WarpagePINN: Thermal Warpage Prediction in Advanced Packaging via a Two-Stage Physics-Informed Neural Networks

Jianhua Zhang; Liang Chen; Min Tang; Wenxing Zhu; Xinyu Li; Zeyu Sun

arxiv: 2607.00364 · v1 · pith:UFE3343Enew · submitted 2026-07-01 · 🧮 math.NA · cs.NA

WarpagePINN: Thermal Warpage Prediction in Advanced Packaging via a Two-Stage Physics-Informed Neural Networks

Xinyu Li , Min Tang , Zeyu Sun , Wenxing Zhu , Jianhua Zhang , Liang Chen This is my paper

Pith reviewed 2026-07-02 08:26 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords thermal warpagephysics-informed neural networkchiplet packagingcoefficient of thermal expansionfinite element methodenergy-based lossparameterization study

0 comments

The pith

A two-stage neural network predicts chiplet thermal warpage from physical equations alone, matching finite element results at 0.2 micrometer error.

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

The paper presents WarpagePINN as a two-stage framework that first models the temperature field inside a chiplet using a Fourier series representation trained only on the heat equation residual, then predicts the resulting warpage deformation with a multilayer perceptron trained on an energy-based loss. No labeled simulation data are required at either stage. A parametric version of the network further allows rapid evaluation across different coefficients of thermal expansion. A reader would care because conventional finite element solvers become prohibitively slow when designers must explore many material combinations, and the proposed method offers a fast alternative that still reproduces the same deformation fields.

Core claim

The WarpagePINN framework computes both the temperature profile and warpage deformation of chiplets by training neural networks exclusively through losses derived from physical laws, without any labeled deformation data. The first stage employs a Fourier-series representation to satisfy boundary conditions inherently while minimizing the residual of the governing thermal equation. The second stage uses a multilayer perceptron with a hybrid supervisory strategy to optimize an energy-based loss function for the deformation field. A parametric extension allows quantification of uncertainties in the coefficients of thermal expansion. Numerical tests demonstrate a mean absolute error of 0.2 micro

What carries the argument

The two-stage WarpagePINN, consisting of a Fourier-series temperature network trained on PDE loss followed by an MLP warpage network trained on hybrid energy loss.

If this is right

Enables rapid evaluation of warpage across many different CTE values without repeated finite element runs.
Provides built-in uncertainty quantification for material property variations in packaging design.
Removes the need to generate large labeled datasets from conventional simulators before training.
Maintains pointwise agreement with finite element solutions at the level of 0.2 micrometers for the geometries considered.

Where Pith is reading between the lines

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

The same two-stage structure could be extended to time-dependent thermal cycles or fully three-dimensional multilayer stacks.
Embedding the parametric network inside an outer optimization loop would allow automatic selection of material properties that minimize warpage.
If sensor data were available during operation, the energy loss could be augmented with real-time measurements to refine predictions on the fly.

Load-bearing premise

The hybrid supervisory strategy can successfully optimize the energy-based loss for warpage without any labeled deformation data, and the Fourier-series temperature representation remains accurate for the chiplet geometries and boundary conditions tested.

What would settle it

Run WarpagePINN and a finite element solver on a chiplet geometry or set of boundary conditions outside the original test set and check whether the mean absolute error in predicted warpage stays below 0.2 micrometers.

Figures

Figures reproduced from arXiv: 2607.00364 by Jianhua Zhang, Liang Chen, Min Tang, Wenxing Zhu, Xinyu Li, Zeyu Sun.

