---
title: Compact Thermal Models (CTMs)
url: https://www.emergentmind.com/topics/compact-thermal-models-ctms
type: topic
---

# Compact Thermal Models (CTMs)

Compact Thermal Models (CTMs) are reduced thermal abstractions that replace a continuous temperature field with an equivalent network of thermal resistances, thermal capacitances, or macromodel states, and are designed for fast repeated evaluation in design-space exploration, electrothermal co-simulation, runtime management, and package/system co-design [2604.03290]. In current research usage, the term covers more than one modeling tradition: equivalent thermal RC networks for chips and packages, analytical on-chip electro-thermal approximations, lumped state-space models with learned parameter maps, and, in specialized nondiffusive regimes, compact transport kernels that compress Boltzmann-level physics into a small constitutive representation [2512.05823] [0710.4759] [2103.16323] [1608.06120].

## 1. Definition and modeling scope

The review literature characterizes CTMs as models that replace a continuous temperature field with an equivalent network of thermal resistances, capacitances, or macromodel states, while abstracting geometry into effective thermal pathways and enabling SPICE-like electrothermal simulation [2604.03290]. This places CTMs at the deployment end of a multiscale hierarchy: atomistic and mesoscopic methods determine fine-scale transport, continuum methods resolve realistic fields, and compact models provide the fast surrogate used in repeated system-level analysis [2604.03290].

The term is nevertheless used with different degrees of literalness. In heterogeneous chiplet analysis, 3D-ICE 4.0 is explicitly framed as a compact thermal modeling framework, but in the sense of a discretized equivalent thermal RC network rather than a tiny few-parameter macromodel [2512.05823]. In sub-100 nm CMOS, the concurrent power-thermal model is best described as a coupled power-temperature analytical approximation for steady-state on-chip electro-thermal estimation, rather than a classical Foster/Cauer package CTM [0710.4759]. Thermal Neural Networks preserve the philosophy of lumped-parameter thermal networks, but learn the dependence of conductances, losses, and capacitances on operating conditions from data [2103.16323]. In quasiballistic heat transport, the compact object is not an RC network at all, but a scalar spatial propagator that acts as a reduced-order nonlocal material law [1608.06120].

| CTM class | Representative form | Typical scope |
|---|---|---|
| Equivalent thermal RC network | \(C\,\dot{\mathbf T}+G\,\mathbf T=\mathbf P+\mathbf b\) | Chip/package thermal maps |
| Analytical electro-thermal approximation | \(T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}\) | Steady-state on-chip estimation |
| Lumped neural state-space CTM | Thermal-node update with learned \(\kappa,\pi,\gamma\) | Real-time temperature estimation |
| Physics-derived convection CTM | Two first-order ODEs | Transient forced laminar convection |
| Nonlocal transport kernel | \(P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]\) | Quasiballistic media |

This multiplicity of usage suggests that “compact” in thermal modeling denotes reduced thermal representation rather than one unique topology. The common denominator is dimensional reduction: a distributed thermal problem is compressed into a smaller model class that preserves the dominant input-output behavior over a specified operating domain.

## 2. Governing principles and canonical formulations

The physical starting point for CTMs in heterogeneous integrated systems is the transient heat equation. The review gives the anisotropic continuum form
\[
\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,
\]
while 3D-ICE 4.0 writes the heterogeneous anisotropic form as
\[
\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (\mathbf{K} \nabla T) + q
\]
with \(K\) or \(\mathbf K\) representing anisotropic thermal conductivity [2604.03290] [2512.05823]. In compact form, this becomes a finite-dimensional thermal balance. The review presents an implied reduced state-space form \(C\,\dot{\mathbf T}+G\,\mathbf T=\mathbf P(t)\), and 3D-ICE 4.0 describes the standard nodal RC balance as
\[
\mathbf{C}\,\frac{d\mathbf{T}}{dt} + \mathbf{G}\,\mathbf{T} = \mathbf{P} + \mathbf{b},
\]
with \(\mathbf G\) assembled from local conductances and \(\mathbf C\) from thermal capacitances [2604.03290] [2512.05823].

