From Physics to Surrogate Intelligence: A Unified Electro-Thermo-Optimization Framework for TSV Networks

Mohamed Gharib; Leonid Popryho; Inna Partin-Vaisband

cs.LGMar 31, 2026

From Physics to Surrogate Intelligence: A Unified Electro-Thermo-Optimization Framework for TSV Networks

Authors: Mohamed Gharib, Leonid Popryho, Inna Partin-Vaisband

Organizations: Department of Electrical and Computer Engineering, University of Illinois Chicago, Chicago, IL 60607, USA

Abstract

High-density through-substrate vias (TSVs) enable 2.5D/3D heterogeneous integration but introduce significant signal-integrity and thermal-reliability challenges due to electrical coupling, insertion loss, and self-heating. Conventional full-wave finite-element method (FEM) simulations provide high accuracy but become computationally prohibitive for large design-space exploration. This work presents a scalable electro--thermal modeling and optimization framework that combines physics-informed analytical modeling, graph neural network (GNN) surrogates, and full-wave sign-off validation. A multi-conductor analytical model computes broadband S-parameters and effective anisotropic thermal conductivities of TSV arrays, achieving $5\%$ -- $10\%$ relative Frobenius error (RFE) across array sizes up to $15\times15$ . A physics-informed GNN surrogate (TSV-PhGNN), trained on analytical data and fine-tuned with HFSS simulations, generalizes to larger arrays with mean RFE below $5\%$ in-distribution. The surrogate is integrated into a multi-objective Pareto optimization framework targeting reflection coefficient, insertion loss, worst-case crosstalk (NEXT/FEXT), and effective thermal conductivity. Millions of TSV configurations can be explored within minutes, enabling exhaustive layout and geometric optimization that would be infeasible using FEM alone. Final designs are validated with Ansys HFSS and Mechanical, showing strong agreement. The proposed framework enables rapid electro--thermal co-design of TSV arrays while reducing per-design evaluation time by more than six orders of magnitude.

Figures & tables

Fig. 1 : Comparison of representative TSV modeling and optimization frameworks in terms of scalability, frequency range, and computational cost.

Fig. 2 : Comparison of the analytical and TSV-PhGNN-based coupled electrothermal simulation frameworks. In both cases, S-parameters are used to extract TSV Joule-heating dissipation, which is then mapped to the ETC-based steady-state thermal solver. In the analytical loop, temperature-dependent resistivity is updated iteratively, whereas in the surrogate loop, the updated temperature is fed back to TSV-PhGNN, until convergence is reached.

Fig. 3 : Overview of the proposed ML-assisted optimization framework. A physics-informed GNN is first pre-trained on large-scale RLCG-based analytical data and then fine-tuned on high-fidelity Ansys HFSS data through Sim-to-Sim transfer learning. The TSV array is modeled as a geometry-, frequency-, and temperature-aware graph, and the trained surrogate is embedded in the coupled electrothermal flow shown in Fig. 2 .

Fig. 4 : Per-sample runtime comparison between the ML surrogate, the analytical solver, and Ansys HFSS versus TSV grid size ( $N\times N$ ).

Fig. 5 : Magnitude comparison (in dB) of selected S-parameters obtained from HFSS full-wave simulations and the proposed analytical model. The corresponding TSV array top view is illustrated in the inset, where red circles denote signal TSVs and blue circles denote ground TSVs, arranged on a uniform pitch grid.

Fig. 6 : Comparison between the proposed homogenized ETC model and high-fidelity FEM-based Ansys Mechanical simulations for fully populated and sparse TSV arrays under natural and forced convection boundary conditions. The percentage above each bar denotes the relative error with respect to the corresponding FEM solution.

Fig. 7 : Total thermal analysis runtime comparison between the full detailed model and the ETC-based model.

arm	split	RFE	MAE NEXT	Worst victim NEXT (dB)	recip residual
Reference (full proposed model)	3–7	0.0288	0.0045	0.89	0.000000
	8–15	0.0331	0.0023	1.07	0.000000
	16–25	0.0982	0.0030	3.19	0.000000
w/o FiLM (concatenation)	3–7	0.0626	0.0092	1.53	0.000000
	8–15	0.1129	0.0087	3.01	0.000000
	16–25	0.1505	0.0060	6.52	0.000000

TABLE VI : Architecture ablation (matched protocol; HFSS-only, identical hyperparameters and seed). “Reference” = full model (FiLM, reciprocity averaging, passivity penalty

\lambda=0.05

); each other arm removes one component. The validation-split label “

N_{1}

–

N_{2}

” denotes the array-size bucket from

N_{1}{\times}N_{1}

through

N_{2}{\times}N_{2}

. “Worst victim NEXT (dB)” = absolute dB error at the largest-coupling NEXT pair; “recip residual” = mean

|S_{ij}-S_{ji}|

on the raw edge head.

