DE-HNN: An effective neural model for Circuit Netlist representation

Zhishang Luo1 Truong Son Hy2 Puoya Tabaghi1 Donghyeon Koh4 Michael Defferrard4

Elahe Rezaei3 Ryan Carey3 Rhett Davis5 Rajeev Jain3 Yusu Wang1

1 University of California San Diego 2 Indiana State University

3 Qualcomm Technologies, Inc. 4 Qualcomm Wireless GmbH 5 North Carolina State
University

###### Abstract

The run-time for optimization tools used in chip design has grown with the
complexity of designs to the point where it can take several days to go
through one design cycle which has become a bottleneck. Designers want fast
tools that can quickly give feedback on a design. Using the input and output
data of the tools from past designs, one can attempt to build a machine
learning model that predicts the outcome of a design in significantly shorter
time than running the tool. The accuracy of such models is affected by the
representation of the design data, which is usually a netlist that describes
the elements of the digital circuit and how they are connected. Graph
representations for the netlist together with graph neural networks have been
investigated for such models. However, the characteristics of netlists pose
several challenges for existing graph learning frameworks, due to the large
number of nodes and the importance of long-range interactions between nodes.
To address these challenges, we represent the netlist as a directed hypergraph
and propose a _Directional Equivariant Hypergraph Neural Network_ (DE-HNN) for
the effective learning of (directed) hypergraphs. Theoretically, we show that
our DE-HNN can universally approximate any node or hyperedge based function
that satisfies certain permutation equivariant and invariant properties
natural for directed hypergraphs. We compare the proposed DE-HNN with several
State-of-the-art (SOTA) machine learning models for (hyper)graphs and
netlists, and show that the DE-HNN significantly outperforms them in
predicting the outcome of optimized place-and-route tools directly from the
input netlists. Our source code and the netlists data used are publicly
available at https://github.com/TILOS-AI-Institute/DEHNN.git.

##  1 Introduction

Chip design is a complicated process involving numerous steps, many of which
involve solving hard optimization problems. Just consider the stage of the
_place and route_ of a synthesized netlist: Here the input is a netlist
consisting of _cells_ and _nets_ , where cells refer to functional units such
as logic gates, and nets refer to connections between cells. The goal is to
produce a layout of this netlist in a specific 2D region, where gates are
placed and connections among them are realized by wires laid out across
multiple layers (called “routed”), all while aiming to optimize multiple key
properties (e.g, minimizing total wirelength and reducing congested
“hotspots”). This place-and-route stage is highly nontrivial to solve for
large netlists, and requires a time-consuming process in practice with
multiple stages and iterations. There therefore arises the need for data-
driven methods to predict properties of a design directly without the time-
consuming place and routing process. To this end, graph neural networks become
natural choices, given that the netlists are often represented as a graph or a
hypergraph. In this paper, we aim to develop an efficient and effective graph
learning architecture to predict post-routing properties (e.g., wirelength or
congestion) for a synthesized netlist accurately. The past decade has
witnessed a tremendous growth of graph learning models. Two most popular
families are (i) message-passing neural networks (MPNNs) (Gilmer et al.,,
2017; Jegelka,, 2022), and (ii) transformer based approaches (e.g, survey
(Müller et al.,, 2023)). However, netlists present several challenges for
existing graph learning architectures: (i) their size can be massive, from
hundreds of thousands to millions of nets, (ii) long-range interactions are
important (e.g., properties of interest might be caused by long paths and
other long-range interactions), and (iii) properties of post-routing netlists
seem to depend on complex information of graph topology beyond simple
statistics such as in-/out-degrees (distributions).

Unfortunately, it is challenging for popular MPNNs to capture long-range
interactions, due to issues such as over-smoothing of graph signals (Chen et
al., 2020a, ) and oversquashing bottlenecks (Topping et al.,, 2022). MPNN’s
ability in capturing graph motifs (e.g cycles and trees) and higher-order
structures is also limited (Xu et al.,, 2019; Jegelka,, 2022; Hy et al.,,
2019). Transformer-based graph neural models appear to be more effective in
capturing long-range interactions (Dwivedi et al.,, 2022; Ngo et al.,, 2023).
However, each transformer layer typically takes time quadratic to the number
of nodes. While there are sparse transformers (Katharopoulos et al.,, 2020;
Tay et al.,, 2022), their representation power is also reduced. Furthermore,
the primary ways for a graph transformer to capture input graph topology have
been either via initial position/structure encoding of nodes, or via the use
of certain pairwise graph distances to “reweight” the attention (Zhang et
al.,, 2023). In general, it is not clear how sensitive a transformer based
model is to features in input graph topology (e.g., specific paths which can
be important to netlists properties).

Our work.  In this paper, similar to (Xie et al.,, 2021), we model a netlist
as a directed hypergraph and present a novel _Directional Equivariant
Hypergraph Neural Networks_ (DE-HNN) for the effective learning of (directed)
hypergraphs. In particular, DE-HNN can be used to predict properties (e.g.,
congestion or net-wirelength) of a post-routed netlist directly from an input
netlist before performing the lengthy place-and-route process. Our DE-HNN
incorporates several new ideas to address the aforementioned challenges posed
by netlists. Our contributions are as follows:

  * •

We advocate for the modeling of a netlist as a directed hypergraph. Indeed, a
net usualy consists of a driver gate/cell $c$, togehter with a set $S$ of
“sinks”; see Section 2 and Figure 1. Recognizing the difference between the
driver gate and sinks in the timing of a routed net, inspired by (Xie et al.,,
2021), we represent a net as a _directed-hyperedge_ $(c,S)$ to separate the
roles of driver and sinks cells.

  * •

We propose a learning model DE-HNN for directed hypergraphs. Theoretically, we
show (Theorem 1) that our DE-HNN can universally approximate any node or
hyperedge based function that satisfies certain permutation equivariant and
invariant properties natural for directed hypergraphs.

  * •

On the practical front, to mitigate the issue of large size of and long-range
interactions in input netlists, we use a hierarchy of virtual nodes (VNs),
which provides additional “bridges” to allow the integration of both local and
global information while still maintaining original graph topology (unless in
a graph pooling approach); see Figure 2 and Section 3.

  * •

To make our initial node features more informative, in addition to using
Laplacian eigenvectors to provide position encoding as in (Kim et al.,, 2022;
Rampášek et al.,, 2022), we also use a topological summary called persistent
homology (Edelsbrunner and Harer,, 2010; Dey and Wang,, 2022), which can be
used to encode the “shape” of graph motif around each node in a multi-scale
manner (e.g., (Zhao et al.,, 2020; Yan et al.,, 2021)).

  * •

We compare our DE-HNN with several SOTA machine learning models for
(hyper)graphs and netlists. Our model significantly outperforms them in
predicting different properties of post-routed netlists directly from input
netlists.

We provide careful ablation studies to demosntrate the utilities of the use of
directed hyperedge, (hierarchical) VNs and persistence-based topological
summaries. Finally, we remark that ML research for chip design currently
suffers from the scarcity of open-source benchmark datasets111Previous netlist
property prediction work sometimes releases the input netlist designs, which
our paper also uses. However, they do not release the resulting properties nor
the post place-and-route netlists due to the use of commercial tools.. We hope
our datasets (which will be made publicly available at
https://github.com/TILOS-AI-Institute/DEHNN.git) can help bridge this gap.
These netlists (of sizes from 400K to 1.3M) can also serve as benchmark for
long-range graph interactions for machine learning researchers.

While in this paper, we design DE-HNN with the goal of netlists representation
and learning, our architecture as well as its theoretical results are general
and applicable to any directed hypergraphs.

Related work on machine learning models for netlists.

Earlier machine learning (ML) approaches for netlist property (e.g,
congestion) prediction assume that the placement of cells (logic gates) are
already given, that is, the input are placed, but not yet routed. They then
convert the input to a 2D image or other 2D grid-based representations to
predict the final congestion map over this region using models such as
convolutional neural networks (Tabrizi et al.,, 2018; Chen et al., 2020b, ;
Xie et al.,, 2018; Al-Hyari et al.,, 2021; Liang et al.,, 2020; Chai et al.,,
2023; Alawieh et al.,, 2020; Chen et al.,, 2022; Wang et al.,, 2022). However,
the placement information is itself very time-consuming to obtain.
Furthermore, representing the input as a 2D image makes it hard to capture
local and global connectivity information in the netlists.

Since a circuit is represented more accurately as a (hyper)graph, recent work
deploys graph neural networks (GNNs) for congestion prediction. The work of
(Kirby et al.,, 2019) constructs a homogeneous graph representation of netlist
in which each node corresponds to a cell, and if two cells are connected by a
net then there exists an edge between those nodes in the graph, and then
applies Graph Attention Networks (GAT) (Veličković et al.,, 2018). Later
follow-up work includes using node embedding computed from partitioned
subgraph (to capture more global graph structure) (Ghose et al.,, 2021) and
using dual graph with both cell and net features (Xie et al.,, 2021). Note
that this convertion of a net to a clique can lead to very large sized
cliques, and also loses net-specific topological information. The SOTA
approach for netlists representation is (Yang et al.,, 2022), which introduces
a heterogeneous (i.e., different edge types) graph construction, called
_circuit graph_ , in which both cells and nets are represented as nodes of a
bipartite graph.

All these approaches still have the issues of capturing long-range interaction
essentially for netlists. As we describe in “Our contributions”, our DE-HNN
deploys a suitable architecture, hierarchical virtual nodes, as well as
informative persistent homology features, to build an effective graph learning
model for netlists (and other directed hypergraphs). Note that similar to
(Kirby et al.,, 2019; Yang et al.,, 2022), our DE-HNN performs learning on
netlists without placement information. If placement informtion is available,
they can easily be added to initial node position encoding.

##  2 Modeling netlists as directed hypergraphs

##### Circuit netlists.

A circuit netlist is a textual representation of electronic components, such
as logical boolean gates, and the connections between them. A (pre-placed,
also called synthesized) netlist ${\mathcal{H}}$ consists of a collection of
cells (logic gates)
${\mathcal{C}}=\\{{\mathsf{c}}_{1},\ldots,{\mathsf{c}}_{n}\\}$, and a set of
nets ${\mathcal{N}}=\\{{\sigma}_{1},\ldots,{\sigma}_{m}\\}$. Each cell (gate)
has a certain number of input pins and an output pin. The number of input pins
is decided by the type of this gate (e.g., an AND gate takes two inputs). For
every gate, its output will flow into the input pins of a collection of other
gates. This information is captured by the concept of net, where a net
${\sigma}\in{\mathcal{N}}$ consists of the output pin of a certain cell,
called its _driver cell_ denoted by ${\mathsf{v}}_{\sigma}\in{\mathcal{C}}$,
together with all those _sink cells_ , denoted by
${\mathsf{S}}_{\sigma}\subseteq{\mathcal{C}}$, where the signal from this
output pin will flow into. In other words, a net can be represented by a tuple
${\sigma}=({{\mathsf{v}}}_{\sigma},{\mathsf{S}}_{\sigma})$. See Figure 1 (a).

Given such a netlist, standard chip design pipelines will first lay it out in
the physical space (_placement_). Then in the _routing stage_ , the connection
from output pin of one cell to the input of other cells are mapped to the
routing channels within the chip’s physical floorplan. Among other properties,
one wishes to minimize the total wirelength of each net, and to reduce routing
“congestions”, which occurs when the number of edges to be routed in a
specific region of the floorplan exceeds the available routing capacity. See
Figure 4 in the Supplement for an illustration of a placed-and-routed netlist.

Directed hypergraphs.  The standard _hypergraph $H$_ is a tuple $(V,\Sigma)$,
where $V$ is a set of nodes, $\Sigma$ is a set of _hyperedge_ , and each
$e\in\Sigma$ is a subset of $V$; i.e., $e\subseteq V$. A _directed hypergraph_
${\overrightarrow{{H}}}$ is a tuple $(V,{\overrightarrow{\Sigma}})$, where
each _directed hyperedge ${\sigma}\in{\overrightarrow{\Sigma}}$_ consists of
an ordered pair ${\sigma}=(v_{\sigma},S_{\sigma})$ with $v_{\sigma}\in V$ and
$S_{\sigma}\subseteq V$. It is easy to see that a netlist
$({\mathcal{C}},{\mathcal{N}})$ can be naturally viewed as a directed
hypergraph where we have: cell $\Leftrightarrow$ node, and net
$\Leftrightarrow$ directed hyperedge. See Figure 1 (b).

In what follows, we often use cells / nodes, as well as nets / directed
hyperedges, interchangeably. In fact, for simplicity, we often use the terms
“nodes” and “nets” as they are more concise. We will also refer to
${\mathsf{v}}_{\sigma}$ and ${\mathsf{S}}_{\sigma}$ from a directed hyperedge
${\sigma}=({\mathsf{v}}_{\sigma},{\mathsf{S}}_{\sigma})$ as its _driver_ and
its _sinks_ , respectively, just like in a net ${\sigma}$.

Given a net $\sigma\in{\overrightarrow{\Sigma}}$, we say that it contains a
node $v\in V$, denoted by $v\in\sigma$, if $v$ is either the driver, or a sink
of $\sigma$. Given a node $v\in V$, we say that a net $\sigma$ is _incident on
$v$_ if $\sigma$ contains $v$. The collection of nets incident to $v$ is
called the _incident-net-set_ of $v$, denoted by
${\mathcal{I}}(v)=\\{\sigma^{\prime}\in{\overrightarrow{\Sigma}}\mid
v\in\sigma^{\prime}\\}$. For example, in Figure 1 (b), the incident-net-set of
$v_{3}$ is
${\mathcal{I}}(v_{3})=\\{{\sigma}_{1},{\sigma}_{2},{\sigma}_{5}\\}$.

|
---|---
(a) | (b)
Figure 1: (a) A netlist with $7$ cells
${\mathcal{C}}=\\{{\mathsf{c}}_{1},\ldots,{\mathsf{c}}_{7}\\}$ and 5 nets. For
example, the output of gate $c_{2}$ flows into cells $c_{3},c_{5},$ and
$c_{7}$, giving rise to the net ${\sigma}=(c_{2},\\{c_{3},c_{5},c_{7}\\})$.
That is, the driver cell of ${\sigma}$ is ${\mathsf{v}}_{{\sigma}}=c_{2}$,
while its sink-set being ${\mathsf{S}}_{\sigma}=\\{c_{3},c_{5},c_{7}\\}$. (b)
The corresponding directed hypergraph with $7$ nodes and $5$ hyperedges. Each
node $v_{i}$ corresponds to cell ${\mathsf{c}}_{i}$, and each hyperedge is
marked as a shaded region.

In the chip design literature, a netlist is oftentimes represented either (1)
as a (directed) graph where two cells are connected if they belong to a common
net; or (2) as a standard hypergraph where a net is a hyperedge consisting of
the union of the driver cell and all sink cells. The former can lead to huge
cliques (as some nets can consist of large number of cells) and also lose
sensitivity to the net topology. In the learning context, the work of Xie et
al., (2021) first separated the role of the driver and sinks of a net in the
representation of a netlist and provided justification for this choice. Their
final representation is still a graph representation that is intuitively a
directed version of the so-called _line graph_ for a hypergraph. However,
converting a hypergraph to a line graph is a lossy process. The directed
hypergraph provides a more informative and cleaner representation: it both
preserves the full net information and differentiates between drivers and
sinks.

##  3 DE-HNN: a neural network for directed hypergraphs

In this section, we first describe a basic neural network (NN) model in
Section 3.1, which we refer to as base-DE-HNN, for the representation learning
of directed hypergraphs. We provide theoretical justification of this model in
Section 3.2. Then in Section 3.3, we describe how to augment this base model
to make it more effective at capturing long-range interactions as well as the
multi-scale graph topology.