Several constitutive details matter disproportionately in modern CTMs. First, anisotropy is central in BEOL, interposer, microbump, and package structures. The review writes
\[
\mathbf q = -K \nabla T,\qquad
K=\begin{bmatrix}
k_x & 0 & 0\\
0 & k_y & 0\\
0 & 0 & k_z
\end{bmatrix},
\]
making clear that directional heat spreading can no longer be reduced safely to one scalar conductivity in many 2.5D/3D stacks [2604.03290]. Second, interfaces introduce temperature jumps. The same review gives thermal boundary resistance in the form
\[
q'' = G (T_1 - T_2) = \frac{T_1 - T_2}{R_K},
\]
together with flux continuity across the interface [2604.03290]. In compact form, this becomes an explicit interface resistor or a contact element between thermal nodes.

At the level of individual compact elements, the review identifies the standard conduction and storage building blocks as
\[
R = \frac{L}{kA}, \qquad C_{th} = \rho C_p V,
\]
while 3D-ICE 4.0 gives direction-specific resistance formulas such as
\[
R_x=\frac{\Delta x}{k_x A_x},\qquad
R_y=\frac{\Delta y}{k_y A_y},\qquad
R_z=\frac{\Delta z}{k_z A_z}
\]
for a Cartesian control volume [2604.03290] [2512.05823]. These relations motivate why compact models for heterogeneous systems are often multi-node rather than single-junction abstractions: once anisotropy, interfaces, lateral spreading, and vertical bottlenecks are important, the dominant thermal physics already has more than one effective degree of freedom.

## 3. RC-network CTMs for heterogeneous ICs and chiplet systems

In contemporary chip thermal analysis, the dominant CTM form remains the equivalent thermal RC network. 3D-ICE 4.0 explicitly places HotSpot, 3D-ICE, and PACT in this category and advances the tradition toward higher fidelity for 2.5D/3D heterogeneous chiplet systems [2512.05823]. Its contribution is not to minimize the number of states, but to preserve material heterogeneity and anisotropy directly from industrial layouts while retaining the sparse-network speed advantage over FEM.

The framework ingests GDSII, polygonizes and merges layout primitives into disjoint shapes, builds an R-tree over bounding boxes, recursively subdivides the domain with a quadtree, computes an overlap ratio
\[
\rho=\frac{A_{\text{ov}}}{\mathrm{Area}(R)},
\]
and generates per-tile equivalent anisotropic models [2512.05823]. This is a significant shift away from the older CTM practice of layer homogenization. The paper reports, for a 4-chiplet example, that a homogenized 3D-ICE 3.1 model predicts a hotspot of \(390.15\,\mathrm K\), whereas preserving detailed material distribution increases the hotspot to \(394.22\,\mathrm K\) [2512.05823]. The implication is that a coarse CTM may remain computationally compact while becoming physically non-conservative for hotspot analysis.

A second innovation is adaptive vertical layer partitioning. For layer \(j\), 3D-ICE 4.0 defines the per-element vertical resistance
\[
R^{\perp}_{j,i} = \frac{h_j}{k^{\perp}_{j,i}\cdot A_{j,i}},
\]
the equivalent vertical resistance
\[
R^{\perp}_{j} = \left(\sum_{i=1}^{N_j} \frac{1}{R^{\perp}_{j,i}}\right)^{-1},
\]
and the stack-wide variance
\[
\mathrm{Var}\!\left(R^{\perp}\right)=\frac{1}{n}\sum_{j=1}^{n}\left(R^{\perp}_j-\bar R^{\perp}\right)^2.
\]
Layers with large \(R^\perp_j\) or large contribution to the variance are subdivided iteratively before simulation [2512.05823]. In the reported 2.5D case, after 8 iterations, TIM and PCB are each subdivided into four sub-layers and the chip into two sub-layers, while the total number of grids per functional layer is held constant; the improved agreement with COMSOL is therefore attributed to better vertical modeling rather than a merely larger model [2512.05823].

The third major refinement is temperature-aware non-uniform grid generation. After a coarse initial solve, local average temperature gradients drive grid sizing through
\[
\mathrm{Gridsize}_{x}=l_x\cdot \alpha \cdot \left(\frac{G_{base}}{G+\epsilon}\right),
\]
with clamping to \([l_{min},l_x]\) [2512.05823]. In the reported benchmark, to achieve RMSE \(<0.3\,\mathrm K\), a uniform grid requires 11,760 grids while the non-uniform grid requires 8,714, a 25.9% reduction; for RMSE \(<0.25\,\mathrm K\), the reduction is 23.3% [2512.05823]. The abstract summarizes the broader performance result as speedups ranging from \(3.61\times\)–\(6.46\times\) over state-of-the-art tools, with grid complexity reduced by more than 23.3% without compromising accuracy [2512.05823].

