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High Performance Graph Convolutional Networks with Applications in Testability Analysis Yuzhe Ma CUHK [email protected] Haoxing Ren NVIDIA [email protected] Brucek Khailany NVIDIA [email protected] Harbinder Sikka NVIDIA [email protected] Lijuan Luo NVIDIA [email protected] Karthikeyan Natarajan NVIDIA [email protected] Bei Yu CUHK [email protected] ABSTRACT Applications of deep learning to electronic design automation (EDA) have recently begun to emerge, although they have mainly been limited to processing of regular structured data such as images. However, many EDA problems require processing irregular struc- tures, and it can be non-trivial to manually extract important fea- tures in such cases. In this paper, a high performance graph con- volutional network (GCN) model is proposed for the purpose of processing irregular graph representations of logic circuits. A GCN classifier is firstly trained to predict observation point candidates in a netlist. The GCN classifier is then used as part of an iterative process to propose observation point insertion based on the clas- sification results. Experimental results show the proposed GCN model has superior accuracy to classical machine learning models on difficult-to-observation nodes prediction. Compared with com- mercial testability analysis tools, the proposed observation point insertion flow achieves similar fault coverage with an 11% reduction in observation points and a 6% reduction in test pattern count. 1 INTRODUCTION Machine learning (ML) and deep learning (DL) has emerged as a powerful technique in many fields. It can derive knowledge from large amounts of data and provide predictions, estimations, or even contents generation. In the field of electronic design automation (EDA), learning approaches have mainly been used to accelerate the design process [1–5]. Among various learning models, convo- lutional neural network (CNN) approaches are widely used. For example, RouteNet [4] leveraged a CNN to predict the routability of a placed design efficiently. Whereas CNNs are discriminative mod- els, generative adversarial networks (GANs) are generative models and have attracted significant attention in recent years. For exam- ple, in [5], a GAN model is developed to perform optical proximity correction (OPC) on lithography masks for better manufacturability. In both example applications, the inputs (layout and mask), can be treated as images in order to build upon prior work using DL for image processing. However, CNN-based approaches often include Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. DAC ’19, June 2–6, 2019, Las Vegas, NV, USA © 2019 Association for Computing Machinery. ACM ISBN 978-1-4503-6725-7/19/06. . . $15.00 https://doi.org/10.1145/3316781.3317838 FC Layers Layer 2 Layer 1 Prediction 0 1 0 0 1 1 Figure 1: Network for node embedding and classification. Layer 1 and Layer 2 generate node embeddings. Nodes are classified through fully-connected (FC) layers. convolution and pooling layers and require data defined on regular grids. There are many EDA applications without grid-based data representations, where alternative approaches are needed. Graphs play a vital role in EDA since they are natural represen- tations for fundamental objects such as netlists and layout. Many problems have previously been solved using graph processing, such as partitioning [6], layer assignment [7], multiple patterning layout decomposition [8, 9] and testability analysis [10]. However, there are few DL-based solutions proposed for graph-based problems in EDA [3]. One reason is that it is not straightforward to generalize CNNs from processing regular grid-based data input to processing graphs. Recently, a number of studies from the DL community have shown how to adopt learning models on graphs, most notably the graph convolutional network (GCN) approach [11–13]. As shown in the example in Figure 1, the essential idea is to obtain an em- bedding for each node by aggregating information from a node’s local neighborhood iteratively such that the node attributes and structural information are encoded. The embeddings can then be leveraged as features in the machine learning tasks. One critical consideration of applying learning-based techniques to EDA is scalability. A typical modern design from a commercial SoC could contain millions of gates, so any model proposed for use in EDA must achieve high throughput on these large graphs. How- ever, previous GCNs [11, 12] did not demonstrate strong scaling capability. In [13], a scalable GCN inference pipeline is proposed for recommendation system in web scale. It relies on a distributed computing framework and is computationally expensive, which makes it difficult to deploy in an EDA toolflow environment. In this paper, a high-performance GCN model is proposed for tackling EDA problems with inputs structured as graphs. With a netlist represented as a graph, the GCN model can generate the embedding for each node automatically using the aggregators and encoders, considering both node attributes and graph structural information. By selecting the aggregators properly and leveraging efficient GPU computation, the GCN model is scalable to process a graph containing millions of nodes and edges efficiently. As a

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Page 1: High Performance Graph Convolutional Networks with Applications …byu/papers/C85-DAC2019-GCN-Test.pdf · inference scheme are presented for efficient GCN training and inference,

High Performance Graph Convolutional Networks withApplications in Testability Analysis

Yuzhe MaCUHK

[email protected]

Haoxing RenNVIDIA

[email protected]

Brucek KhailanyNVIDIA

[email protected]

Harbinder SikkaNVIDIA

[email protected]

Lijuan LuoNVIDIA

[email protected]

Karthikeyan NatarajanNVIDIA

[email protected]

Bei YuCUHK

[email protected]

ABSTRACTApplications of deep learning to electronic design automation (EDA)have recently begun to emerge, although they have mainly beenlimited to processing of regular structured data such as images.However, many EDA problems require processing irregular struc-tures, and it can be non-trivial to manually extract important fea-tures in such cases. In this paper, a high performance graph con-volutional network (GCN) model is proposed for the purpose ofprocessing irregular graph representations of logic circuits. A GCNclassifier is firstly trained to predict observation point candidatesin a netlist. The GCN classifier is then used as part of an iterativeprocess to propose observation point insertion based on the clas-sification results. Experimental results show the proposed GCNmodel has superior accuracy to classical machine learning modelson difficult-to-observation nodes prediction. Compared with com-mercial testability analysis tools, the proposed observation pointinsertion flow achieves similar fault coverage with an 11% reductionin observation points and a 6% reduction in test pattern count.

