINN [ 44 ] , Causal-PINN [ 56 ] , and DCGD [ 25 ]

\alpha_{\theta}

rows coincide because all three collapse to the same near-trivial constant; see Section 5.2 and Appendix C .

Figure 4: Homogeneous-setting reconstructions: depth-wise $z$ -slices show NeFTY isolating defect outlines at the correct depth while Grid Opt. yields a noisy field and PINN variants converge to near-uniform diffusivity.

Figure 5: Layered-setting reconstructions: NeFTY recovers defects against the stratified bulk, whereas Grid Opt. exhibits ringing artifacts and PINN variants again collapse to featureless fields.

Table 8

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Category	Parameter	Value
Network architecture	MLP depth $L_{\mathrm{MLP}}$	10
	MLP hidden width $W_{\mathrm{MLP}}$	512
	Encoding bandwidth $N$	12
	Skip-connection layer	4
	Output activation	scaled sigmoid
Domain and grid	Domain size $(L,L,H)$	$10.0\times 10.0\times 1.0$

Appendix

Table 5: Hyperparameter configuration for NeFTY on the synthetic benchmark.

	Homogeneous								Layered Composite
	$1$ Defect		$2$ Defects		$3$ Defects		$4$ Defects		$3$ Layers		$4$ Layers
Method	PSNR	IoU	PSNR	IoU	PSNR	IoU	PSNR	IoU	PSNR	IoU	PSNR	IoU
Supervised
U-Net (Full)	$27.98$	$0.72$	$24.99$	$0.72$	$22.97$	$0.69$	$20.84$	$0.66$	$20.81$	$0.67$	$19.26$	$0.68$
U-Net (Sound-Only)	$18.96$	$0.00$	$15.95$	$0.00$	$13.19$	$0.00$	$11.23$	$0.00$	$15.62$	$0.00$	$15.22$	$0.00$
Label-free

Appendix

Table 6: Robustness breakdown of Table 1 . PSNR

\uparrow

and IoU

\uparrow

stratified by defect count (Homogeneous,

1

4

defects) and layer count (Layered Composite,

3

and

4

layers, each with

1

4

defects). The supervised U-Net (Full) upper bound is included here, as it consumes the volumetric labels the label-free baselines never see. Best label-free entry per column in bold .

	Homogeneous								Layered Composite
	$1$ Defect		$2$ Defects		$3$ Defects		$4$ Defects		$3$ Layers		$4$ Layers
Method	MSE $\downarrow$	PSNR $\uparrow$	MSE $\downarrow$	PSNR $\uparrow$	MSE $\downarrow$	PSNR $\uparrow$	MSE $\downarrow$	PSNR $\uparrow$	MSE $\downarrow$	PSNR $\uparrow$	MSE $\downarrow$	PSNR $\uparrow$
Grid Opt.	$4.82$	$75.51$	$7.02$	$71.56$	$9.14$	$71.03$	$29.84$	$67.36$	$15.94$	$71.14$	$6.61$	$71.95$
PINN	$43.89$	$63.04$	$45.10$	$62.82$	$52.12$	$62.20$	$52.65$	$62.19$	$54.28$	$62.05$	$55.39$	$61.95$
NeFTY (Ours)	$\mathbf{0.50}$	$\mathbf{82.33}$	$\mathbf{0.52}$	$\mathbf{82.26}$	$\mathbf{0.73}$	$\mathbf{81.34}$	$\mathbf{0.56}$	$\mathbf{82.17}$	$\mathbf{0.54}$	$\mathbf{82.10}$	$\mathbf{0.50}$	$\mathbf{82.42}$

Appendix

Table 7: Surface-temperature reconstruction error. For each method we re-simulate the temperature with the recovered

\alpha_{\theta}

(or directly read off

T_{\phi}

for the soft-PINN family) and compare to the ground-truth surface frames. Comparing with the volumetric IoU of Table 6 surfaces the data-fit paradox of Section 5.2 : PINN attains a surface PSNR above

62

dB while its IoU stays at

\sim 0.01

. MSE is scaled by

10^{-4}

Figure 7: Surface-temperature error maps over time for a representative three-defect specimen. NeFTY (Row 2) attains the lowest residual error throughout the decay; Grid Opt. (Row 3) and PINN (Row 4) exhibit structured residuals that persist and diffuse outward, signaling that their reconstructed interiors are inconsistent with the observed boundary thermograms.