Fig. 8 : Coupled electrothermal trajectories for the representative $4{\times}4$ testcase. GNN+ETC, HSPICE+ETC, and the Ansys HFSS $\leftrightarrow$ Mechanical reference are compared for: (a) $\overline{T}_{\mathrm{sub}}$ , (b) $T_{\mathrm{die}}$ , (c) $P_{\mathrm{die}}(T)$ , and (d) $|\Delta\overline{T}|$ convergence.

Parameters	Symbol	Value
TSV radius	$r_{\mathrm{TSV}}$	$5~\mu\mathrm{m}$
TSV height	$h_{\mathrm{TSV}}$	$100~\mu\mathrm{m}$
Pitch between adjacent TSVs	$p$	$60~\mu\mathrm{m}$
Thickness of SiO 2 layer	$t_{\mathrm{ox}}$	$0.5~\mu\mathrm{m}$
Copper conductivity	$\sigma_{\mathrm{Cu}}$	$5.8\times 10^{7}~\mathrm{S/m}$
Silicon conductivity	$\sigma_{\mathrm{Si}}$	$10~\mathrm{S/m}$

TABLE VII : TSV Array Physical and Geometrical Parameters.

Panel of Figure 11 — Fig. 9 : Near-end (NEXT) and far-end (FEXT) crosstalk at 15 GHz for the worst-case victim TSV, comparing the proposed signal–ground assignment with the state-of-the-art configuration in [ 22 ] . Insets show the corresponding TSV layouts. Bars denote the proposed design and line markers the results reported in [ 22 ] . The horizontal dashed lines indicate the worst-case total crosstalk, showing an improvement of (a) 11 dB in the 12-signal ( $5\times 5$ ) array and (b) 18 dB in the 9-signal ( $5\times 5$ ) array.

Fig. 10 : Results of a design space exploration 9-15 signal TSVs in a $5\times 5$ array. Near-end (NEXT) and far-end (FEXT) crosstalk at 15 GHz for the worst-case victim TSV in the optimized signal–ground assignment (14 signal TSVs). The inset shows the corresponding TSV layout.

$N$	Search space $\|\mathcal{S}\|$	NSGA-II HV mean [min, max]	Random HV mean [min, max]	Ratio
$7\!\times\!7$	${}_{49}C_{24}$ $\approx 6.5{\times}10^{13}$	0.311 [0.288, 0.345]	0.173 [0.076, 0.255]	$1.79\times$
$9\!\times\!9$	${}_{81}C_{39}$ $\approx 8.1{\times}10^{22}$	0.431 [0.287, 0.541]	0.137 [0.109, 0.187]	$3.15\times$
$11\!\times\!11$	${}_{121}C_{58}$ $\approx 4.4{\times}10^{35}$	0.797 [0.718, 0.862]	0.453 [0.106, 0.685]	$1.76\times$

TABLE VIII : NSGA-II versus random-search hypervolume at

10{,}000

surrogate evaluations (3 seeds). Search space

|\mathcal{S}|={}_{N^{2}}C_{n_{\text{sig}}}

n_{\text{sig}}=\mathrm{round}(0.48\,N^{2})

; objectives:

-\overline{|S_{21}|}

\max|S_{11}|

\max|\text{NEXT}|

\max|\text{FEXT}|

in dB.

