A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications

Keith D. Humfeld; Navid Zobeiry

REVIEW 4 major objections 5 minor 1 cited by

A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications

T0 review · 4 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read A physics-informed neural network can solve the heat equation with convective boundaries and predict heat transfer beyond its training window.

desk verdict A solid applied PINN paper with one unquantified load-bearing extrapolation claim; worth refereeing, but not citable until that claim is checked. read the letter →

arxiv 2010.02011 v1 pith:7K6UQKFZ submitted 2020-09-28 cs.LG

Navid Zobeiry , Keith D. Humfeld This is my paper · ORCID

classification cs.LG

keywords physics-informedneuralnetworksheattransferequationconvectiveboundaryconditionsfeatureengineeringseparationofvariablesfiniteelementvalidationmanufacturingprocesssimulationnearreal-timesurrogatemodels

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 proposes a physics-informed neural network (PINN) as a fast surrogate for solving the heat conduction equation with convective boundary conditions, the kind of problem that arises when composite parts are heated in ovens. The central claim is that by building the loss function from the PDE, boundary conditions, and initial condition, and by engineering the first layer to mimic the separation-of-variables form $e^{at}\sin(bx)$, the trained network can predict temperatures accurately even beyond the time range it was trained on. That extrapolation matters because one trained model can then answer many boundary-condition scenarios in near real time, replacing repeated finite element simulations. The claim is supported by 1D and 2D comparisons with finite element results, and by a comparison showing that a plain neural network without engineered features fails to extrapolate.

What carries the argument

The load-bearing mechanism is the engineered feature layer: two pre-layers, one applying a trainable exponential activation to the time input and one applying a trainable sine activation to the position input, are multiplied term-by-term to produce features of the form $e^{at+a_0}\sin(bx+b_0)$. This mirrors the analytic separation-of-variables solution of the heat equation without convection, a sum of exponentially decaying sinusoidal modes, so the network only needs to learn weights and biases rather than an unstructured representation. It is this structure that lets predictions continue past the training time window; the paper contrasts it with a plain dense network that matches finite elements inside the training zone but diverges outside it. The adaptive normalization scheme and the ELU (exponential linear unit) activation function are supporting machinery that make training on PDE residuals practical.

What would settle it

Train the same PINN on the first 15 minutes of a multi-ramp air temperature profile and compare its 15-to-30 minute prediction with a finite element reference; if the discrepancy grows with the number of air-temperature ramps or with temperature-dependent conductivity, the beyond-training-zone claim is restricted to the simple constant-property, single-ramp setting.

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Extended reading notes

Core claim

On its own terms, the discovery is that a PINN whose first layer is built from terms of the form $e^{at+a_0}\sin(bx+b_0)$ can represent the solution of the heat equation with convective boundary conditions well enough to predict heat transfer beyond the training zone. The network is trained with no pre-generated solution data; the loss is the mean squared residual of the heat equation, the two convection boundary equations, and the initial condition, with adaptive normalization factors that keep the four loss terms balanced. Once trained with the heat transfer coefficients $h_1$ and $h_2$ as inputs, the same network evaluates any combination of boundary conditions almost instantly. The paper demonstrates this for a 10 mm, a 20 mm, and a 30 mm composite part in 1D, and for a 60 mm by 20 mm part in 2D, with maximum deviation under 0.97 $^\circ$C in the reported 1D case.

Load-bearing premise

The load-bearing premise is that the temperature field is well described by a sum of exponential-in-time times sinusoidal-in-space modes, so the engineered feature layer only extrapolates reliably when material properties are constant and the air-temperature and convection conditions stay within the family represented by the training set.

Editorial extensions

If this is right

A trained PINN gives a temperature solution for any combination of the two convective heat transfer coefficients in near real time, removing the need to rerun a finite element analysis for each new boundary condition.
The same architecture extends to 2D and 3D by adding position inputs and corresponding PDE and boundary residuals, at the cost of more training epochs and larger batches.
Material properties such as thermal diffusivity and conductivity can be added as inputs, so one model could potentially cover multiple materials.
The air temperature profile is encoded in the training procedure, so a new profile requires a new model unless profile parameters are added as inputs.
Because the network is trained on pointwise residuals rather than fixed data, it can be evaluated at any spatial location, not just at finite element nodes.