These developments align with the review’s broader argument that CTMs in heterogeneous 2.5D/3D systems must preserve inter-tier coupling, lateral chiplet crosstalk, hotspot superposition, interface resistance, and package context if they are to remain credible deployment models rather than oversimplified surrogates [2604.03290].

## 4. Analytical, electro-thermal, and data-driven compact models

Not all CTMs are explicit RC meshes. A distinct line of work uses compact analytical formulas to collapse heat spreading and electrothermal coupling into closed-form approximations. For sub-100 nm digital ICs, the concurrent power-thermal model introduces a compact analytical framework for leakage and steady-state temperature estimation that avoids full SPICE simulation, numerical solution of the heat diffusion PDE, and, in key places, iterative numerical solving of stack node voltages [0710.4759]. On the thermal side, the central approximation for a rectangular heat source is
\[
T(x,y)=\min\{T_0,\;T_{\text{line}}(x,y)\},
\]
with superposition over multiple rectangles through
\[
T(x,y)= \sum_{i=1}^{M} \min\{T_i,\;T_{\text{line}}(x-x_i,\;y-y_i)\}.
\]
The model is explicitly steady-state, on-chip, and analytical; it is therefore adjacent to, rather than identical with, classical package-level RC CTMs [0710.4759].

A second branch is grey-box learning. Thermal Neural Networks retain the lumped-parameter thermal network structure but learn inverse capacitances, conductances, and losses from data. The thermal-node update is written as
\[
\hat\vartheta_i[k+1] = \hat\vartheta_i[k] + T_s\,\kappa_i[k]
\left(
\pi_i[k]
+ \sum_{j\neq i}(\hat\vartheta_j[k]-\hat\vartheta_i[k])\gamma_{i,j}[k]
+ \sum_{j=1}^{n}(\tilde\vartheta_j[k]-\hat\vartheta_i[k])\gamma_{i,j}[k]
\right),
\]
with the state vector itself equal to the physical temperatures [2103.16323]. In the motor dataset studied, a small TNN with 64 parameters achieved a mean squared error of \(3.18~\mathrm{K}^2\) and a worst-case error of \(5.84~\mathrm K\), outperforming the previous LPTN baseline while keeping a compact state dimension [2103.16323]. This is a CTM in the strong sense of a low-dimensional thermal state model, but with parameter maps inferred statistically rather than from geometry.

A third variant appears in transient forced laminar convection. There the compact-modeling claim is methodological: transient convection should not be modeled by a time-varying thermal resistance, because a transient process contains both a static/resistive part and a dynamic/storage part [2510.03411]. The derived compact pattern is a pair of first-order dynamics associated with bulk-fluid temperature and near-wall temperature, implemented as ODEs of the form
\[
\tau \frac{d\theta}{dt}+\theta = R\,q_m\,r(t).
\]
The two characteristic times are associated with \(\tau_1\sim z/Pe\) for the bulk fluid and
\[
\tau_2=\frac{z/Pe}{1+\pi^2 z/Pe}
\]
for the wall-to-bulk branch [2510.03411]. In this usage, the CTM is a rational low-order transfer structure extracted analytically from the governing PDE.

## 5. Nonlocal CTMs for quasiballistic transport

A broader but technically important extension of CTM methodology arises when Fourier diffusion itself ceases to be the appropriate constitutive law. In multidimensional quasiballistic heat transport, the compact representation is not a resistor-capacitor network but a scalar spatial propagator \(\psi(\|\vec{\xi}\|)\) such that
\[
P(\vec{\xi}, s) = \frac{1}{s + \psi(\|\vec{\xi}\|)}.
\]
Within the isotropic Poissonian-flight formulation, once this single scalar function is known, the full bulk spatiotemporal thermal response in transformed space is known [1608.06120].

This model is explicitly proposed because extraction of “effective” Fourier conductivities from TTG, TDTR, FDTR, spot-size, or nanograting measurements yields geometry- and protocol-dependent surrogates rather than intrinsic descriptors of the medium [1608.06120]. In the diffusive limit,
\[
\psi(\zeta)\to D\zeta^2,
\]
recovering the Gaussian Fourier heat kernel. Outside that limit, the paper distinguishes diffusive/Brownian scaling \(\psi\sim \zeta^2\), quasiballistic alloy Lévy-like scaling \(\psi\sim \zeta^\alpha\) with \(1<\alpha<2\), and quasiballistic single-crystal Cauchy-like scaling \(\psi\sim \zeta\) [1608.06120].