###  3.1 Base-DE-HNN

Our base-DE-HNN uses message-passing mechanisms. However, it differs from
standard MPNN (message-passing neural networks) (Gilmer et al.,, 2017) in how
the messages are aggregated and updated, so as to process node and net based
features, as well as to respect the direction of hyperedges.

More specifically, base-DE-HNN consists of $L$ layers. Consider an input
directed hypergraph ${\overrightarrow{{H}}}=(V,{\overrightarrow{\Sigma}})$.
For the $\ell$-th layer, each node $v\in V$ (resp. each net/directed hyperedge
${\sigma}\in{\overrightarrow{\Sigma}}$) will maintain a node feature (resp.
net feature) denoted as $m^{\ell}(v)$ (resp. $M^{\ell}({\sigma})$). For
simplicity, assume that
$m^{\ell}(v),M^{\ell}({\sigma})\in\mathbb{R}^{d_{\ell}}$ are
$d_{\ell}$-dimensional vectors. Assume first that our final goal is to predict
net properties. The base-DE-HNN will compute $m^{\ell}(v)$ and
$M^{\ell}({\sigma})$ using feature representations from the ($\ell-1$)-th
layers by the following two steps:

[Node Update]: First, the features of each node (cell) $v\in V$ is updated
using features of the set of those nets containing it, that is, via the
features of those nets in the incident-net-set ${\mathcal{I}}(v)$ of $v$ as
follows:

$m^{\ell}(v)=\mathrm{Agg}^{\ell}_{{\sigma}\rightarrow
v}({\\{\\!\\{}M^{\ell-1}(\sigma^{\prime}){\\}\\!\\}}_{\sigma^{\prime}\in{\mathcal{I}}(v)}),$
(1)

where ${\\{\\!\\{}\cdot{\\}\\!\\}}$ denotes a _multiset_ as some neighboring
nets could have identical feature representations. The function
$\mathrm{Agg}_{{\sigma}\rightarrow v}$ operates on a multiset and should be
_invariant_ to the order of neighboring nets of $v$ in ${\mathcal{I}}(v)$.
Similar to the Deep Set architecture (Zaheer et al.,, 2017) which can hancle
such _permutation invariance_ in multisets, we implement the function
$\mathrm{Agg}_{{\sigma}\rightarrow v}$ by:

$m^{\ell}(v)=\sum_{\sigma^{\prime}\in{\mathcal{I}}(v)}\text{MLP}_{1}^{\ell}\big{(}M^{\ell-1}(\sigma^{\prime})\big{)},$
(2)

where $\text{MLP}_{1}$ stands for a multi-layer perceptron. For example, the
update of node feature for $v_{4}$ in Figure 1 (b) is
$m^{\ell}(v_{4})=\text{MLP}_{1}^{\ell}\big{(}M^{\ell-1}(\sigma_{1})\big{)}+\text{MLP}_{1}^{\ell}\big{(}M^{\ell-1}(\sigma_{3})\big{)}$
as ${\mathcal{I}}(v_{4})=\\{{\sigma}_{1},{\sigma}_{3}\\}$. It is easy to see
that the update in Eqn (2) satisfies the needed permutation invariance.

[Net Update]: Next, the features of each net
${\sigma}=({\mathsf{v}}_{\sigma},{\mathsf{S}}_{\sigma})$ is updated using the
new node features for those nodes contained in ${\sigma}$. Since the net
(hyperedge) ${\sigma}$ is directed, we wish to separate the roles of its
driver node ${\mathsf{v}}_{\sigma}$ and the set of sinks
${\mathsf{S}}_{\sigma}$, that is,

$M^{\ell}({\sigma})=\mathrm{Agg}^{\ell}_{v\rightarrow{\sigma}}(m^{\ell}({\mathsf{v}}_{\sigma}),{\\{\\!\\{}m^{\ell}(v^{\prime}){\\}\\!\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}})$
(3)

However, the update should not depend on the ordering of nodes in the sink set
${\mathsf{S}}_{\sigma}$, i.e., $\mathrm{Agg}_{v\rightarrow{\sigma}}$ needs to
be _permutation invariant_ w.r.t. its second parameter, the multiset
${\\{\\!\\{}m^{\ell}(v^{\prime})){\\}\\!\\}}_{v^{\prime}\in{\mathsf{S}}(\sigma)}$.
We use the following to implement Eqn (3) to guarantee the needed permutation
invariance of $\mathrm{Agg}_{v\rightarrow{\sigma}}$ w.r.t. the ordering of
nodes in ${\mathsf{S}}_{\sigma}$:

$\displaystyle M^{\ell}({\sigma})=$
$\displaystyle\,\text{MLP}^{\ell}_{3}\bigg{[}m^{\ell}({\mathsf{v}}_{{\sigma}})\oplus\bigg{(}\sum_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\text{MLP}^{\ell}_{2}(m^{\ell}(v^{\prime}))\bigg{)}\bigg{]}$
(4)

where $\text{MLP}_{2}$ and $\text{MLP}_{3}$ are multi-layer perceptrons, and
$\oplus$ denotes vector concatenation. For example, in Figure 1, the update of
${\sigma}_{1}=(v_{1},\\{v_{3},v_{4}\\})$ is
$M^{\ell}({\sigma}_{1})=\text{MLP}^{\ell}_{3}\big{(}m^{\ell}(v_{1})\oplus(\text{MLP}^{\ell}_{2}(m^{\ell}(v_{3}))+\text{MLP}^{\ell}_{2}(m^{\ell}(v_{4}))\big{)}$.

We note that if the target task is to predict node-level features (instead of
net-level features), then we will apply dual update rules where the roles of
nets and nodes will be swapped. Finally, in our implementation, we also add
residuals (i.e, node / net features from the previous level) to each node /
net features during the updates. There are $L$ such layers, and in the end, if
the given task is a regression task, then another linear layer $\psi$ is
applied to the values $m^{L}(\cdot)$ (resp. $M^{L}(\cdot)$) to generate the
final node-based (resp. net-based) regression function. During the training
process, the loss function in this case is the standard mean squared error;
that is, if it is a node-based regression with ground-true function
$\mathrm{y}:V\to\mathbb{R}$, then we have:

$\mathcal{L}(\bm{\theta})=\frac{1}{|V|}\sum_{v\in
V}[\psi(m^{L}(v))-\mathrm{y}(v)]^{2},$ (5)

where $\bm{\theta}$ denotes the set of all learnable parameters in the entire
base-DE-HNN model. Node or net classification tasks are handled similarly but
with the cross-entropy loss.

###  3.2 Theoretical analysis of DE-HNN

We now provide some theoretical guarantee of our base-DE-HNN model. For
simplicity, in what follows we assume that our target (regression) functions
are net-valued functions222The net-valued function is more natural for
properties such as net wirelength and congestion. (or simply _net-functions_)
of the form $F:{\overrightarrow{\Sigma}}\to\mathbb{R}$. Our main result below
holds for node-valued functions via a symmetric argument.

Let us consider one update stage at a fixed layer $\ell\in[1,L]$. From the
net-function perspective, the goal of the update stage is the following: At
the beginning of this stage, we start with input net features
$\mu_{\sigma}:=M^{\ell-1}({\sigma})$, for all
${\sigma}\in{\overrightarrow{\Sigma}}$. In the end, we obtain a set of new
features $\mu^{*}_{\sigma}:=M^{\ell}({\sigma})$ for each
${\sigma}\in{\overrightarrow{\Sigma}}$. If we view this feature update as a
function on net features, then ideally, for any fixed
${\sigma}\in{\overrightarrow{\Sigma}}$, its new feature should depend on the
features of all its _neighboring nets_ which are those nets that share at
least one node with ${\sigma}$. However, note that the neighbors of ${\sigma}$
are naturally classified to two families:

(type-A)

those in the set ${\mathcal{I}}({\mathsf{v}}_{\sigma})$ (recall
${\mathcal{I}}(v)$ is the set of nets that contains node $v$), which are
connected to ${\sigma}$ via the driver node ${\mathsf{v}}_{\sigma}$ of
${\sigma}$; and

(type-B)

those in
$\bigcup_{v^{\prime}\in{\mathsf{S}}_{\sigma}}{\mathcal{I}}(v^{\prime})$, where
each set ${\mathcal{I}}(v^{\prime})$ are those nets connected to ${\sigma}$
via a sink node $v^{\prime}$ of ${\sigma}$ (i.e.,
$v^{\prime}\in{\mathsf{S}}_{\sigma}$). Note that
$\Big{\\{}{\mathcal{I}}(v^{\prime})\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}$
is a set of sets of neighboring (type-B) nets of ${\sigma}$.

For example, in Figure 1, for $\sigma_{5}=\\{v_{3},\\{v_{5},v_{7}\\}\\}$, its
(type-A) neighbors are
${\mathcal{I}}(v_{3})=\\{{\sigma}_{1},{\sigma}_{2},{\sigma}_{5}\\}$, while
(type-B) is a set of sets
$\\{\\{{\sigma}_{2},{\sigma}_{4},{\sigma}_{5}\\},\\{{\sigma}_{2},{\sigma}_{5}\\}\\}$.
It is natural to model the desired update function for the directed hyper-
edges as follows, which differentiate (type-A) and (type-B) neighbors of
${\sigma}$:

$\displaystyle\mu^{*}_{\sigma}$
$\displaystyle=\mathcal{F}\Big{(}{\\{\\!\\{}\mu_{{\sigma}^{\prime}}{\\}\\!\\}}_{{\sigma}^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})},\Big{\\{}\\!\Big{\\{}{\\{\\!\\{}\mu_{{\sigma}^{\prime}}{\\}\\!\\}}_{{\sigma}^{\prime}\in{\mathcal{I}}(v^{\prime})}\Big{\\}}\\!\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\Big{)}$
(6)

Note that the first variable,
${\\{\\!\\{}\mu_{{\sigma}^{\prime}}{\\}\\!\\}}_{{\sigma}^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})}$,
consists of the set of input feature representations of (type-A) neighbors of
${\sigma}$, while the second variable,
$\Big{\\{}\\!\Big{\\{}{\\{\\!\\{}\mu_{{\sigma}^{\prime}}{\\}\\!\\}}_{{\sigma}^{\prime}\in{\mathcal{I}}(v^{\prime})}\Big{\\}}\\!\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}$,
is a set of sets333For concision, we use “set” instead of “multiset” here. (of
net features), constituting multisets of feature representations of those
(type-B) neighbors of ${\sigma}$. The function $\mathcal{F}$ should be
invariant not only to the order within each individual set ${\mathcal{I}}(u)$
for some cell $u$, but also invariant to the order of nodes in the sink-set
${\mathsf{S}}_{\sigma}$, i.e., the order of sets
$\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}(v^{\prime})}$’s
within the outer-set in the second variable of ${\mathcal{F}}$. We refer to
such a function $\mathcal{F}$ as nested-permutation invariant.

Now recall that in our base-DE-HNN, at each layer we perform update of node-
feature by features of its incident nets as described in Eqn (2), followed by
the update of net-features by the new features of those nodes contained in
${\sigma}$ as specified in Eqn (4). In other words, one can think of the
update of net features from $M^{\ell-1}(\cdot)$ to $M^{\ell}(\cdot)$ to be the
composition of updates in Eqns (2) and (4). By construction, it is easy to see
that the composition of these two udpate steps indeed gives rise to a nested-
permutation invariant function. However, can any nested-permutation invariant
net-function be represented (or approximated) by such a composition of two
steps? A priori, the answer to this opposite direction is not clear at all as
the iterative updates factored through the node features appears more
restrictive. Our main theorem below shows that the answer is in fact positive.

######  Theorem 1 (Simplified).

Let ${\mathcal{F}}$ be any continuous, nested-permutation invariant, net-value
function as in Eqn (6). For simplicity, assume both input nets and output of
$M$ take values in a compact set $\mathcal{B}\subset\mathbb{R}^{d}$, a
connected compact subset of $\mathbb{R}^{d}$. Then we have that
${\mathcal{F}}$ can be rewritten as the following sum-decomposition

$\displaystyle\forall\sigma:~{}$
$\displaystyle{\mathcal{F}}\Big{(}\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})},\Big{\\{}\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}(v^{\prime})}\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\Big{)}$
$\displaystyle=~{}$
$\displaystyle\rho\Big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})}\phi_{1}(\mu_{\sigma^{\prime}}),\sum_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\phi_{2}\big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}(v^{\prime})}\phi_{1}(\mu_{\sigma^{\prime}})\big{)}\Big{)}$
(7)

where $\phi_{1}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d^{\prime}}$,
$\phi_{2}:\mathbb{R}^{d^{\prime}}\rightarrow\mathbb{R}^{d^{\prime\prime}}$,
and $\rho$ are continuous functions.

Recall that for the $\ell$-th layer, the input feature for any net
${\sigma}^{\prime}$ is $\mu_{{\sigma}}=M^{\ell-1}({\sigma}^{\prime})$. Now
compare the right-hand side of Eqn (7) with Eqns (2) and (4): it is easy to
see that by using $\text{MLP}_{1}$ from Eqn (2) to approximate the continuous
function $\phi_{1}$, and using $\text{MLP}_{2}$ and $\text{MLP}_{3}$ from Eqn
(4) to approximate continuous functions $\phi_{2}$ and $\rho$ respectively, we
then have that our iterative updates using Eqns (2) and (4) approximate any
desired update of the net features via a nested-permutation invariant function
as in Eqn (6). In other words, the iterative message-passing update steps in
our base-DE-HNN provides an universal approximation of the desired nested-
permutation invariant update functions over nets. The proof of this theorem
and more discussions can be found in the Supplement.

###  3.3 Augmenting base-DE-HNN to DE-HNN

The base-DE-HNN provides an effective way to update features for both nodes
and hyperedges in an iterative manner. We now describe further augmentation
strategies for the resulting DE-HNN to capture long-range interactions as well
as to be more senstive to the multi-scale graph topology.

##### Hierarchy of virtual nodes.

The standard message-passing GNNs have difficulty to capture long-range
interaction due to issues such as over-smoothing, over-squashing and under-
reaching. Most GNNs used in practice have few layers. Transformer-type models
for graphs may alleviate the issue (Dwivedi et al.,, 2022), but each self-
attention layer requires quadratic computation, which does not scale to large
graphs. Furthermore, it is not clear how to effectively encode graph
structures for transformers, even with the use of initial position encoding
(Müller et al.,, 2023) or reweighting based on graph distances (Zhang et al.,,
2023). Intuitively, we want to keep the simple and efficient message-passing
framework over the input (very sparse) hypergraph, but also have a way to
propagate long-range information among nodes that are far away from each other
(in terms of graph distances). We will do so via the use of _virtual nodes_.

Specifically, a _virtual node (VN)_ is an additional node we add that is
connected to all input nodes. This effectively reduces the graph diameter to
$2$, and has been a popular strategy in graph learning literature; e.g.
(Gilmer et al.,, 2017). However, note that messages cannot be directly passed
between two nodes. Instead, information of all nodes have to be aggregated at
the virtual node before being passed back to all nodes. Nevertheless, (Cai et
al.,, 2023) shows that a message-passing GNN augmented by a single VN can
already approximate Lineaer Transformer (Katharopoulos et al.,, 2020) and
Performer (Choromanski et al.,, 2021) (as well as general transformers to some
extent) and bring significant empirical gains.

Figure 2: A two-level hierarchy of virtual nodes (VNs).

While adding a single VN will only linearly increase the number of edges, the
VN itself will have a high degree and potentially become a computatoinal
bottleneck. Furthermore, since the features of all nodes have to be aggregated
at the virtual node, the aggregated messages will lose sensitivity to
individual node features. The benefits of adding a single VN thus diminishes
as the graph becomes larger. We instead propose to use a hierarchy of VNs as
follows (see Figure 2).

