Compact Thermal Models (CTMs)

• Compact Thermal Models are reduced abstractions that replace continuous temperature distributions with equivalent networks of thermal resistances and capacitances for efficient simulation.
• They integrate diverse approaches including RC networks, analytical approximations, state-space neural models, and nonlocal kernels to address various thermal transport regimes.
• CTMs enable rapid electrothermal co-simulation and design-space exploration by balancing model fidelity with computational speed, validated against high-fidelity simulations and measurements.

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 (Barua et al., 26 Mar 2026). 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 (Zhu et al., 5 Dec 2025, 0710.4759, Kirchgässner et al., 2021, Vermeersch, 2016).

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 (Barua et al., 26 Mar 2026). 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 (Barua et al., 26 Mar 2026).

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 (Zhu et al., 5 Dec 2025). 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 (Kirchgässner et al., 2021). 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 (Vermeersch, 2016).

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 (Barua et al., 26 Mar 2026, Zhu et al., 5 Dec 2025). 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 $T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$0 assembled from local conductances and $T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$1 from thermal capacitances (Barua et al., 26 Mar 2026, Zhu et al., 5 Dec 2025).

Several constitutive details matter disproportionately in modern CTMs. First, anisotropy is central in BEOL, interposer, microbump, and package structures. The review writes

$T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$2

making clear that directional heat spreading can no longer be reduced safely to one scalar conductivity in many 2.5D/3D stacks (Barua et al., 26 Mar 2026). Second, interfaces introduce temperature jumps. The same review gives thermal boundary resistance in the form

$T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$3

together with flux continuity across the interface (Barua et al., 26 Mar 2026). 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

$T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$4

while 3D-ICE 4.0 gives direction-specific resistance formulas such as

$T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$5

for a Cartesian control volume (Barua et al., 26 Mar 2026, Zhu et al., 5 Dec 2025). 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 (Zhu et al., 5 Dec 2025). 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

$T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$6

and generates per-tile equivalent anisotropic models (Zhu et al., 5 Dec 2025). 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 $T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$7, whereas preserving detailed material distribution increases the hotspot to $T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$8 (Zhu et al., 5 Dec 2025). 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 $T(x,y)=\min\{T_0,T_{\text{line}}(x,y)\}$9, 3D-ICE 4.0 defines the per-element vertical resistance

$\kappa,\pi,\gamma$0

the equivalent vertical resistance

$\kappa,\pi,\gamma$1

and the stack-wide variance

$\kappa,\pi,\gamma$2

Layers with large $\kappa,\pi,\gamma$3 or large contribution to the variance are subdivided iteratively before simulation (Zhu et al., 5 Dec 2025). 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 (Zhu et al., 5 Dec 2025).

The third major refinement is temperature-aware non-uniform grid generation. After a coarse initial solve, local average temperature gradients drive grid sizing through

$\kappa,\pi,\gamma$4

with clamping to $\kappa,\pi,\gamma$5 (Zhu et al., 5 Dec 2025). In the reported benchmark, to achieve RMSE $\kappa,\pi,\gamma$6, a uniform grid requires 11,760 grids while the non-uniform grid requires 8,714, a 25.9% reduction; for RMSE $\kappa,\pi,\gamma$7, the reduction is 23.3% (Zhu et al., 5 Dec 2025). The abstract summarizes the broader performance result as speedups ranging from $\kappa,\pi,\gamma$8–$\kappa,\pi,\gamma$9 over state-of-the-art tools, with grid complexity reduced by more than 23.3% without compromising accuracy (Zhu et al., 5 Dec 2025).

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 (Barua et al., 26 Mar 2026).

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

$P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$0

with superposition over multiple rectangles through

$P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$1

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

$P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$2

with the state vector itself equal to the physical temperatures (Kirchgässner et al., 2021). In the motor dataset studied, a small TNN with 64 parameters achieved a mean squared error of $P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$3 and a worst-case error of $P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$4, outperforming the previous LPTN baseline while keeping a compact state dimension (Kirchgässner et al., 2021). 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 (Sabry et al., 3 Oct 2025). 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

$P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$5

The two characteristic times are associated with $P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$6 for the bulk fluid and

$P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$7

for the wall-to-bulk branch (Sabry et al., 3 Oct 2025). 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 $P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$8 such that

$P(\vec{\xi},s)=1/[s+\psi(\|\vec{\xi}\|)]$9

Within the isotropic Poissonian-flight formulation, once this single scalar function is known, the full bulk spatiotemporal thermal response in transformed space is known (Vermeersch, 2016).

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 (Vermeersch, 2016). In the diffusive limit,

$$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$0

recovering the Gaussian Fourier heat kernel. Outside that limit, the paper distinguishes diffusive/Brownian scaling $$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$1, quasiballistic alloy Lévy-like scaling $$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$2 with $$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$3, and quasiballistic single-crystal Cauchy-like scaling $$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$4 (Vermeersch, 2016).

The compactness lies in low-parameter propagators. For alloys, the tempered Lévy form

$$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$5

compresses the full isotropic bulk quasiballistic kernel into three parameters $$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$6. For single crystals, the log-tempered Cauchy form

$$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$7

provides an analogous three-parameter description $$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$8 (Vermeersch, 2016). 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 (Vermeersch, 2016).

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 (Barua et al., 26 Mar 2026). 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 (Barua et al., 26 Mar 2026).

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 (Zhu et al., 5 Dec 2025). The on-chip analytical electrothermal model is compared against SPICE for leakage and against self-heating measurements on 0.35 $$\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K \nabla T) + Q,$$9 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 (Kirchgässner et al., 2021). 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 (Sabry et al., 3 Oct 2025). The quasiballistic kernel model highlights a different limitation: finite experimental $$\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (\mathbf{K} \nabla T) + q$$0-windows can make several propagator forms nearly indistinguishable, so extrapolation outside the probed band is model-based rather than directly identified (Vermeersch, 2016).

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 (Barua et al., 26 Mar 2026). 3D-ICE 4.0 remains a discretized RC solver with heuristic refinement policies, not a highly reduced macro-model (Zhu et al., 5 Dec 2025). 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 (Kirchgässner et al., 2021). The quasiballistic kernel relies on isotropy and targets the weakly quasiballistic regime $$\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (\mathbf{K} \nabla T) + q$$1 rather than fully ballistic transport (Vermeersch, 2016). The convection CTM is derived for laminar forced convection in a parallel-plate channel with constant properties and simplified inlet conditions (Sabry et al., 3 Oct 2025).

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 (Barua et al., 26 Mar 2026). 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.