Reading between the lines

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

Going beyond the paper: the feature-engineering idea of embedding the dominant analytic modes of a parabolic PDE into the first layer should carry over to other diffusion-type problems, such as moisture diffusion or neutron diffusion, whenever the eigenfunction family is known.
Not claimed in the paper is the inverse use: because the PINN maps $h_1,h_2$ to temperatures, feeding it measured thermocouple data could identify effective heat transfer coefficients in an oven in near real time; the paper only gestures at feedback loops.
The beyond-training claim is demonstrated for time extrapolation; a natural extension would test whether the engineered features also extrapolate in the boundary-condition dimension, for example $h$ values far outside the training range, where the separation-of-variables structure may offer less protection.
Because the feature layer encodes constant-property physics, the method's extrapolation should degrade for temperature-dependent conductivity or phase change; the paper does not test that regime.

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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

Read this if you work on PINNs as fast surrogates for process simulation. The genuinely new part is narrow: engineered feature layer using separation-of-variables modes exp(at)sin(bx), h1/h2 as network inputs, and an adaptive loss-weighting scheme. The 1D validation against FE is concrete and quantitative (max deviation 0.97 C). The loss setup—PDE residual plus convective BC residuals—is standard PINN and is internally consistent.

Where it gets soft: the paper's central claim, that engineered features enable prediction beyond the training zone, rests on Figure 7, a qualitative comparison with no error metric in the extrapolated interval. The stress-test note is right that the feature layer does not force the full network to stay in the modal solution family; with six ELU hidden layers, the trained function can do anything between training points, and nothing in the loss constrains t>15 min. So this claim is plausible but load-bearing and unquantified. The 2D result has no error metrics. The adaptive normalization scheme is described heuristically (threshold 0.01, update every 100 epochs), which is fine as engineering, but it is not analyzed. No code or data is provided, so the numerical results can't be independently checked.

What is not a problem: the PDE and BC residuals are correctly written, the FE comparison is reasonable, and the limitation statement is honest—it explicitly says a new model is needed for each air-temperature profile, which narrows but also sharpens the contribution. The citation pattern is normal, with appropriate credit to Raissi et al.

Bottom line: this is a plausible application paper for manufacturing/PINN practitioners, not a methodological breakthrough. It deserves a serious referee, because the industrial use case is real and the quantitative 1D result is there, but acceptance should be conditional on supplying the extrapolation errors, 2D metrics, and code/data. I would not cite it yet; the headline claim needs to survive an actual check.

Referee Report

4 major / 5 minor

Summary. The paper presents a physics-informed neural network (PINN) for solving one- and two-dimensional conductive heat transfer problems with convective boundary conditions, motivated by oven heating in manufacturing. The loss function combines residuals of the heat equation, the convective boundary conditions, and the initial condition, with an adaptive normalization scheme to balance the loss terms. The authors also engineer a first layer of features of the form exp(at) sin(bx) suggested by the separation-of-variables solution of the no-convection heat equation. They compare PINN predictions with finite element results for 1D and 2D cases, report a maximum deviation of 0.97 degrees Celsius in the 1D training-window validation, and claim that the engineered features enable prediction beyond the training time interval. The paper further demonstrates that a PINN taking heat transfer coefficients as inputs can evaluate multiple boundary-condition scenarios at low computational cost.

Significance. If the central claim holds, the contribution is a useful fast surrogate for linear heat conduction with constant material properties and convective boundary conditions, relevant to process control and Industry 4.0 applications. The training is genuinely physics-informed in the sense that the loss is defined from PDE/BC/IC residuals rather than from precomputed target outputs, so the validation against external finite element results is not circular. The quantitative 1D result (maximum deviation 0.97 C against FE) is a concrete strength. However, the paper's headline capability, extrapolation beyond the training zone, currently rests on a single qualitative figure with no error metric, and the proposed mechanism does not by itself guarantee the claimed temporal generalization.

major comments (4)

