Organizations: 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 · independent researcher
Thermal-aware optimization of multi-die 3D integrated circuits evaluates many designs, each a costly heat-equation solve. Operator-learning surrogates replace this solve with a fast forward pass, ideally trained from physics alone, without labeled data. DeepOHeat-v1 made such surrogates fast and trustworthy, but only on low-contrast geometries. High-contrast multi-die stacks break it in two ways: discontinuous conductivities make the continuous physics loss ill-defined at material interfaces, and ill-conditioning (κ2(Ah)≈6×104) puts the discretized strong-form loss beyond first-order optimization. We propose DeepOHeat-v2 to overcome both. First, we train on a discretized physics loss that handles the discontinuities natively; its energy form reduces the prediction-space loss-Hessian conditioning from κ2 to κ, and a matrix-preconditioned optimizer cuts the mean peak temperature error from over 30 K to 0.55 K. Second, because optimization leaves the training distribution, we propose a self-improving framework: a hotspot trust gate sends flagged placements to a reference solver, and the surrogate incrementally retrains on the refined solutions, keeping an update only when it improves held-out validation error. On a multi-die benchmark, the surrogate-true peak gap on the returned design falls from 1.12 K to 0.11 K, matching a solve-at-every-step optimizer while running 56× faster.
Figures & tables
Fig. 1: Distribution shift along the search. Peak temperatures of 1000 random placements (the pretraining distribution, mean ≈405 K, broad) and of 1000 placements visited along an optimization trajectory (mean ≈391 K, concentrated in the low-temperature tail). The optimizer drives the design into a region the random pretraining distribution rarely samples, so the surrogate is evaluated far from where most of its training examples lie.
Fig. 2: The DeepOHeat-v2 self-improving loop. The optimizer proposes a placement; the surrogate predicts its temperature field; the trust gate either accepts the prediction or sends it to a warm-started GMRES refinement. Refined solutions accumulate in an online buffer that periodically retrains the surrogate, so the surrogate grows more accurate along the trajectory the search follows.
Fig. 3: The F2F chiplet benchmark: two bulk-plus-active-Si dies vertically stacked and bonded face-to-face on an organic substrate, cooled by a top cold plate and weak bottom board conduction, with copper TSVs passing through the silicon layers.
#
Layer
Thickness
k (W/m ⋅ K)
1
Substrate (BT/FR4)
500μ m
0.5
2
Die 1 bulk Si (+ TSV Cu)
45μ m
140/400
3
Die 1 active Si (+ TSV)
5μ m
140/400
4
Hybrid bond (Cu–Cu)
3μ m
100
5
Die 2 active Si (+ TSV)
5μ m
140/400
6
Die 2 bulk Si (+ TSV Cu)
45μ m
140/400
TABLE I: Layer stack of the F2F chiplet benchmark (bottom → top); TIM is the thermal interface material.
Die
Block
Power (W)
1
PHY
1.0
1
PLL
0.6
1
SRAM0
0.08
1
SRAM1
0.08
1
Analog
0.08
2
CPU0
0.7
TABLE II: Power blocks on the two dies ( ≈3.5 W total). PHY, PLL, and the two CPU cores are hotspots; the rest are background blocks.
Quantity
Value
Trust-window length ( ∣W∣ )
100
Hotspot cells ( m )
100
Target flag rate ( ftarget )
0.30
Refinement rel. tolerance ( ϵ )
5×10−1
Loss weights ( λe,λd )
0.1,1
Incremental LRs ( ηmuon,ηadam )
5×10−6,1×10−6
TABLE III: DeepOHeat-v2 self-improving-loop hyperparameters, fixed across all runs.
Loss form
Optimizer
Peak temp. error (K)
Relative L2
MAE (K)
Train (min)
Supervised MSE
Adam
9.81
1.28×10−2
3.87
20.0
Strong-form physics
Adam
30.88
2.91×10−2
8.76
9.5
Strong-form physics
Muon
∼2.2×1011†
5.84×108
∼2.2×1011
11.8
Strong-form physics
Muon 2
∼3.6×1010†
1.18×108
∼4.4×1010
11.9
Energy-form physics
Adam
1.05
6.79×10−4
0.179
11.7
Energy-form physics
Muon
0.639
4.18×10−4
0.112
13.3
TABLE IV: Pretraining results on the n=100 held-out set after 40,000 epochs: mean peak temperature error, relative L2 field error, field mean-absolute error (MAE), and wall-clock training time. † training diverged; values are the error magnitude at termination.
Fig. 4: Energy + Muon 2 surrogate predictions on 10 random held-out placements. Rows, top to bottom: block placement (Die 1 solid fill, Die 2 dashed overlay) with reference and predicted peak temperatures; reference field Tref at the Die-1 active layer; surrogate prediction T at the same layer (shared color scale); absolute residual ∣T−Tref∣ . The hotspot under PHY is captured consistently; residuals concentrate at block edges where in-plane gradients are steepest.