1 INTRODUCTIONMachine learning (ML) and deep learning (DL) has emerged as apowerful technique in many fields. It can derive knowledge fromlarge amounts of data and provide predictions, estimations, or evencontents generation. In the field of electronic design automation(EDA), learning approaches have mainly been used to acceleratethe design process [1–5]. Among various learning models, convo-lutional neural network (CNN) approaches are widely used. Forexample, RouteNet [4] leveraged a CNN to predict the routability ofa placed design efficiently. Whereas CNNs are discriminative mod-els, generative adversarial networks (GANs) are generative modelsand have attracted significant attention in recent years. For exam-ple, in [5], a GAN model is developed to perform optical proximitycorrection (OPC) on lithography masks for better manufacturability.In both example applications, the inputs (layout and mask), can betreated as images in order to build upon prior work using DL forimage processing. However, CNN-based approaches often include

Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for components of this work owned by others than ACMmust be honored. Abstracting with credit is permitted. To copy otherwise, or republish,to post on servers or to redistribute to lists, requires prior specific permission and/or afee. Request permissions from [email protected] ’19, June 2–6, 2019, Las Vegas, NV, USA© 2019 Association for Computing Machinery.ACM ISBN 978-1-4503-6725-7/19/06. . . $15.00https://doi.org/10.1145/3316781.3317838

FC LayersLayer 2Layer 1Prediction

010

0

1 1

Figure 1: Network for node embedding and classification.Layer 1 and Layer 2 generate node embeddings. Nodes areclassified through fully-connected (FC) layers.

convolution and pooling layers and require data defined on regulargrids. There are many EDA applications without grid-based datarepresentations, where alternative approaches are needed.

Graphs play a vital role in EDA since they are natural represen-tations for fundamental objects such as netlists and layout. Manyproblems have previously been solved using graph processing, suchas partitioning [6], layer assignment [7], multiple patterning layoutdecomposition [8, 9] and testability analysis [10]. However, thereare few DL-based solutions proposed for graph-based problems inEDA [3]. One reason is that it is not straightforward to generalizeCNNs from processing regular grid-based data input to processinggraphs. Recently, a number of studies from the DL community haveshown how to adopt learning models on graphs, most notably thegraph convolutional network (GCN) approach [11–13]. As shownin the example in Figure 1, the essential idea is to obtain an em-bedding for each node by aggregating information from a node’slocal neighborhood iteratively such that the node attributes andstructural information are encoded. The embeddings can then beleveraged as features in the machine learning tasks.

One critical consideration of applying learning-based techniquesto EDA is scalability. A typical modern design from a commercialSoC could contain millions of gates, so any model proposed for usein EDA must achieve high throughput on these large graphs. How-ever, previous GCNs [11, 12] did not demonstrate strong scalingcapability. In [13], a scalable GCN inference pipeline is proposedfor recommendation system in web scale. It relies on a distributedcomputing framework and is computationally expensive, whichmakes it difficult to deploy in an EDA toolflow environment.

In this paper, a high-performance GCN model is proposed fortackling EDA problems with inputs structured as graphs. With anetlist represented as a graph, the GCN model can generate theembedding for each node automatically using the aggregators andencoders, considering both node attributes and graph structuralinformation. By selecting the aggregators properly and leveragingefficient GPU computation, the GCN model is scalable to processa graph containing millions of nodes and edges efficiently. As a

https://doi.org/10.1145/3316781.3317838
Page 2: High Performance Graph Convolutional Networks with Applications …byu/papers/C85-DAC2019-GCN-Test.pdf · inference scheme are presented for efficient GCN training and inference,

demonstration, the proposed GCN model is applied to testabilityanalysis for improving circuits testability, which is cast as an im-balanced classification problem and an observation point insertionproblem. The proposed GCN model provides fast and accurateprediction, and iteratively inserts observation points to improvetestability effectively. The main contributions are summarized asfollows:

• A methodology using GCNs for netlist representation andmodeling is proposed.• A parallel training scheme with multiple GPUs and a fastinference scheme are presented for efficient GCN trainingand inference, which can scale to millions of standard cells.• A GCN classifier is trained to predict observation point can-didates in a netlist, and the proposed GCN classifier is inte-grated in an iterative observation point insertion process.• Experimental results indicate the GCN outperforms con-ventional machine learning models in terms of classifica-tion accuracy, and the proposed flow can achieve superiortestability results to conventional testability analysis tool onindustrial designs.

The rest of this paper is organized as follows. In Section 2, pre-liminary material about GCN and testability analysis is introduced.The proposed GCN model is presented in Section 3. Section 4 de-scribes how to integrate the GCN classifier to observation pointsinsertion flow. The experimental results are reported in Section 5,and Section 6 concludes the paper.

2 PRELIMINARIES2.1 Node Embedding and GCNGraph-based learning is a new approach to machine learning witha wide range of applications [14]. One advantage is the graph struc-ture itself can reveal relevant information. Node embedding andclassification is one of the most important problems in graph-basedlearning. Before a machine learning model can be trained, a rep-resentation of the node must be obtained first, which is known asembedding. The embedding is a vector with a specific dimension,and can be fed to conventional machine learning models for pre-diction. Previous approaches exploring node embedding problemscan be classified into two categories. The first class of approachesare based on heuristics to encode the structural information [15].A more recent approach is data-driven, which learns node embed-dings automatically [12, 13, 16].

Graph convolution networks are one of these data-driven ap-proaches. Within these data-driven approaches, they can be furtherclassified into transductive and inductive. Transductive approachesdirectly optimize the embedding for each node, thus they requireall nodes to be present during training, and hence cannot generalizeto unseen graphs [16]. Inductive approaches generate node embed-dings through learning a set of functions to aggregate the structuralinformation and node attributes, which make the learned modelindependent from training graphs. Therefore, inductive models canbe applied to unseen data [12, 13].

2.2 Test Points InsertionTest point insertion (TPI) is a broadly used approach in design-for-testability (DFT) to modify a circuit and improve its testability,

Module 1

Module 2

(a)

Module 1

OP

CP1 CP2

01

Module 2I

(b)

Figure 2: (a) Original circuit. Module 1 is unobservable. Mod-ule 2 is uncontrollable; (b) Insert test points to the circuit.Set (CP1, CP2) to (0, 1) and (1, 1) will set line I to 0 and 1,respectively; Set CP2 to 0 is the normal operation mode.

which involves adding extra control points (CPs) or observationpoints (OPs) to the circuit. An example of TPI is given in Figure 2.CPs can be used for setting signal lines to desired logic values,while OPs are added as scan cells to make a node observable. Thereare several issues that needed to be considered when performingTPI. On one hand, the optimal test point placement problem isNP-complete [17]. Numerous TPI methods have been proposed toinvestigate the efficiency and performance of TPI. Based on theruntime, these methods can be categorized into exact fault simu-lation [18], approximate measurements [19] and simulation-basedmethods [20]. On the other hand, inserting test points may degradethe performance of a design in terms of area, power and timing.The ultimate goal of TPI is to achieve high fault coverage with lessperformance degradation. Previous works explored beneficial trade-offs between testability improvement and performance degradation[21–25], among which CPs insertion is considered in [24] and OPsinsertion is considered in [25]. Touba et al. [22] consider insert-ing both CPs and OPs. The approach investigated in this paper isgeneric and can be applied to both CPs insertion and OPs insertion.