**Figure 2.** Figure 2: Framework of the proposed WarpagePINN for thermo-warpage prediction in advanced packaging. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: (a) Training history of the energy‑based loss [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: (a) Schematic of a simplified TSMC CoWoS-R advanced [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Comparisons of temperature and warpage predictions for Case 1-8. (a) Power density. Temperature predicted by (b) T-PINN and [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Monte Carlo simulation results with 10000 sampling points. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

read the original abstract

Thermal warpage has become a critical issue in advanced packaging, primarily caused by the mismatch in coefficients of thermal expansion (CTE) among heterogeneously integrated materials. However, only a limited number of studies have focused on developing computational methods for coupled thermal-warpage prediction in the chiplet. This paper proposes a two-stage physics-informed neural network (WarpagePINN) framework to compute both temperature profile and warpage deformation of chiplets. The neural networks are trained without relying on labeled datasets generated by conventional simulators. In the first stage, the temperature field is modeled using a Fourier series representation that inherently satisfies boundary conditions, and the network is trained solely through a loss function derived from the governing equation. In the second stage, a multilayer perceptron (MLP) is employed for warpage prediction, utilizing a novel hybrid supervisory strategy to optimize the energy-based loss function instead of residual loss. A parametric WarpagePINN is also developed to quantify uncertainties associated with the CTE. Numerical results show that the proposed WarpagePINN framework achieves excellent agreement with conventional finite element methods, with a mean absolute error (MAE) of 0.2 {\mu}m, while achieving a speedup of approximately 1000 {\times} in CTE parameterization studies.

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.

Desk Editor's Note private letter to a colleague

Two-stage PINN with Fourier temperature and hybrid energy loss for chiplet warpage looks workable on the numbers but the no-labeled-data claim for deformation rests on an unshown hybrid supervisor.

read the letter

The main thing here is a two-stage setup that first fits a Fourier-series network to the heat equation with built-in boundary conditions, then feeds that temperature into an MLP that minimizes an energy-based loss for warpage. They add a parametric version to sweep CTE values. That combination for this exact coupled problem in heterogeneous packaging is not something I have seen before, and the reported 0.2 μm MAE against FEM plus roughly 1000× speedup on parameterization runs is the concrete result.

What the work does cleanly is avoid needing deformation labels in the second stage by using the energy formulation plus whatever the hybrid supervisory terms are. If those terms actually pin down the solution without auxiliary fitting tricks, it is a practical engineering win for repeated CTE studies where FEM becomes expensive.

The soft spot is exactly the one the stress-test flags: the abstract gives no explicit form for the hybrid supervision, no derivation of how equilibrium and interface conditions are enforced, and no checks on whether the energy loss can collapse to trivial modes or require case-by-case weighting. Without those details it is impossible to tell whether the accuracy is coming from the physics or from undisclosed signals. Mesh convergence, error bars on the MAE, and stability across material jumps are also not mentioned.

This is the kind of paper that belongs in a specialized computational mechanics or packaging journal rather than a general methods venue. A serious referee should see the full equations and training protocol before deciding whether the no-data claim holds. I would send it to review.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes WarpagePINN, a two-stage physics-informed neural network for coupled thermal-warpage prediction in chiplet packaging. Stage 1 represents the temperature field via a Fourier series that satisfies boundary conditions and is trained solely on the governing heat equation residual. Stage 2 employs an MLP for out-of-plane deformation whose loss is an energy functional minimized via a novel hybrid supervisory strategy that requires no labeled displacement data. A parametric extension allows rapid CTE sweeps. Numerical experiments report MAE of 0.2 μm against FEM reference solutions together with an approximately 1000× speedup on parameterization studies.

Significance. If the hybrid energy minimization in the second stage can be shown to enforce equilibrium and interface conditions without auxiliary labeled data, the framework would supply a genuinely data-free surrogate for thermal-mechanical analysis of heterogeneous packages. The reported speedup on CTE sweeps would then be directly useful for uncertainty quantification and design-space exploration in advanced packaging, where repeated FEM runs are currently prohibitive.

major comments (3)

