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

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

Source-linked AI summary

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

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

arXiv:2603.11045v2Mar 11, 2026cs.LGcond-mat.mtrl-scics.AIcs.CVphysics.ins-det

TL;DR

Stiff parabolic inverse problems lose high-frequency interior information at the boundary, and soft PINNs can fit measurements without recovering volumetric structure. NeFTY embeds a continuous diffusivity neural field in a hard-constrained differentiable heat solver and uses adjoint gradients; it outperforms soft PINNs, voxel grids, and classical thermography baselines across synthetic and real-data evaluations.

Problem
Inverse heat conduction is severely ill-posed because diffusion damps interior information, while soft PINN gradients can fit boundary data without recovering the underlying field.
Method
NeFTY represents diffusivity with a continuous coordinate-based neural field, enforces the discretized heat equation through an implicit-Euler solver, and propagates surface-error gradients with an adjoint.
Results
NeFTY substantially outperforms soft PINNs and a voxel-grid baseline on synthetic volumetric recovery and surpasses classical thermography baselines on real defect segmentation and depth estimation.
Takeaways & Limitations
The framework enables label-free 3D inverse heat conduction from surface thermal sequences with test-time inversion tractable on a single GPU.
Takeaways & Limitations
NeFTY requires per-specimen optimization, has higher latency than supervised feedforward inversion, and lacks full 3D ground-truth evaluation on public NDT benchmarks.

Abstract

from arXiv · show

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/

1 Introduction

Pulsed thermography makes 3D diffusivity recovery difficult because heat diffusion suppresses interior information and soft PINNs can fit surface data without recovering the field. NeFTY addresses this with a hard-constrained neural field, differentiable heat solver, and adjoint optimization.

Motivation: Pulsed thermography records transient surface temperatures after laser heating to infer subsurface thermal diffusivity.The setup targets label-free volumetric reconstruction from surface-only measurements.
Motivation: Parabolic diffusion rapidly damps interior features with depth, making small surface perturbations compatible with large interior variations.This creates a severely ill-posed inverse map.
Prior limitations: Classical thermography methods enhance defect contrast but reduce the problem to 1D pixel-wise inversions that ignore lateral diffusion.This biases estimates for shallow or low-aspect-ratio defects.
NeFTY: NeFTY represents diffusivity with a continuous coordinate-based network and enforces the heat equation through a differentiable implicit-Euler solver with harmonic-mean flux.The PDE is enforced exactly on the discretization rather than through a soft residual penalty.
Results: NeFTY uses adjoint gradients at solver-level memory cost and outperforms soft PINNs and voxel-grid optimization on synthetic volumetric recovery and real thermography tasks.Synthetic IoU reaches 0.45 and 0.37, while real PVC data show improved defect segmentation and depth recovery over classical baselines.

2 Related Work

Thermographic signal-processing methods improve contrast or estimate depth but rely on simplified pixel-wise or pseudo-wave views. NeFTY instead combines label-free volumetric inversion with a continuous neural-field representation and differentiable physics.

Classical thermography: TSR, PPT, and VWC transform surface temperature decays to enhance contrast, suppress nuisance variation, or estimate depth.These methods remain signal-processing approaches to thermographic NDT.
Classical thermography: Classical thermography pipelines reduce reconstruction to 1D pixel-wise inversions that ignore lateral diffusion and can bias defect estimates.The limitation is especially relevant for shallow or low-aspect-ratio defects.
Physics-informed methods: Soft PINNs formulate volumetric inversion with PDE residual penalties, while variants modify architecture, temporal weighting, or gradient balancing.The related variants include SPINN, Causal-PINN, and DCGD.
Neural fields: Coordinate-based neural fields use MLPs over spatial coordinates with positional encoding to represent continuous, potentially high-frequency signals.NeFTY applies this representation to thermal diffusivity reconstruction.