2.3 Problem FormulationIn this work, we focus on applying observation points insertion(OPI) to improve the testability of a design. From the perspectiveof a machine learning model, finding the location where the obser-vation points should be inserted in a circuit can be cast as a binaryclassification problem. For each gate in a design, the problem iswhether to add an observation point on the output port or not. Ifhistorical data on a sufficient number of designs can be obtained,a classifier can be trained and applied to other designs. Consider-ing that the netlist can be easily represented as graph, GCN is anappropriate approach for this application.

Problem1 (ObservationPoints Insertion). Given a set of netlistswith all the nodes labeled as either difficult-to-observe or easy-to-observe. The objective is to train a classifier and adopt it to find a setof locations where the observation points should be inserted, whichcan maximize fault coverage and minimize observation points num-ber and test pattern number.

3 GCN FOR NODE CLASSIFICATION3.1 Dataset GenerationA netlist is represented as a directed graph in which each nodecorresponds to a cell and each edge is a wire. The source nodesand sink nodes correspond to primary input and primary output,respectively. Each node has an attribute which is a four dimensionalfeature vector [LL,C0,C1,O]. LL denotes logic level of the corre-sponding gate. [C0,C1,O] are SCOAP measurements [26], which

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Encoding Encoding Encoding Encoding

1 2 3 4[4 ⇥ 4]

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1 2 5[3 ⇥ 4]

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1 3 6[3 ⇥ 4]

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1 4[2 ⇥ 4]

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Aggregation Aggregation Aggregation Aggregation

d = 2<latexit sha1_base64="jpxVY4aTErt9hJMB0trgCd4dSrA=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4ut4KkkvehFKHjxWMG2QhvKZrNpl242YXcilNAf4cWDIl79Pd78N27bHLT1hYWHd2bYmTdIpTDout9OaWNza3unvFvZ2z84PKoen3RNkmnGOyyRiX4MqOFSKN5BgZI/pprTOJC8F0xu5/XeE9dGJOoBpyn3YzpSIhKMorV69ZDckGZ9WK25DXchsg5eATUo1B5WvwZhwrKYK2SSGtP33BT9nGoUTPJZZZAZnlI2oSPet6hozI2fL9adkQvrhCRKtH0KycL9PZHT2JhpHNjOmOLYrNbm5n+1fobRtZ8LlWbIFVt+FGWSYELmt5NQaM5QTi1QpoXdlbAx1ZShTahiQ/BWT16HbrPhWb5v1lrNIo4ynME5XIIHV9CCO2hDBxhM4Ble4c1JnRfn3flYtpacYuYU/sj5/AEa744J</latexit><latexit sha1_base64="jpxVY4aTErt9hJMB0trgCd4dSrA=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4ut4KkkvehFKHjxWMG2QhvKZrNpl242YXcilNAf4cWDIl79Pd78N27bHLT1hYWHd2bYmTdIpTDout9OaWNza3unvFvZ2z84PKoen3RNkmnGOyyRiX4MqOFSKN5BgZI/pprTOJC8F0xu5/XeE9dGJOoBpyn3YzpSIhKMorV69ZDckGZ9WK25DXchsg5eATUo1B5WvwZhwrKYK2SSGtP33BT9nGoUTPJZZZAZnlI2oSPet6hozI2fL9adkQvrhCRKtH0KycL9PZHT2JhpHNjOmOLYrNbm5n+1fobRtZ8LlWbIFVt+FGWSYELmt5NQaM5QTi1QpoXdlbAx1ZShTahiQ/BWT16HbrPhWb5v1lrNIo4ynME5XIIHV9CCO2hDBxhM4Ble4c1JnRfn3flYtpacYuYU/sj5/AEa744J</latexit><latexit sha1_base64="jpxVY4aTErt9hJMB0trgCd4dSrA=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4ut4KkkvehFKHjxWMG2QhvKZrNpl242YXcilNAf4cWDIl79Pd78N27bHLT1hYWHd2bYmTdIpTDout9OaWNza3unvFvZ2z84PKoen3RNkmnGOyyRiX4MqOFSKN5BgZI/pprTOJC8F0xu5/XeE9dGJOoBpyn3YzpSIhKMorV69ZDckGZ9WK25DXchsg5eATUo1B5WvwZhwrKYK2SSGtP33BT9nGoUTPJZZZAZnlI2oSPet6hozI2fL9adkQvrhCRKtH0KycL9PZHT2JhpHNjOmOLYrNbm5n+1fobRtZ8LlWbIFVt+FGWSYELmt5NQaM5QTi1QpoXdlbAx1ZShTahiQ/BWT16HbrPhWb5v1lrNIo4ynME5XIIHV9CCO2hDBxhM4Ble4c1JnRfn3flYtpacYuYU/sj5/AEa744J</latexit><latexit sha1_base64="jpxVY4aTErt9hJMB0trgCd4dSrA=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4ut4KkkvehFKHjxWMG2QhvKZrNpl242YXcilNAf4cWDIl79Pd78N27bHLT1hYWHd2bYmTdIpTDout9OaWNza3unvFvZ2z84PKoen3RNkmnGOyyRiX4MqOFSKN5BgZI/pprTOJC8F0xu5/XeE9dGJOoBpyn3YzpSIhKMorV69ZDckGZ9WK25DXchsg5eATUo1B5WvwZhwrKYK2SSGtP33BT9nGoUTPJZZZAZnlI2oSPet6hozI2fL9adkQvrhCRKtH0KycL9PZHT2JhpHNjOmOLYrNbm5n+1fobRtZ8LlWbIFVt+FGWSYELmt5NQaM5QTi1QpoXdlbAx1ZShTahiQ/BWT16HbrPhWb5v1lrNIo4ynME5XIIHV9CCO2hDBxhM4Ble4c1JnRfn3flYtpacYuYU/sj5/AEa744J</latexit>