Figure 8: Training dynamics of the four PINN variants compared to NeFTY (representative homogeneous specimen, identical $y$ -axis scale across panels). Solid blue: PINN data loss; red: PDE residual; dashed blue: NeFTY data loss. Across all four variants the PDE residual is orders of magnitude larger than the data term and highly noisy; the optimizer drives the data loss down while the diffusivity network saturates at a near-trivial solution. NeFTY’s data loss (dashed) reaches orders of magnitude below all PINN data losses, since the hard solver couples surface fitting and diffusivity recovery directly.

	Homogeneous				Layered
Variant	MSE ( $10^{-4}$ ) $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	IoU $\uparrow$	MSE ( $10^{-4}$ ) $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	IoU $\uparrow$
Base	$127.4\pm 92.2$	$0.47\pm 3.17$	$0.28\pm 0.05$	$0.03\pm 0.02$	$221.6\pm 130.5$	$2.89\pm 1.88$	$0.24\pm 0.05$	$0.02\pm 0.02$
+ PE	$126.1\pm 59.4$	$4.17\pm 1.36$	$0.12\pm 0.02$	$0.09\pm 0.03$	$87.16\pm 29.71$	$6.15\pm 1.11$	$0.13\pm 0.02$	$0.09\pm 0.03$
+ PE, FA	$29.80\pm 4.78$	$8.33\pm 0.59$	$0.16\pm 0.02$	$0.14\pm 0.03$	$44.33\pm 15.94$	$8.81\pm 0.87$	$0.18\pm 0.03$	$0.14\pm 0.03$
+ PE, FA, $\sigma$	$31.43\pm 8.58$	$9.19\pm 0.94$	$0.26\pm 0.03$	$0.18\pm 0.04$	$35.60\pm 6.07$	$9.27\pm 0.82$	$0.24\pm 0.04$	$0.14\pm 0.04$
+ PE, FA, $\sigma$ , HM	$21.01\pm 5.67$	$10.95\pm 0.94$	$0.36\pm 0.03$	$0.24\pm 0.05$	$23.09\pm 4.77$	$11.26\pm 0.84$	$0.33\pm 0.05$	$0.22\pm 0.06$

Appendix

Table 8: Cumulative ablation in full. Each row adds one component to the previous configuration, mean

\pm

95\%

CI across the

1{,}000

-sample synthetic benchmark. The compact summary version is Table 3 in the main body.

Method	Params	Colloc.	Iters	ms/iter	Wall-clock	Peak GPU
PINN [ 44 ]	$4.91$ M	$24{,}576$	$22{,}000$	$128$	$47.0$ min	$10.4$ GB
Causal-PINN [ 56 ]	$4.91$ M	$24{,}576$	$22{,}000$	$144$	$52.7$ min	$10.4$ GB
DCGD [ 25 ]	$4.91$ M	$24{,}576$	$22{,}000$	$159$	$58.3$ min	$11.7$ GB
SPINN [ 8 ]	$2.64$ M	$24{,}576$	$22{,}000$	$49.5$	$18.2$ min	$\phantom{0}3.9$ GB
Grid Opt.	$0.07$ M	—	$10{,}000$	$47.8$	$\phantom{0}8.0$ min	$\phantom{0}1.2$ GB
NeFTY (Ours)	$\mathbf{2.44}$ M	—	$\mathbf{10{,}000}$	$57.5$	$\mathbf{9.6}$ min	$\phantom{0}\mathbf{4.3}$ GB

Appendix

Table 9: Training-level wall-clock and memory per specimen on a single NVIDIA RTX PRO 6000 (96 GB) , the training-level companion to the solver-level Table 4 in the main body. Wall-clock includes the one-time torch.compile overhead (Grid Opt.