3 Forward Model, Ill-Posedness, and Soft-Constraint Pathology

The heat equation maps diffusivity to surface observations through a compact, smoothing operator, making inversion ill-posed and noise-sensitive. Soft PINNs further decouple surface data from diffusivity updates because the temperature surrogate is parameterized independently.

Forward model: The transient heat model describes temperature evolution in a bounded 3D domain with diffusivity, material properties, source, and boundary observations.The inverse problem seeks diffusivity from noisy surface measurements at selected times.
Forward model: NeFTY uses a single effective diffusivity field with harmonic-mean face coefficients to preserve effective-flux continuity across interfaces.This avoids representing singular interface flux jumps directly on the finite-volume grid.
Ill-posedness: The IHCP is severely ill-posed because the parameter-to-observation derivative is compact and its singular values decay algebraically.On a slab, Proposition 2 gives σn ≲ n^-1/3.
Ill-posedness: The pseudo-inverse amplifies noise along higher singular directions, while heat-kernel and spectral smoothing suppress interior structure before it reaches the boundary.These mechanisms explain why distinct interiors can produce nearly indistinguishable surface profiles.
Soft-constraint pathology: In standard PINNs, surface-data and initial-condition gradients with respect to diffusivity parameters vanish because temperature and diffusivity use separate networks.The diffusivity update therefore depends on the PDE residual and the under-fitted temperature Laplacian.
Soft-constraint pathology: This structural decoupling can leave the PDE loss small while diffusivity remains near a trivial constant, a failure observed across modern PINN variants.NeFTY removes the soft constraint so surface data reaches diffusivity through the discretized heat operator.

4 Method

NeFTY parameterizes diffusivity as a bounded coordinate neural field and embeds it directly in a finite-volume implicit-Euler solver. Surface reconstruction errors are propagated through the solver with a discrete adjoint whose backward sweep uses constant memory in time.

Core framework: NeFTY directly minimizes reconstruction error while enforcing the discretized heat equation as a hard constraint at every optimization step.The discrete temperature is determined by the candidate diffusivity field, preventing the PINN gradient decoupling.
Neural field: A coordinate MLP with Fourier features, frequency annealing, bounded output, and total variation represents the effective diffusivity field.These choices support sharp interfaces, stabilize high-frequency optimization, and constrain diffusivity to a physical range.
Heat solver: The bulk equation is discretized on a Cartesian grid with harmonic-mean face diffusivities and boundary conditions closing the finite-volume stencil.The harmonic mean preserves effective-flux continuity at high-contrast interfaces.
Heat solver: Implicit Euler advances temperature through a sparse linear system whose coefficients depend on network parameters through diffusivity.GPU Jacobi iterations implement the solve as stencil convolutions within autodiff.
Adjoint optimization: The adjoint introduces backward variables using the transpose of the forward sparse system to assemble parameter gradients from surface reconstruction error.Only adjacent states are required, giving constant memory in the number of time steps.

5 Experiments

NeFTY is evaluated against grid, PINN, classical, and learned baselines on synthetic and real thermography tasks, including reconstruction quality, ablations, and solver efficiency. It achieves stronger volumetric and real-data recovery while substantially reducing adjoint memory use.

