Physics-Informed Surrogates for Temperature Prediction of Multi-Tracks in Laser Powder Bed Fusion

Henning Wessels; Hesameddin Safari

REVIEW 3 major objections 5 minor 36 references

Physics-Informed Surrogates for Temperature Prediction of Multi-Tracks in Laser Powder Bed Fusion

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A track-by-track PINN surrogate predicts multi-track laser powder bed fusion temperatures within 2.5% error.

desk verdict A useful application paper showing sequential per-track PINNs beat an operator network on a small multi-track LPBF benchmark, but the unvalidated FD ground truth means the absolute error numbers are only as good as that baseline. read the letter →

arxiv 2502.01820 v1 pith:KDSXRWF4 submitted 2025-02-03 cs.CE

Hesameddin Safari , Henning Wessels This is my paper · ORCID

classification cs.CE

keywords physics-informedneuralnetworksoperatorlearninglaserpowderbedfusionmulti-trackscanpathstemperaturepredictionsurrogatemodelingDeepONetsequentialPINN

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that multi-track laser powder bed fusion thermal histories can be predicted by a chain of physics-informed neural networks, one trained per track, instead of by a single neural operator trained on entire scan paths. The chained surrogate keeps mean absolute percentage error below 2.5% and melt pool dimension errors below 5% for all six orderings of three parallel tracks, while cutting training time by a factor of 8.5 relative to the enriched DeepONet it is compared against. That matters because scan-path planning is one of the most expensive parts of laser powder bed fusion simulation, and a cheap parametric surrogate opens the door to path optimization and real-time monitoring. The authors verify the surrogates against finite-difference solutions of the three-dimensional heat equation, not against experiments.

What carries the argument

The load-bearing mechanism is the sequential PINN decomposition: a scenario with n tracks is split into n time intervals, a dedicated PINN is trained for every track in every predecessor context (15 networks for n = 3), and the trained networks are stored and recalled in order to assemble any complete path. The initial condition for each track is the end-state of its predecessor, which is what keeps the stitched solution physically consistent. Accuracy near the moving laser comes from adaptive clustering of collocation points around the heat source, so the networks resolve the sharp thermal gradients that dominate melt pool dimensions.

What would settle it

Run the same six three-track paths with a grid-converged finite-difference or finite-element thermal model, or with instrumented single-bead and multi-bead experiments, and recompute the surrogate's MAPE and melt pool errors: if the errors exceed the reported 2.5% and 5% thresholds, the central claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the sequential PINN is a scalable surrogate where the one-shot physics-informed operator fails: when the laser path is treated as an input function, the enriched DeepONet's error grows to a MAPE of 6.81% and melt pool errors of 8-10% on six three-track paths, whereas decomposing each path into tracks and chaining track-level PINNs brings MAPE to 2.49% and melt pool errors below 5%. The chain works because each track network takes the final temperature field of its predecessor as its initial condition, preserving continuity of the thermal history across track boundaries. The same chained model also tracks the cooling and consolidation phase with MAPE below 5% over time, which the authors identify as relevant for microstructure predictions.

Load-bearing premise

The finite-difference solver that generates the reference temperatures is an accurate and converged representation of the true physical process, yet no mesh-convergence, analytical-benchmark, or experimental validation of that solver is reported.