  * •

Given an input hypergraph
${\overrightarrow{{H}}}=(V,{\overrightarrow{\Sigma}})$, we first use Metis
(Karypis and Kumar,, 1998) to partition its node set to $k$ disjoint subsets
$V=V_{1}\cup V_{2}\cup\cdots\cup V_{k}$. In our experiments, we keep the sizes
of $V_{i}$s roughly balanced, and therefore the value of $k$ varies as input
graph size changes.

  * •

We introduce a VN ${\omega}_{i}$ for each subset $V_{i}$, for $i\in[1,k]$, and
${\omega}_{i}$ is connected to all nodes in $V_{i}$. These VNs are _first-
level VNs_. Note that the total number of edges added this way is only
$n=|V|$.

  * •

We further create a _super-VN_ , denoted by ${\omega}_{0}$, which connects to
all the first-level VNs. This introduce an additional $k$ number of edges.

In what follows, we refer to nodes from input hypergraphs as _standard nodes_.
During the updating of node/edge features, we will use _heterogeneous updates_
, where the aggregation functions at the first-level VNs and super-VN are
different from that at standard nodes. Different from graph pooling, the use
of additional VNs keep the original hypergraph topology, while they act as
additional “bridges” and allow information flow both at global (among far-
apart nodes via VNs) and local scales (along original edges).

As the input hypergraph becomes even larger, we could use multiple layers in
our hierarchy of VNs. Nevertheless, we observe that two layers already yield
good performance in our current test cases. If nodes are placed, one could
partition the node set by spatial locations (e.g., by decomposing the
rectangular region) instead of using Metis.

##### Initial positional / structural encodings.

Finally, we aim to encode meaningful multi-scale features for each node /
hyperedge. To this end, for each node, we add the Laplacian positional
encoding consisting of the function value of this node from the first $s$
($s=10$ in our experiments) eigenvectors of the undirected version of input
hypergraph.

Furthermore, to better capture the “shape” of the neighborhood of each node
$v$ in a more discriminative and multi-scale manner, similar to (Yan et al.,,
2021; Zhao et al.,, 2020), we use the so-called extended persistence diagram
(PD) summary induced by the shortest path distance function within the $r$-hop
neighborhood of each node ($r=6$ in our experiments) as an initial _structural
encoding_ for each node. Note that it is known (Tian and Wang,, 2019) that PDs
constructed this way can encode rich graph information of the $r$-hop
neighborhood of each node $v$ (viewed as a local motif around $v$), such as
clustering coefficients, number of nodes at distance $a\leq r$ to $v$, number
of independent cycles, etc. See the Supplement for more details. Indeed, as
our ablation study and results in Supplement show, adding PDs improve _both
our and previous graph learning models_.

##  4 Experiments

###  4.1 Datasets

We used 12 of the Superblue circuits from (Viswanathan et al.,, 2011, 2012) to
evaluate our proposed models and baselines. The Superblue circuits are some of
the most complex yet publicly available circuits used in previous work about
VLSI placement and routing. The size of these netlists range from 400K to 1.3M
nodes, with similar number of nets. Note that these hypergraphs are very
sparse, although there are a very small fraction of nets with large sizes. See
the Supplement for more detailed statistics of these netlists, including the
size of each design as well as the statistics of target properties.

We pulled our designs from the RosettaStone (Kahng et al.,, 2022) repository’s
Skywater 130 nm technology (Edwards,, 2020) benchmark set. We then used a
commercial physical design tool to perform placement optimization for each
design with a utilization ratio of 0.7. We exported global-routing congestion
information in the form of demand and capacity for each global-route cell
(GRC) in each routing layer.

###  4.2 Setup

##### Baselines.

We compare with a range of SOTA baselines: (i) Graph Convolutional Networks
(GCN) (Kipf and Welling,, 2017); (ii) a SOTA variant of Graph Attention
Networks (GATv2) (Brody et al.,, 2022); (iii) Hypergraphs Message Passing
Neural Network (HMPNN) (Heydari and Livi,, 2022); (iv) Hypergraph (Neural)
Networks with Hyperedge Neurons (HNHN) (Dong et al.,, 2020); (v) A multiset
function framework for Hypergraph Neural Network (AllSet) (Chien et al.,,
2022); (v) the SOTA graph-based model specifically defined for netlist
property predictions, NetlistGNN (Yang et al.,, 2022). See the Supplement for
more details on their architecture, including model complexity. We also
compare with the hypergraph convolutional operator (HyperConv) (Bai et al.,,
2021) and Linear Transformers (Katharopoulos et al.,, 2020). As both models’
performances on average are not as good as other baseline models, we include
their results only in the Supplement.

We compare these baselines with our DE-HNN model, which we refer to as full-
DE-HNN in results to differentiate with base-DE-HNN model, whose performance
we also include for reference. Note that base-DE-HNN is without persistence
diagrams (PDs) as initial features nor virtual nodes (VNs). We implement our
DE-HNN and the baselines with PyTorch (Paszke et al.,, 2019) and PyTorch-
geometric (Fey and Lenssen,, 2019).

Features.  For each cell, its initial features include: type, width, height of
gate, degree, degree distribution of a local neighborhood (summarized into a
vector), top-10 Laplacian eigenvectors, and persistence diagram (PD) features
(see Supplement). For all methods other than Linear Transformer, the initial
net features are the nets’ degrees. For Linear Transformer, net features are
obtained by averaging the features of those nodes contained in it so as to
provide more topology information. (see Supplement). For a fair comparison,
the same initial cell/net features are used for all models (other than our
base-DE-HNN, which do not have PDs as it is a base model without any
augmentation). One might wonder whether adding persistence diagrams (PDs)
features are beneficial for both our method and baselines. Indeed as we show
later in Ablation study and in Supplement, using PDs improve model performance
for both our methods and baselines. For example, it improves NetlistGNN by
$3.6\%$ on average in terms of demand regression in the single-design setting.

Prediction tasks.  We test three different tasks in our experiments, including
both net- and node-based tasks. These tasks cover the types of experiments
performed by previous netlist prediction approaches.

  * •

Net-based wirelength regression: We use half-perimeter wirelength (HPWL), a
common estimator used for wirelength calculation, to calculate the wirelength
of each net (Mirhoseini et al.,, 2021). Similar to (Yang et al.,, 2022), we
take the $\log_{2}$ of the wirelength to reduce the range.

  * •

Net-based demand regression: We predict the net-based demand. Congestion
happens when demands exceeds capacity. There is no concensus on how to define
congestion (difference or ratio) and we thus directly predict demand.

  * •

Cell-based congestion classification: Similar to (Yang et al.,, 2022) and
(Wang et al.,, 2022), we classify the cell-based congestion values (computed
as the ratio of demand/capacity) into (a) [0, 0.9], _not-congested_ ; and (b)
[0.9, $\inf$]; _congested_.

Our experiments have two settings:

  * •

Single-design: We train and evaluate on each individual design separately. For
each graph in a design, we apply 4-fold cross-validation to randomly split the
nodes into 4 subsamples with same sizes (25%/25%/25%/25%). We report the
average performance across all 12 designs with cross-validation applied to
each design. The distribution of target values and the average performance
from cross-validation for each design can be found in the Supplement.

  * •

Cross-design: We aim to evaluate the ability of the models to generalize to
unseen netlist topologies. Following previous work (Yang et al.,, 2022), we
use 10 designs, Superblue1,2,3,5,6,7,9,11,14,16, for training, superblue18 for
validation, and superblue19 for testing.

###  4.3 Results

Table 1: Net-based wirelength regression. Last row “Improvement” refers to the
improvement of our full DE-HNN model over the best baseline approach for each
metric.

Model | Single-Design | Cross-Design
---|---|---
|  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$ |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
GCN | $1.762$ | $1.276$ | $0.750$ | $1.691$ | $1.276$ | $0.746$
GATv2 | $1.812$ | $1.330$ | $0.687$ | $1.717$ | $1.281$ | $0.737$
AllSet | $1.718$ | $1.264$ | $0.760$ | $1.837$ | $1.348$ | $0.695$
HMPNN | $1.841$ | $1.368$ | $0.710$ | $1.785$ | $1.335$ | $0.710$
HNHN | $1.852$ | $1.368$ | $0.717$ | $1.754$ | $1.333$ | $0.701$
NetlistGNN | $1.773$ | $1.320$ | $0.740$ | $1.762$ | $1.324$ | $0.718$
base DE-HNN | $1.751$ | $1.269$ | $0.748$ | $1.731$ | $1.291$ | $0.730$
full DE-HNN | $\bm{1.689}$ | $\bm{1.245}$ | $\bm{0.770}$ | $\bm{1.677}$ | $\bm{1.242}$ | $\bm{0.754}$
Improvement | $\bm{1.7\%}$ | $\bm{1.6\%}$ | $\bm{1.3\%}$ | $\bm{1.9\%}$ | $\bm{2.6\%}$ | $\bm{1.8\%}$

Table 2: Net-based demand regression for each design.

Model | Single-Design | Cross-Design
---|---|---
|  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$ |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
GCN | $9.321$ | $6.163$ | $0.570$ | $6.571$ | $5.024$ | $0.365$
GATv2 | $9.342$ | $6.118$ | $0.561$ | $6.623$ | $5.137$ | $0.363$
AllSet | $9.072$ | $5.745$ | $0.632$ | $6.120$ | $4.820$ | $0.345$
HMPNN | $9.342$ | $6.118$ | $0.561$ | $6.979$ | $5.356$ | $0.306$
HNHN | $9.119$ | $5.885$ | $0.594$ | $6.390$ | $4.870$ | $0.358$
NetlistGNN | $9.063$ | $5.839$ | $0.623$ | $8.328$ | $6.839$ | $0.367$
base DE-HNN | $8.997$ | $5.764$ | $0.630$ | $6.778$ | $5.085$ | $0.337$
full DE-HNN | $\bm{8.381}$ | $\bm{5.334}$ | $\bm{0.683}$ | $\bm{6.037}$ | $\bm{4.670}$ | $\bm{0.372}$
Improvement | $\bm{7.5\%}$ | $\bm{7.2\%}$ | $\bm{8.1\%}$ | $\bm{1.4\%}$ | $\bm{4.1\%}$ | $\bm{1.4\%}$

Table 3: Cell-based congestion classification.

Model | Single-Design | Cross-Design
---|---|---
|  Precision $\uparrow$ |  Recall $\uparrow$ |  F_score $\uparrow$ |  Precision $\uparrow$ |  Recall $\uparrow$ |  F_score $\uparrow$
GCN | $0.761$ | $0.857$ | $0.802$ | $0.633$ | $0.997$ | $0.773$
GATv2 | $0.810$ | $0.864$ | $0.835$ | $0.630$ | $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{0.999}$ | $0.765$
AllSet | $0.782$ | $0.837$ | $0.804$ | $0.645$ | $0.964$ | $0.773$
HMPNN | $0.774$ | $0.826$ | $0.792$ | $0.633$ | $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{0.999}$ | $0.772$
HNHN | $0.792$ | $0.869$ | $0.826$ | $0.648$ | $0.939$ | $0.767$
NetlistGNN | $0.812$ | $0.860$ | $0.831$ | $0.647$ | $0.953$ | $0.771$
base DE-HNN | $0.824$ | $0.860$ | $0.840$ | $0.653$ | $0.990$ | $0.774$
full DE-HNN | $\bm{0.833}$ | $\bm{0.876}$ | $\bm{0.853}$ | $\bm{0.660}$ | $0.986$ | $\bm{0.780}$
Improvement | $\bm{2.6\%}$ | $\bm{0.8\%}$ | $\bm{2.2\%}$ | $\bm{1.7\%}$ | $\bm{-}$ | $\bm{1.0\%}$

The test performance of all baselines and our methods (base-DE-HNN and full-
DE-HNN) are shown in Tables 1, 2 and 3. For the regression tasks, to be
comprehensive, we use three metrics: the Mean Squared Error (MSE), the Mean
Average Error (MAE), and the Pearson correlation (Pearson). For classification
tasks, we report the Precision, Recall, and F-Score. Note that for MSE and
MAE, the smaller the value is the better; while for Pearson, Precision,
Recall, and F-Score, the larger the better. In all tables, we highlight the
best performance results in red, while the best among all baseline models are
colored blue (unless a baseline result is the best, in which case it will be
red-colored). In the last row of these tables, we show the Improvement of our
full-DE-HNN model over the best of all baselines for each metric; note that
our improvement over any individual baseline can only be better than this
improvement. Our full-DE-HNN model outperforms all baselines, sometimes
significantly. For example, compared to NetlistGNN (Yang et al.,, 2022), the
previous SOTA for netlist representatin learning, our improvement on average
is around $5.3\%$ for wirelength prediction, and $8.6\%$ for demand
prediction, both in terms of MAE. See the Supplement for the full results,
including the test performances for each design in the single-design setting.

Ablation study.  We carried out an ablation study to understand the effects of
the different strategies employed in our DE-HNN. In particular, the factors we
wish to test are: (a) the use of direction in modeling nets, (b) the use of
persistence diagrams (PDs) as features, and (c) the use of single and
hierarchical VNs (virtual nodes). To this end, we compare the performance of
the following versions: (a) base-E-HNN stands for treating a net as a standard
hyperedge (thus no direction), and using neither PD features nor VNs. (b)
base-DE-HNN is the base model for directed hypergraph (described in Section
3.1) with neither PDs nor VNs. In other words, the difference between base-DE-
HNN and base-E-HNN shows the effect of adding directions. (c) base-DE-HNN+PD
is the base model with only PDs. Hence the difference between (c) and (b) is
to show the effect of PDs. (d) base-DE-HNN+PD+single VN is the base model with
PD and a single global VN. (e) Finally, full-DE-HNN is our full model with PDs
and a two-level hierarchy of VNs. The results for net-based demand regression
and cell-based congestion classification are shown in Figure 3. Full results
are found in the Supplements, Other metrics and tasks show a similar behavior.
For example, for demand prediction, from base-E-HNN, to adding directions,
PDs, single VN, and finally two-level VNs, performance improves over the
previous version by $2.5\%$, $2.6\%$, $1.0\%$ and $3.4\%$, respectively.
Overall, full-DE-HNN improves over the base-DE-HNN by around $6.8\%$, while
its improvement over base-E-HNN (base model with no direction) is $9.2\%$.

Figure 3: Ablation study for net-based demand regression (left, RMSE) and
cell-based congestion classification (right, F-score).

##  5 Conclusion

In this paper, we presented an effective model for representation learning on
directed hypergraphs. We considered learning on netlists, the hypergraph
representation of circuits in chip design. This has great importance in
practice but so far ML approaches for netlist representation learning has been
limited, partly due to their huge sizes, long-range interactions, as well as
the scarsity of benchmark datasets. We introduced several strategies to
capture long-range interactions, graph motifs, and to consider direction in a
hyperedge. Our model significantly outperforms a range of SOTA methods over a
collection of chip designs. Our datasets will be publicly available, which we
hope will facilitate further research on ML for chip design applications,
pushing the ability of methods to capture long-range interactions. Finally,
while we significantly outperformed other approaches, we note that in general,
improvements in the cross-design setting are less prominent, potentially due
to large variations in circuit designs. It will be interesting to further
explore this direction.

#### Acknowledgements

This work is partially supported by NSF under grants CCF-2112665 and
CCF-2310411, as well as by a gift fund from QualComm. The first, second and
last authors would like to extend our sincere thanks to Prof. Andrew B. Kahng
for the many insightful discussions and contributions in the early stage of
this work, especially in exploring and experimenting the use of topological
summaries for property prediction of netlists.