K2 = 128<latexit sha1_base64="X6qJQxdJCNDB4/kqt0E4zH3odo8=">AAAB8HicbZDLSgMxFIbP1Futt6pLN8FWcFVmZmM3QsGN4KaCvUg7lEyaaUOTzJBkhDL0Kdy4UMStj+POtzFtZ6GtPwQ+/nMOOecPE860cd1vp7CxubW9U9wt7e0fHB6Vj0/aOk4VoS0S81h1Q6wpZ5K2DDOcdhNFsQg57YSTm3m980SVZrF8MNOEBgKPJIsYwcZaj9W7gX/t+fXqoFxxa+5CaB28HCqQqzkof/WHMUkFlYZwrHXPcxMTZFgZRjidlfqppgkmEzyiPYsSC6qDbLHwDF1YZ4iiWNknDVq4vycyLLSeitB2CmzGerU2N/+r9VIT1YOMySQ1VJLlR1HKkYnR/Ho0ZIoSw6cWMFHM7orIGCtMjM2oZEPwVk9eh7Zf8yzf+5WGn8dRhDM4h0vw4AoacAtNaAEBAc/wCm+Ocl6cd+dj2Vpw8plT+CPn8wdc846+</latexit><latexit sha1_base64="X6qJQxdJCNDB4/kqt0E4zH3odo8=">AAAB8HicbZDLSgMxFIbP1Futt6pLN8FWcFVmZmM3QsGN4KaCvUg7lEyaaUOTzJBkhDL0Kdy4UMStj+POtzFtZ6GtPwQ+/nMOOecPE860cd1vp7CxubW9U9wt7e0fHB6Vj0/aOk4VoS0S81h1Q6wpZ5K2DDOcdhNFsQg57YSTm3m980SVZrF8MNOEBgKPJIsYwcZaj9W7gX/t+fXqoFxxa+5CaB28HCqQqzkof/WHMUkFlYZwrHXPcxMTZFgZRjidlfqppgkmEzyiPYsSC6qDbLHwDF1YZ4iiWNknDVq4vycyLLSeitB2CmzGerU2N/+r9VIT1YOMySQ1VJLlR1HKkYnR/Ho0ZIoSw6cWMFHM7orIGCtMjM2oZEPwVk9eh7Zf8yzf+5WGn8dRhDM4h0vw4AoacAtNaAEBAc/wCm+Ocl6cd+dj2Vpw8plT+CPn8wdc846+</latexit><latexit sha1_base64="X6qJQxdJCNDB4/kqt0E4zH3odo8=">AAAB8HicbZDLSgMxFIbP1Futt6pLN8FWcFVmZmM3QsGN4KaCvUg7lEyaaUOTzJBkhDL0Kdy4UMStj+POtzFtZ6GtPwQ+/nMOOecPE860cd1vp7CxubW9U9wt7e0fHB6Vj0/aOk4VoS0S81h1Q6wpZ5K2DDOcdhNFsQg57YSTm3m980SVZrF8MNOEBgKPJIsYwcZaj9W7gX/t+fXqoFxxa+5CaB28HCqQqzkof/WHMUkFlYZwrHXPcxMTZFgZRjidlfqppgkmEzyiPYsSC6qDbLHwDF1YZ4iiWNknDVq4vycyLLSeitB2CmzGerU2N/+r9VIT1YOMySQ1VJLlR1HKkYnR/Ho0ZIoSw6cWMFHM7orIGCtMjM2oZEPwVk9eh7Zf8yzf+5WGn8dRhDM4h0vw4AoacAtNaAEBAc/wCm+Ocl6cd+dj2Vpw8plT+CPn8wdc846+</latexit><latexit sha1_base64="X6qJQxdJCNDB4/kqt0E4zH3odo8=">AAAB8HicbZDLSgMxFIbP1Futt6pLN8FWcFVmZmM3QsGN4KaCvUg7lEyaaUOTzJBkhDL0Kdy4UMStj+POtzFtZ6GtPwQ+/nMOOecPE860cd1vp7CxubW9U9wt7e0fHB6Vj0/aOk4VoS0S81h1Q6wpZ5K2DDOcdhNFsQg57YSTm3m980SVZrF8MNOEBgKPJIsYwcZaj9W7gX/t+fXqoFxxa+5CaB28HCqQqzkof/WHMUkFlYZwrHXPcxMTZFgZRjidlfqppgkmEzyiPYsSC6qDbLHwDF1YZ4iiWNknDVq4vycyLLSeitB2CmzGerU2N/+r9VIT1YOMySQ1VJLlR1HKkYnR/Ho0ZIoSw6cWMFHM7orIGCtMjM2oZEPwVk9eh7Zf8yzf+5WGn8dRhDM4h0vw4AoacAtNaAEBAc/wCm+Ocl6cd+dj2Vpw8plT+CPn8wdc846+</latexit>

d = 1<latexit sha1_base64="q6zaCrIEPIh1Iyb0c1SQe3TZnL8=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4ut4KkkvehFKHjxWMG2QhvKZjNpl242YXcjlNAf4cWDIl79Pd78N27bHLT1hYWHd2bYmTdIBdfGdb+d0sbm1vZOebeyt39weFQ9PunqJFMMOywRiXoMqEbBJXYMNwIfU4U0DgT2gsntvN57QqV5Ih/MNEU/piPJI86osVavHpIb4tWH1ZrbcBci6+AVUINC7WH1axAmLItRGiao1n3PTY2fU2U4EzirDDKNKWUTOsK+RUlj1H6+WHdGLqwTkihR9klDFu7viZzGWk/jwHbG1Iz1am1u/lfrZya69nMu08ygZMuPokwQk5D57STkCpkRUwuUKW53JWxMFWXGJlSxIXirJ69Dt9nwLN83a61mEUcZzuAcLsGDK2jBHbShAwwm8Ayv8Oakzovz7nwsW0tOMXMKf+R8/gAZao4I</latexit><latexit sha1_base64="q6zaCrIEPIh1Iyb0c1SQe3TZnL8=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4ut4KkkvehFKHjxWMG2QhvKZjNpl242YXcjlNAf4cWDIl79Pd78N27bHLT1hYWHd2bYmTdIBdfGdb+d0sbm1vZOebeyt39weFQ9PunqJFMMOywRiXoMqEbBJXYMNwIfU4U0DgT2gsntvN57QqV5Ih/MNEU/piPJI86osVavHpIb4tWH1ZrbcBci6+AVUINC7WH1axAmLItRGiao1n3PTY2fU2U4EzirDDKNKWUTOsK+RUlj1H6+WHdGLqwTkihR9klDFu7viZzGWk/jwHbG1Iz1am1u/lfrZya69nMu08ygZMuPokwQk5D57STkCpkRUwuUKW53JWxMFWXGJlSxIXirJ69Dt9nwLN83a61mEUcZzuAcLsGDK2jBHbShAwwm8Ayv8Oakzovz7nwsW0tOMXMKf+R8/gAZao4I</latexit><latexit sha1_base64="q6zaCrIEPIh1Iyb0c1SQe3TZnL8=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4ut4KkkvehFKHjxWMG2QhvKZjNpl242YXcjlNAf4cWDIl79Pd78N27bHLT1hYWHd2bYmTdIBdfGdb+d0sbm1vZOebeyt39weFQ9PunqJFMMOywRiXoMqEbBJXYMNwIfU4U0DgT2gsntvN57QqV5Ih/MNEU/piPJI86osVavHpIb4tWH1ZrbcBci6+AVUINC7WH1axAmLItRGiao1n3PTY2fU2U4EzirDDKNKWUTOsK+RUlj1H6+WHdGLqwTkihR9klDFu7viZzGWk/jwHbG1Iz1am1u/lfrZya69nMu08ygZMuPokwQk5D57STkCpkRUwuUKW53JWxMFWXGJlSxIXirJ69Dt9nwLN83a61mEUcZzuAcLsGDK2jBHbShAwwm8Ayv8Oakzovz7nwsW0tOMXMKf+R8/gAZao4I</latexit><latexit sha1_base64="q6zaCrIEPIh1Iyb0c1SQe3TZnL8=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4ut4KkkvehFKHjxWMG2QhvKZjNpl242YXcjlNAf4cWDIl79Pd78N27bHLT1hYWHd2bYmTdIBdfGdb+d0sbm1vZOebeyt39weFQ9PunqJFMMOywRiXoMqEbBJXYMNwIfU4U0DgT2gsntvN57QqV5Ih/MNEU/piPJI86osVavHpIb4tWH1ZrbcBci6+AVUINC7WH1axAmLItRGiao1n3PTY2fU2U4EzirDDKNKWUTOsK+RUlj1H6+WHdGLqwTkihR9klDFu7viZzGWk/jwHbG1Iz1am1u/lfrZya69nMu08ygZMuPokwQk5D57STkCpkRUwuUKW53JWxMFWXGJlSxIXirJ69Dt9nwLN83a61mEUcZzuAcLsGDK2jBHbShAwwm8Ayv8Oakzovz7nwsW0tOMXMKf+R8/gAZao4I</latexit>