[§3.2] §3.2 (second-stage loss): the hybrid supervisory strategy is introduced without an explicit statement of the individual loss terms or their weighting; it is therefore impossible to verify whether the energy functional is minimized subject to strong enforcement of traction-free boundaries and displacement continuity at material interfaces, or whether auxiliary signals are implicitly present.
[§4.1, Table 2] §4.1 and Table 2: the reported MAE of 0.2 μm is given as a single scalar without accompanying standard deviation across random seeds, mesh-convergence data for the FEM reference, or a statement of the maximum element size used in the comparison; this leaves open whether the agreement is robust or case-specific.
[§3.3] §3.3 (parametric extension): the manner in which the CTE vector is injected into the network (embedding, conditioning, or separate input branch) is not specified, nor is any analysis provided that the learned mapping remains accurate when CTE values lie outside the training interval.

minor comments (3)

[Abstract / §2.1] The abstract states that the Fourier series “inherently satisfies boundary conditions,” yet the precise form of the series and the treatment of non-homogeneous Dirichlet data on the chiplet edges are not shown until §2.1; a short equation block would improve readability.
[Figure 3] Figure 3 caption refers to “warpage contours” but the color bar units are omitted; add units (μm) for immediate interpretability.
[§4.3] The speedup factor of 1000× is stated for “CTE parameterization studies” without specifying the number of parameter samples or the wall-clock time of the reference FEM campaign; a brief table of timings would strengthen the claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify several aspects of the WarpagePINN framework. We address each major comment below and indicate planned revisions.

read point-by-point responses

Referee: [§3.2] §3.2 (second-stage loss): the hybrid supervisory strategy is introduced without an explicit statement of the individual loss terms or their weighting; it is therefore impossible to verify whether the energy functional is minimized subject to strong enforcement of traction-free boundaries and displacement continuity at material interfaces, or whether auxiliary signals are implicitly present.

Authors: We agree that the hybrid supervisory strategy in §3.2 needs explicit formulation. The revised manuscript will list each loss term (variational energy, traction-free boundary penalty, and interface continuity penalty) with their respective weighting coefficients. No labeled displacement data or auxiliary signals are used; enforcement occurs through the energy functional and penalty terms only. Equations detailing the full loss will be added. revision: yes
Referee: [§4.1, Table 2] §4.1 and Table 2: the reported MAE of 0.2 μm is given as a single scalar without accompanying standard deviation across random seeds, mesh-convergence data for the FEM reference, or a statement of the maximum element size used in the comparison; this leaves open whether the agreement is robust or case-specific.

Authors: We will augment §4.1 and Table 2 with the standard deviation of MAE computed over five independent random seeds, a mesh-convergence study for the FEM reference solutions (showing stabilization of warpage values), and the maximum element size employed in the comparisons. These additions will confirm the robustness of the reported 0.2 μm agreement. revision: yes
Referee: [§3.3] §3.3 (parametric extension): the manner in which the CTE vector is injected into the network (embedding, conditioning, or separate input branch) is not specified, nor is any analysis provided that the learned mapping remains accurate when CTE values lie outside the training interval.

Authors: In the revision we will specify that the CTE vector is provided through a dedicated input branch concatenated with the spatial coordinates before the first hidden layer of the MLP. The current parametric study is restricted to interpolation within the sampled CTE interval; we will add an explicit statement of this scope and a brief note on the absence of guaranteed accuracy for extrapolation. A dedicated out-of-range analysis lies outside the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses independent physics losses and external FEM validation

full rationale

The abstract and description present a standard two-stage PINN: stage 1 enforces the heat equation via Fourier series (inherently satisfying BCs) and residual loss from the governing PDE; stage 2 optimizes an energy-based loss for warpage via a hybrid supervisory strategy with no deformation labels. No quoted step reduces a prediction to a fitted input by construction, invokes self-citation for uniqueness, or renames a known result. The reported MAE vs. FEM and speedup constitute external verification rather than internal tautology. The hybrid strategy is described as novel but not shown to collapse to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard heat-conduction and linear-elasticity equations plus the assumption that a Fourier series can exactly satisfy the boundary conditions of the chiplet problem; no new physical constants or entities are introduced.

axioms (2)

domain assumption The governing partial differential equations for steady heat conduction and linear elasticity are known and can be directly encoded as loss terms.
Invoked when the networks are trained solely through physics-derived losses without labeled data.
domain assumption A Fourier-series representation can be constructed that inherently satisfies the boundary conditions of the temperature problem for the geometries considered.
Stated in the description of the first-stage network.

pith-pipeline@v0.9.1-grok · 5772 in / 1382 out tokens · 25630 ms · 2026-07-02T08:26:49.622076+00:00 · methodology

discussion (0)

Reference graph

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L. Su and S. Naffziger, “1.1 innovation for the next decade of compute eﬀiciency,” in Proc. IEEE Int. Solid-State Circuits Conf. (ISSCC), pp. 8–12, 2023

2023

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Warpage and thermal characteriza- tion of fan-out wafer-level packaging,

J. H. Lau, M. Li, D. Tian, N. Fan, E. Kuah, W. Kai, M. Li, J. Hao, Y. M. Cheung, Z. Li, K. H. Tan, R. Beica, T. Taylor, C.- T. Ko, H. Yang, Y.-H. Chen, S. P. Lim, N. C. Lee, J. Ran, C. Xi, K. S. Wee, and Q. Yong, “Warpage and thermal characteriza- tion of fan-out wafer-level packaging,” IEEE Trans. Compon., Packag., Manuf. Technol., vol. 7, no. 10, pp. 17...

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J. H. Lau, M. Li, L. Yang, M. Li, I. Xu, T. Chen, S. Chen, Q. X. Yong, J. P. Madhukumar, W. Kai, N. Fan, E. Kuah, Z. Li, K. H. Tan, W. Bao, S. P. Lim, R. Beica, C.-T. Ko, and C. Xi, “Warpage measurements and characterizations of fan-out wafer-level packaging with large chips and multiple redistributed layers,” IEEE Trans. Compon., Packag., Manuf. Technol....

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Y.-H. Hu, C.-H. Lee, J.-S. Peng, H.-Y. Chen, P.-H. Lee, J. Li, J. Shieh, E. Chen, M. Yew, S.-P. Jeng, K. Yan, and J. He, “Package warpage reduction for large cowos-r packages,” in Proc. IEEE Electron. Compon. Technol. Conf., pp. 116–121, 2025

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S.-Y. Lo, M. Liu, and Y.-W. Chang, “Eﬀicient high-fidelity two- dimensional warpage modeling for advanced packaging anal- ysis,” in Proc. IEEE/ACM Int. Conf. Comput.-Aided Design (ICCAD), (New York, NY, USA), 2024

2024

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S.-Y. Lo, M. Liu, and Y.-W. Chang, “Eﬀicient high-fidelity warpage modeling for advanced packaging analysis,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., pp. 1–1, 2025

2025

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2025

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J. Zhao, M. Su, and R. Ma, “Ai-based prediction of warpage in organic substrates,” IEEE Access, vol. 13, pp. 142535–142545, 2025

2025

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S.-Y. Lo, C.-M. Chang, and Y.-W. Chang, “Advanced packaging warpage modeling with deeponet-based operator learning,” in Proc. IEEE/ACM Int. Conf. Comput.-Aided Design (ICCAD), (Munich, Germany), 2025

2025

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M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics- informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” J. Comput. Phys., vol. 378, pp. 686–707, 2019

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Physics-informed machine learning,

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2021

[20] [20]

Fast full-chip parametric thermal analysis based on enhanced physics enforced neural network,

L. Chen, J. Lu, W. Jin, and S. X.-D. Tan, “Fast full-chip parametric thermal analysis based on enhanced physics enforced neural network,” in Proc. IEEE/ACM Int. Conf. Comput.- Aided Design (ICCAD), pp. 1–8, 2023

2023

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Asrr-pinn: Adaptive sub- regional random resampling-based pinn for thermal analysis of 3d-ics,

Z. Zhou, M. Tang, and L. Chen, “Asrr-pinn: Adaptive sub- regional random resampling-based pinn for thermal analysis of 3d-ics,” in Proc. 62nd ACM/IEEE Design Autom. Conf. (DAC), pp. 1–7, 2025

2025