## References

  * Adams et al., (2017) Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., Chepushtanova, S., Hanson, E., Motta, F., and Ziegelmeier, L. (2017).  Persistence images: A stable vector representation of persistent homology.  Journal of Machine Learning Research, 18(8):1–35.
  * Al-Hyari et al., (2021) Al-Hyari, A., Szentimrey, H., Shamli, A., Martin, T., Gréwal, G., and Areibi, S. (2021).  A deep learning framework to predict routability for fpga circuit placement.  ACM Trans. Reconfigurable Technol. Syst., 14(3).
  * Alawieh et al., (2020) Alawieh, M. B., Li, W., Lin, Y., Singhal, L., Iyer, M. A., and Pan, D. Z. (2020).  High-definition routing congestion prediction for large-scale fpgas.  In Asia and South Pacific Design Automation Conference, pages 26–31.
  * Bai et al., (2021) Bai, S., Zhang, F., and Torr, P. H. (2021).  Hypergraph convolution and hypergraph attention.  Pattern Recognition, 110:107637.
  * Brody et al., (2022) Brody, S., Alon, U., and Yahav, E. (2022).  How attentive are graph attention networks?  In Int. Conf. on Learning Representations.
  * Cai et al., (2023) Cai, C., Hy, T. S., Yu, R., and Wang, Y. (2023).  On the connection between MPNN and graph transformer.  In Int. Conf. on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 3408–3430. PMLR.
  * Chai et al., (2023) Chai, Z., Zhao, Y., Liu, W., Lin, Y., Wang, R., and Huang, R. (2023).  Circuitnet: An open-source dataset for machine learning in vlsi cad applications with improved domain-specific evaluation metric and learning strategies.  IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, pages 1–1.
  * (8) Chen, D., Lin, Y., Li, W., Li, P., Zhou, J., and Sun, X. (2020a).  Measuring and relieving the over-smoothing problem for graph neural networks from the topological view.  Proceedings of the AAAI Conference on Artificial Intelligence, 34(04):3438–3445.
  * (9) Chen, J., Kuang, J., Zhao, G., Huang, D. J. H., and Young, E. F. Y. (2020b).  Pros: A plug-in for routability optimization applied in the state-of-the-art commercial eda tool using deep learning.  In Int. Conf on Computer-Aided Design.
  * Chen et al., (2022) Chen, X., Di, Z., Wu, W., Wu, Q., Shi, J., and Feng, Q. (2022).  Detailed routing short violation prediction using graph-based deep learning model.  IEEE Trans. Circuits and Systems II: Express Briefs, 69(2):564–568.
  * Chien et al., (2022) Chien, E., Pan, C., Peng, J., and Milenkovic, O. (2022).  You are allset: A multiset function framework for hypergraph neural networks.  In International Conference on Learning Representations.
  * Choromanski et al., (2021) Choromanski, K. M., Likhosherstov, V., Dohan, D., Song, X., Gane, A., Sarlos, T., Hawkins, P., Davis, J. Q., Mohiuddin, A., Kaiser, L., Belanger, D. B., Colwell, L. J., and Weller, A. (2021).  Rethinking attention with performers.  In Int. Conf. on Learning Representations.
  * Cohen-Steiner et al., (2009) Cohen-Steiner, D., Edelsbrunner, H., and Harer, J. (2009).  Extending persistence using poincaré and lefschetz duality.  Foundations of Computational Mathematics, 9(1):79–103.
  * Dey and Wang, (2022) Dey, T. K. and Wang, Y. (2022).  Computational Topology for Data Analysis.  Cambridge University Press.  452 pages.
  * Dong et al., (2020) Dong, Y., Sawin, W., and Bengio, Y. (2020).  Hnhn: Hypergraph networks with hyperedge neurons.
  * Dwivedi et al., (2022) Dwivedi, V. P., Rampášek, L., Galkin, M., Parviz, A., Wolf, G., Luu, A. T., and Beaini, D. (2022).  Long range graph benchmark.  In Advances in Neural Information Processing Systems, volume 35.
  * Edelsbrunner and Harer, (2010) Edelsbrunner, H. and Harer, J. (2010).  Computational Topology : an Introduction.  American Mathematical Society.
  * Edwards, (2020) Edwards, R. T. (2020).  Google/skywater and the promise of the open pdk.  In Workshop on Open-Source EDA Technology.
  * Fey and Lenssen, (2019) Fey, M. and Lenssen, J. E. (2019).  Fast graph representation learning with pytorch geometric.  Int. Conf. on Learning Representations.
  * Fitzpatrick, (1989) Fitzpatrick, P. (1989).  Klaus deimling, nonlinear functional analysis.
  * Ghose et al., (2021) Ghose, A., Zhang, V., Zhang, Y., Li, D., Liu, W., and Coates, M. (2021).  Generalizable cross-graph embedding for gnn-based congestion prediction.  In IEEE/ACM Int. Conf. Computer Aided Design, ICCAD, page 1–9.
  * Gilmer et al., (2017) Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. (2017).  Neural message passing for quantum chemistry.  In Int. Conf. on Machine Learning, ICML, page 1263–1272.
  * Heydari and Livi, (2022) Heydari, S. and Livi, L. (2022).  Message Passing Neural Networks for Hypergraphs, page 583–592.  Springer Nature Switzerland.
  * Hy et al., (2019) Hy, T. S., Trivedi, S., Pan, H., Anderson, B. M., and Kondor, R. (2019).  Covariant compositional networks for learning graphs.  In Proc. Int. Workshop on Mining and Learning with Graphs (MLG).
  * Jegelka, (2022) Jegelka, S. (2022).  Theory of graph neural networks: Representation and learning.  Int. Congress of Mathematicians, ICM.
  * Kahng et al., (2022) Kahng, A. B., Kim, M., Kim, S., and Woo, M. (2022).  Rosettastone: Connecting the past, present, and future of physical design research.  IEEE Design and Test, 39(5):70–78.
  * Karypis et al., (1999) Karypis, G., Aggarwal, R., Kumar, V., and Shekhar, S. (1999).  Multilevel hypergraph partitioning: applications in vlsi domain.  IEEE Trans. Very Large Scale Integr. Syst., 7(1):69–79.
  * Karypis and Kumar, (1998) Karypis, G. and Kumar, V. (1998).  A fast and high quality multilevel scheme for partitioning irregular graphs.  SIAM Journal on Scientific Computing, 20(1):359–392.
  * Katharopoulos et al., (2020) Katharopoulos, A., Vyas, A., Pappas, N., and Fleuret, F. (2020).  Transformers are RNNs: Fast autoregressive transformers with linear attention.  In Int. Conf. on Machine Learning, ICML.
  * Kim et al., (2022) Kim, J., Nguyen, D., Min, S., Cho, S., Lee, M., Lee, H., and Hong, S. (2022).  Pure transformers are powerful graph learners.  In Advances in Neural Information Processing Systems, volume 35.
  * Kipf and Welling, (2017) Kipf, T. N. and Welling, M. (2017).  Semi-supervised classification with graph convolutional networks.  In Int. Conf. on Learning Representations.
  * Kirby et al., (2019) Kirby, R., Godil, S., Roy, R., and Catanzaro, B. (2019).  Congestionnet: Routing congestion prediction using deep graph neural networks.  In IFIP/IEEE Int. Conf. Very Large Scale Integration, VLSI-SoC, pages 217–222.
  * Liang et al., (2020) Liang, R., Xiang, H., Pandey, D., Reddy, L., Ramji, S., Nam, G.-J., and Hu, J. (2020).  Drc hotspot prediction at sub-10nm process nodes using customized convolutional network.  In Int. Symposium on Physical Design, ISPD, page 135–142.
  * Mirhoseini et al., (2021) Mirhoseini, A., Goldie, A., Yazgan, M., Jiang, J. W., Songhori, E. M., Wang, S., Lee, Y.-J., Johnson, E., Pathak, O., Nazi, A., Pak, J., Tong, A., Srinivasa, K., Hang, W., Tuncer, E., Le, Q. V., Laudon, J., Ho, R., Carpenter, R., and Dean, J. (2021).  A graph placement methodology for fast chip design.  Nature, 594:207 – 212.
  * Müller et al., (2023) Müller, L., Morris, C., Galkin, M., and Rampášek, L. (2023).  Attending to Graph Transformers.  Arxiv preprint.
  * Ngo et al., (2023) Ngo, N. K., Hy, T. S., and Kondor, R. (2023).  Multiresolution graph transformers and wavelet positional encoding for learning long-range and hierarchical structures.  The Journal of Chemical Physics, 159(3):034109.
  * Paszke et al., (2019) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., and Chintala, S. (2019).  Pytorch: An imperative style, high-performance deep learning library.  In Advances in Neural Information Processing Systems, volume 32.
  * Rampášek et al., (2022) Rampášek, L., Galkin, M., Dwivedi, V. P., Luu, A. T., Wolf, G., and Beaini, D. (2022).  Recipe for a general, powerful, scalable graph transformer.  In Advances in Neural Information Processing Systems, volume 35, pages 14501–14515.
  * Sutherland, (2009) Sutherland, W. A. (2009).  Introduction to metric and topological spaces.  Oxford University Press.
  * Tabaghi and Wang, (2023) Tabaghi, P. and Wang, Y. (2023).  Universal Representation of Permutation-Invariant Functions on Vectors and Tensors.  arXiv preprint arXiv:2310.13829.
  * Tabrizi et al., (2018) Tabrizi, A. F., Rakai, L., Darav, N. K., Bustany, I., Behjat, L., Xu, S., and Kennings, A. (2018).  A machine learning framework to identify detailed routing short violations from a placed netlist.  In ACM/ESDA/IEEE Design Automation Conference, DAC), pages 1–6.
  * Tay et al., (2022) Tay, Y., Dehghani, M., Bahri, D., and Metzler, D. (2022).  Efficient transformers: A survey.  ACM Comput. Surv., 55(6).
  * Tian and Wang, (2019) Tian, M. and Wang, Y. (2019).  A limit theorem for the $1$st Betti number of layer-$1$ subgraphs in random graphs.  ArXiv preprint.
  * Topping et al., (2022) Topping, J., Giovanni, F. D., Chamberlain, B. P., Dong, X., and Bronstein, M. M. (2022).  Understanding over-squashing and bottlenecks on graphs via curvature.  In Int. Conf. on Learning Representations.
  * Veličković et al., (2018) Veličković, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., and Bengio, Y. (2018).  Graph attention networks.  In Int. Conf. on Learning Representations.
  * Viswanathan et al., (2012) Viswanathan, N., Alpert, C., Sze, C., Li, Z., and Wei, Y. (2012).  The DAC 2012 routability-driven placement contest and benchmark suite.  In Annual Design Automation Conference, DAC, page 774–782.
  * Viswanathan et al., (2011) Viswanathan, N., Alpert, C. J., Sze, C., Li, Z., Nam, G.-J., and Roy, J. A. (2011).  The ISPD-2011 routability-driven placement contest and benchmark suite.  In Int. Symposium on Physical Design, ISPD, page 141–146.
  * Wagstaff et al., (2019) Wagstaff, E., Fuchs, F., Engelcke, M., Posner, I., and Osborne, M. A. (2019).  On the limitations of representing functions on sets.  In Int. Conf. on Machine Learning, pages 6487–6494. PMLR.
  * Wang et al., (2022) Wang, B., Shen, G., Li, D., Hao, J., Liu, W., Huang, Y., Wu, H., Lin, Y., Chen, G., and Heng, P. A. (2022).  Lhnn: Lattice hypergraph neural network for vlsi congestion prediction.  In ACM/IEEE Design Automation Conference, DAC, page 1297–1302.
  * Xie et al., (2018) Xie, Z., Huang, Y.-H., Fang, G.-Q., Ren, H., Fang, S.-Y., Chen, Y., and Hu, J. (2018).  Routenet: Routability prediction for mixed-size designs using convolutional neural network.  In IEEE/ACM Int. Conf on Computer-Aided Design, ICCAD, pages 1–8.
  * Xie et al., (2021) Xie, Z., Liang, R., Xu, X., Hu, J., Duan, Y., and Chen, Y. (2021).  Net2: A graph attention network method customized for pre-placement net length estimation.  In Asia and South Pacific Design Automation Conference, ASPDAC, page 671–677.
  * Xu et al., (2019) Xu, K., Hu, W., Leskovec, J., and Jegelka, S. (2019).  How powerful are graph neural networks?  In Int. Conf. on Learning Representations.
  * Yan et al., (2021) Yan, Z., Ma, T., Gao, L., Tang, Z., and Chen, C. (2021).  Link prediction with persistent homology: An interactive view.  In Int. Conf. on Machine Learning, pages 11659–11669. PMLR.
  * Yang et al., (2022) Yang, S., Yang, Z., Li, D., Zhang, Y., Zhang, Z., Song, G., and Hao, J. (2022).  Versatile multi-stage graph neural network for circuit representation.  In Advances in Neural Information Processing Systems, volume 35.
  * Zaheer et al., (2017) Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhutdinov, R. R., and Smola, A. J. (2017).  Deep sets.  In Advances in Neural Information Processing Systems, volume 30.
  * Zhang et al., (2023) Zhang, B., Luo, S., Wang, L., and He, D. (2023).  Rethinking the expressive power of GNNs via graph biconnectivity.  In The Eleventh Int. Conf. on Learning Representations.
  * Zhao et al., (2020) Zhao, Q., Ye, Z., Chen, C., and Wang, Y. (2020).  Persistence enhanced graph neural network.  In Int. Conf. on Artificial Intelligence and Statistics, pages 2896–2906. PMLR.

Supplement of the AISTATS submission #1797,
Title: “DE-HNN: An effective neural model for Circuit Netlist representation”

##  Appendix A More Background

Figure 4: Visulization of a circuit netlits in the post place-and-route stage:
Each cell ($c_{i}$) is positioned within the physical layout of the chip and
interconnected with other components following the net maps ($\sigma_{i}$).

###  A.1 Circuit netlists

A circuit netlist is a textual representation of electronic components, such
as logical boolean gates, and the connections between them. Figure 4 provides
an example illustrating a circuit netlist consisting of five components
interconnected by five nets. After laying out the netlist in the physical
space (placement), during the routing stage, the edges of the netlist are
mapped to the routing channels within the chip’s physical floorplan (indicated
by dashed lines in Figure 4). Routing congestion occurs when the number of
edges to be routed in a specific region of the floorplan exceeds the available
routing capacity.

###  A.2 Persistence homology based features

Persistent homology is one of the most important development in the field of
topological data analysis in the past two decades. It can encode meaningful
topological features in a multi-scale manner and has already been applied to
numerous applications; see books (Edelsbrunner and Harer,, 2010; Dey and
Wang,, 2022). Intuitively, given a domain $X$, a $p$-th homology class (or
informally, a $p$-th homological feature) essentially captures a family of
equivalent $p$-dimensional “holes” in $X$; for example, 0-, 1- and
2-dimensional homology captures connected components, equivalent classes of
loops and of 2-dimensional voids, respectively. The $p$-th homology group
$\mathrm{H}_{p}(X)$ (using $\mathbb{Z}_{2}$ coefficients) characterizes the
space of $p$-th homological features of $X$, and its rank
$rank(\mathrm{H}_{p}(X))$ corresponds to the number of independent $p$-th
homological features. For example, $rank(\mathrm{H}_{0}(X))$ denotes the
number of connected components of space $X$. If the domain $X$ is a graph,
then its $1$-st homology group is homeomorphic the space of cycles (loops),
and $rank(\mathrm{H}_{1}(X))$ is simply the number of independent cycles.