Editorial extensions

If this is right

Scan-path optimization for laser powder bed fusion becomes feasible with a surrogate that holds melt pool errors below 5%, avoiding thousands of full finite-difference or finite-element solves.
Physics-informed surrogates are no longer restricted to single tracks: the sequential formulation extends the parametric idea to path-dependent processes without the accuracy collapse seen in the one-shot DeepONet.
Reusing stored track networks cuts training time from 17 hours to 2 hours for the six-path test case, so adding a new scan order costs only the recall of already-trained track models.
Because the chain resolves the cooling phase, the surrogate can feed microstructure models or support process control decisions that depend on thermal history, not just peak temperature.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A boundary the authors leave implicit: the number of stored PINNs in Eq. (20) grows factorially with the number of tracks, so for many-track parts the benefit of reuse shrinks unless tracks share common predecessors or the approach is paired with transfer learning.
The same chaining idea applies naturally to multi-layer builds: each new layer could start from the previous layer's final temperature field, which is the direct next test of whether error accumulation stays bounded.
A stronger validation the paper does not attempt would compare the chained surrogate's melt pool widths and depths against high-speed imaging or cross-section metallography, since the reference itself is an unvalidated finite-difference solve.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take on arXiv:2502.01820. The genuinely new bit is the multi-track LPBF setting and the sequential-PINN trick: break a path into track stages, train one PINN per stage with the predecessor's end state as initial condition, then chain them. It's simple, and it works for the six three-track paths they test: MAPE under 2.5% and melt-pool dimensions under 5%, while PI-EnDeepONet degrades to 6.8% MAPE and 8-10% melt-pool errors. That comparison is the paper's real contribution—a clear demonstration that the enriched DeepONet does not handle moving-source, multi-track inputs well, and that a staged composition of standard PINNs is a practical alternative.

The methodology is mostly sound. The PINN training is unsupervised with physics losses, so the reported errors are not fits to the benchmark. The material model with temperature-dependent conductivity and apparent heat capacity is standard, and the FD scheme is independent. I think the paper is honest: they state the PI-EnDeepONet failure and acknowledge error accumulation in multi-track sequences.

The soft spots, in order of severity. The finite-difference reference is never validated. No grid sizes or time steps are given, no mesh-convergence study, no comparison to an analytic solution or experimental data. Explicit Euler with a moving Gaussian source and strong temperature-dependent coefficients can be diffusive or under-resolved. So the absolute accuracy numbers (2.49%, etc.) are only as trustworthy as that baseline; if the FD is biased near the melt pool, both surrogates inherit the bias. The relative comparison between the two methods is less affected, but the absolute claims should be softened. Second, there are no multiple-seed runs, so we don't know how stable the PINN results are. Third, the "curse of dimensionality" explanation for PI-EnDeepONet's failure is asserted, not demonstrated with scaling tests. Fourth, the sequential approach grows combinatorially: 15 networks for 3 tracks, and roughly factorial for n tracks. Calling it "scalable" is generous; it works for small n.

Minor: the abstract says "melting and consolidation" but the model is conduction-only with apparent heat capacity—fine as a limitation, but the wording overreaches.

Net: this is a solid application paper for the AM surrogate community. The main comparison is useful and likely reproducible, but the unvalidated benchmark and missing seed variance should be addressed before publication. I would send it to review.

Referee Report

3 major / 5 minor

Summary. This manuscript develops physics-informed surrogates for the three-dimensional temperature field in laser powder bed fusion, with emphasis on multi-track scanning paths. The authors solve the transient heat equation in an unsupervised manner using a PINN and a PI-EnDeepONet, and they propose a sequential PINN that trains one small network per track transition and assembles full multi-track solutions by recalling the trained networks. Single-track results show MAPE below 2% for both architectures; for six three-track paths, the sequential PINN achieves a MAPE of 2.49% and melt-pool errors below 5%, while reducing training time by a factor of 8.5 relative to PI-EnDeepONet. All accuracy claims are measured against a finite-difference solution of the heat equation.

Significance. If the reported accuracy holds, the sequential-PINN idea is a pragmatic contribution: it replaces one hard multi-track operator-learning problem with many small, Markovian track-transition problems, uses no labeled data, and cuts training time substantially for the n=3 case. The training is unsupervised, with physics-based losses, and the FD reference is independent, so there is no circular fitting of the benchmark. The paper also names the error-accumulation limitation of the sequential approach explicitly. However, the single-run nature of the experiments and the absence of any validation of the FD reference mean that the quantitative conclusions should be read as promising demonstrations rather than established accuracy guarantees.

major comments (3)