[Section 4, PINN versus NN; Figure 7] The central claim that engineered features enable prediction beyond the training zone is supported only by a qualitative visual comparison in Figure 7, with no error metric reported for the extrapolated interval from 15 to 30 minutes. In light of the paper's own limitation statement in Section 5 that a new model is required for each air-temperature profile, the authors should quantify the extrapolation error within the scope of the claim: report maximum and mean absolute deviations versus FE for both the top-surface and midplane temperature histories over the interval 15-30 minutes, and ideally for more than one boundary-condition and thickness setting. Without such quantitative evidence, the load-bearing claim of the abstract is unsubstantiated.
[Section 3, Physics-informed NN Architecture and Engineered Features] Equation (10) introduces engineered features exp(at) sin(bx), but the trained function is not structurally restricted to a linear combination of these modes because the features feed into six ELU hidden layers with 32 nodes each. Satisfying the PDE/BC/IC losses on the training window [0, 15] minutes does not by itself constrain the network output for t > 15 minutes, since the loss is never evaluated there. Thus the proposed mechanism does not logically guarantee temporal extrapolation; the quantitative test suggested for Figure 7 is necessary to determine whether the observed extrapolation is robust or an artifact of the single displayed case.
[Section 5, Figure 9] The 2D extension is validated only by a qualitative heat-map comparison with FE, with no numerical error measure. Since the authors present the 2D case as a validated extension and use it to support the generality of the method, a quantitative comparison is required, for example the maximum and mean absolute temperature differences over the domain at the three reported time snapshots.
[Section 3, Adaptive Normalization Factors] The adaptive normalization scheme is described heuristically, and the paper does not state conditions under which the normalization factors converge to unity or avoid oscillation, nor does it report the specific threshold and update interval used in the experiments. Because the method's success depends on balancing the four loss terms, please include a representative plot of the normalization factors during training and report the chosen threshold and update interval in the implementation details.

minor comments (5)

[Equation (8)] The subscripts in the loss expression are inconsistent: the first boundary-condition term uses N_BC0 in the denominator, the second term uses N_PDE, and the third uses N_BC2; these should all be the number of points for the corresponding loss term.
[Section 4, PINN with Convective BCs as Inputs] The claim that the extended PINN was trained 'to similar accuracy' as the two-input network is not accompanied by any quantitative comparison; please report the validation errors for both models.
[Figure 8] The subplots in Figure 8 lack labeled axes and a color scale in the caption, which makes it difficult to assess the quantitative accuracy of the boundary-condition sweep.
[Section 4, PINN versus NN] The discussion of Figure 6 states that the NN 'trained faster' and reached a 'stable' loss, but no criterion for stability or numerical loss values is given; please clarify what is meant by stable and report final loss values for both architectures.
[Section 4, PINN versus FE] The sentence 'the geometry was discretized into 10 elements in each directions' contains a typo ('directions' should be 'direction'), and the FE time-step sensitivity discussion is not reported, so it is unclear how the 5-second time discretization was justified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PINN's predictions are validated against external FE benchmarks and trained from PDE residuals, not from the target outputs they later predict.

full rationale

This paper does not exhibit circular reasoning in the sense of a claimed derivation that reduces to its own inputs. The central workflow is: define a physics-informed loss from the heat equation, convective boundary conditions, and initial condition (Equations 5-8); train a neural network to minimize that residual loss; then validate the trained solution against independently computed finite-element results (Figures 5, 7-9). The training signal is the PDE/BC/IC residual, not the FE temperatures that are later used for comparison, so the validation is not a self-fulfilling fit. The engineered feature layer in Equation (10) is introduced as an explicit ansatz based on the separation-of-variables solution of the conduction equation, and the paper acknowledges that convective cases do not have that exact analytic form. Using such an ansatz is a modeling choice, not circularity: it does not define the target prediction out of the input data, and the beyond-training behavior is checked against FE calculations rather than asserted from the feature construction. The paper also contains several self-citations, but none is load-bearing in the derivation: they support background statements about composite processing and prior ML usage, and the key validation targets are external FE models. The extrapolation claim is empirically supported only by a qualitative comparison in Figure 7 and would benefit from quantitative error metrics, but that is a correctness/evidence-strength concern, not a circularity defect. Overall, the derivation chain is self-contained with respect to its stated inputs and validated against external benchmarks.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard heat-transfer assumptions, the separation-of-variables feature ansatz, the heuristic adaptive normalization, the ELU-based architecture, and trust in the finite element reference. No new physical entities are introduced. The feature ansatz is the least externally justified item because it is asserted to remain valid for convective boundary conditions without direct verification.

free parameters (7)