Persistent homology is a modern extension of homology, where instead of a
single space $X$, we now inspect a sequence of growing spaces $X_{1}\subseteq
X_{2}\subseteq\cdots X_{m}=X$, called a _filtration_ 444Note that persistent
homology has been extended for much broader families of filtrations then the
growing sequence we describe here; see (Edelsbrunner and Harer,, 2010; Dey and
Wang,, 2022).. Intuitively, we can view this filtration as an evolution of the
space $X$. As the space grows, we track its corresponding topological features
(as captured by the homology groups introduced above). Sometimes new
topological features (e.g., a new component, or a hole) will appear, and
sometimes they will disappear (e.g, a hole is filled and thus “disappear”).
The _persistent homology (PH)_ captures such creation and death of topological
features, and outputs a feature summary called _persistent diagram (PD)_ ,
consisting of the birth-time and death-time of classes of topological features
during this evolution. In short, the PDs provide a multiscale summary for the
topological features of $X$ from the perspective of filtration $X_{1}\subseteq
X_{2}\subseteq\cdots\subseteq X_{m}$. In the past few years, there have been a
series of approaches to vectorize PDs or to kernelize them for ML
applications; see Chapter 13 of (Dey and Wang,, 2022). In our work, we use the
so-called _persistence images_ (Adams et al.,, 2017) to convert a PD to a
finite dimensional vector to be used as part of the node feature.

Persistence diagram (PD) induced from Netlists. We compute the persistence
diagram (PD) based on the following directed graph representation of the
netlist (instead of using the directed hypergraph representation, for easier
computation of persistence).

In particular, given a netlist, we create a directed graph $G=(V,E)$ where
each node in $V$ corresponds to a cell in the netlist, while a directed edge
is formed between the driver cell of some net with each of the sink in that
net; that is, each net $\sigma=({\mathsf{v}}_{\sigma},{\mathsf{S}}_{\sigma})$
gives rise to a set of directed edges
$\\{({\mathsf{v}}_{\sigma},v)\\}_{v\in{\mathsf{S}}_{\sigma}}$. We refer to
this graph as the _star-graph_ induced by the input netlist.

Now, for each node $v$ in the star-graph $G$, we create its $k$-ring _out-flow
neighborhood $G^{in}_{v}$_ which is simply the subgraph in $G$ spanned by all
nodes reachable from $v$ by a path of at most $k$ hops. Symmetrically, the
$k$-ring _in-flow neighborhood $G^{out}_{v}$_ of $v$ is the subgraph of $G$
spanned by all nodes such that $v$ is reachable within $k$ hops. These two
$G_{v}$s form the _local motifs_ around $v$. We wish to obtain a feature
vector summary for $G^{in}_{v}$ and $G^{out}_{v}$. To this end, similar to
(Yan et al.,, 2021; Zhao et al.,, 2020), we use the so-called _$0$ th extended
persistence diagram_ $PD^{*}_{v}$ (Cohen-Steiner et al.,, 2009) of $G^{*}_{v}$
(where $*\in\\{in,out\\}$) induced by the shortest path distance function to
$v$ as its summary. For our specific graph settig, using results of (Tian and
Wang,, 2019), one can show that $PD^{*}_{v}$ computed this way, is a concise
summary that encodes rich information around $v$, including the number of
triangles incident to $v$, clustering coefficient of $v$, number of nodes at
distance $r\leq k$ away from $v$, number of crossing edges from distance-$r$
nodes to distance ($r+1$) nodes, certain “shortest” system of cycle basis
passing through $v$ of $G^{*}_{v}$, and so on. Finally, each PD (for in-flow
neighborhood and out-flow neighborhood of $v$) is vectorized to persistent
images as in (Adams et al.,, 2017), and we further vectorize (flatten) each
image matrix all together to generate a new vector as the persistent
representation vector for each node.

###  A.3 Metis Partitioning

Metis (Karypis and Kumar,, 1998) is currently State-of-the-art (SOTA)
algorithm in chip design applications to provide balanced k-way partitions for
large graphs efficiently. We choose Metis due to its practical performance,
that it produces multiple clusters of balanced sizes, and the ease it is to
control the size of the clusters. Besides Metis, hMetis (Karypis et al.,,
1999) developed on the base of Metis for hypergraph parititioning is an
alternative choice. In our paper we used Metis in case we need partitioning
for other graph based methods we compare our method with, and Metis will give
us consistent partitioning. Nevertheless, we expect that hMetis can be used
without much impact to the performance of our pipeline.

##  Appendix B Theoretical Analysis

###  B.1 Preliminaries

Let $\mathbb{D}$ be a domain, such as $\mathbb{Q},\mathbb{R}$, and
$\mathbb{R}^{d}$. Consider the set function
$f:2^{\mathbb{D}}\rightarrow\mathrm{codom}(f)$ where $\mathbb{D}$ is a
countable domain. Then, we have

$\displaystyle\forall X\subseteq\mathbb{D}:f(X)=\rho\circ\Phi(X),\ \
\Phi(X)=\sum_{x\in X}\phi(x),$ (8)

where $\phi:\mathbb{D}\rightarrow\mathrm{codom}(\phi)\subset\mathbb{R}$, and
$\rho:\mathrm{codom}(\phi)\rightarrow\mathrm{codom}(f)$. This is the so-called
sum-decomposable representation of $f$ via $\mathbb{R}$ in terms of
$(\phi,\rho)$ basis functions. We refer to ambient space of
$\mathrm{codom}(\phi)$ (in Eqn (8), it is $\mathbb{R}$) as the model’s latent
space (Zaheer et al.,, 2017). There is an extended sum-decomposable
representation of $f$ on multisets whose elements are drawn from an
uncountable domain (e.g., $\mathbb{R}^{d}$) by restricting the size of input
multisets. Recently, Tabaghi and Wang, (2023) proposed an encoder
$\phi:\mathbb{R}^{d}\rightarrow\mathrm{codom}(\phi)\subset\mathbb{R}^{2dN}$
such that for any continuous multiset function
$f:\mathbb{X}_{N,\mathcal{B}}\rightarrow\mathrm{codom}(f)$ we have,

$\forall X\in\mathbb{X}_{N,\mathcal{B}}:f(X)=\rho\circ\Phi(X),$

where $\mathbb{X}_{N,\mathcal{B}}$ is the set of multisets of size $N$ with
elements drawn from $\mathcal{B}$ — a compact subset of $\mathbb{R}^{d}$ — and
$\Phi(X)=\sum_{x\in X}\phi(x)$. The latent space dimension of this
representation is $2dN$. As a result of this representation,
$\Phi:\mathbb{X}_{N,\mathcal{B}}\rightarrow\mathrm{codom}(\Phi)$ is an
injective map where $\mathrm{codom}(\Phi)\subset\mathbb{R}^{2dN}$.
Furthermore, Tabaghi and Wang, (2023) show that $\rho$ is continuous over the
latent space $\mathbb{R}^{2dN}$.

At their core, sum-decomposable models rely on a bijection between multisets
and encoded features, that is, $X=\alpha\circ\Phi(X)$ for any multiset
$X\in\mathbb{X}_{N,\mathcal{B}}$ and a bijective map $\alpha$. In the
following proposition, we first generalize this result to multisets with
varied sizes.

######  Proposition 1.

Let $\mathcal{B}$ be a connected compact subset of $\mathbb{R}^{d}$. There
exists a function $\phi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{2dN}$ such that

$\forall X\in\mathbb{X}_{\leq N,\mathcal{B}}:X=\alpha\bigg{(}\sum_{x\in
X}\phi(x)\bigg{)}=\alpha\circ\Phi(X)$

where $\mathbb{X}_{\leq N,\mathcal{B}}$ is all multisets of size $\leq N$ with
elements from $\mathcal{B}$, $\alpha$ is a continuous map over
$\mathbb{R}^{2dN}$.

###### Proof.

As mentioned earlier, Tabaghi and Wang, (2023) proposed a continuous encoding
function $\phi^{\prime}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{2dN}$ such that
$\Phi^{\prime}(x)=\sum_{x\in X}\phi^{\prime}(x)$ is an injective map over
multisets with exactly $N$ elements, that is,
$(\Phi^{\prime})^{-1}\circ\Phi^{\prime}(X)=X$ where elements of multiset $X$
is drawn from $\mathbb{R}^{d}$ and $|X|=N$ 555Note that we trivially changed
the notation from $\phi$ to $\phi^{\prime}$..

######  Lemma 1.

The function $\Phi^{\prime}$ is a homeomorphism.

###### Proof.

The function $\Phi^{\prime}$ is continous and injective by contruction. We
want to show that $(\Phi^{\prime})^{-1}$ is continuous over
$\mathrm{codom}(\Phi^{\prime})=\\{\Phi^{\prime}(X):X\in\mathbb{X}_{N,\mathcal{B}}\\}$
where $\mathcal{B}$ is a compact and connected subset of $\mathbb{R}^{d}$. The
set $\mathbb{X}_{N,\mathcal{B}}$ is compact; refer to Lemma 5 in (Tabaghi and
Wang,, 2023). Also, $\mathrm{codom}(\Phi^{\prime})\subseteq\mathbb{R}^{2dN}$
forms a metric space with $\ell_{2}$ metric; Hence, it is also a Hausdorff
space. By the inverse function theorem, $\Phi^{\prime}$ — a continuous
bijection from a compact space to a Hausdorff space — has a continuous
inverse; see Proposition 13.26 in (Sutherland,, 2009). ∎

To extend the result of Lemma 1 to multisets of variable sizes, we follow the
same proof sketch as the one used for the one-dimensional case (Wagstaff et
al.,, 2019). In particular, let
$x_{\circ}\in\mathbb{R}^{d}\setminus\mathcal{B}$ where
$\inf_{x\in\mathcal{B}}\|x-x_{\circ}\|_{2}>0$. Then, we define
$\phi(x)=\phi^{\prime}(x)-\phi^{\prime}(x_{\circ})$. For any multiset with
$M\leq N$ elements from $\mathcal{B}$, say $X$, we have

$\displaystyle\forall X\in\mathbb{X}_{\leq N,\mathcal{B}}:\Phi(X)$
$\displaystyle=\sum_{x\in X}\phi(x)=\sum_{x\in
X}\phi^{\prime}(x)-M\phi^{\prime}(x_{\circ})$
$\displaystyle=\Phi^{\prime}(X\cup\\{\\{\underbrace{x_{\circ},\ldots,x_{\circ}}_{N-M}\\}\\})+\mathrm{const},$

where $\mathrm{const}=-N\phi^{\prime}(x_{\circ})$. This is in the sum-
decomposable form. Since $\Phi^{\prime}$ is an injective map over
$\mathbb{X}_{N,\mathcal{B}}$, $\Phi$ is also an injective map over
$\mathbb{X}_{\leq N,\mathcal{B}}$. Specifically, we can derive a closed-form
expression for its inverse as follows:

$\displaystyle\Big{(}{\Phi^{\prime}}^{-1}\circ(\Phi(X)-\mathrm{const}\big{)}\Big{)}\cap\mathcal{B}$
$\displaystyle=\Big{(}{\Phi^{\prime}}^{-1}\circ\Phi^{\prime}(X\cup\\{\\{\underbrace{x_{\circ},\ldots,x_{\circ}}_{N-M}\\}\\})\Big{)}\cap\mathcal{B}$
$\displaystyle=(X\cup\\{\\{\underbrace{x_{\circ},\ldots,x_{\circ}}_{N-M}\\}\\})\cap\mathcal{B}$
$\displaystyle=X.$

In other words, we have
$\Phi^{-1}(U)={\Phi^{\prime}}^{-1}\big{(}U-\mathrm{const}\big{)}\cap\mathcal{B}$
for all $U\in\mathrm{codom}(\Phi)=\\{\Phi(X):X\in\mathbb{X}_{\leq
N,\mathcal{B}}\\}$.

The function $\Phi:\mathbb{X}_{\leq
N,\mathcal{B}}\rightarrow\mathrm{codom}(\Phi)$ is a continuous bijection. Its
domain is compact because it is a finite union of comapct spaces, that is,
$\mathbb{X}_{\leq N,\mathcal{B}}=\cup_{n\in[N]}\mathbb{X}_{n,\mathcal{B}}$.
Furthermore, its codomain forms a metric space with $\ell_{2}$ distance;
Hence, it is also a Hausdorff space. Therefore, by an inverse function
theorem, $\Phi$ has a continuous inverse; (Sutherland,, 2009). If we let
$\alpha={\Phi}^{-1}$, we arrive at the Proposition’s statement. ∎

######  Remark 1.

Proposition 1 gurantees the continuity of $\alpha$ over a compact set
$\mathrm{codom}(\Phi)\subset\mathbb{R}^{2dN}$. This function can be
continuously extended to the ambient space $\mathbb{R}^{2dN}$; refer to the
continuous extension theorem (Fitzpatrick,, 1989).

###  B.2 Proof of Theorem 1

As discussed in the main text, the general net-value function is nested-
permutation invariant and take the form of Eqn (6) in the main text. In what
follows, we provide the detailed version of Theorem 1 and show that such
nested-permutation invariant function adapts a nested sum-decomposition
aligned with our DE-HNN’s node and net update rules.

[Detailed]theoremgoldbach Let ${\mathcal{F}}$ be any continuous, nested-
permutation invariant, net-value function as in Eqn (6). For simplicity,
assume both input nets and output of $M$ take values in a compact set
$\mathcal{B}\subset\mathbb{R}^{d}$, a connected compact subset of
$\mathbb{R}^{d}$. We also let $N_{v}=\max\\{|{\mathcal{I}}(v)|:\forall v\\}$
and $N_{{\sigma}}=\max\\{|{\mathsf{S}}({\sigma})|:\forall{\sigma}\\}$. Then we
have that ${\mathcal{F}}$ can be expressed as the following sum-decomposition:

$\displaystyle\forall\sigma:{\mathcal{F}}\Big{(}\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})},\Big{\\{}\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}(v^{\prime})}\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\Big{)}=\rho\Big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})}\phi_{1}(\mu_{\sigma^{\prime}}),\sum_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\phi_{2}\big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}(v^{\prime})}\phi_{1}(\mu_{\sigma^{\prime}})\big{)}\Big{)},$

where $\phi_{1}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d^{\prime}(d,N_{v})}$,
$\phi_{2}:\mathbb{R}^{d^{\prime}(d,N_{v})}\rightarrow\mathbb{R}^{d^{\prime\prime}(d,N_{v},N_{{\sigma}})}$,
and
$\rho:\mathbb{R}^{d^{\prime}(d,N_{v})}\times\mathbb{R}^{d^{\prime\prime}(d,N_{v},N_{\sigma})}\rightarrow\mathbb{R}^{d}$
are continuous functions.

###### Proof.

The general net-value function takes the following form:

$\displaystyle\forall\sigma:~{}$
$\displaystyle{\mathcal{F}}\Big{(}\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})},\Big{\\{}\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}(v^{\prime})}\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\Big{)}.$

We assume ${\mathcal{F}}$ is a continuous function that takes values in
$\mathcal{B}$, a connected compact subset of $\mathbb{R}^{d}$, and
$N_{v}=\max\\{|{\mathcal{I}}(v)|:\forall v\\}$. From Proposition 1, there
exist continuous functions $\phi_{1}$ and $\alpha_{1}$ such that

$\forall X\in\mathbb{X}_{\leq N_{v},\mathcal{B}}:X=\alpha_{1}(\sum_{x\in
X}\phi_{1}(x)),$

where $\mathrm{codom}(\phi_{1})\subseteq\mathbb{R}^{2dN_{v}}$, that is, the
latent dimension depends on $d$ and $N_{v}$ which we denote as
$d^{{}^{\prime}}(d,N_{v})$. Therefore, we apply the result in Proposition 1 to
sets of net values (of size at most $N_{v}$) and express ${\mathcal{F}}$
follows:

$\displaystyle\forall{\sigma}:{\mathcal{F}}\Big{(}\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})},\Big{\\{}\\{\mu_{{\sigma}^{\prime}}\\}_{{\sigma}^{\prime}\in{\mathcal{I}}(v^{\prime})}\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\Big{)}$
$\displaystyle=\mathcal{F}\Big{(}\alpha_{1}\big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})}\phi_{1}(\mu_{\sigma}^{\prime})\big{)},\Big{\\{}\alpha_{1}\big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}(v^{\prime})}\phi_{1}(\sigma^{\prime})\big{)}\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\Big{)}$
$\displaystyle=\mathcal{F}_{1}\Big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})}\phi_{1}(\mu_{\sigma}^{\prime}),\Big{\\{}\sum_{\sigma^{\prime}\in{\mathcal{I}}(v^{\prime})}\phi_{1}(\sigma^{\prime})\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\Big{)}$

where
$\mathcal{F}_{1}(x,Y)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\mathcal{F}(\alpha_{1}(x),\\{\alpha_{1}(y)\\}_{y\in
Y})$ for all $x\in\mathrm{codom}(\Phi_{1})=\\{\Phi_{1}(Y)=\sum_{y\in
Y}\phi_{1}(y):Y\in\mathbb{X}_{\leq N_{v},\mathcal{B}}\\}$ and any set $Y$ with
elements in $\mathrm{codom}(\Phi_{1})$. Since both $\alpha_{1}$ and
$\mathcal{F}$ are continuous functions, then $\mathcal{F}_{1}$ is also a
continuous function. 666We can use the matching distance to define a metric on
multisets; refer to (Tabaghi and Wang,, 2023) for details.