\sim 6

s, NeFTY

\sim 11

s). Iterations are reported at convergence or at the saturation plateau; collocation columns apply only to the PINN family.

Figure 9: Mean radial power spectrum of the recovered diffusivity field, averaged across the $1{,}000$ -sample homogeneous benchmark. NeFTY (green) tracks the ground-truth spectrum (black) across all spatial frequencies up to the Nyquist limit; Grid Opt. tracks at low frequencies but drops off above the mid range; the four PINN variants collapse to near-zero power.

Figure 10: Single-defect reconstruction (homogeneous bulk). NeFTY recovers a clean defect outline at the correct depth; Grid Opt. exhibits ringing artifacts; the four PINN variants converge to featureless fields.

Figure 11: Two-defect reconstruction. NeFTY separates two laterally adjacent defects without merging their thermal signatures, while Grid Opt. blurs them together.

Figure 12: Four-defect reconstruction. NeFTY isolates each of four defects at the highest density of the synthetic benchmark; the Sound-Only U-Net ghosts all defects, in line with its zero IoU in Table 6 .

Figure 13: Four-layer composite reconstruction. The bulk diffusivity steps with depth (visible as changing background intensity in the ground-truth row); NeFTY correctly localizes the embedded defects against this stratified background.

Figure 14: Failure mode: shallow defects close to the heat source. NeFTY correctly localizes the central defects but introduces a low-diffusivity artifact along the back-face boundary of the lateral plane.

Method	Abs Rel $\downarrow$	RMSE (mm) $\downarrow$	$\delta<1.25$ $\uparrow$	$\delta<1.25^{2}$ $\uparrow$	$\delta<1.25^{3}$ $\uparrow$
Swin-UNETR [ 21 ]	$3.131$	$1.983$	$0.052$	$0.126$	$0.190$
PPT [ 32 ]	$3.416$	$2.075$	$0.005$	$0.006$	$0.073$
TSR [ 50 ]	$13.366$	$7.655$	$0.002$	$0.002$	$0.002$
SPINN [ 8 ]	$6.737$	$3.913$	$0.002$	$0.002$	$0.004$
Grid Opt.	$0.941$	$0.983$	$0.104$	$0.399$	$0.614$
NeFTY (Ours)	$\mathbf{0.465}$	$\mathbf{0.541}$	$\mathbf{0.385}$	$\mathbf{0.621}$	$\mathbf{0.743}$

Appendix

Table 10: Real PVC 2.5D depth recovery, full metric breakdown. Companion to Table 2 in the main body, reporting RMSE in millimetres and the full

\delta<1.25^{k}

threshold accuracies for

k\in\{1,2,3\}

alongside Abs Rel. NeFTY is best label-free on every metric.

Figure 15: Real PVC reconstruction on a second held-out specimen (companion to Figure 6 ). Row 1: defect masks; Row 2: predicted depth (greyscale, darker is deeper). NeFTY recovers both the defect layout and the depth ordering from the surface IR sequence, in contrast to PPT/TSR which produce coarse depth and SPINN which collapses to a near-uniform field.

Figure 16: Gaussian Diffusion Rate Validation. ( Left ) Temporal evolution of the temperature variance $\sigma^{2}$ along the $x$ and $y$ directions, measured on the surface for a diffusing Gaussian heat source with constant diffusivity $\alpha=0.1$ . Both $\sigma_{x}^{2}(t)$ and $\sigma_{y}^{2}(t)$ grow linearly over time and closely follow the analytical prediction $\sigma^{2}(t)=\sigma_{0}^{2}+2\alpha t$ . ( Right ) Comparison between the measured average slope $\mathrm{d}\sigma^{2}/\mathrm{d}t$ and the theoretical value $2\alpha$ , showing a relative error of $0.16\%$ . This result quantitatively confirms that the proposed simulator reproduces the correct diffusion rate of the heat equation.