Adam learning rate = 0.0001
Chosen by grid search; used for all training runs (Section 3).
batch size = 150 per loss term
Chosen by grid search; sample count per loss term per epoch (Section 3).
training epochs = 100,000 in most cases
Default training length; loss curves show convergence but no early-stopping criterion is given (Section 3, Figure 6).
network architecture = 6 hidden layers x 32 nodes
Selected by grid search; used for both NN and PINN (Section 3).
number of engineered features = 32
Number of exp(t) times sin(x) modes in the feature layer (Section 3).
adaptive normalization threshold and update interval = threshold 0.01, interval 100 epochs
Heuristic parameters controlling rescaling of loss terms (Section 3).
training point density near kinks = not specified
The paper increases point density near t=0 and the air-temperature kink but gives no quantitative densities (Section 3, Prediction Accuracy around Boundary Condition Kinks).

assumptions (6)

domain assumption Heat equation with constant thermal properties and no heat generation governs the temperature field (Equations 1 and 2).
The method is scoped to conduction-only oven heating; phase changes, heat generation, and property variation are excluded.
domain assumption Convective boundary conditions with spatially uniform heat transfer coefficients h1 and h2 (Equation 4) describe the surface heat exchange.
This is the paper's application setting; the model treats h1 and h2 as inputs.
domain assumption The separation-of-variables mode family exp(at) sin(bx) spans the solution space for the tested convective problems (Equations 9 and 10).
The feature layer is based on the no-convection analytic form; its adequacy with convection is asserted, not proven.
standard math A neural network with ELU activations and 6 hidden layers can approximate the solution to the required accuracy.
Universal approximation and training success are assumed; no approximation error bounds are given.
ad hoc to paper The adaptive normalization scheme converges to a balanced loss landscape.
The scheme is described heuristically with no convergence analysis.
domain assumption The in-house finite element code with 10 elements and 5-second steps is an accurate reference.
The FE results are treated as ground truth; mesh and time-step convergence are stated but not shown.

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Cite this review

Pith. "Pith review of A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications." pith.science (2026). https://pith.science/paper/7K6UQKFZ

@misc{pith2026201002011,
  author       = {Pith},
  title        = {Pith review of: A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7K6UQKFZ}},
  note         = {Machine review of arXiv:2010.02011}
}

read the original abstract

A physics-informed neural network is developed to solve conductive heat transfer partial differential equation (PDE), along with convective heat transfer PDEs as boundary conditions (BCs), in manufacturing and engineering applications where parts are heated in ovens. Since convective coefficients are typically unknown, current analysis approaches based on trial and error finite element (FE) simulations are slow. The loss function is defined based on errors to satisfy PDE, BCs and initial condition. An adaptive normalizing scheme is developed to reduce loss terms simultaneously. In addition, theory of heat transfer is used for feature engineering. The predictions for 1D and 2D cases are validated by comparing with FE results. It is shown that using engineered features, heat transfer beyond the training zone can be predicted. Trained model allows for fast evaluation of a range of BCs to develop feedback loops, realizing Industry 4.0 concept of active manufacturing control based on sensor data.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

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Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. U. S. A. 115, 8505–8510. https://doi.org/10.1073/pnas.1718942115 Incropera, F.P., DeWitt, D.P., Bergman, T.L., Lavine, A.S.,

work page doi:10.1073/pnas.1718942115

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https://doi.org/10.1063/1.4946894 Erol, S., Schuhmacher, A., Sihn, W.,

work page doi:10.1063/1.4946894

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work page arXiv 2017

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work page doi:10.1016/j.jmsy.2018.01.011 2018

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DGM: A deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339–1364. https://doi.org/10.1016/j.jcp.2018.08.029 Wagner, N., Rondinelli, J.M.,

work page doi:10.1016/j.jcp.2018.08.029 2018

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work page doi:10.1103/physrevfluids.2.034603

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arXiv Prepr

An Iterative Scientific Machine Learning Approach for Discovery of Theories Underlying Physical Phenomena. arXiv Prepr. arXiv:1909. Zobeiry, N., Park, J., Poursartip, A., 2019a. An infrared thermography-based method for the evaluation of the thermal response of tooling for composites manufacturing. J. Compos. Mater. 53, 1277–1290. https://doi.org/10.1177/...

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