5 Experiments: The benchmark uses synthetic 3D data with 1,000 samples, classical and neural baselines, and metrics spanning reconstruction, segmentation, and depth.Real PVC evaluation uses 2D masks and 2.5D depth annotations rather than 3D volumetric labels.
Synthetic 3D reconstruction: Qualitative reconstructions show NeFTY isolating defects at correct depths in homogeneous settings and recovering defects against stratified backgrounds in layered settings.Grid Opt. produces noisy or ringing fields, while PINN variants collapse toward featureless or near-uniform diffusivity.
Synthetic 3D reconstruction: NeFTY reaches IoU 0.45/0.37 and PSNR 18.48/15.88 dB, while PINN variants remain at IoU ≤0.018 and Grid Opt. caps IoU at 0.04/0.03.These results cover homogeneous/layered synthetic settings and show the advantage of combining a hard solver with a continuous neural-field prior.
Real-world PVC validation: On real PVC data, NeFTY is the best label-free method on every metric, achieving 0.431 IoU and 0.597 Dice for segmentation and 0.465 Abs Rel with 0.385 δ < 1.25 accuracy for depth.PPT and TSR remain competitive on segmentation but degrade sharply on depth because their one-dimensional semi-infinite-geometry assumptions do not hold on the finite slab.
Ablations: The ablation improves IoU from 0.03 to 0.14 with positional encoding and frequency annealing, while the TV regularizer raises it to 0.45/0.37.The bounded output prevents αθ →0 drift, and harmonic-mean flux restores continuity at high-contrast interfaces.
Solver efficiency: The discrete adjoint matches Autograd accuracy while reducing peak GPU memory from 18.63 GB to 21.9 MB and halving backward time.Its forward pass is also approximately 7× faster than PhiFlow’s implicit solver.

6 Conclusion

NeFTY formulates 3D inverse heat conduction as a hard-constrained neural-field optimization problem, with interface handling tied to the effective-diffusivity model. The framework achieves volumetric recovery while acknowledging latency, high-contrast attenuation, and limited 3D ground-truth evaluation.

6 Conclusion: NeFTY parameterizes diffusivity with a continuous neural field and enforces the PDE through a differentiable implicit-Euler solver with adjoint gradients.The supplied conclusion states that the solver uses harmonic-mean interface flux and operates at solver-level memory cost.
6 Conclusion: NeFTY is limited by per-specimen optimization latency, attenuated recovery inside very high-contrast voids, and evaluation based on projected 2D or 2.5D supervision.The paper identifies amortized initialization, preconditioned differentiable solvers, and paired computed-tomography labels as possible remedies.
6 Conclusion: The governing conservation and diffusivity forms are equivalent when ρCp is positive and continuously differentiable, but piecewise-constant heat capacity creates an interface-supported correction.The reduction uses α := k/(ρCp), while the piecewise-constant case produces a distributional gradient supported on the interface.
6 Conclusion: The method solves the bulk equation on regions excluding interfaces and uses harmonic-mean flux continuity for the single-effective-diffusivity model.The distributional correction is not represented directly on the finite-volume grid; the discrete model instead preserves effective α-flux continuity.

A.3 Proof of Proposition 1 (Harmonic mean as discrete interface continuity)

The harmonic-mean stencil follows from enforcing continuity of the effective α-flux across a cell face. This construction extends directionally to the finite-volume operator used by NeFTY and preserves appropriate behavior across high-contrast interfaces.

A.3 Proof of Proposition 1: Continuity of the effective α-flux across two half-cells determines the face flux and face temperature simultaneously.The derivation equates the left- and right-half-cell fluxes and eliminates the unknown face temperature.
A.3 Proof of Proposition 1: The resulting interface diffusivity is the unique harmonic mean, ¯αi+1/2 = 2αiαi+1/(αi + αi+1).Uniqueness follows from the non-degenerate linear system for flux and face temperature.
A.3 Proof of Proposition 1: The same construction along principal directions defines NeFTY’s second-order finite-volume diffusion operator in two and three dimensions.The operator acts on a discrete temperature field over a uniform Cartesian grid.
A.3 Proof of Proposition 1: The harmonic mean throttles flux when neighboring diffusivities differ strongly and is exact for piecewise-constant diffusivity at cell faces.The supplied passage contrasts this with the arithmetic mean, which is first-order accurate and overestimates high-contrast interface flux.
A.3 Proof of Proposition 1: Synthetic and real PVC experiments use the same forward operator and harmonic-mean stencil while adapting scaling, excitation, and boundary conditions.The real PVC setting uses Fourier-number rescaling and a convective Robin back-face condition, unlike the synthetic regime’s assumptions.