Figure 17: Effect of Diffusivity Magnitude on Surface Heat Diffusion. Surface temperature evolution for a defect-free domain under high diffusivity (

\alpha=1.0

, top row) and low diffusivity (

\alpha=0.1

, bottom row), shown at matched time steps. With identical spatial and temporal discretization, the high-diffusivity case exhibits substantially faster spatial spreading and a more rapid decay of peak temperature, while the low-diffusivity case retains a localized, high-contrast heat profile. These results qualitatively illustrate the expected dependence of diffusion dynamics on the thermal diffusivity parameter

\alpha

Figure 18: Surface Temperature Evolution under Constant and Spatially Varying Diffusivity. Surface temperature maps at selected time steps for heat diffusion with constant diffusivity ( $\alpha=0.1$ , top row) and spatially varying diffusivity with an embedded low-diffusivity defect (bottom row). In the homogeneous case, the heat source spreads isotropically and preserves Gaussian symmetry over time. In contrast, the presence of a defect locally impedes heat propagation, leading to asymmetric temperature distributions and delayed diffusion in the defect region (outlined by the dashed circle). These results demonstrate that the simulator correctly captures both uniform and heterogeneous diffusion behavior.

Figure 19: Surface Temperature Profiles under Different Diffusivities. One-dimensional cross-sections of the surface temperature along the

x

direction at the midline (

y=\mathrm{middle}

), comparing high diffusivity (

\alpha=1.0

) and low diffusivity (

\alpha=0.1

). ( Left ) At the initial time step, both cases exhibit identical Gaussian profiles, confirming consistent initialization. ( Right ) At the final time step, the high-diffusivity case shows a significantly broader and lower-amplitude profile, reflecting faster spatial spreading of heat, while the low-diffusivity case retains a sharper peak. These results qualitatively validate the expected dependence of diffusion dynamics on the thermal diffusivity parameter

\alpha

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Intelligent Maintenance and Operations Systems, ´Ecole polytechnique f´ed´erale de Lausanne, Station 18, Lausanne, 1015, Vaud, Switzerland. · Schindler Lab EPFL, Schindler Holding AG, Station 18, Lausanne, 1015, Vaud, Switzerland.

Thermal Imaging 3D Reconstruction Materials Property Prediction Physics-Informed ML Inverse Problems

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Inferring latent physical properties from sensory observations is a fundamental challenge in machine perception. Among available sensing modalities, thermal imaging is particularly promising because temperature evolution is directly governed by heat-transfer physics and therefore encodes information about underlying thermophysical properties of a scene. Recovering spatially resolved thermophysical properties from thermal observations could transform applications ranging from digital twins and infrastructure monitoring to robotics and scientific imaging. However, existing thermal scene reconstruction methods can recover temperature fields in complex 3D environments without identifying the thermophyiscal properties that govern thermal evolution, whereas inverse methods provide physically interpretable parameter estimation but typically rely on simplified geometries and controlled experimental conditions. Here we introduce ThermoField, a framework that unifies thermal scene reconstruction and thermophysical parameter estimation through differentiable heat-transfer simulation. The proposed framework represents these quantities as spatially varying neural fields and constrains them through scene geometry, governing heat-transfer physics, and temporal thermal observations. We demonstrate that ThermoField jointly reconstructs geometry, estimates spatially varying thermal diffusivity, and predicts thermal evolution under previously unseen environmental conditions. By integrating neural scene representations with differentiable heat-transfer solver, the framework enables physically interpretable parameter inference in complex 3D scenes. Our results establish a bridge between thermal scene reconstruction and inverse heat-transfer analysis, providing a unified approach for geometry reconstruction, thermophysical property estimation, and predictive thermal simulation from thermal observations.

Abstract

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