Next, we can apply the result of Proposition 1 to $\mathcal{F}_{1}$. The
elements of the set
$\Big{\\{}\sum_{\sigma^{\prime}\in{\mathcal{I}}(v^{\prime})}\phi_{1}(\sigma^{\prime})\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}$
take values in $\mathrm{codom}(\Phi_{1})$ — a connected compact subset of
$\mathbb{R}^{d^{\prime}(d,N_{v})}$. If we assume $|{\mathsf{S}}_{\sigma}|\leq
N_{{\sigma}}$ for all ${\sigma}$, Proposition 1 claims that there exist
continuous functions $\alpha_{2}$ and $\phi_{2}$ such that the following holds
true:

$\displaystyle\forall{\sigma}:\mathcal{F}_{1}\Big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})}\phi_{1}(\mu_{\sigma}^{\prime}),\Big{\\{}\sum_{\sigma^{\prime}\in{\mathcal{I}}(v^{\prime})}\phi_{1}(\sigma^{\prime})\Big{\\}}_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\Big{)}$
$\displaystyle=\mathcal{F}_{1}\Bigg{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})}\phi_{1}(\mu_{\sigma}^{\prime}),\alpha_{2}\Big{(}\sum_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\phi_{2}\big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}(v^{\prime})}\phi_{1}(\sigma^{\prime})\big{)}\Big{)}\Bigg{)}$
$\displaystyle=\rho\Bigg{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}({\mathsf{v}}_{\sigma})}\phi_{1}(\mu_{\sigma}^{\prime}),\sum_{v^{\prime}\in{\mathsf{S}}_{\sigma}}\phi_{2}\big{(}\sum_{\sigma^{\prime}\in{\mathcal{I}}(v^{\prime})}\phi_{1}(\sigma^{\prime})\big{)}\Bigg{)}$

where
$\rho(x,y)\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{F}_{1}(x,\alpha_{2}(y))$
for all $x\in\mathrm{codom}(\Phi_{1})$ and
$y\in\mathrm{codom}(\Phi_{2})\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\sum_{z\in
Z}\phi_{2}(z):Z\in\mathrm{codom}(\Phi_{1}),|Z|\leq N_{{\sigma}}\\}$. The
function $\rho$ is a composition of continuous functions $\mathcal{F}_{1}$ and
$\alpha_{2}$. Therefore, it is a continuous function. Also, we have
$\mathrm{codom}(\phi_{2})\subseteq\mathbb{R}^{2d^{\prime}(d,N_{v})N_{{\sigma}}}$,
that is, the latent dimension depends on $d,N_{v},$ and $N_{\sigma}$ which we
denote as $d^{\prime\prime}(d,N_{v},N_{\sigma})$. Finally, functions
$\phi_{2}$ and $\rho$ are continuous over compact domains
$\mathrm{codom}(\Phi_{1})$, and
$\mathrm{codom}(\Phi_{1})\times\mathrm{codom}(\Phi_{2})$. Therefore, they can
be continuously extended to $\mathbb{R}^{d^{\prime}(d,N_{v})}$ and
$\mathbb{R}^{d^{\prime}(d,N_{v})}\times\mathbb{R}^{d^{\prime\prime}(d,N_{v},N_{\sigma})}$.
∎

##  Appendix C Experimental details

###  C.1 Dataset Statistics

We report the sizes of each design and their cell/net-degrees distribution in
Table 5, and net-based demand/wirelength distributions in the Table 6. We can
see that (hyper)graphs in our dataset are large and sparse, ranging from 400K
to 1.3M nodes. with few outliers that have high degrees. We also summarize the
more detailed distributions of net-based demand, net-based wirelength, and
cell-based congestion in figure5.

###  C.2 Experiment Setup

We engineer the input features as Cell features and Net features. None of the
following features contained placement information or are computed from
placement information. We engineer cell features as follows:

  * •

We analyze the statistics in a design including the minimum and maximum of all
cells’ width, height. Then, we normalize all these quantities to be in the
range $[0,1]$. We concatenate them with the cell’s discrete orientation and
cell’s degree (number of nets each cell connecting to), that results into the
cell’s input feature vector.

  * •

We calculate the top-10 eigenvectors of the graph Laplacian of the
heterogenous graph as the positional encoding (denoted as LapPE) for every
cell and net.

  * •

We compute the persistence diagram (denoted as PD) for each cell-node $v$ on a
directed graph $G=(V,E)$, as we mentioned earlier in A.2, as a common cell-
node input feature vector for all baselines, to capture the local topological
information.

  * •

We also compute the degree distribution for each cell-node $v$ based on their
$k$-ring _undirected_ _neighborhood $G_{v}$_ in the star-graph we introduced
in A.2 to compute the persistence diagrams. However, we ignore the direction
when we compute the degree distribution.

For the net features, for all models other than linear Transformer, we
initialize the features of each net
${\sigma}=({\mathsf{v}}_{\sigma},{\mathsf{S}}_{\sigma})$ as the net’s degree
(i.e. number of cells each net connecting to). For Linear Transformer, in
order to provide better topology information, we instead initialize the
features of each net ${\sigma}=({\mathsf{v}}_{\sigma},{\mathsf{S}}_{\sigma})$
(denoted as $M({\sigma})$) as the average of all the features of the nodes
this net contains, computed as,

$M({\sigma})=\frac{1}{|{\mathsf{S}}_{\sigma}|+1}(m({\mathsf{v}}_{\sigma})+\sum_{v^{\prime}\in{\mathsf{S}}_{\sigma}}m(v^{\prime})).$
(9)

As also described in the main text, for GCN and GATv2, we use the bipartite
graph representation of the input netlist, where there are two types of graph
nodes: those corresponding to cells and those corresponding nets. If a cell is
contained in a net, then there is an edge between their respective graph
nodes. We use heterogenous message passing, where nodes corresponding to cells
use a different message passing mechanism from nodes corresponding to nets.

For GCN, GATv2, HyperConv, and all other HNN based models, we used 4 layers
with 64 dimension each layer as the setting. For Linear Transformer, we used 2
layers with 64 dimension each layer as the setting. For NetlistGNN we used 4
layers with node dimension 64 and net dimension 64 as the setting. For
E-HNN/DE-HNN based models, we used 3 layers with node dimension 64.

We report the number of parameters, training time per epoch and total training
epochs in the Table 4. We aim to use similar number of parameters, although
currently full-DE-HNNdoes have the highest number of parameters, with
NetlistGNN having the second largest. Interestingly, while they all take
similar number of epochs to converge, our DE-HNN in fact is much faster per
epoch, so overall takes less time to train. We used two NVIDIA A100-SXM4-80GB
GPUs on Linux CentOS Stream system (Ver.8) for both single-design and cross-
design experiments.

Model | GCN | GATv2 | HyperConv | Lin. Transformer | NetlistGNN | AllSet | HMPNN | HNHN | base-DE-HNN | full-DE-HNN
---|---|---|---|---|---|---|---|---|---|---
Num. of parameters | 218113 | 251905 | 218625 | 203201 | 364743 | 348289 | 235137 | 217729 | 272165 | 383105
Time per epoch (single design) | 2.77 | 2.91 | 1.06 | 3.72 | 1.13 | 0.79 | 0.69 | 0.62 | 0.49 | 0.65
Total num. of epochs (single design) | 878 | 775 | 670 | 636 | 716 | 689 | 691 | 699 | 673 | 810
Time per epoch (cross design) | 33.20 | 36.28 | 12.28 | 55.37 | 16.51 | 11.35 | 4.14 | 5.37 | 5.26 | 5.50
Total num. of epochs (cross design) | 478 | 482 | 473 | 471 | 455 | 460 | 472 | 468 | 479 | 473

Table 4: Comparison of Model complexity for all the models we used. We compare the number of parameters, training time per epoch for single design (on average), total training epoches to converge for single design (on average), trainig time per epoch for cross design, and total training epoches to converge for cross design.  Design | Cells | Nets | Cell-degree | Net-degree
---|---|---|---|---
|  |  | Min | Max | Mean | STD | Min | Max | Mean | STD
Superblue1 | 797,938 | 821,523 | 0.0 | 1243 | 3.70 | 5.66 | 0.0 | 140605 | 3.58 | 155.85
Superblue2 | 951,166 | 985,117 | 0.0 | 1317 | 3.51 | 5.66 | 0.0 | 190487 | 3.39 | 192.22
Superblue3 | 901,254 | 925,667 | 0.0 | 2245 | 3.65 | 5.23 | 0.0 | 168630 | 3.55 | 175.91
Superblue5 | 727,341 | 803,681 | 0.0 | 1381 | 3.46 | 5.82 | 0.0 | 114259 | 3.13 | 128.03
Superblue6 | 998,122 | 1049,225 | 0.0 | 1689 | 3.57 | 4.03 | 0.0 | 179410 | 3.39 | 175.69
Superblue7 | 1319,052 | 1339,522 | 0.0 | 849 | 3.91 | 3.03 | 0.0 | 265765 | 3.85 | 230.24
Superblue9 | 810,812 | 830.308 | 0.0 | 1265 | 3.83 | 4.78 | 0.0 | 129541 | 3.74 | 202.42
Superblue11 | 923,355 | 954,144 | 0.0 | 1983 | 3.74 | 6.97 | 0.0 | 203194 | 3.63 | 294.36
Superblue14 | 604,921 | 627,036 | 0.0 | 1023 | 3.90 | 4.66 | 0.0 | 167911 | 3.76 | 300.30
Superblue16 | 671,284 | 696,983 | 0.0 | 1016 | 3.77 | 6.12 | 0.0 | 140741 | 3.63 | 238.58
Superblue18 | 459,495 | 468,888 | 0.0 | 1192 | 4.22 | 4.30 | 0.0 | 102047 | 4.14 | 150.74
Superblue19 | 495,234 | 510,258 | 0.0 | 1507 | 3.58 | 6.02 | 0.0 | 94682 | 3.48 | 135.31
Table 5: Dataset details & statistics. 1st column in the table shows the name
of the design, 2nd column and 3rd column show the number of cells and nets in
each design, 4th-7th columns show the distribution of cell-degrees for each
design and 8th-11th columns show the distribution of net-degrees for each
design.

Design | Net-based demand | Net-based wirelength | Cell-based congestion
---|---|---|---
| Min | Max | Mean | STD | Min | Max | Mean | STD | Min | Max | Mean | STD
Superblue1 | 0.0 | 103.50 | 26.24 | 8.01 | 8.91 | 23.92 | 14.80 | 2.45 | 0.0 | 12.00 | 1.21 | 0.55
Superblue2 | 0.0 | 139.37 | 26.61 | 8.52 | 8.91 | 24.61 | 15.21 | 2.69 | 0.0 | 5.00 | 0.70 | 0.45
Superblue3 | 0.0 | 103.50 | 24.07 | 7.31 | 8.91 | 23.83 | 14.91 | 2.45 | 0.0 | 5.19 | 0.90 | 0.47
Superblue5 | 0.0 | 1203.25 | 41.62 | 32.00 | 8.91 | 24.06 | 15.68 | 2.96 | 0.0 | 78.92 | 1.27 | 0.94
Superblue6 | 0.0 | 980.33 | 33.83 | 27.45 | 8.91 | 23.83 | 14.88 | 2.57 | 0.0 | 74.15 | 1.35 | 1.15
Superblue7 | 0.0 | 495.25 | 23.89 | 5.34 | 8.91 | 23.93 | 14.89 | 2.48 | 0.0 | 43.38 | 1.09 | 0.33
Superblue9 | 0.0 | 79.50 | 22.47 | 8.02 | 8.91 | 23.83 | 14.71 | 2.42 | 0.0 | 5.0 | 0.74 | 0.46
Superblue11 | 0.0 | 75.00 | 20.05 | 6.54 | 8.91 | 24.17 | 15.11 | 2.38 | 0.0 | 8.0 | 0.93 | 0.36
Superblue14 | 0.0 | 401.41 | 23.42 | 9.11 | 8.91 | 23.67 | 15.06 | 2.48 | 0.0 | 46.85 | 1.06 | 0.65
Superblue16 | 0.0 | 1091.0 | 28.96 | 14.09 | 8.91 | 23.51 | 15.01 | 2.54 | 0.0 | 65.53 | 1.29 | 0.91
Superblue18 | 0.0 | 50.0 | 20.39 | 4.13 | 8.91 | 23.26 | 14.98 | 2.28 | 0.0 | 4.0 | 0.91 | 0.27
Superblue19 | 0.0 | 87.83 | 23.05 | 6.40 | 8.91 | 23.53 | 14.86 | 2.36 | 0.0 | 6.0 | 0.96 | 0.50

Table 6: Dataset details & statistics. This table shows the distribution of
net-based demands for each design, distribution of net-based logged wirelength
for each design and the distribution of cell-based congestion values for each
design. Remind that cell-based congestion, as we described in the paper, we
classify the cell-based congestion values (computed as the ratio of
demand/capacity) into (a) [0, 0.9], _not-congested_ ; and (b) [0.9, $\inf$];
_congested_.

Figure 5: Net-based wirelength, net-based demand and cell-based congestion
distributions of each design.