K1 = 64<latexit sha1_base64="IHHu0N0VArtRsMtnZvyN3BWYoEM=">AAAB8XicbZBNSwMxEIZn/az1q+rRS7AVPJXdIupFKHgRvFSwH9guJZvOtqHZ7JJkhVL6L7x4UMSr/8ab/8a03YO2vhB4eGeGzLxBIrg2rvvtrKyurW9s5rby2zu7e/uFg8OGjlPFsM5iEatWQDUKLrFuuBHYShTSKBDYDIY303rzCZXmsXwwowT9iPYlDzmjxlqPpbuuR67JxXmpWyi6ZXcmsgxeBkXIVOsWvjq9mKURSsME1brtuYnxx1QZzgRO8p1UY0LZkPaxbVHSCLU/nm08IafW6ZEwVvZJQ2bu74kxjbQeRYHtjKgZ6MXa1Pyv1k5NeOWPuUxSg5LNPwpTQUxMpueTHlfIjBhZoExxuythA6ooMzakvA3BWzx5GRqVsmf5vlKsVrI4cnAMJ3AGHlxCFW6hBnVgIOEZXuHN0c6L8+58zFtXnGzmCP7I+fwBmIqO1g==</latexit><latexit sha1_base64="IHHu0N0VArtRsMtnZvyN3BWYoEM=">AAAB8XicbZBNSwMxEIZn/az1q+rRS7AVPJXdIupFKHgRvFSwH9guJZvOtqHZ7JJkhVL6L7x4UMSr/8ab/8a03YO2vhB4eGeGzLxBIrg2rvvtrKyurW9s5rby2zu7e/uFg8OGjlPFsM5iEatWQDUKLrFuuBHYShTSKBDYDIY303rzCZXmsXwwowT9iPYlDzmjxlqPpbuuR67JxXmpWyi6ZXcmsgxeBkXIVOsWvjq9mKURSsME1brtuYnxx1QZzgRO8p1UY0LZkPaxbVHSCLU/nm08IafW6ZEwVvZJQ2bu74kxjbQeRYHtjKgZ6MXa1Pyv1k5NeOWPuUxSg5LNPwpTQUxMpueTHlfIjBhZoExxuythA6ooMzakvA3BWzx5GRqVsmf5vlKsVrI4cnAMJ3AGHlxCFW6hBnVgIOEZXuHN0c6L8+58zFtXnGzmCP7I+fwBmIqO1g==</latexit><latexit sha1_base64="IHHu0N0VArtRsMtnZvyN3BWYoEM=">AAAB8XicbZBNSwMxEIZn/az1q+rRS7AVPJXdIupFKHgRvFSwH9guJZvOtqHZ7JJkhVL6L7x4UMSr/8ab/8a03YO2vhB4eGeGzLxBIrg2rvvtrKyurW9s5rby2zu7e/uFg8OGjlPFsM5iEatWQDUKLrFuuBHYShTSKBDYDIY303rzCZXmsXwwowT9iPYlDzmjxlqPpbuuR67JxXmpWyi6ZXcmsgxeBkXIVOsWvjq9mKURSsME1brtuYnxx1QZzgRO8p1UY0LZkPaxbVHSCLU/nm08IafW6ZEwVvZJQ2bu74kxjbQeRYHtjKgZ6MXa1Pyv1k5NeOWPuUxSg5LNPwpTQUxMpueTHlfIjBhZoExxuythA6ooMzakvA3BWzx5GRqVsmf5vlKsVrI4cnAMJ3AGHlxCFW6hBnVgIOEZXuHN0c6L8+58zFtXnGzmCP7I+fwBmIqO1g==</latexit><latexit sha1_base64="IHHu0N0VArtRsMtnZvyN3BWYoEM=">AAAB8XicbZBNSwMxEIZn/az1q+rRS7AVPJXdIupFKHgRvFSwH9guJZvOtqHZ7JJkhVL6L7x4UMSr/8ab/8a03YO2vhB4eGeGzLxBIrg2rvvtrKyurW9s5rby2zu7e/uFg8OGjlPFsM5iEatWQDUKLrFuuBHYShTSKBDYDIY303rzCZXmsXwwowT9iPYlDzmjxlqPpbuuR67JxXmpWyi6ZXcmsgxeBkXIVOsWvjq9mKURSsME1brtuYnxx1QZzgRO8p1UY0LZkPaxbVHSCLU/nm08IafW6ZEwVvZJQ2bu74kxjbQeRYHtjKgZ6MXa1Pyv1k5NeOWPuUxSg5LNPwpTQUxMpueTHlfIjBhZoExxuythA6ooMzakvA3BWzx5GRqVsmf5vlKsVrI4cnAMJ3AGHlxCFW6hBnVgIOEZXuHN0c6L8+58zFtXnGzmCP7I+fwBmIqO1g==</latexit>

(b)

Figure 3: An illustration to compute the embedding for anode with D = 2. (a) Graph; (b) Procedure to compute theembedding for node 1.

correspond to controllability-0, controllability-1 and observability,respectively. Every node also has a binary label. ‘0’ (negative) meanseasy-to-observe and ‘1’ (positive) means difficult-to-observe. Labelscan be obtained from commercial DFT tools. Given the graphs withnode attributes and node labels, a GCNmodel can be trained, whichwill be introduced later.

