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

Xinling Yu, Yixing Li, Ziyue Liu, Xin Ai, Zhiyu Zeng, Hai Li, Zheng Zhang

arXiv:2608.16080v1cs.LGphysics.data-an

TL;DR

High-contrast multi-die stacks challenge physics-only thermal surrogates because discontinuous interfaces and ill-conditioning hinder training. DeepOHeat-v2 combines discretized energy-form training, matrix-preconditioned optimization, and self-improvement from solver-verified placements, reducing the returned-design peak-temperature gap to 0.11 K while running about 56× faster than solving at every step.

  • Problem

    High-contrast multi-die stacks challenge physics-informed surrogates because smooth networks cannot represent nonsmooth interface fields, while changing TSV layouts make interface-specific architectures impractical.

  • Method

    DeepOHeat-v2 combines discretized energy-form physics training, matrix-preconditioned optimization, and trust-gated retraining on verified solver solutions.

  • Results

    0.11 K was the surrogate–true peak-temperature gap at the returned design, with the integrated loop running approximately 56× faster than solving at every step.

  • Takeaways & Limitations

    The framework delivers oracle-quality returned-design predictions for thermal placement optimization without solving at every optimization step.

  • Takeaways & Limitations

    The surrogate architecture is tied to axis-aligned, grid-resolved structures, so round, tapered, off-grid, irregular, or non-Manhattan geometries require architectural or mesh changes.

Abstract

from arXiv · show

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 ($κ_2(A_h) \approx 6 \times 10^4$) 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 $κ^2$ 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 $56\times$ faster.

I. INTRODUCTION … 1) Operator Learning for Thermal Simulation:

DeepOHeat-v2 targets the computational cost and reliability challenges of thermal-aware 3D-IC optimization by replacing repeated heat-equation solves with a physics-trained neural operator. It extends DeepOHeat-v1 to heterogeneous stacks by handling discontinuous conductivities and improving surrogate reliability during optimization.

  • I. INTRODUCTION: 3D-ICs concentrate power and lengthen thermal paths, making heat dissipation difficult and degrading timing, reliability, lifetime, and leakage.
  • 1) Operator Learning for Thermal Simulation:: Operator learning approximates the design-to-temperature solution map, enabling a new configuration’s temperature field from one forward pass instead of a fresh thermal solve.
  • I. INTRODUCTION: DeepOHeat-v1’s residual-based refinement recovered solver-level accuracy, but unreliable confidence signals and discarded refined solutions prevented the surrogate from improving.
  • I. INTRODUCTION: DeepOHeat-v2 trains on a discretized physics loss that handles layer and TSV conductivity discontinuities natively, without interface-condition terms or TSV homogenization.
  • I. INTRODUCTION: The energy-form loss reduces prediction-space loss-Hessian conditioning from κ^2 to κ, addressing the ill-conditioning that stalls first-order optimization.
  • A. Thermal-Aware Design Optimization: Every optimization step requires evaluating peak temperature, and each evaluation entails a full heat-equation solve that dominates optimization cost.
  • B. The DeepOHeat-v1 Framework: DeepOHeat-v1 provided a fast, trustworthy surrogate and a hybrid optimization loop, but its continuous strong-form loss fails across discontinuous conductivities in realistic multi-die stacks.

2) Separable Operator Network with ChebyKAN Trunks: · 3) Data-Free Physics-Informed Training:

DeepOHeat uses a separable operator network with ChebyKAN trunks to make full-grid physics-informed training feasible at chip resolution. It trains directly from governing equations and boundary conditions through automatic differentiation, without simulation or labeled solver data.