###  C.3 More experimental results

| net-based wirelength regression
---|---
Model |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
base E-HNN | $1.818$ | $1.344$ | $0.731$
base DE-HNN | $1.751$ | $1.269$ | $0.748$
base DE-HNN+PD | $1.731$ | $1.257$ | $0.754$
base DE-HNN+PD+S. VN | $1.724$ | $1.253$ | $0.758$
full DE-HNN | $1.689$ | $1.245$ | $0.770$
net-based demand regression
---
RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
$9.228$ | $5.959$ | $0.591$
$8.997$ | $5.764$ | $0.630$
$8.765$ | $5.526$ | $0.647$
$8.687$ | $5.519$ | $0.658$
$8.381$ | $5.334$ | $0.683$
cell-based congestion classification
---
Precision $\uparrow$ |  Recall $\uparrow$ |  F_score $\uparrow$
$0.816$ | $0.864$ | $0.829$
$0.824$ | $0.860$ | $0.840$
$0.830$ | $0.869$ | $0.847$
$0.832$ | $0.874$ | $0.851$
$0.833$ | $0.876$ | $0.853$

Table 7: Ablation Study: Full experimental results to show the improvements
from base E-HNN to full DE-HNN.

| net-based wirelength regression
---|---
Model |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
NetlistGNN with no PD | $1.818$ | $1.344$ | $0.731$
NetlistGNN+PD | $1.773$ | $1.320$ | $0.740$
Improvement | 2.5% | 1.8% | 1.2%
GCN with no PD | $1.809$ | $1.326$ | $0.735$
GCN+PD | $1.762$ | $1.276$ | $0.750$
Improvement | 1.9% | 3.6% | 5.2%
GATv2 with no PD | $1.920$ | $1.401$ | $0.659$
GATv2+PD | $1.812$ | $1.330$ | $0.687$
Improvement | 0.7% | 0.6% | 1.6%
net-based demand regression |
---|---
RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$ |
$9.237$ | $6.060$ | $0.592$ |
$9.063$ | $5.839$ | $0.623$ |
1.9% | 3.6% | 5.2% |
$9.698$ | $6.453$ | $0.547$ |
$9.321$ | $6.163$ | $0.570$ |
3.9% | 4.5% | 4.2% |
$9.710$ | $6.392$ | $0.539$ |
$9.342$ | $6.118$ | $0.561$ |
3.8% | 4.3% | 4.1% |
cell-based congestion classification
---
Precision $\uparrow$ |  Recall $\uparrow$ |  F_score $\uparrow$
$0.806$ | $0.855$ | $0.818$
$0.812$ | $0.860$ | $0.831$
0.7% | 0.6% | 1.6%
$0.746$ | $0.837$ | $0.784$
$0.761$ | $0.857$ | $0.802$
2.0% | 2.4% | 2.3%
$0.802$ | $0.856$ | $0.811$
$0.810$ | $0.864$ | $0.835$
1.0% | 1.0% | 3.0%

Table 8: Ablation Study: the effect of using persistence diagrams (PDs) to
three best-performing baselines. For each method, the 3rd row shows the
percentage of improvement after using PD as part of the input features. Note
that in the main text, the results we reported are those baselines+PD.

In the main text, we have reported plots for ablation studies of different
strategies used in our model. Here in Table 7 we report detailed numbers for
all three tasks: net-based wirelength regression, net-based demand regression,
and cell-based congestion classification across all the designs.

Besides ablation study over our models, as we mentioned in main paper, adding
the persistence diagrams (PDs) features are beneficial for both our method and
baselines. We have already shown the benefits for our model in the ablation
studies in the main text and above. Here in Table 8, we show its benefits to
the three best performing baseline models: NetlistGNN, GCN, GATv2. As we can
see, using PDs improve all these models. Note that in the main text, when we
report results of these baseline models, we are already reporting results with
PDs.

We show more detailed experimental results of single-design experiments for
net-based wirelength regression in Table 9, net-based regression in Table 10
and cell-based congestion classification in Table 11 for each design. We also
show more results of cross-design experiments for net-based wirelength
regression, net-based demand regression and cell-based congestion
classification in Table 13. Similar to how we report the results in the main
paper, we highlight the models’ results with the best performance in red, and
highlight the models (other than our models) that with second best performance
as blue. Interestingly, in the single design case, it appears that the methods
AllSet and NetlistGNN often perform the best among the baselines for net-
length and net-based demand regression tasks, while for the cell-based
congestion classification, other models (e.g, GATv2) sometimes perform the
best among baselines. In cross-design experiments, GCN and GATv2 sometimes
emerge as winners. In the cases that a baseline model is better than our
model, we highlight that baseline model’s results in red instead.

##### Preliminary exploration with placement information.

In this paper, we focus on the case when our input do not have placement
information (i.e, coordinates of cells). We also conducted some preliminary
experiments to examine the effect of adding cell placement information to
initial node features, and use placement information based partitioning
instead of Metis. We first partition the nodes in netlist circuit (after
placement) into $k$ bounding boxes $V_{i}$ with fixed width and height as
$0.8$ millimeters. The number $k$ is depending on the width and height of each
netlist circuit. With coordinates of cells added and bounding boxes based
partitioning (for full-DE-HNN), we rerun both single-design and cross-design
net-based demand regression experiments on superblue19. See Table 12 below.
Placement information helps to improve the performance on average around 10%
for MAE and RMSE, and the improvement is more substantial in terms of Pearson
Correlation. We aim to explore more about how to effectively leverage
placement information in the future works.

Design | Model |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
---|---|---|---|---
Superblue1 | GCN | 1.731 | 1.243 | 0.751
GATv2 | 1.742 | 1.253 | 0.742
HyperConv | 1.988 | 1.413 | 0.645
AllSet | 1.700 | 1.235 | 0.757
HMPNN | 1.779 | 1.291 | 0.727
HNHN | 1.813 | 1.319 | 0.720
NetlistGNN | 1.761 | 1.276 | 0.735
base DE-HNN | 1.774 | 1.274 | 0.731
full DE-HNN | 1.657 | 1.203 | 0.771
| Improvement | 2.5$\%$ | 2.6$\%$ | 1.8$\%$
Superblue2 | GCN | 2.011 | 1.487 | 0.714
GATv2 | 1.931 | 1.485 | 0.723
HyperConv | 2.217 | 1.661 | 0.641
AllSet | 1.817 | 1.353 | 0.775
HMPNN | 1.981 | 1.480 | 0.723
HNHN | 1.992 | 1.488 | 0.724
NetlistGNN | 1.857 | 1.410 | 0.748
base DE-HNN | 1.933 | 1.438 | 0.741
full DE-HNN | 1.810 | 1.344 | 0.778
| Improvement | 0.4$\%$ | 0.7$\%$ | 0.4$\%$
Superblue3 | GCN | 1.741 | 1.273 | 0.738
GATv2 | 1.716 | 1.270 | 0.748
HyperConv | 1.941 | 1.438 | 0.661
AllSet | 1.685 | 1.239 | 0.759
HMPNN | 1.752 | 1.300 | 0.734
HNHN | 1.769 | 1.303 | 0.729
NetlistGNN | 1.689 | 1.239 | 0.754
base DE-HNN | 1.765 | 1.271 | 0.741
full DE-HNN | 1.679 | 1.234 | 0.761
| Improvement | 0.4$\%$ | 0.4$\%$ | 0.3$\%$
Superblue5 | GCN | 1.892 | 1.428 | 0.801
GATv2 | 1.909 | 1.438 | 0.793
HyperConv | 2.114 | 1.606 | 0.742
AllSet | 1.842 | 1.361 | 0.810
HMPNN | 2.224 | 1.754 | 0.706
HNHN | 2.069 | 1.562 | 0.756
NetlistGNN | 1.915 | 1.406 | 0.790
base DE-HNN | 1.881 | 1.393 | 0.799
full DE-HNN | 1.795 | 1.330 | 0.822
| Improvement | 2.6$\%$ | 2.3$\%$ | 1.5$\%$
Superblue6 | GCN | 1.811 | 1.341 | 0.740
GATv2 | 1.782 | 1.322 | 0.751
HyperConv | 1.928 | 1.435 | 0.702
AllSet | 1.733 | 1.260 | 0.766
HMPNN | 1.851 | 1.360 | 0.730
HNHN | 1.859 | 1.365 | 0.727
NetlistGNN | 1.940 | 1.452 | 0.725
base DE-HNN | 1.786 | 1.296 | 0.749
full DE-HNN | 1.689 | 1.233 | 0.780
| Improvement | 2.5$\%$ | 2.1$\%$ | 1.8$\%$
Superblue7 | GCN | 1.842 | 1.315 | 0.710
GATv2 | 1.847 | 1.371 | 0.699
HyperConv | 2.010 | 1.479 | 0.628
AllSet | 1.753 | 1.295 | 0.733
HMPNN | 1.889 | 1.409 | 0.681
HNHN | 1.858 | 1.370 | 0.694
NetlistGNN | 1.801 | 1.316 | 0.721
base DE-HNN | 1.760 | 1.291 | 0.701
full DE-HNN | 1.719 | 1.267 | 0.747
| Improvement | 1.9$\%$ | 2.2$\%$ | 1.9$\%$
Design | Model |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
---|---|---|---|---
Superblue9 | GCN | 1.819 | 1.338 | 0.697
GATv2 | 1.853 | 1.381 | 0.683
HyperConv | 1.966 | 1.435 | 0.639
AllSet | 1.684 | 1.238 | 0.750
HMPNN | 1.827 | 1.355 | 0.695
HNHN | 1.821 | 1.335 | 0.701
NetlistGNN | 1.805 | 1.325 | 0.701
base DE-HNN | 1.765 | 1.287 | 0.720
full DE-HNN | 1.663 | 1.226 | 0.757
| Improvement | 1.2$\%$ | 1.0$\%$ | 0.9$\%$
Superblue11 | GCN | 1.831 | 1.356 | 0.694
GATv2 | 1.814 | 1.349 | 0.702
HyperConv | 1.986 | 1.468 | 0.633
AllSet | 1.717 | 1.274 | 0.740
HMPNN | 1.830 | 1.369 | 0.696
HNHN | 1.834 | 1.353 | 0.697
NetlistGNN | 1.780 | 1.291 | 0.707
base DE-HNN | 1.741 | 1.290 | 0.731
full DE-HNN | 1.690 | 1.257 | 0.751
| Improvement | 1.6$\%$ | 1.3$\%$ | 1.5$\%$
Superblue14 | GCN | 1.792 | 1.331 | 0.739
GATv2 | 1.794 | 1.361 | 0.738
HyperConv | 2.116 | 1.547 | 0.605
AllSet | 1.756 | 1.299 | 0.750
HMPNN | 1.873 | 1.415 | 0.707
HNHN | 1.895 | 1.404 | 0.703
NetlistGNN | 1.812 | 1.363 | 0.731
base DE-HNN | 1.816 | 1.336 | 0.730
full DE-HNN | 1.728 | 1.282 | 0.760
| Improvement | 1.6$\%$ | 1.3$\%$ | 1.3$\%$
Superblue16 | GCN | 1.763 | 1.265 | 0.741
GATv2 | 1.741 | 1.267 | 0.751
HyperConv | 2.047 | 1.446 | 0.639
AllSet | 1.688 | 1.207 | 0.772
HMPNN | 1.896 | 1.362 | 0.701
HNHN | 1.816 | 1.312 | 0.731
NetlistGNN | 1.773 | 1.274 | 0.736
base DE-HNN | 1.705 | 1.218 | 0.767
full DE-HNN | 1.656 | 1.194 | 0.782
| Improvement | 1.9$\%$ | 1.1$\%$ | 1.3$\%$
Superblue18 | GCN | 1.635 | 1.249 | 0.739
GATv2 | 1.701 | 1.246 | 0.714
HyperConv | 1.937 | 1.345 | 0.681
AllSet | 1.664 | 1.246 | 0.730
HMPNN | 1.769 | 1.335 | 0.686
HNHN | 1.752 | 1.314 | 0.696
NetlistGNN | 1.625 | 1.213 | 0.752
base DE-HNN | 1.653 | 1.263 | 0.735
full DE-HNN | 1.632 | 1.219 | 0.743
| Improvement | 0.4$\%$ | 0.5$\%$ | 1.2$\%$
Superblue19 | GCN | 1.637 | 1.214 | 0.762
GATv2 | 1.634 | 1.198 | 0.765
HyperConv | 1.910 | 1.405 | 0.666
AllSet | 1.582 | 1.171 | 0.783
HMPNN | 1.705 | 1.255 | 0.739
HNHN | 1.752 | 1.288 | 0.727
NetlistGNN | 1.635 | 1.205 | 0.765
base DE-HNN | 1.641 | 1.208 | 0.763
full DE-HNN | 1.557 | 1.153 | 0.792
| Improvement | 1.6$\%$ | 1.5$\%$ | 1.1$\%$

Table 9: Average results of single-design net-based hpwl(wirelength)
regression for each design, based on 4-fold cross validations. Last row
“Improvement” refers to the improvement of our full DE-HNN model over the best
baseline approach for each metric.

Design | Model |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
---|---|---|---|---
Superblue1 | GCN | 6.469 | 4.979 | 0.595
GATv2 | 6.409 | 4.964 | 0.612
HyperConv | 6.662 | 4.951 | 0.224
AllSet | 6.100 | 4.587 | 0.650
HMPNN | 6.770 | 5.206 | 0.541
HNHN | 6.394 | 4.825 | 0.610
Lin. Transformers | 7.991 | 6.046 | 0.089
NetlistGNN | 6.039 | 4.623 | 0.660
base DE-HNN | 6.093 | 4.670 | 0.653
full DE-HNN | 5.674 | 4.263 | 0.709
Improvement | 6.0$\%$ | 7.1$\%$ | 7.4$\%$
Superblue2 | GCN | 6.556 | 5.245 | 0.641
GATv2 | 6.736 | 5.288 | 0.616
HyperConv | 7.654 | 6.151 | 0.374
AllSet | 6.430 | 4.981 | 0.659
HMPNN | 6.982 | 5.501 | 0.579
HNHN | 6.699 | 5.217 | 0.625
Lin. Transformers | 8.251 | 6.356 | 0.313
NetlistGNN | 6.259 | 4.932 | 0.682
base DE-HNN | 6.399 | 5.035 | 0.663
full DE-HNN | 5.966 | 4.637 | 0.718
Improvement | 4.7$\%$ | 6.0$\%$ | 5.3$\%$
Superblue3 | GCN | 5.789 | 4.512 | 0.612
GATv2 | 5.837 | 4.558 | 0.565
HyperConv | 6.468 | 5.149 | 0.265
AllSet | 5.265 | 4.014 | 0.695
HMPNN | 6.022 | 4.713 | 0.572
HNHN | 5.686 | 4.338 | 0.631
Lin. Transformers | 7.046 | 5.493 | 0.264
NetlistGNN | 5.414 | 4.214 | 0.673
base DE-HNN | 5.423 | 4.188 | 0.670
full DE-HNN | 5.041 | 3.837 | 0.725
Improvement | 4.3$\%$ | 4.4$\%$ | 4.3$\%$
Superblue5 | GCN | 27.169 | 14.867 | 0.504
GATv2 | 27.343 | 14.754 | 0.508
HyperConv | 29.563 | 15.817 | 0.107
AllSet | 27.881 | 14.632 | 0.490
HMPNN | 28.753 | 15.485 | 0.439
HNHN | 28.314 | 15.309 | 0.473
Lin. Transformers | 32.614 | 16.889 | 0.076
NetlistGNN | 27.586 | 14.470 | 0.515
base DE-HNN | 27.205 | 14.129 | 0.536
full DE-HNN | 26.684 | 13.512 | 0.565
Improvement | 1.8$\%$ | 6.6$\%$ | 9.7$\%$
Superblue6 | GCN | 21.963 | 12.692 | 0.607
GATv2 | 21.492 | 12.119 | 0.629
HyperConv | 25.615 | 13.356 | 0.128
AllSet | 17.945 | 10.156 | 0.759
HMPNN | 21.868 | 12.633 | 0.611
HNHN | 17.735 | 10.094 | 0.767
Lin. Transformers | 28.807 | 14.583 | 0.119
NetlistGNN | 20.238 | 11.696 | 0.697
base DE-HNN | 19.935 | 11.227 | 0.694
full DE-HNN | 16.946 | 9.680 | 0.790
Improvement | 4.4$\%$ | 4.1$\%$ | 3.0$\%$
Superblue7 | GCN | 4.243 | 3.064 | 0.600
GATv2 | 4.403 | 3.247 | 0.561
HyperConv | 4.689 | 3.327 | 0.225
AllSet | 4.201 | 2.991 | 0.621
HMPNN | 4.527 | 3.246 | 0.541
HNHN | 4.458 | 3.165 | 0.557
Lin. Transformers | 5.245 | 3.752 | 0.139
NetlistGNN | 4.115 | 2.986 | 0.634
base DE-HNN | 4.110 | 2.957 | 0.631
full DE-HNN | 3.971 | 2.860 | 0.662
Improvement | 3.5$\%$ | 4.2$\%$ | 4.4$\%$
Design | Model |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
---|---|---|---|---
Superblue9 | GCN | 6.871 | 5.156 | 0.520
GATv2 | 6.893 | 5.139 | 0.512
HyperConv | 7.014 | 5.241 | 0.289
AllSet | 6.184 | 4.576 | 0.640
HMPNN | 7.152 | 5.367 | 0.467
HNHN | 6.544 | 4.884 | 0.582
Lin. Transformers | 8.007 | 5.934 | 0.092
NetlistGNN | 6.511 | 4.796 | 0.589
base DE-HNN | 2.990 | 2.228 | 0.696
full DE-HNN | 5.685 | 4.237 | 0.709
Improvement | 8.1$\%$ | 7.4$\%$ | 10.8$\%$
Superblue11 | GCN | 5.693 | 4.224 | 0.502
GATv2 | 5.684 | 4.203 | 0.504
HyperConv | 6.243 | 4.523 | 0.105
AllSet | 5.115 | 3.792 | 0.625
HMPNN | 5.974 | 4.402 | 0.418
HNHN | 5.277 | 3.882 | 0.592
Lin. Transformers | 6.576 | 4.678 | 0.034
NetlistGNN | 5.176 | 3.830 | 0.617
base DE-HNN | 5.214 | 3.855 | 0.608
full DE-HNN | 4.918 | 3.677 | 0.666
Improvement | 3.9$\%$ | 3.0$\%$ | 6.6$\%$
Superblue14 | GCN | 7.261 | 4.827 | 0.584
GATv2 | 7.370 | 4.825 | 0.545
HyperConv | 8.210 | 5.241 | 0.210
AllSet | 7.162 | 4.565 | 0.619
HMPNN | 7.687 | 4.992 | 0.540
HNHN | 7.327 | 4.693 | 0.597
Lin. Transformers | 8.853 | 6.168 | 0.176
NetlistGNN | 6.872 | 4.444 | 0.642
base DE-HNN | 6.874 | 4.453 | 0.639
full DE-HNN | 6.533 | 4.211 | 0.684
Improvement | 4.9$\%$ | 5.2$\%$ | 6.5$\%$
Superblue16 | GCN | 11.774 | 8.242 | 0.391
GATv2 | 12.853 | 8.283 | 0.377
HyperConv | 16.501 | 9.486 | 0.175
AllSet | 12.558 | 7.837 | 0.469
HMPNN | 13.539 | 8.582 | 0.312
HNHN | 12.720 | 8.082 | 0.446
Lin. Transformers | 14.020 | 8.827 | 0.003
NetlistGNN | 12.982 | 8.385 | 0.353
base DE-HNN | 12.282 | 7.946 | 0.465
full DE-HNN | 11.867 | 7.644 | 0.520
Improvement | - | 2.5$\%$ | 10.9$\%$
Superblue18 | GCN | 3.061 | 2.262 | 0.681
GATv2 | 3.102 | 2.285 | 0.672
HyperConv | 4.013 | 2.915 | 0.255
AllSet | 3.057 | 2.294 | 0.674
HMPNN | 3.246 | 2.446 | 0.624
HNHN | 3.208 | 2.377 | 0.637
Lin. Transformers | 4.090 | 2.913 | 0.154
NetlistGNN | 2.882 | 2.173 | 0.726
base DE-HNN | 2.990 | 2.228 | 0.696
full DE-HNN | 2.855 | 2.136 | 0.730
Improvement | 0.9$\%$ | 1.7$\%$ | 0.6$\%$
Superblue19 | GCN | 5.034 | 3.734 | 0.616
GATv2 | 4.949 | 3.691 | 0.636
HyperConv | 5.746 | 3.974 | 0.312
AllSet | 4.682 | 3.474 | 0.685
HMPNN | 5.294 | 3.980 | 0.571
HNHN | 5.063 | 3.750 | 0.620
Lin. Transformers | 6.315 | 4.565 | 0.127
NetlistGNN | 4.683 | 3.520 | 0.681
base DE-HNN | 4.946 | 3.720 | 0.632
full DE-HNN | 4.429 | 3.317 | 0.723
Improvement | 5.4$\%$ | 4.5$\%$ | 5.5$\%$