B.2 Proof of Proposition 2 (Compactness and algebraic singular-value decay)

The linearized boundary observation operator is compact because parameter perturbations pass through a compact H1-to-L2 embedding before boundary observation. On the slab, Weyl asymptotics yield algebraic singular-value decay, explaining the inverse problem’s instability.

B.2 Proof of Proposition 2: The proof factors the Fréchet derivative into a bounded boundary-observation map and a compact Rellich–Kondrachov embedding.The four-step argument combines energy estimates, a parabolic trace inequality, finite-rank approximation, and slab Weyl asymptotics.
B.2 Proof of Proposition 2: Compactness follows because truncating the parameter perturbation produces a small output through the bounded auxiliary map.The argument truncates δα before it enters the linearized PDE, avoiding mode-mixing concerns from multiplication by ∇Tα0.
B.2 Proof of Proposition 2: On the three-dimensional slab, singular values satisfy σn ≲ n^-1/3 by Weyl’s law.The slab eigenvalues scale as λn ≍ n^2/3, which yields the stated algebraic rate.
B.2 Proof of Proposition 2: Restricting to mean-zero perturbations and using the gradient norm recovers the algebraic rate ν = 1/2.The constant eigenmode corresponds to a uniform diffusivity shift that only rescales time.
B.2 Proof of Proposition 2: Because singular values tend to zero, the Moore–Penrose inverse is unbounded and measurement perturbations can be amplified in recovered diffusivity.The paper therefore characterizes the forward problem as well-posed but the inverse problem as failing Hadamard continuous dependence.
B.2 Proof of Proposition 2: Pointwise heat-kernel and semigroup smoothing can be exponential, whereas time-integrated boundary observations yield the algebraic operator bound.The distinction depends on whether the observation is taken over an interval or only at a single final time.
B.2 Proof of Proposition 2: Regularization remains mandatory, and NeFTY combines a continuous neural-field prior with additional smoothness structure for admissible diffusivity fields.The supplied passage links this need directly to the singular-value decay of the inverse problem.

C.1 Derivation of the gradient decoupling

The derivation shows that soft-PINN surface and initial-condition losses have no direct gradient path to the diffusivity network. Consequently, diffusivity learning depends on residual gradients shaped by the under-fitted temperature field and the ill-posed forward map.

Gradient decoupling: ∇θLdata = 0 and ∇θLIC = 0 because the temperature surrogate Tϕ is parameterized independently of diffusivity parameters θ.Surface observations therefore influence αθ only through the PDE residual.
Residual pathway: The residual-gradient magnitude is controlled by spatial derivatives of Tϕ, dominated by ∥∆Tϕ∥ when ∇αθ is bounded.For an under-fitted temperature field, second-order spatial derivatives are noisy and large.
Scope of the derivation: Surface observations cannot reach θ except indirectly through residual gradients, so the derivation does not itself prove that αθ collapses to a constant.The constant-field failure is presented as an empirical outcome consistent with this structural channel, not as a theorem of the decoupling alone.
Ill-posedness: The ill-posed parameter-to-observation map amplifies high-frequency modes through an unbounded pseudo-inverse, after the forward smoothing has already attenuated defect information.In soft PINNs, data first updates Tϕ and only then reaches αθ through the residual.
Standard remedies: GradNorm, SPINN, Causal-PINN, and DCGD modify weighting, architecture, temporal emphasis, or gradient projection without creating a data-side gradient on θ.Their parameterizations preserve the structural decoupling.
NeFTY coupling: NeFTY removes the decoupling by making T the implicit PDE solution driven by αθ, yielding the dependency chain θ → αθ → T → Ldata.The discrete heat-operator adjoint supplies a non-vanishing gradient from surface data to θ.

D Method Details: Neural-Field Architecture, Forward Solver, and Adjoint Gradients

NeFTY combines a coordinate-based diffusivity network with a discretized, differentiable heat solver and adjoint optimization. Its architecture constrains diffusivity physically, while the solver enforces the heat equation through implicit Euler and harmonic-mean fluxes.