  • 2) Separable Operator Network with ChebyKAN Trunks:: DeepOHeat represents the surrogate as a DeepONet whose branch encodes configuration u into coefficients and whose trunk maps spatial coordinates to basis functions.The predicted temperature field is formed by the inner product of branch coefficients and trunk basis functions.
  • 2) Separable Operator Network with ChebyKAN Trunks:: DeepOHeat-v1 replaces one trunk over y = (y1, y2, y3) with separate trunks for the spatial axes.This separable design changes the trunk structure from a single three-dimensional network to one network per coordinate axis.
  • 2) Separable Operator Network with ChebyKAN Trunks:: N1 + N2 + N3 trunk evaluations and an outer product replace N1N2N3 pointwise evaluations for full-grid prediction.Combined with forward-mode differentiation of the physics loss, this makes physics-informed training feasible at full-chip resolution.
  • 2) Separable Operator Network with ChebyKAN Trunks:: ChebyKAN trunks use learnable univariate edge functions to mitigate the spectral bias of plain MLP trunks and resolve spatial structure.The architecture uses Chebyshev Kolmogorov–Arnold networks for accuracy.
  • 3) Data-Free Physics-Informed Training:: The surrogate is trained directly from the governing equations, defining an interior residual for each candidate field T and configuration u.Training configurations are used to evaluate the physics residuals rather than to provide simulation labels.
  • 3) Data-Free Physics-Informed Training:: The continuous physics-informed loss penalizes interior and boundary residuals of predicted fields at collocation points, with λb > 0 balancing the terms.The boundary operator collects the conditions in Eq. (3), and the loss is minimized over training configurations.
  • 3) Data-Free Physics-Informed Training:: Automatic differentiation applies all derivatives to the network output, so training requires no labeled solver data and is data-free.This is the stated reason DeepOHeat-v1 is data-free.

4) Hybrid Optimization for Trustworthiness: … A. An FVM-Discretized Physics Loss

DeepOHeat-v2 combines finite-volume physics training with hybrid verification and refinement to address discontinuous interfaces, severe ill-conditioning, and surrogate errors on high-contrast multi-die stacks. The method represents interface physics by construction while retaining a data-free discretized residual formulation.

  • 4) Hybrid Optimization for Trustworthiness:: The verify-and-refine loop accepts predictions with small discretized residuals and solves flagged cases with GMRES warm-started from the predicted field.Residual evaluation costs one sparse matrix–vector product, and warm starts require fewer iterations than cold solves.
  • C. Challenges of High-Contrast Multi-Die Stacks: An ∼800× conductivity range in the benchmark multi-die stack creates discontinuous interfaces and challenges that DeepOHeat-v1 cannot handle.The stack includes stacked dies, copper TSVs, and an organic substrate with conductivities orders of magnitude apart.
  • C. Challenges of High-Contrast Multi-Die Stacks: DeepOHeat-v2’s training method fixes the interface and conditioning challenges, while its self-improving framework addresses the hybrid loop’s third challenge.The paper separates the training contribution from the self-improving contribution.
  • A. An FVM-Discretized Physics Loss: The continuous strong-form loss is singular at discontinuous interfaces because its interface term lacks a finite value unless the predicted normal derivative vanishes.This would force a smooth network to carry no normal flux across interfaces, contradicting conducting heat.
  • A. An FVM-Discretized Physics Loss: The physical temperature is C0 but not C1 at material interfaces, placing it outside the hypothesis class of any C1 network.Continuity of normal heat flux forces an abrupt slope change despite temperature continuity.
  • A. An FVM-Discretized Physics Loss: FVM with harmonic-mean face conductivities produces an SPD system whose flux continuity handles discontinuities by construction, without requiring solved fields for training.The data-free squared residual is assembled directly from Ah and bh coefficients.
  • A. An FVM-Discretized Physics Loss: κ2(Ah) ≈6.02×104 on the benchmark, while the squared residual’s Hessian reaches condition number κ2(Ah)2 ≈3.6 × 109.The squared residual therefore puts the loss beyond first-order optimization.

B. A Reformulation with the Energy Form

The section replaces the squared strong-form residual with a discrete energy-form loss for discontinuous finite-volume systems. This changes prediction- and parameter-space conditioning from κ^2 to κ without increasing evaluation cost, making first-order operator learning feasible at high contrast.

  • Energy-form loss: The energy-form loss replaces the squared residual while preserving the same linear-system solution as its unique minimizer.It is a strictly convex quadratic whose curvature is Ah rather than A_h^2.
  • Conditioning improvement: κ^2 → κ: The energy form reduces prediction-space Hessian conditioning from κ2(A_h)^2 to κ2(A_h).This conditioning gain is the central result of Theorem 4.
  • Parameter-space conditioning: The κ^2 → κ reduction carries into parameter-space Gauss–Newton conditioning near a good fit, with the common Jacobian factor κ(J)^2 canceling in comparison.The energy loss adds no computational cost because both formulations use the same matrix-free stencil.
  • Matrix-free implementation: Matrix-free evaluation uses local six-neighbor face-flux stencils and a Robin boundary term, with face conductances precomputed from the geometry.Neither loss requires explicitly forming A_h; evaluation only applies A_h to the predicted field.