Fig. 5: Catch-rate operating curves for rhot vs. the global relative residual rrel on the surrogate pretrained for 40,000 epochs ( n=100 held-out placements). Each curve plots the fraction of bad predictions caught against the flag-rate budget; the diagonal is the random baseline. (a) Bad =∣ΔTpeak∣>0.5 K ( P=45 positives). (b) Bad =∣ΔTpeak∣>1 K ( P=12 positives). rhot dominates rrel across the budget range.
Fig. 6: Online stability of DeepOHeat-v2 on the integrated run. Left: model-selection trace; for each incremental-training event the 10 most-recent validation samples are plotted at (Gθpre error, Gθpost error ) , with points below the diagonal improved and above regressed. Event 1 reverts to Gθpre (red), events 2–3 keep Gθpost (blue); the per-event scale drops from ∼1 K to ∼0.2 K as SA cools. Right: anti-forgetting; the post-pipeline held-out mean peak temperature error stays below the pretrained baseline, so the loop does not forget the broad distribution.
Configuration
Predicted peak (K)
True/oracle peak (K)
Gap (K)
Surrogate-only
387.67
388.79
1.12
DeepOHeat-v1
387.25
388.03
0.78
DeepOHeat-v2
387.15
387.04
0.11
Oracle SA (ref)
—
387.36
0.00
TABLE V: Predicted vs. true peak on the integrated run from a random initial placement. The gap is the surrogate–true difference at the returned placement; Oracle SA is the non-competing zero-gap reference.
Configuration
True peak (K)
FVM solves in SA
Wall
Surrogate-only
388.79
0
40 s
DeepOHeat-v1
388.03
∼300
335 s
DeepOHeat-v2
387.04
150
292 s
Oracle SA (ref)
387.36
1000
4.51 h
TABLE VI: True (oracle) peak, FVM solves issued during SA, and wall time on the integrated run; “FVM solves in SA” excludes the single final verification solve.
Fig. 7: Optimized placements and oracle FVM temperature fields at the Die-1 active layer for the four configurations on the integrated run. Row 1: placement with Die 1 solid outlines and Die 2 dashed overlays. Row 2: oracle FVM field, shared color scale. DeepOHeat-v2 matches Oracle SA in placement structure and, to within the SA basin scatter ( ∼0.3 K), in peak temperature.
Accurate thermal analysis of heterogeneous 2.5D/3D-IC packages is essential yet computationally prohibitive. A single full-package FEM simulation can take hours, while AI-based surrogates treat the entire stack as a monolithic prediction target and must be retrained whenever the die count or topology changes. To address this limitation, this work proposes Domain-Decomposed AI-Accelerated Iterative Solver for Thermal Analysis (DAIST), a composable thermal solver that decomposes the global package simulation into block-level subdomain problems, replaces subdomain solvers with neural operators, and couples them through iterative exchanges of interfacial temperature and heat flux. This local-to-global architecture eliminates the topology lock-in of monolithic models: block-level neural operators can be directly reused in unseen package assemblies without retraining. The iterative coupling strategy further provides a controllable accuracy-runtime tradeoff, where the iteration budget can be adjusted to trade accuracy for runtime. Evaluated on a multi-chiplet system and an advanced packaging system, DAIST achieves up to 178× speedup over traditional FEM solvers with mean temperature errors of 0.068% and 0.323%, respectively, while demonstrating cross-topology reuse of block-level models across structurally distinct package assemblies.
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 15×15. 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.
Mohamed Gharib, Leonid Popryho, Inna Partin-Vaisband
Department of Electrical and Computer Engineering, University of Illinois Chicago, Chicago, IL 60607, USA
Topology optimization (TO) remains computationally intensive due to repeated finite element analysis (FEA) evaluations required at each iteration. While neural network-based surrogates offer potential acceleration, existing approaches often suffer from gradient inconsistency between predicted objectives and sensitivities, leading to optimization instability. This work presents KATOsuper, an objective-agnostic framework that couples neural-reparameterized topology optimization with a Sensitivity-Consistent Fourier Neural Operator (SC-FNO). The framework employs the forward_split architecture, which derives deployed sensitivities via automatic differentiation through the predicted objective field and thereby preserves consistency between the predicted objective and the gradient used for optimization. The case studies include three 2D benchmark problems and three 3D structures considering compliance or stress minimization. A physics-informed multi-channel input encoding with Fourier position embedding enables resolution-invariant learning, supporting zero-shot extrapolation beyond the training resolution, with useful performance at moderate scaling factors and topology-preserving exploration at up to 64x without retraining. The framework extends to 3D through KATO3D, featuring novel KANConv3D blocks with learnable B-spline activations. KATOsuper demonstrates 15--110x deployment-time speedup over MATLAB baselines while maintaining competitive optimality, with the clearest gains observed in complex 3D and stress-optimization cases. The insight that sensitivity direction matters more than magnitude enables robust optimization even with approximate physics evaluation, extensible to other differentiable physics-driven design objectives.
Shengyu Yan, Jasmin Jelovica
Department of Mechanical Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada · Department of Civil Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada