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.
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