[Section 2.1, Eq. (5); Tables 2 and 3] The finite-difference solution is the sole reference for all accuracy metrics in Tables 2 and 3, yet the scheme is never validated and its discretization parameters are not reported. Eq. (5) is an explicit Euler update with temperature-dependent conductivity and an apparent-heat-capacity treatment of latent heat; such schemes have restrictive stability limits, and under-resolution near the moving Gaussian source would directly bias the peak temperatures and melt-pool dimensions that the surrogates are compared against. Please report the grid spacings and time step actually used, add a spatial and temporal mesh-convergence study, and validate the converged FD solution against an independent benchmark (an analytical solution for a Gaussian heat source, a well-established FEM code, or experimental thermocouple/IR data). Without this, the MAPE values are errors with respect to a possibly biased numerical baseline rather than to the true thermal field.
[Section 3.3, Tables 2 and 3] All accuracy and training-time comparisons are based on a single training run per model. Physics-informed networks are sensitive to initialization and collocation-point sampling; the differences between PINN (MAPE 1.44%) and PI-EnDeepONet (1.77%) in single-track cases, and the melt-pool errors of 2.5-5%, may be within run-to-run variance. Report results over multiple random seeds, including the mean and standard deviation of MAPE and melt-pool errors, so that the claimed superiority of the sequential PINN over PI-EnDeepONet is statistically supported.
[Section 3.3 and Section 4] The paper attributes the PI-EnDeepONet degradation in multi-track scenarios to the 'curse of dimensionality' but provides no diagnostic evidence. The branch-net input does grow with the number of tracks, but the error increase could equally be caused by insufficient collocation points, branch/trunk capacity, the fixed weighting scheme of Eq. (14), or optimization difficulty. Please add a controlled experiment (for example, varying the input discretization for a fixed path, or keeping the number of tracks fixed while changing the input size) to isolate the cause, and soften the conclusion accordingly if the evidence is inconclusive.

minor comments (5)

[Abstract and Section 4] The abstract and conclusion state that the surrogate analyzes the effect of 'scanning paths and laser parameters,' but only the tool path is varied in the experiments; laser power, scan speed, and spot size are fixed in Table 1. Please restrict the parametric claims to tool-path variation or add experiments that vary laser parameters.
[Section 3.3, Figures 7 and 8] The text refers to 'Fig. 8 illustrates the temperature contours' when the contour plot is actually Figure 7; the temperature-history plot is Figure 8. Correct the cross-references.
[Table 1 and Eq. (5)] Table 1 lists a convection heat transfer coefficient of 10 W/(m2K), but the governing equation in Eq. (1) and the FD update in Eq. (5) neglect convective losses. Please clarify how the convective boundary condition is imposed in the finite-difference solver and in the PINN loss terms, or remove the unused parameter.
[Section 2.3, Eq. (20)] The number of required PINNs in Eq. (20) grows like the sum of falling factorials, exceeding the number of scenarios for n>2. The word 'scalable' in the conclusion should therefore be qualified as 'scalable relative to PI-EnDeepONet for the n=3 case' rather than suggesting that the approach extends to large multi-track problems without further changes.
[General] No code or data are provided for the finite-difference reference or the trained networks. Releasing the FD solver and the training configuration would materially aid reproducibility, given that all benchmark metrics depend on the reference implementation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the surrogate claims rest on unsupervised physics-based training and an independent finite-difference comparison, not on fitted inputs or self-citations.

full rationale

Score 0. The central claims are not circular. The PINN and PI-EnDeepONet are trained without labeled finite-difference data: their losses (Eqs. 7-13) enforce initial conditions, boundary conditions, and PDE residuals, and the loss weights (Eq. 14) are computed from initial loss values rather than from any benchmark output. The finite-difference solver of Eq. (5) is used only as a post-training validation baseline, and the MAPE in Eq. (22) is an evaluation metric applied after training, not a training objective. The sequential PINN decomposes multi-track scenarios into per-track problems whose initial conditions are inherited from previously trained networks; this is a compositional construction, not a fit to the reference data. There is no load-bearing self-citation chain: the author-overlapping references ([1], [2], [25]) support peripheral points about POD limitations, data-assisted training, and model-calibration reviews, while the material properties and adaptive clustering are taken from the external reference [11]. The acknowledged limitations, namely the unvalidated finite-difference baseline and possible error accumulation across tracks, are validation and correctness concerns rather than circularity. No specific equation or fitted parameter was found to reduce to the paper's own inputs, so no circular step is identified.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The model rests on standard heat conduction assumptions and a heat-source parameterization borrowed from prior work. No new physical entities are introduced. The free parameters are network scaling choices, not physical constants fit to data.

free parameters (2)

Network architecture hyperparameters = 5 hidden layers x 64 neurons (full/trunk), branch 2x64, sine(pi*beta*x)
Chosen by hand and standard for PINNs; they influence accuracy but are not fitted to the physical data.
Loss weighting factors lambda_IC, lambda_BC, lambda_PDE = Set from initial loss values via Eq. (14)
These normalize the loss terms; they are computed from the initial state of training, not from target outputs, but they are still manually specified scaling choices.

assumptions (4)

domain assumption Heat equation with temperature-dependent conductivity and a moving Gaussian volumetric heat source (Eqs. 2 and 4) models the LPBF thermal process.
Neglects convection, radiation, melt pool flow, and evaporation; this model is assumed accurate for the surrogate target.
domain assumption Apparent heat capacity cp(T) in Eq. (21) accounts for latent heat of fusion.
This smooths phase change into a modified heat capacity and is taken from prior literature without independent calibration.
domain assumption Explicit Euler finite-difference solution (Eq. 5) is a converged and accurate reference.
No convergence study or comparison to experiments or analytical solutions is provided; if the reference is inaccurate, all reported errors are invalid.
domain assumption Dirichlet and Neumann boundary conditions (Eq. 3) fully describe the domain boundaries.
The convection coefficient listed in Table 1 does not appear in the loss terms, so the boundary model is incomplete or inconsistent.

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Cite this review

Pith. "Pith review of Physics-Informed Surrogates for Temperature Prediction of Multi-Tracks in Laser Powder Bed Fusion." pith.science (2026). https://pith.science/paper/KDSXRWF4

@misc{pith2026250201820,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Surrogates for Temperature Prediction of Multi-Tracks in Laser Powder Bed Fusion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDSXRWF4}},
  note         = {Machine review of arXiv:2502.01820}
}

read the original abstract

Modeling plays a critical role in additive manufacturing (AM), enabling a deeper understanding of underlying processes. Parametric solutions for such models are of great importance, enabling the optimization of production processes and considerable cost reductions. However, the complexity of the problem and diversity of spatio-temporal scales involved in the process pose significant challenges for traditional numerical methods. Surrogate models offer a powerful alternative by accelerating simulations and facilitating real-time monitoring and control. The present study presents an operator learning approach that relies on the deep operator network (DeepONet) and physics-informed neural networks (PINN) to predict the three-dimensional temperature distribution during melting and consolidation in laser powder bed fusion (LPBF). Parametric solutions for both single-track and multi-track scenarios with respect to tool path are obtained. To address the challenges in obtaining parametric solutions for multi-track scenarios using DeepONet architecture, a sequential PINN approach is proposed to efficiently manage the increased training complexity inherent in those scenarios. The accuracy and consistency of the model are verified against finite-difference computations. The developed surrogate allows us to efficiently analyze the effect of scanning paths and laser parameters on the thermal history.

Figures

Figures reproduced from arXiv: 2502.01820 by the authors.

**Figure 2.** Schematic representation of the En-DeepONet with physics-informed loss functions. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** (a) Outline of multiple tracks on a workpiece. (b)Tree structure representation of breaking [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Schematic representation of the computational domain and boundary conditions. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Comparison of the calculated temperature and meltpool dimensions for all single-track sce [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Comparison of the calculated temperature and meltpool dimensions for all multi-track scenarios [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: Comparison between the PINN predictions and finite-difference solutions for all paths in both [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: (a) Comparison of the temperature history at the center of the meltpool during the cooling [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

Discussion (0). Continue with ORCID to comment.

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