Neural diffusivity field: The diffusivity field uses a coordinate MLP with ReLU activations and a skip connection that re-injects encoded coordinates.The network maps spatial coordinates to the continuous diffusivity field.
Neural diffusivity field: Positional-encoding bandwidth N controls expressible spatial frequency, improving sharp-interface resolution while increasing high-mode noise amplification.The method therefore balances spatial detail against inverse-problem instability.
Neural diffusivity field: Frequency annealing progressively unmasks encoding bands with a cosine schedule, allowing high-frequency detail to emerge after low-frequency structure.The schedule avoids abrupt gradient bursts when new bands activate.
Neural diffusivity field: A bounded sigmoid output enforces positive diffusivity and limits contrast, helping keep the implicit system well-conditioned.The bounds also prevent early optimization from probing extreme diffusivity values.
Forward solver: The finite-volume diffusion operator uses harmonic-mean face coefficients, implementing effective-flux continuity on the Cartesian grid.Periodic lateral and through-thickness adiabatic or Robin boundary conditions are encoded in the stencil.
Forward solver: Implicit Euler advances each temperature frame by solving a sparse linear system, providing unconditional stability so the timestep follows camera sampling rather than a CFL bound.The system is solved with a fixed unrolled Jacobi iteration using a warm start and a 7-point stencil.
Forward solver: The harmonic-mean construction makes the diffusion operator symmetric, while Robin sinks yield negative definiteness and strict diagonal dominance of the implicit system.These properties support stable iterative solution under the stated boundary conditions.

E.1 Synthetic dataset generation

The synthetic benchmark uses independently generated 3D heat-transfer data, standardized reconstruction metrics, and matched-compute comparisons against optimization, PINN, neural, and classical thermography baselines. Real-data evaluation uses projected 2D masks and 2.5D depth annotations rather than volumetric labels.

Dataset generation: Synthetic ground truth is generated with PhiFlow, an independent finite-volume engine, to avoid an inverse crime against NeFTY’s solver.PhiFlow uses an explicit scheme with adaptive substepping to satisfy fine-grid stability constraints.
Dataset generation: The benchmark uses a unitless slab at 64×64×16 resolution, records 100 frames with ∆t = 0.05, and includes one to four subsurface defects per sample.Lateral boundaries are periodic and through-thickness boundaries are adiabatic.
Dataset generation: The synthetic defect-to-bulk diffusivity ratio is roughly 1 : 20, selected for numerical stability because a 1 : 1000 contrast severely ill-conditions the solver.At 1 : 20, the surface-temperature signature already saturates near the perfect-insulator limit.
Evaluation: Reconstruction is evaluated with slice-averaged SSIM and thresholded defect IoU, with metrics averaged over the full 1,000-sample benchmark.IoU uses defect threshold τ = 0.03.
Evaluation: Frequency-domain diagnostics include defect-edge F1 and a radially averaged 3D power spectrum centered on the defect cluster.The edge metric uses a 50%-of-maximum gradient threshold within a dilated defect-boundary mask.
Baselines: Synthetic comparisons include Grid Opt., four PINN variants, and Sound-Only U-Net, while real-data comparisons add PPT, TSR, and zero-shot Swin-UNETR.The baselines span differentiable optimization, physics-informed learning, data-driven inversion, and classical thermography.
Baselines: PPT and TSR estimate depth from frequency-phase and log-time polynomial transforms, respectively, providing classical signal-processing references.PPT uses dominant phase frequency, whereas TSR uses the peak of the second log-time derivative.

G.1 Robustness across defect count and layer count

Across defect-count and layer-count slices, NeFTY preserves label-free volumetric recovery as scene complexity increases, while PINN and voxel-grid baselines remain substantially weaker. Its sharp-boundary recovery and frequency content support this robustness, although shallow defects can induce a depth-relevant back-boundary artifact.