C. Optimization with a Matrix-Valued Preconditioner · IV. SELF-IMPROVING OPERATOR-LEARNING FRAMEWORK · A. Online Adaptation: Incremental Training and Model Selection

DeepOHeat-v2 combines matrix-aware preconditioning with online self-improvement to address ill-conditioned optimization and distribution shift during thermal-placement search. A trust-gated refinement loop incrementally retrains and validates the surrogate, then runs it alone after a fixed adaptation budget.

  • C. Optimization with a Matrix-Valued Preconditioner: Muon2 preconditions matrix-shaped branch-network updates jointly across directions, while Adam trains the ChebyKAN trunk tensors and one-dimensional parameters.Muon approximates the orthogonal polar factor with a degree-5 Newton–Schulz iteration; the configuration applies it to 2D MLP branch weights.
  • C. Optimization with a Matrix-Valued Preconditioner: The pretraining study finds that Adam plateaus on the energy loss, Muon removes most conditioning, and Muon2 reaches the lowest error.
  • IV. SELF-IMPROVING OPERATOR-LEARNING FRAMEWORK: The optimizer shifts the search toward low-peak placements that are rare in the random pretraining distribution, creating a distribution gap for surrogate evaluation.Figure 1 contrasts broad random-placement temperatures with concentrated temperatures visited along an optimization trajectory.
  • IV. SELF-IMPROVING OPERATOR-LEARNING FRAMEWORK: A trust gate sends flagged trajectory placements to a solver, and periodic retraining on refined solutions improves the surrogate where the search travels.Refinement and retraining occur only during a fixed number of early rounds; later search uses the adapted surrogate alone.
  • A. Online Adaptation: Incremental Training and Model Selection: Incremental training combines the fixed pretraining set with an online buffer of solver-refined placements while carrying forward the pretrained architecture, optimizer, and state.The loss retains energy terms on original and refined data and adds a data-MSE term on refined labels.
  • A. Online Adaptation: Incremental Training and Model Selection: Each adaptation event evaluates the pre-update model, updated model, and weight average on held-out trajectory data, deploying the candidate with minimum mean absolute peak-temperature error.Including the pre-update model ensures the deployed model is never worse on the held-out validation slice.
  • A. Online Adaptation: Incremental Training and Model Selection: After the fixed number of adaptation events, proposals use the surrogate alone without further solver calls or retraining.The limit prevents indefinite refinement because the trust gate flags a fixed fraction of proposals by design.

B. When to Call the Non-AI Solver: The Hotspot Trust Gate

The hotspot trust gate replaces DeepOHeat-v1’s unreliable global-residual ranking with a hotspot-localized residual and an adaptive sliding-window threshold. This targets rare high-error placements while adapting to residual-scale changes during search and retraining.

  • Why the global residual fails: ρ = −0.018 shows that v1’s global relative residual orders held-out predictions essentially at random by peak error.The global bound also evaluates to ≈1.6×10^5 K because it carries κ_2(A_h) = 6.02×10^4, far above observed sub-Kelvin errors.
  • Hotspot-localized residual: The gate ranks predictions using the mean residual magnitude over the top-m hottest predicted cells.This hotspot-localized residual measures error where the gate needs it rather than across the whole field.
  • Hotspot-localized residual: One sparse matrix–vector product computes the hotspot-localized residual against the same b_h used for surrogate training.A local bound replaces the global norm with (A_h^-1)_x*,x*, which does not grow with κ_2(A_h).
  • Adaptive sliding-window flagging: A FIFO window W of recent hotspot-residual values sets the threshold relative to residuals currently observed.The threshold adapts because residual scale shrinks as the search narrows and shifts after surrogate updates.

C. How to Call the Non-AI Solver: Warm-Started GMRES Refinement … A. The F2F Chiplet Benchmark

The optimization run uses warm-started AMG-GMRES refinement for flagged proposals, then incrementally retrains and selects surrogate updates while solver events remain active. The F2F benchmark is a high-contrast, ill-conditioned two-die chiplet stack with asymmetric cooling, TSVs, and movable power blocks.