Parameters	Symbol	Value
TSV radius	$r_{\mathrm{TSV}}$	$2\rightarrow 6~\mu\mathrm{m}$
TSV height	$h_{\mathrm{TSV}}$	$60\rightarrow 100~\mu\mathrm{m}$
Pitch between adjacent TSVs	$p$	$20\rightarrow 60~\mu\mathrm{m}$
Thickness of SiO 2 layer	$t_{\mathrm{ox}}$	$0.5\rightarrow 3~\mu\mathrm{m}$

TABLE IX : Geometrical Parameters of TSV Array Sweep

Parameter	Best Crosstalk	Best Thermal	Best Insertion	Best Reflection
$r_{\mathrm{TSV}}$ ( $\mu$ m)	3.56	2.63	2.03	2.69
$p$ ( $\mu$ m)	20.15	53.43	29.66	20.09
$h_{\mathrm{TSV}}$ ( $\mu$ m)	62.20	60.04	60.74	60.17
$t_{\mathrm{ox}}$ ( $\mu$ m)	2.72	0.90	2.98	2.98
$\overline{S_{21}}$ (dB)	-0.083	-0.171	-0.066	-0.071
$\max S_{11}$ (dB)	-35.29	-35.74	-35.30	-45.75

TABLE X : Pareto-Optimal TSV Designs at 15 GHz

Electronic Design Automation

Electronic Design Automation uses algorithms and software to design, verify, optimize, and manufacture electronic circuits and systems.

See all 213 papers

Mesh-Native Physics-Informed Graph Surrogates for TCAD-in-the-Loop Design Space Exploration
Sep 2, 2026Leonid Popryho, Ayoub Sadeghi, Inna Partin-VaisbandPDE Surrogate Modeling Physics-Informed ML
University of Illinois Chicago · Chicago, IL, USA
AI for Science with GPT-6 Astra: Thermal Design and Electrothermal Analysis of 2D CFET
Sep 15, 2026Min-Hui Kim, Khushi Sharma, Sarah Zhang +1
Graduate School of Semiconductor Materials and Devices, Ulsan National Institute of Science and Technology (UNIST), Ulsan 44919, Republic of Korea · Materials Science and Engineering, National University of Singapore, Singapore · Materials Science and Engineering, Cornell University, Ithaca, NY, USA +1
DeepOHeat-v2: Self-Improving Operator Learning for Fast and Trustworthy Thermal Optimization in 3D-IC Design
Aug 17, 2026Xinling Yu, Yixing Li, Ziyue Liu +4PDE Operator Learning Surrogate-Assisted Optimization
Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106 USA · Cadence Design Systems, Austin, TX 78759 USA · Department of Computer Science, University of California, Santa Barbara, CA 93106 USA +1

Sep 2, 2026J/K move · Enter open · S save

Mesh-Native Physics-Informed Graph Surrogates for TCAD-in-the-Loop Design Space Exploration

Leonid Popryho, Ayoub Sadeghi, Inna Partin-Vaisband

University of Illinois Chicago · Chicago, IL, USA

PDE Surrogate Modeling Physics-Informed ML Physics-Informed GNNs

Open paper AskPDF arXiv

High-fidelity TCAD simulation of drift-diffusion transport remains the workhorse of emerging FinFET device design, but it is computationally expensive, especially for 3D structures where runtime escalates steeply with mesh complexity. This sharply limits multi-objective design space exploration. Existing machine-learning surrogates map a fixed set of design parameters to a few scalar device metrics, discarding the underlying physics and losing transferability across device geometries and families. A physics-informed graph attention network (GAT) surrogate is proposed. It operates directly on the tetrahedral TCAD mesh and predicts, at every mesh node, the electrostatic potential together with the electron and hole quasi-Fermi levels, the fundamental unknowns of the drift-diffusion system. Training combines a data loss with finite-volume current-continuity residuals, embedding carrier-transport physics into the objective. Operating on the mesh as a graph, the surrogate inherits size generalization: a model trained on few-fin meshes applies unchanged to substantially larger arrays, bounded at inference only by GPU memory. Per-node uncertainty from a deep ensemble drives an active-learning loop that screens large candidate pools in seconds and forwards only the most informative designs for full simulation. Benchmarked against Sentaurus Device on multi-fin tri-gate FinFETs, the surrogate reproduces the three drift-diffusion fields with sub-volt per-field RMSE and reaches a per-design throughput orders of magnitude higher than the full simulator. The advantage grows with device size: on large multi-fin arrays that are prohibitively slow to simulate directly, inference still completes in under a second per device, enabling Pareto-front exploration across device scales infeasible for direct TCAD sweeps.

Abstract

Electronic Design Automation

Explore similar work

Mesh-Native Physics-Informed Graph Surrogates for TCAD-in-the-Loop Design Space Exploration

AI for Science with GPT-6 Astra: Thermal Design and Electrothermal Analysis of 2D CFET

DeepOHeat-v2: Self-Improving Operator Learning for Fast and Trustworthy Thermal Optimization in 3D-IC Design