CN108614903B - Correlation modeling method and device for simulation data of integrated circuit - Google Patents

Correlation modeling method and device for simulation data of integrated circuit Download PDF

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CN108614903B
CN108614903B CN201611134827.7A CN201611134827A CN108614903B CN 108614903 B CN108614903 B CN 108614903B CN 201611134827 A CN201611134827 A CN 201611134827A CN 108614903 B CN108614903 B CN 108614903B
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曾璇
朱恒亮
李昕
曾溦
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Fudan University
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Abstract

The invention belongs to the field of integrated circuit design automation, and particularly relates to an integrated circuit simulation data correlation modeling method and device based on correlation clustering and covariance contraction technology, wherein the method comprises the following steps: according to the method, circuit simulation data required by modeling are firstly obtained, an original multivariate normal distribution is constructed according to the data, and the distribution is corrected through correlation clustering and covariance contraction technologies to obtain a correlation model represented by the corrected multivariate normal distribution. The invention improves the reliability and the accuracy of the correlation model, so that the correlation model of the circuit simulation data can be suitable for circuits of any scale, and the algorithm developed by utilizing the model can be ensured in the aspects of accuracy and efficiency.

Description

Correlation modeling method and device for simulation data of integrated circuit
Technical Field
The invention belongs to the field of integrated circuit design automation, and particularly relates to modeling of correlation of integrated circuit simulation data. In particular to a correlation modeling method and a device for simulation data of an integrated circuit based on correlation clustering and covariance contraction technology.
Background
The prior art discloses that as integrated circuits are inevitably affected by process conditions during the manufacturing process, physical parameters of each transistor, such as doping concentration, gate oxide thickness, surface charge, etc., can fluctuate randomly, and thus, the electrical characteristics of the transistors, such as threshold voltage, leakage current, etc., and the circuits composed of these transistors, also have certain randomness in their behavior (such as delay, gain, etc.). As integrated circuit process technology has advanced and technology nodes of semiconductor processes have entered the nanometer scale, these deviations have become increasingly severe and non-negligible. It is believed that these deviations must be accounted for in the integrated circuit design and verification process.
Aiming at the current situations of small size, complex circuit and various process parameters of the current device, the most common tool in the circuit design and verification stage in the industry at present is Monte Carlo simulation, namely, a plurality of circuit samples with randomly changed process parameters are generated through circuit analysis software, and circuit simulation is carried out on each sample one by one under various different working conditions (such as temperature, power supply voltage, circuit load and the like) so as to judge whether the circuit meets the design requirement when the process fluctuates. Practice shows that the Monte Carlo simulation fully considers the complex relationship between the transistor process parameters and the circuit performance, the data reliability is good, the algorithm complexity is irrelevant to the number of the process parameters, but if Monte Carlo simulation is carried out on all the circuit performances under various working conditions one by one, a large amount of research and development time is usually consumed.
However, for a certain circuit sample, the deviation of the value from the standard value under different working conditions usually shows a certain correlation, because the process parameters are already determined. In addition, for most analog circuits and digital-analog hybrid circuits, there is also a certain correlation between different circuit performances. If the correlations can be reasonably utilized, the times of actual circuit simulation can be reduced, and the efficiency of circuit design and verification is improved.
Studies have shown that statistically, the correlation of multivariate data is characterized by the covariance matrix of the data, and the strength of the correlation of two dimensions in the data is determined by the absolute value of the corresponding term in the correlation coefficient matrix. For a given set of multivariate data, the correlation in any two dimensions can be determined simply by unbiased estimation or maximum likelihood estimation of the covariance matrix of the multivariate data; however, due to the limitation of simulation cost, simulation data of an integrated circuit often has the characteristics of high dimensionality and few samples, and an estimated value of the covariance matrix obtained under the condition is unstable and has a large error, and cannot be directly used for computer processing.
Based on the problems in the prior art, the inventor of the present application intends to provide an accurate and universal simulation data correlation model, and in particular, relates to a correlation clustering and covariance contraction technology-based integrated circuit simulation data correlation modeling method and device.
The prior art related to the present invention is:
[1] A. Y. Ng, M. I. Jordan, and Y. Weiss, “On spectral clustering: Analysis and an algorithm,” Advances in Neural Information Processing Systems, 2:849-856, 2002.
[2] J. Zhang, G. Sudre, X. Li, W. Wang, D. J. Weber, and A. Bagic, “Cluster linear discriminant analysis for MEG-based brain computer interfaces,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 19, no. 3, pp. 221– 231, 2011.
[3] J. Schäfer and K. Strimmer, “A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics,” Stat. Appl. Genet. Molec. Biol., vol. 4, no. 32, 2005.
[4] O. Ledoit and M. Wolf, “A well-conditioned estimator for large-dimensional covariance matrices,” J. Multivar. Anal., vol. 88, no. 2, pp. 365–411, 2004.
[5] O. Ledoit and M. Wolf, “Honey, I shrunk the sample covariance matrix," J. Portfolio. Manage., vol. 30, no. 4, pp. 110–119, 2004。
disclosure of Invention
The invention aims to provide an accurate and universal simulation data correlation model based on the problems in the prior art, and particularly relates to an integrated circuit simulation data correlation modeling method and device based on correlation clustering and covariance contraction technology.
The correlation clustering and covariance contraction technology-based correlation modeling method and device for integrated circuit simulation data comprise two parts of original correlation model establishment and model correction, and fig. 1 shows the overall framework of the invention.