  • C. How to Call the Non-AI Solver: Warm-Started GMRES Refinement: Flagged proposals run inherited GMRES refinement on AhT = bh from the warm start T0 = bT with a Ruge–Stüben AMG V-cycle preconditioner.Because the warm prediction is already within a few K of the true field, refinement uses a deliberately loose relative tolerance ϵ.
  • D. Algorithm: During each optimization run, proposals pass through a percentile trust gate; flagged cases are refined, buffered, and used for incremental-training events with model selection.The refined peak replaces the surrogate peak in acceptance testing, while unflagged proposals retain the surrogate estimate.
  • D. Algorithm: After the event budget is exhausted, search continues on the adapted surrogate alone, followed by one tight FVM verification of the best-so-far placement.This final verification is distinct from the warm-started refinements used during the active solver phase.
  • A. The F2F Chiplet Benchmark: The benchmark is a 3×3 mm face-to-face stack of two bulk-plus-active-silicon dies bonded through hybrid Cu–Cu and mounted on an organic substrate.Cooling uses a top water cold plate with h = 5000 W/m2K and weak bottom board conduction with h = 50 W/m2K.
  • A. The F2F Chiplet Benchmark: Ten power blocks dissipate ≈3.5 W in total, and their same-die x,y positions define the configuration subject to non-overlap constraints.Cross-die overlap is allowed; the blocks include four hotspots and six lower-power background blocks.
  • A. The F2F Chiplet Benchmark: 6.02×10^4 is the Lanczos estimate of κ2(Ah), placing the F2F benchmark far beyond DeepOHeat-v1’s demonstrated conditioning regime.The challenge arises from high conductivity contrast in the discretized operator.

B. Experimental Setup · C. Pretraining: Loss Form and Optimizer · D. Confidence-Gate Validation

The paper evaluates DeepOHeat-v2 on a fixed multi-die benchmark, establishing energy-form Muon2 pretraining and validating a trust gate that identifies high-error placements. These components support accurate surrogate predictions and targeted refinement during optimization.

  • B. Experimental Setup: Experiments use 100 held-out random placements to test the loss/optimizer recipe, confidence-gate catch rate, and full simulated-annealing loop.The study uses a single fixed protocol and evaluates the complete loop from a random initial placement.
  • B. Experimental Setup: Four configurations isolate per-proposal cost policies: surrogate-only, DeepOHeat-v1, DeepOHeat-v2, and oracle simulated annealing with a tight solve at every step.DeepOHeat-v1 adds flagging and refinement without updating the surrogate, while DeepOHeat-v2 adds online adaptation; oracle SA is a ground-truth upper bound.
  • C. Pretraining: Loss Form and Optimizer: Pretraining uses 10,000 unlabeled power maps without temperature labels, comparing strong-form and energy-form losses with Adam, Muon, and Muon2.A supervised control uses 100 labeled pairs and pure MSE; all combinations are evaluated on 100 held-out placements.
  • C. Pretraining: Loss Form and Optimizer: 1.05 → 0.64 → 0.55 K: within the energy form, Muon reduces error 39% below Adam and Muon2 provides another 14% reduction.Muon2 is selected as the default optimizer because it adds restoration of the smallest singular directions to Muon’s joint rescaling.
  • C. Pretraining: Loss Form and Optimizer: About 30 K: under the strong-form loss, Adam plateaus far from optimum while both Muon variants diverge at κ2 ≈ 3.6×10^9.The energy form avoids this failure because its lower conditioning preserves the optimizer’s contraction.
  • C. Pretraining: Loss Form and Optimizer: 0.55 K mean peak temperature error: the energy+Muon2 surrogate closely reproduces held-out reference fields and localizes hotspots.The largest residuals are confined to block edges, and this pretrained surrogate is used in the remaining experiments.
  • D. Confidence-Gate Validation: ρ = +0.642 versus ρ = −0.018: rhot ranks predictions by true error better than rrel, catching 64.4% of >0.5 K errors and 91.7% of >1 K errors at a 30% flag budget.rrel remains near the 30% random baseline, catching 24.4% and 33.3%, respectively.
  • D. Confidence-Gate Validation: 12% of held-out placements have peak temperature error >1 K and 2% exceed 2 K, confirming a substantive high-error tail that requires solver-based identification.Which individual placements belong to this tail cannot be known without a solve.