3.2 Node Embedding StructuresTo classify a node in the graph, the neural network first generatesits node embedding which is not only based on its own attributesbut also the structural information of the local neighborhood. Then,a classification model takes the node embedding as input and pre-dicts a label. To achieve this, three kinds of modules are included inour GCN model, which are aggregator, encoder and classifier. Ag-gregators and encoders are used to generate the node embeddingsby exploiting node attributes and the neighborhood information.The classifier predicts the label for each node in the graph basedon its embedding.

Next, we introduce how the node embedding is generated usingaggregators and encoders. Essentially, an aggregator or an encodercan be interpreted as a layer of the GCN. Each of them performsa specific operation on the node. An aggregator gathers the fea-ture information from the node’s neighbors using an aggregationfunction Aдд(·). An encoder is applied to propagate informationbetween different layers using a weight matrix. The embeddingcomputation process, i.e., aggregation and encoding, is performediteratively. Fully-connected layers are used as classifier which takesthe node embedding as input, and predicts the label for the node.

Suppose that the network is trained and all the weights are ob-tained. Given a graph G(V,E) and node attributes {x (v) : ∀v ∈ V},the node embeddings {e(v) : ∀v ∈ V} are generated as in Algo-rithm 1. Since the node embedding is expected to aggregate theinformation in local neighborhood, a depth D is specified to indi-cate the “radius” of the neighborhood region of a node. The initialrepresentation [LL,C0,C1,O] is set as the node attributes (line 1).There are two loops involved. In each step of the outer loop, therepresentation of each node is updated through aggregation andencoding. More specifically, in the d-th iteration of the outer loop,every node first aggregates information from its neighbors using ag-gregation function Aдд(·) which takes the representations of node

Algorithm 1 Node Embedding Computation

Input: Graph G(V,E); node attributes {x (v) : ∀v ∈ V}; Searchdepth D; non-linear activation function σ (·); Weight matricesWd of encoders Ed ,d = 1, ...,D;

Output: Embedding of for each node e(v)D ,∀v ∈ V.1: e(v)0 ← x (v),∀v ∈ V;2: for d = 1, ...,D do3: for all v ∈ V do4: Compute д(v)d based on Equation (1); ▷ Aggregation

5: e(v)d ← σ (Wd · д(v)d ); ▷ Encoding

6: end for7: end for

Stage-1 Stage-2 Stage-3

Positive pointNegative point

Decision boundary

+-

+- +-

Figure 4: An example of three-stage GCN classification.

v and its neighbors generated at (d − 1)-th iteration as input, andgenerates a new representation for node v , denoted by д(v)d (line 4).We use a weighted sum function as the aggregation function. As-sume predecessors (PR) and successors (SU) have different weights.The aggregation Aдд(·) can be formulated explicitly as:

д(v)d = e(v)d−1 +wpr ×∑

u ∈PR(v)e(u)d−1 +wsu ×

∑u ∈SU(v)

e(u)d−1, (1)

where wpr and wsu are weights for predecessors and successors,respectively, and they are the same in each step of outer loop. Next,a non-linear transformation is performed to encode the aggregatedrepresentation using a weight matrixWd ∈ R

Kd−1×Kd and an acti-vation function σ (·) (line 5). Kd is the dimension of the embeddingafter d-th step and K0 is 4 which is the initial attribute dimension.A concrete example is shown in Figure 3 which illustrates the pro-cedure of computing node embedding with D = 2. Essentially, afterd iterations, the embedding of a node combines the information ofits d-hop neighborhood.

When maximum depth D is reached, the final embeddings areobtained and fed to the fully-connected layers for classification.Parameters that need to be trained includewpr ,wsu ,W1, . . . ,WDand parameters in FC layers. All the parameters in the network canbe trained end-to-end.

Different from other transductive approaches which cannot gen-eralize to unseen graphs [11, 16], the entire classification procedurefor each node is only based on its neighborhood information andthe learned parameters and can be shared across different graphs.

3.3 Multi-stage ClassificationFor a typical design, it is common to have many more negativenodes than positive nodes, which is not desireable for trainingmachine learning models. Training a single classification modelcan lead to poor overall performance since significant bias wouldbe introduced towards the majority class. To tackle this imbalanceissue, we developed a multi-stage GCN for this problem. In eachstage, a GCN is trained and only filters out negative cases with highconfidence, and passes the remaining nodes to the next stage, which

Page 4: High Performance Graph Convolutional Networks with Applications …byu/papers/C85-DAC2019-GCN-Test.pdf · inference scheme are presented for efficient GCN training and inference,

is illustrated in Figure 4. This is achieved by imposing a large weighton the positive nodes such that the penalty of misclassifying themwould be large. In this way, most positive points remain on the rightside of the decision boundary until negative points are substantiallyreduced. After a few stages, the remaining nodes should becomerelatively balanced and a network can make the final predictions.

3.4 Efficient Training and Inference3.4.1 Inference. Making the GCN model scalable to large

graphs is a critical problem, especially because fast inference isdesired. The computation shown in Algorithm 1 is an iterative pro-cess. However, it would be inefficient since the neighborhoods ofdifferent nodes may have overlap, thus there are many duplicatedcomputations [12, 13]. Here we introduce another approach to in-ference computation that enables our GCN to process millions ofnodes efficiently. The key idea is to leverage the adjacency matrixof the graph, denoted byA ∈ RN×N . N is the total number of nodesin the graph. A matrix Ed ∈ RN×Kd can also be obtained, in whichthe v-th row represents the embedding of node v after the d-thiteration, i.e., Ed [v, :] = e(v)d . Take the graph in Figure 3(a) as anexample. The weighted sum aggregation in iteration d is equivalentto Equation (2).