The compactness lies in low-parameter propagators. For alloys, the tempered Lévy form
\[
\psi(\zeta)=\frac{D\zeta^2}{\left(1+r_{\mathrm{LF}}^2\zeta^2\right)^{1-\alpha/2}}
\]
compresses the full isotropic bulk quasiballistic kernel into three parameters \((D,\alpha,r_{\mathrm{LF}})\). For single crystals, the log-tempered Cauchy form
\[
\psi(\zeta)=\frac{D\zeta^2}{\ln 2}\ln\!\left[1+\left(1+r_{\mathrm{CF}}^{\,b}\zeta^b\right)^{-1/b}\right]
\]
provides an analogous three-parameter description \((D,r_{\mathrm{CF}},b)\) [1608.06120]. Practical demonstrations are given for raw TTG data on GaAs and collective TDTR fitting on InGaAs. The paper’s own classification is careful: this is a compact thermal model in the broader reduced-order sense, but not a circuit CTM for arbitrary packages or multilayer devices [1608.06120].

## 6. Validation, limitations, and emerging directions

Across the literature, CTMs are treated as calibrated surrogates rather than self-sufficient substitutes for full thermal field models. The multiscale review recommends a hierarchical calibration workflow: constrain interface properties first, calibrate layer and package parameters next, and only then fit compact or surrogate models [2604.03290]. The same review insists on validation against both high-fidelity continuum simulations and measurements, using observables such as hotspot temperature, transient response, inter-tier gradients, and package-level distributions [2604.03290].

The validation modalities vary by CTM class. 3D-ICE 4.0 is validated against PACT and COMSOL, with explicit emphasis on capturing both lateral and vertical heat flows in heterogeneous chiplet systems [2512.05823]. The on-chip analytical electrothermal model is compared against SPICE for leakage and against self-heating measurements on 0.35 \(\mu\mathrm m\) nMOS transistors for thermal resistance [0710.4759]. Thermal Neural Networks use cross-validation, a held-out generalization set, and repeated random seeds, with physically interpretable temperatures as states [2103.16323]. The transient convection model is checked against CFD and reports excellent agreement at very small times and at very large times or steady state, with bounded intermediate-time error attributable to the first-order Padé approximation [2510.03411]. The quasiballistic kernel model highlights a different limitation: finite experimental \(\zeta\)-windows can make several propagator forms nearly indistinguishable, so extrapolation outside the probed band is model-based rather than directly identified [1608.06120].

The limitations are correspondingly diverse. The review emphasizes boundary-condition dependence, interface variability, anisotropy, and the pitfalls of decoupled electrical/thermal analysis in 2.5D/3D systems [2604.03290]. 3D-ICE 4.0 remains a discretized RC solver with heuristic refinement policies, not a highly reduced macro-model [2512.05823]. The sub-100 nm analytical model is steady-state and abstracts away transient thermal dynamics [0710.4759]. TNNs depend on representative training data and do not provide a formal proof of stability or passivity [2103.16323]. The quasiballistic kernel relies on isotropy and targets the weakly quasiballistic regime \(|s|\tau\ll 1\) rather than fully ballistic transport [1608.06120]. The convection CTM is derived for laminar forced convection in a parallel-plate channel with constant properties and simplified inlet conditions [2510.03411].

Current research directions follow directly from these constraints. The review calls for interface-aware compact models, multi-fidelity workflows that combine CTMs with local refinement, online model updating using in situ sensing, AI- or physics-informed machine learning that augments rather than replaces compact modeling, and uncertainty-aware CTMs that propagate variability in TBR, TIM properties, boundary conditions, and inferred power maps [2604.03290]. A plausible implication is that future CTMs will be less uniformly “compact” in the classical few-parameter sense and more deliberately matched to the dominant thermal physics of the intended operating domain: sparse RC networks for layout-faithful chiplet analysis, structured state-space models for real-time estimation, and nonlocal constitutive kernels where Fourier diffusion is no longer an adequate abstraction.

Source: https://www.emergentmind.com/topics/compact-thermal-models-ctms