E. Online Stability of the Adaptation Loop · F. Integrated Run: Optimization Quality and Cost · VI. CONCLUSION

DeepOHeat-v2 stabilizes online adaptation through validation-based model selection and anti-forgetting, then delivers near-oracle integrated optimization quality at substantially lower cost. Its conclusion identifies geometric limits of the current architecture and motivates reusing solver queries as training signals for broader solver-in-the-loop optimization.

  • E. Online Stability of the Adaptation Loop: Event 1 reverted the update when validation error rose to 0.700 K versus the pretrained model’s 0.601 K, while events 2–3 retained the post-update model.The per-event error scale dropped from ∼1 K to ∼0.2 K as validation focused on later simulated-annealing placements.
  • E. Online Stability of the Adaptation Loop: 0.499 K broad-random-set error after adaptation improved on the pretrained 0.551 K, demonstrating no loss of accuracy outside the search trajectory.The energy term preserves broad-distribution accuracy while the data-MSE term fits the trajectory.
  • F. Integrated Run: Optimization Quality and Cost: 0.11 K surrogate–true peak gap was achieved at the returned design in 292 s, about 56× faster than solving at every optimization step.The loop used a bounded number of finite-volume-method solves.
  • F. Integrated Run: Optimization Quality and Cost: 1.12 K, 0.78 K, and 0.11 K were the surrogate–true gaps for surrogate-only, DeepOHeat-v1, and DeepOHeat-v2, respectively.The gap measures prediction error at the returned placement, not placement-quality ranking.
  • F. Integrated Run: Optimization Quality and Cost: 292 s matched Oracle SA’s placement quality versus 4.51 h, with all four configurations within ∼1.8 K on absolute true peak temperature.DeepOHeat-v2’s 387.04 K true peak was 0.32 K below Oracle SA’s 387.36 K because stochastic search trajectories settled in different basins.
  • VI. CONCLUSION: DeepOHeat-v2 addresses data-free optimization by combining trainability on high-contrast multi-die stacks with accuracy maintenance as search placements leave the training distribution.The integrated loop drives the returned-design surrogate–true gap to 0.11 K at approximately 56× the speed of solving at every step.
  • VI. CONCLUSION: The separable operator architecture is limited to axis-aligned, grid-resolved structures, making round, tapered, off-grid, irregular, or non-Manhattan geometries require architectural or discretization changes.The matrix-free FVM loss can train non-separable or mesh-agnostic surrogates without second-order automatic differentiation, paired with geometry-conforming discretization.
  • VI. CONCLUSION: Solver queries already paid for during optimization can be reused as training signals, improving the surrogate where the search concentrates and reducing later solver queries.The conclusion presents this adapt-during-search principle as transferable beyond thermal analysis and the current surrogate.

APPENDIX A OPERATOR CONDITIONING · A. Conditioning of the Continuous Operator · B. Parameter-Space Transfer of the Conditioning Reduction

Appendix A shows that high-contrast thermal operators are intrinsically ill-conditioned, with continuous-operator bounds matching the measured discrete condition number. It also establishes that the conditioning reduction transfers to trainable parameter space without changing the Jacobian contribution.

  • A. Conditioning of the Continuous Operator: The weak formulation defines the continuous heat-conduction operator L on V = H1(Ω), with conductivity, volumetric source, ambient temperature, and Robin boundary coefficients.The operator is induced by the bilinear form a(·, ·) and the linear functional ℓ.
  • A. Conditioning of the Continuous Operator: The continuous condition number is bounded below by the conductivity contrast kmax/kmin, using test functions concentrated in high-k and low-k regions.The Rayleigh-quotient ratio scales as kmax/kmin.
  • A. Conditioning of the Continuous Operator: The upper operator bound follows from continuity, controlled by kmax and the largest film coefficient ¯h.The continuity estimate uses the trace theorem for the boundary term.
  • A. Conditioning of the Continuous Operator: Robin boundary penalization restores coercivity by controlling the full H1 norm and removing the constant mode from the null space.This relies on a Robin–Poincaré inequality rather than a homogeneous Poincaré inequality.
  • A. Conditioning of the Continuous Operator: κ(L) ≳4–8×10^4 matches the measured κ2(Ah) = 6.02×10^4 for kmax/kmin = 800 and c0(Ω) ∼50–100.The result indicates that discrete conditioning is intrinsic to material contrast, not a mesh artifact, and mesh refinement does not reduce it.
  • B. Parameter-Space Transfer of the Conditioning Reduction: The prediction-space loss curvature pulls back to parameter space through the network output Jacobian J on directions that change the prediction.The parameter-space analysis formalizes transfer of the conditioning result to the optimizer’s variables.
  • B. Parameter-Space Transfer of the Conditioning Reduction: The conditioning bound tightens by the same factor relative to κ2(Ah), while the Jacobian factor κ(J)^2 is identical for both losses.Rewriting the loss changes the operator-conditioning contribution but leaves κ(J)^2 unchanged.