Gd = A · Ed−1 =

1 2 3 4 5 6

1 1 w1 w1 w1 0 02 w2 1 0 0 w1 03 w2 0 1 0 0 w24 w2 0 0 1 0 05 0 w2 0 0 1 06 0 0 w1 0 0 1

×

e(1)d−1

e(2)d−1

e(3)d−1

e(4)d−1

e(5)d−1

e(6)d−1

(2)

HereA ∈ R6×6,Gd ∈ R6×Kd−1 , and the v-th row is the represen-

tation for node v after aggregation in the d-th iteration. The innerloop in Algorithm 1 (line 3 – line 6) can be simply formulated as

Ed = σ (Gd ·Wd ) = σ ((A · Ed−1) ·Wd ). (3)

Then, all the computation can be formulated as a series of matrixmultiplications which can be efficiently computed, and duplicatedcomputation can be avoided. One potential issue is the dimension ofadjacency matrix A is N × N , which is extremely large and cannotbe stored in memory directly. However, we can exploit the factthat the A is a sparse matrix. For every design in our benchmarks,the sparsity is higher than 99.95%. Then A can be represented ina compressed sparse format, e.g., coordinate (COO) format whichstores a list of (value, row_index, column_index) tuple. Thematrix can be stored in the memory to enable the matrix multipli-cation. For instance, the COO representation of A in Equation (3)is represented as

value: [1, w1, w1, w1, w2, 1, w1,w2,1, w2,w2,1, w2, 1, w1,1]r_index: [1, 2, 3, 4, 1, 2, 5, 1, 3, 6, 1, 4, 2, 5, 3, 6]c_index: [1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6]

,

where each column is a tuple indicating the value and indices ofa non-zero element in the matrix. Moreover, the COO format isvery efficient for incremental matrix construction which facilitatesgraph modifications in our flow.

Training data:

Output Evaluate

GradientGPU1 GPU2

OutputFigure 5: Parallel training with multiple GPUs.

Predicted-0

Predicted-1

OP

Fan-in cone

a

(a)

a

(b)

Figure 6: An illustration to compute the impact for anOP. (a)Prediction on original graph with 5 positives in fan-in coneof a; (b) Prediction on graph after inserting an observationpoint with 1 positive. The impact of node a is 5 − 1 = 4.

3.4.2 Training. The strategy proposed for inference can alsoaccelerate the GCN training process. In practice, the training setmay contain many graphs. A straightforward method is to composemultiple graphs into a single graph, but the memory of a singleGPU may not be sufficient to hold all intermediate results in thiscase. To overcome this bottleneck, we leverage a parallel trainingscheme with multiple GPUs shown in Figure 5. Our scheme can beseen as a variant of a conventional data-parallel scheme. Conven-tionally, a batch of input data is split into equal chunks and eachGPU processes a chunk. With our GCN approach, the input of onegraph includes an adjacency matrix and node representation matrixwhich cannot be split. Therefore, we separate multiple graphs intoindividual graph. Each GPU processes one graph, and all of the out-put is gathered to calculate the loss and then do back-propagationto update the model.

4 ITERATIVE OP INSERTIONAfter a multi-stage GCN model is trained, it can identify difficult-to-observe nodes in a netlist, which can be used as guidance forobservation point insertion. However, not every difficult-to-observenode has the same impact for improving the observability since it ispossible that adding an observe point may improve the observabilityof other nodes in its fan-in cone. In order to minimize the numberof insertion points, we must select the observation point locationswith largest impact. Next we develop a flow to identify whichlocations are more significant using the trained classifier. We definethe impact of each location as the positive prediction reduction ina local neighborhood after inserting an observation point. Figure 6gives an example of impact calculation. An iterative flow for pointsinsertion is developed, which is shown in Figure 7. In each iteration,every positive prediction is evaluated to get its impact. Finally, alist containing observation points location is obtained. Then theyare sorted based on their impact and select the top ranked locations.Next, we modify the graph and perform inference for prediction.The positive predictions will become the candidates in the nextiteration. The exit condition is there are no positive predictions left.

Page 5: High Performance Graph Convolutional Networks with Applications …byu/papers/C85-DAC2019-GCN-Test.pdf · inference scheme are presented for efficient GCN training and inference,

Table 1: Statistics of benchmarks

Design #Nodes #Edges #POS #NEG

B1 1384264 2102622 8894 1375370B2 1456453 2182639 9755 1446698B3 1416382 2137364 9043 1407338B4 1397586 2124516 8978 1388608

Inserting observation points modifies the netlist, thus the graphshould be modified, including the graph structure and node at-tributes. Inserting one observation point to a target node v corre-sponds to adding a new node p to the graph and adding an edgefrom the target nodev to new node p. Essentially, the adjacency ma-trixA and initial embedding matrix E0 should be updated. A criticalproblem in this iterative flow is how to update the graph efficiently.In our flow Figure 7, the graph can be updated incrementally.A canbe incrementally updated by adding a column and a row, and settingcorresponding entries aswpr orwsu , which can be done efficientlyunder COO format by appending 3 tuples (wpr ,p,v), (wsu ,v,p)and (1,p,p). E0 is updated by appending attribute of new node pwhich is set to [0,1,1,0]. Then only the attributes of the nodes in thefan-in cone of the new node should be updated based on SCOAPmethod [26]. Since this GCN model is inductive, the updatedA andE0 are directly fed to the model for prediction.

5 EXPERIMENTAL RESULTSThe experiments are performed on 4 industrial designs implementedin 12nm technology. Statistics of designs are summarized in Table 1.#POS and #NEG indicate the number of difficult-to-observe nodesand easy-to-observe nodes, respectively. The labels are obtainedfrom commercial DFT tools. The GCN is implemented with PyTorchand trained on a Linux machine with 32 cores and 4 NVIDIA TeslaV100 GPUs. The total memory used in the training is 64GB.

Search depth is an important hyper-parameter that may affectthe performance of a GCN. On one hand, increasing search depthcan cover a larger neighborhood such that more information canbe involved. On the other hand, too large region may lead to over-fitting. We select the search depth by monitoring and comparingthe training and testing accuracy among different settings on thesearch depth. K1, K2 and K3 are set as 32, 64 and 128, respectively.The Rectified linear unit (ReLU) function ReLU(x) = max(x , 0) [27]is used as the activation function. Four FC layers are consistent,whose dimensions are 64, 64, 128 and 2. Figure 8 shows the recordof training accuracy and testing accuracy during learning for 300epochs with search depth 1, 2 and 3. It can be seen that the per-formance of GCN improves as the search depth increases. Searchdepth D = 3 is used for all the remaining experiments. The entirenetwork consists of 3 aggregators, 3 encoders (with ReLU layer)and 4 FC layers. We use cross-entropy function as loss function andstochastic gradient descent for optimization.