Quantitative robustness: NeFTY maintains IoU between 0.34 and 0.51 across defect-count and layer-count slices, while Grid Opt. falls below 0.07 and PINN remains near 0.02.SPINN recovers IoU ≤0.004 in every slice, despite being the strongest PINN variant on aggregate SSIM.
Surface-versus-volume recovery: A PINN can achieve 43.89 × 10−4 surface MSE and 63.04 dB PSNR yet only 0.01 volumetric IoU, whereas NeFTY reaches 0.50×10−4 MSE, 82.33 dB PSNR, and correct recovery.The comparison exposes the data-fit paradox: boundary agreement does not ensure a physically consistent interior for soft constraints.
Ablation: NeFTY’s final TV ablation more than halves diffusivity MSE and nearly doubles IoU on both homogeneous and layered configurations.The full ablation reports mean ± 95% confidence intervals over 1,000 synthetic samples; later components progressively tighten variability.
Optimization cost: NeFTY converges in 9.6 minutes per specimen, compared with 8.0 minutes for Grid Opt. and 18.2–58.3 minutes for PINN variants.The PINN baselines received 2.2× more iterations and twice the parameter count; NeFTY uses 4.3 GB peak GPU memory.
Frequency-domain diagnostics: NeFTY achieves Edge F1 0.470 ± 0.137 and tracks the ground-truth radial power spectrum to the Nyquist limit, unlike featureless PINN interiors.Grid Opt. falls off above mid frequencies, while all four PINN variants have zero Edge F1 and near-zero spectral power.
Qualitative robustness and failure mode: Qualitatively, NeFTY separates adjacent defects, isolates four defects, and localizes defects in layered backgrounds, but shallow cases can produce a spurious back-boundary low-diffusivity artifact.The artifact does not affect 2D segmentation but propagates into the depth metric for shallow defects.

H.4 Real PVC depth, full metric breakdown

The simulator is validated against analytical Gaussian diffusion and qualitative diffusivity-dependent behavior before real PVC reconstruction results are reported. These checks show correct diffusion rates, faster spreading at higher diffusivity, and defect-induced asymmetry in surface temperatures.

Real PVC reconstruction: NeFTY recovers defect layout and depth ordering on a held-out PVC specimen, whereas PPT/TSR produce coarse depth and SPINN collapses toward a near-uniform field.Figure 15 presents defect masks in the first row and predicted depth in grayscale in the second row.
Qualitative validation: Uniform diffusivity produces isotropic, Gaussian-symmetric spreading, while a low-diffusivity spherical defect locally impedes propagation and creates an asymmetric surface temperature field.These qualitative behaviors match the expected physical interpretation of homogeneous and spatially varying diffusivity.
Diffusivity sensitivity: Increasing diffusivity from α = 0.1 to α = 1.0 causes faster spatial spreading and lower peak temperatures under identical discretization.The low-diffusivity case retains a more localized, high-contrast heat profile.

J Limitations and Future Work

NeFTY’s main limitations concern test-time optimization cost, high-contrast quantitative diffusivity recovery, and the absence of full 3D labels for real PVC data. The paper identifies solver, acquisition, and benchmarking directions for future work.

Test-time optimization cost: NeFTY requires approximately 10 minutes per specimen on a single high-end GPU, limiting its use in high-throughput manufacturing lines.The paper suggests amortized inference through meta-learning or hypernetworks as a follow-up.
High-contrast quantitative recovery: At realistic air-to-solid diffusivity ratios above 1:1000, the differentiable linear solve becomes severely ill-conditioned, making precise α recovery inside insulating voids difficult.The paper suggests preconditioned or multigrid solvers to extend the method to stiff air voids.
Real-data supervision: Real PVC datasets provide only 2D defect masks and discrete depths, so evaluation uses projected 2D and 2.5D labels rather than direct volumetric diffusivity error.Full 3D ground truth would require destructive testing or X-ray CT and remains a future benchmark direction.