APPENDIX B THE HOTSPOT-LOCALIZED BOUND AND OPERATOR · A. Derivation of the Bound

The appendix derives a hotspot-localized a-posteriori bound that replaces the global inverse-operator norm with a local hotspot quantity, avoiding dependence on κ2(A_h). Under hotspot coincidence and hotspot-row diagonal dominance, the bound is about 92× tighter in coefficient terms, though remains numerically loose.

  • A. Derivation of the Bound: The localized bound replaces the global norm ∥A_h^-1∥2 in the error prefactor with the hotspot diagonal entry (A_h^-1)_x*,x*, which does not scale with κ2(A_h).This identifies the local operator quantity controlling the bound instead of the global inverse-operator norm.
  • A. Derivation of the Bound: The proposition assumes hotspot coincidence, arg max_y b^T(y) = arg max_y T(y), so x* is also the reference hotspot.This assumption equates the surrogate and reference hotspot locations for interpreting |e(x*)| as peak temperature error.
  • A. Derivation of the Bound: The second assumption is hotspot-row diagonal dominance, requiring the hotspot diagonal entry (A_h^-1)_x*,x* to dominate the corresponding row entries.Together with hotspot coincidence, this supports the signal-term bound on the localized region.
  • A. Derivation of the Bound: The proof splits the hotspot row of e := b^T − T = A_h^-1r into contributions on and off S_m, bounding the localized signal and tail separately.The signal term uses diagonal dominance, while the tail applies Cauchy–Schwarz to the off-region row.
  • A. Derivation of the Bound: The proof combines the signal and tail bounds to obtain the localized bound, and verifies assumptions (A1) and (A2) numerically below.The derivation concludes by adding the two components from the hotspot-row decomposition.
  • A. Derivation of the Bound: ≈1.7×10^3 K versus ≈1.6×10^5 K: the localized bound is about 92× tighter than the global relative-residual bound.The reported operator norms are ∥A_h^-1∥2 = 3.4 × 10^6 K/W and ∥(A_h^-1)_x*,·/∈S_m∥2 = 3.69 × 10^4 K/W.
  • A. Derivation of the Bound: 0.5–3 K: both localized and global bounds are numerically loose relative to observed errors, serving instead to identify the operator quantity controlling error.The hotspot residual indicator’s detection power is established empirically through Spearman correlation and catch rate in Section V-D.

B. Operator Numerics and Assumption Verification

The assembled operator has severe estimated conditioning, with the strong-form loss-Hessian condition derived from it. The two assumptions underlying Proposition 7 are checked on a held-out set using hotspot coincidence and hotspot-row behavior.

  • Conditioning estimate: 6.02×10^4 is the estimated condition number κ2(Ah), obtained from largest and smallest eigenvalues using forward Lanczos and AMG-preconditioned inverse iteration.The estimate converges to three significant figures within a few hundred Lanczos steps.
  • Conditioning estimate: 3.6×10^9 is the derived strong-form loss-Hessian conditioning κ2, computed as κ2(Ah)^2 rather than measured separately.This value follows from κ2(Ah) = 6.02×10^4 as in Section III-A.
  • Verification of A1 and A2: A1 is checked on the same n = 100 held-out placements through hotspot coincidence or inclusion of the reference hotspot in the predicted top-m set Sm.Most placements have top-1 coincidence; remaining nearmisses are absorbed by top-m aggregation and reported through the catch-rate curve.
  • Verification of A1 and A2: A2 is checked through the hotspot-row diagonal behavior and tail-row norm, consistent with a thermal-spreading kernel that decays away from the source.The quantities (Ah^-1)x∗,x∗ = 298 K/W and the tail-row norm are read from the same computation.
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