NetlistTrained GCN

Model

Prediction Satisfied?

Impact Evaluation

OP Insertion

ENDY

N

Figure 7: Iterative flow for observation points insertion.

0 100 200 300708090100

Epochs

Accuracy(%)

(a) Training accurary

0 100 200 300708090100

Epochs

Accuracy(%)

D=1D=2D=3

(b) Testing accuracy

Figure 8: Performance with different search depth D.Table 2: Accuracy Comparison on Balanced Dataset

Design LR RF SVM MLP GCNB1 0.778 0.790 0.813 0.860 0.928B2 0.767 0.785 0.809 0.845 0.929B3 0.779 0.793 0.814 0.856 0.930B4 0.782 0.801 0.821 0.862 0.935

Average 0.777 0.792 0.814 0.856 0.931

We compare the classification performance between our pro-posed GCN and various classical machine learning models, includ-ing logistic regression (LR), random forest (RF), support vectormachine (SVM) and Multi-layer perceptron (MLP). Considering asingle classifier may not have good performance on the benchmarkthat is highly imbalanced, we compare the performance on bal-anced datasets by sampling a subset within the entire dataset. Sinceother classical machine learning models require handcrafted fea-tures with a fixed dimension, we integrate neighborhood featuresby collecting the features of the nodes in the fan-in cone and fan-out cone. 500 nodes in fan-in cone and 500 nodes in fan-out coneare collected. Starting from the target node, breadth-first-searchis performed to collect the nodes in each cone. Every time a nodeis visited, the feature of this node is concatenated to the currentfeature vector. Therefore, the dimension of the feature vector fortraditional learning models is (500 + 500 + 1) × 4 = 4004.

For each design, we generate a balanced dataset by using all thepositive nodes and sampling the same number of negative nodesrandomly. Each time we use three designs for training and theremaining one for testing. For the MLP approach, the configurationof the network is the same as the classifier module in GCN. Theaccuracy comparison is presented in Table 2. GCN achieves 93.1%accuracy on average, outperforming other models on all designs.

We have shown that a GCN can obtain significantly better per-formance in distinguishing positive nodes and negative nodes thanclassical learning models. However, the performance a single GCNis not satisfying if it is directly trained and tested on original imbal-anced dataset. To validate the advantage of the multi-stage GCN,we compare the performance between the multi-stage GCN andsingle GCN. In highly imbalanced classification, F1-score is com-monly used since accuracy would be misleading. The training andtesting schemes are the same as before. Each time we use threedesigns for training and the remaining design for testing. 3 stagesare applied and the prediction results of each stage are combinedto obtain the F1-score on the entire dataset. The results comparisonis shown in Figure 9. The multi-stage GCN achieves much higherF1-scores than single GCN on all these imbalanced datasets.

By taking the advantage of the sparse representation of theadjacency matrix and matrix multiplication-based computation,

Page 6: High Performance Graph Convolutional Networks with Applications …byu/papers/C85-DAC2019-GCN-Test.pdf · inference scheme are presented for efficient GCN training and inference,

B1 B2 B3 B40

0.20.40.6

F1-Score

GCN-S GCN-M

Figure 9: F1-Score compari-son.

103 104 105 10610−2100102104

Number of nodes

Inferencetim

e(s) [12] Ours

Figure 10: Scalability vs. [12].Table 3: Testability results comparison

Design Industrial Tool GCN-Flow#OPs #PAs Coverage #OPs #PAs Coverage

B1 6063 1991 99.31% 5801 1687 99.31%B2 6513 2009 99.39% 5736 2215 99.38%B3 6063 2026 99.29% 4585 1845 99.29%B4 6063 2083 99.30% 5896 1854 99.31%

Average 6176 2027 99.32% 5505 1900 99.32%Ratio 1.00 1.00 1.00 0.89 0.94 1.00

the proposed GCN can scale to process designs with millions ofcells. To demonstrate the scalability of our acceleration strategy,we compare the inference runtime between two approaches ongraphs with different sizes, ranging from 103 nodes to 106 nodes.We leverage the released implementation of [12] for comparison,which uses recursion-based computation. The inference runtimecomparison between two approaches is shown in Figure 10. For adesign with 1 million cells, the recursive computation takes morethan one hour to complete the inference. With our accelerationstrategy, the inference runtime is only 1.5 seconds, which is threeorders of magnitude speedup compared to [12].

Finally we show the testability results of applying the multi-stage GCN and the iterative test points insertion flow. We use thetestability analysis results from a commercial testability analysistool as a baseline. 3 metrics are used for evaluation, including thetotal number of OPs inserted, the fault coverage and the number oftest patterns required. The output list of our proposed iterative GCN-based flow is fed to the same commercial tool to get a test reportfrom which we can get the pattern number and fault coverage fora fair comparison. The results are shown in Table 3. Column ‘#OPs’represents the total number of OPs inserted. ‘#PAs’ represents thenumber of test patterns. ‘Coverage’ represents fault coverage. It canbe seen that the OPs recommended by the proposed GCN modeland iterative flow outperform the industrial tool, achieving 11%reduction on the number of inserted OPs and 6% reduction on testpatterns without any degradation on fault coverage.

6 CONCLUSIONIn this paper, a GCN-based methodology is proposed for netlistrepresentation in EDA field. The methodology is applied to iden-tification of difficult-to-observe points in a netlist, in which GCNshows superior performance to classical learning models. A multi-stage GCN is developed to handle the imbalanced classificationproblem, which achieves significantly higher F1-score than singleGCN model. A parallel training scheme is developed to enable largescale training and a fast inference scheme is proposed to enhancethe scalability of the GCN model. Based on the GCN model and aniterative observation points insertion flow, we have achieved better

testability on industrial designs compared to a state-of-the-art com-mercial tool. We believe the GCN-based methodology is genericand can be applied to other EDA problems, and hope this paperwill inspire more future research in this area.

7 ACKNOWLEDGMENTSThis work is supported in part by The Research Grants Council ofHong Kong SAR (Project No. CUHK24209017).

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[9] B. Yu, K. Yuan, D. Ding, and D. Z. Pan, “Layout decomposition for triple patterninglithography,” IEEE TCAD, vol. 34, no. 3, pp. 433–446, March 2015.

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