Table 10: Results of single-design net-based demand regression for each
design.

Design | Model |  Precision $\uparrow$ |  Recall $\uparrow$ |  F_score $\uparrow$
---|---|---|---|---
Superblue1 | GCN | 0.839 | 0.944 | 0.888
GATv2 | 0.867 | 0.944 | 0.904
HyperConv | 0.873 | 0.966 | 0.917
AllSet | 0.880 | 0.955 | 0.916
HMPNN | 0.866 | 0.968 | 0.916
HNHN | 0.868 | 0.969 | 0.916
Lin. Transformers | 0.853 | 0.941 | 0.895
NetlistGNN | 0.862 | 0.936 | 0.920
| base DE-HNN | 0.876 | 0.967 | 0.920
|  full DE-HNN | 0.885 | 0.969 | 0.925
| Improvement | 1.6$\%$ | 0.5$\%$ | 0.5$\%$
Superblue2 | GCN | 0.741 | 0.657 | 0.697
GATv2 | 0.782 | 0.739 | 0.760
HyperConv | 0.779 | 0.706 | 0.741
AllSet | 0.727 | 0.664 | 0.694
HMPNN | 0.730 | 0.587 | 0.649
HNHN | 0.718 | 0.633 | 0.670
Lin. Transformers | 0.752 | 0.530 | 0.621
NetlistGNN | 0.765 | 0.614 | 0.682
| base DE-HNN | 0.796 | 0.717 | 0.755
|  full DE-HNN | 0.797 | 0.767 | 0.782
| Improvement | 1.2$\%$ | 9.7$\%$ | 2.9$\%$
Superblue3 | GCN | 0.731 | 0.837 | 0.780
GATv2 | 0.768 | 0.840 | 0.798
HyperConv | 0.770 | 0.815 | 0.792
AllSet | 0.728 | 0.773 | 0.747
HMPNN | 0.711 | 0.777 | 0.739
HNHN | 0.706 | 0.777 | 0.737
Lin. Transformers | 0.749 | 0.757 | 0.753
NetlistGNN | 0.786 | 0.814 | 0.799
| base DE-HNN | 0.791 | 0.819 | 0.805
|  full DE-HNN | 0.817 | 0.816 | 0.816
| Improvement | 0.7$\%$ | - | 2.1$\%$
Superblue5 | GCN | 0.745 | 0.932 | 0.834
GATv2 | 0.783 | 0.923 | 0.848
HyperConv | 0.827 | 0.935 | 0.878
AllSet | 0.795 | 0.939 | 0.861
HMPNN | 0.788 | 0.932 | 0.853
HNHN | 0.786 | 0.932 | 0.852
Lin. Transformers | 0.798 | 0.911 | 0.851
NetlistGNN | 0.844 | 0.933 | 0.885
| base DE-HNN | 0.842 | 0.938 | 0.887
|  full DE-HNN | 0.852 | 0.940 | 0.894
| Improvement | 4.9$\%$ | 1.3$\%$ | 1.0$\%$
Superblue6 | GCN | 0.837 | 0.921 | 0.877
GATv2 | 0.876 | 0.920 | 0.897
HyperConv | 0.851 | 0.916 | 0.891
AllSet | 0.817 | 0.940 | 0.874
HMPNN | 0.809 | 0.965 | 0.879
HNHN | 0.815 | 0.947 | 0.875
Lin. Transformers | 0.833 | 0.906 | 0.868
NetlistGNN | 0.819 | 0.928 | 0.889
| base DE-HNN | 0.859 | 0.928 | 0.892
|  full DE-HNN | 0.885 | 0.930 | 0.906
| Improvement | 0.7$\%$ | - | 1.0$\%$
Superblue7 | GCN | 0.839 | 0.980 | 0.904
GATv2 | 0.863 | 0.981 | 0.920
HyperConv | 0.899 | 0.967 | 0.932
AllSet | 0.888 | 0.956 | 0.921
HMPNN | 0.874 | 0.962 | 0.916
HNHN | 0.875 | 0.955 | 0.913
Lin. Transformers | 0.792 | 0.870 | 0.891
NetlistGNN | 0.868 | 0.918 | 0.923
| base DE-HNN | 0.900 | 0.938 | 0.887
|  full DE-HNN | 0.908 | 0.969 | 0.937
| Improvement | 5.6$\%$ | - | 0.5$\%$
Design | Model |  Precision $\uparrow$ |  Recall $\uparrow$ |  F_score $\uparrow$
---|---|---|---|---
Superblue9 | GCN | 0.684 | 0.556 | 0.613
GATv2 | 0.719 | 0.613 | 0.666
HyperConv | 0.716 | 0.605 | 0.656
AllSet | 0.675 | 0.505 | 0.577
HMPNN | 0.612 | 0.418 | 0.495
HNHN | 0.664 | 0.447 | 0.529
Lin. Transformers | 0.649 | 0.551 | 0.596
NetlistGNN | 0.778 | 0.568 | 0.656
| base DE-HNN | 0.740 | 0.647 | 0.690
|  full DE-HNN | 0.695 | 0.653 | 0.673
| Improvement | - | 6.5$\%$ | 1.1$\%$
Superblue11 | GCN | 0.634 | 0.896 | 0.743
GATv2 | 0.706 | 0.844 | 0.769
HyperConv | 0.644 | 0.910 | 0.755
AllSet | 0.620 | 0.946 | 0.749
HMPNN | 0.627 | 0.927 | 0.748
HNHN | 0.628 | 0.897 | 0.738
Lin. Transformers | 0.671 | 0.691 | 0.680
NetlistGNN | 0.691 | 0.914 | 0.787
| base DE-HNN | 0.677 | 0.863 | 0.759
|  full DE-HNN | 0.719 | 0.850 | 0.789
| Improvement | 1.1$\%$ | - | 0.3$\%$
Superblue14 | GCN | 0.763 | 0.891 | 0.822
GATv2 | 0.834 | 0.889 | 0.860
HyperConv | 0.830 | 0.886 | 0.857
AllSet | 0.809 | 0.863 | 0.835
HMPNN | 0.819 | 0.858 | 0.838
HNHN | 0.800 | 0.855 | 0.826
Lin. Transformers | 0.735 | 0.764 | 0.749
NetlistGNN | 0.827 | 0.870 | 0.787
| base DE-HNN | 0.856 | 0.876 | 0.866
|  full DE-HNN | 0.856 | 0.902 | 0.878
| Improvement | 3.7$\%$ | 10.6$\%$ | 2.1$\%$
Superblue16 | GCN | 0.713 | 0.926 | 0.807
GATv2 | 0.864 | 0.928 | 0.894
HyperConv | 0.855 | 0.833 | 0.844
AllSet | 0.844 | 0.827 | 0.833
HMPNN | 0.838 | 0.821 | 0.829
HNHN | 0.831 | 0.819 | 0.822
Lin. Transformers | 0.810 | 0.831 | 0.815
NetlistGNN | 0.779 | 0.912 | 0.865
| base DE-HNN | 0.874 | 0.823 | 0.847
|  full DE-HNN | 0.895 | 0.910 | 0.903
| Improvement | 8.5$\%$ | - | 1.0$\%$
Superblue18 | GCN | 0.779 | 0.845 | 0.811
GATv2 | 0.798 | 0.848 | 0.822
HyperConv | 0.790 | 0.888 | 0.836
AllSet | 0.753 | 0.892 | 0.816
HMPNN | 0.763 | 0.888 | 0.821
HNHN | 0.749 | 0.881 | 0.810
Lin. Transformers | 0.775 | 0.852 | 0.812
NetlistGNN | 0.868 | 0.939 | 0.902
| base DE-HNN | 0.798 | 0.890 | 0.842
|  full DE-HNN | 0.807 | 0.886 | 0.845
| Improvement | - | - | -
Superblue19 | GCN | 0.812 | 0.894 | 0.851
GATv2 | 0.857 | 0.899 | 0.878
HyperConv | 0.879 | 0.877 | 0.878
AllSet | 0.870 | 0.866 | 0.868
HMPNN | 0.861 | 0.862 | 0.862
HNHN | 0.865 | 0.835 | 0.848
Lin. Transformers | 0.809 | 0.880 | 0.843
NetlistGNN | 0.869 | 0.946 | 0.856
| base DE-HNN | 0.883 | 0.885 | 0.884
|  full DE-HNN | 0.895 | 0.910 | 0.903
| Improvement | 0.4$\%$ | - | 2.8$\%$

Table 11: Results of single-design cell-based congestion classification for
each design.

| Single-Design without placement | Single-Design with placement | Cross-Design without placement | Cross-Design with placement
---|---|---|---|---
Model | RMSE | MAE | Pearson | RMSE (imp.) | MAE (imp.) | Pearson (imp.) | RMSE | MAE | Pearson | RMSE (imp.) | MAE (imp.) | Pearson (imp.)
GCN | 5.034 | 3.734 | 0.616 | 4.495 (10.7%) | 3.334 (10.7%) | 0.717 (16.4%) | 6.571 | 5.024 | 0.365 | 6.126 (6.8%) | 4.709 (6.3%) | 0.440 (20.5%)
GATv2 | 4.949 | 3.691 | 0.636 | 4.382 (11.4%) | 3.112 (15.7%) | 0.758 (19.2%) | 6.623 | 5.137 | 0.363 | 5.812 (10.8%) | 4.695 (8.6%) | 0.442 (21.8%)
full DE-HNN | 4.429 | 3.317 | 0.723 | 4.005 (9.6%) | 2.987 (10.0%) | 0.785 (8.6%) | 6.037 | 4.670 | 0.372 | 5.795 (4.0%) | 4.337 (7.1%) | 0.452 (21.5%)

Table 12: Results of net-based demand regression for Superblue19. For each
metric, the (imp.) refers to the improvements when placement information
added.

|  | Wirelength Regression
---|---|---
Design | Model |  RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
Superblue19 | GCN | 1.691 | 1.276 | 0.746
GATv2 | 1.717 | 1.281 | 0.737
Lin. Transformer | 2.159 | 1.588 | 0.521
NetlistGNN | 1.762 | 1.324 | 0.718
HyperConv | 2.390 | 1.788 | 0.558
Allset | 1.837 | 1.348 | 0.695
HMPNN | 1.785 | 1.335 | 0.710
HNHN | 1.754 | 1.333 | 0.701
| base DE-HNN | 1.731 | 1.291 | 0.730
|  full DE-HNN | 1.677 | 1.242 | 0.754
| Improvement | 1.9$\%$ | 2.6$\%$ | 1.8$\%$
Demand Regression
---
RMSE $\downarrow$ |  MAE $\downarrow$ |  Pearson $\uparrow$
6.571 | 5.024 | 0.365
6.623 | 5.137 | 0.363
6.564 | 4.819 | 0.086
8.328 | 6.839 | 0.367
8.569 | 5.294 | 0.241
6.120 | 4.820 | 0.345
6.979 | 5.356 | 0.306
6.390 | 4.870 | 0.358
6.778 | 5.085 | 0.337
6.037 | 4.670 | 0.372
1.4$\%$ | 4.1$\%$ | 1.4$\%$
Congestion Classification
---
Precision $\uparrow$ |  Recall $\uparrow$ |  F_score $\uparrow$
0.633 | 0.997 | 0.773
0.630 | 0.999 | 0.765
0.618 | 0.859 | 0.772
0.647 | 0.953 | 0.771
0.655 | 0.923 | 0.778
0.645 | 0.964 | 0.773
0.633 | 0.999 | 0.773
0.648 | 0.939 | 0.767
0.653 | 0.990 | 0.774
0.660 | 0.986 | 0.780
0.7$\%$ | - | 0.3$\%$

Table 13: Results of cross-design net-based hpwl(wirelength) regression, net-
based demand regression and cell-based congestion classification for different
netlist design, including comparisons with other HNN models.