The invention adopts the following technical scheme for solving the technical problems in the prior art:
based on simulation data of a small amount of integrated circuit samples, the correlation of performance fluctuation caused by process parameter fluctuation under different working conditions (such as temperature, power supply voltage, circuit load and the like) is described through correlation clustering and covariance contraction technology to obtain a correlation model, and the correlation model can be used for proposing or improving a correlation algorithm of the computer aided design of the integrated circuit, so that the efficiency of circuit design and verification is improved.
The modeling method of the invention comprises the following steps:
step 1, inputting simulation data: according to the circuit network table and the technological parameters of each circuit sample, carrying out complete circuit simulation on the circuit samples to obtain simulation data;
step 2, establishing an original correlation model: establishing an original multivariate normal distribution by taking the circuit sample in the simulation data obtained in the step 1 as a sample and taking the circuit expression as the dimensionality of the sample;
step 3, correcting the correlation model: aiming at the reduction of the reliability of the correlation model caused by the fact that the dimensionality is smaller than the number of samples in practical application, the original correlation model needs to be corrected, and the specific scheme is selective contraction of a correlation coefficient matrix, and the method comprises the following steps:
step 3a, clustering the dimension with larger correlation (measured by the absolute value of the correlation coefficient) in the original model obtained in the step 2, wherein the dimension belonging to the same cluster does not exceed a given value;
step 3b, constructing a target correlation coefficient matrix, and keeping correlation coefficients between every two dimensionalities belonging to the same cluster according to the clustering result obtained in the step 3 a; and setting the correlation coefficient of the dimensionality belonging to different clusters to be 0. Thereby obtaining a target correlation coefficient matrix;
step 3c, calculating an optimal shrinkage factor, wherein the shrinkage factor is such that the expectation of the square error of the correlation coefficient matrix obtained thereby is minimum;
step 3d, calculating a corrected correlation coefficient matrix and a corresponding covariance matrix according to the target correlation coefficient matrix obtained in the step 3b and the shrinkage coefficient obtained in the step 3 c;
and 4, outputting the correlation model. The model is a multivariate normal distribution and is uniquely determined by the mean vector obtained in the step (2) and the covariance matrix obtained in the step (3 d);
in the invention, the relativity of performance fluctuation caused by the fluctuation of circuit process parameters under different working conditions (such as temperature, power supply voltage, circuit load and the like) is characterized by using a multivariate normal distribution. Correlation clustering and covariance contraction techniques are used to ensure reliability and accuracy of the model in high-dimensional, small-sample cases.
The result shows that the parameter yield analysis result performed by the method has extremely small error compared with the traditional Monte Carlo method, and meanwhile, the analysis efficiency is greatly improved.
Compared with the prior art, the invention has the beneficial effects that:
1. the covariance contraction technology is used for solving the influence of common high-dimensionality and small sample phenomena of integrated circuit simulation data on the reliability and accuracy of a simulation data correlation model, so that the correlation model can be used for circuits of any scale.
2. By using a hierarchical relevance clustering mode, the reliability and the accuracy of a relevance model are improved while the relevance information on strong relevant dimensions is kept, so that the accuracy and the efficiency of an algorithm developed by using the model are ensured.
Drawings
Figure 1 shows the general framework of the invention.
Detailed Description
To make the above objects, features and advantages of the present invention more comprehensible, the present invention is applied to fast estimation of yield of integrated circuit parameters, and compared with the existing correlation modeling method and the existing covariance shrinkage technique.
Example 1
The basic principle of the rapid analysis of the parameter yield is that on the basis of Monte Carlo simulation, the correlation of simulation data is utilized, the statistical distribution of other simulation data is predicted through part of known simulation data, and unnecessary circuit simulation is identified and reduced, so that the purpose of improving the analysis efficiency is achievednUsing each circuit sample for correlation modeling, wherein the dimension of simulation data is 432, and the optimal parameter combination of a correlation modeling algorithm is (n, maxSize 0) The method is generated by cross validation, and the rest steps and methods in the analysis algorithm of the parameter yield are consistent except for the modeling method of the correlation:
in this embodiment, the correlation modeling of the simulation data of the integrated circuit is performed according to the following method:
step 1, inputting simulation data. And carrying out complete circuit simulation on the circuit samples according to the circuit network table and the process parameters of each circuit sample to obtain simulation data. The simulation data required for modeling included all circuit performance values for each sample. Wherein, different working conditions and different circuit indexes are all regarded as different performance values. If used, thenA plurality of circuit samples, each circuit sample havingDThe performance value is the simulation data inputn×DIs used for the two-dimensional matrix of (1).
And 2, establishing an original correlation model. And (3) establishing an original multivariate normal distribution by taking the circuit sample in the simulation data obtained in the step (1) as a sample and taking the circuit expression as the dimension of the sample. Let us remembersSecond of the circuit sampleiAn expression value ofx i (s)（s = 1, 2, …, n; i = 1, 2, …, D) Then the original correlation model is distributed by multivariate normal distribution N D (
Figure DEST_PATH_IMAGE001
,
Figure 573696DEST_PATH_IMAGE002
) Is given in
Figure 553154DEST_PATH_IMAGE001
=[
Figure DEST_PATH_IMAGE003
1, …,
Figure 208257DEST_PATH_IMAGE003
D ]T,
Figure 915051DEST_PATH_IMAGE004
, i = 1, …, D,
Figure 791740DEST_PATH_IMAGE002
=[
Figure DEST_PATH_IMAGE005
Figure 427252DEST_PATH_IMAGE006
] D D×,
Figure DEST_PATH_IMAGE007
i, j = 1, …, D.
And 3, correcting the correlation model. The correlation model obtained in step 2 is only in the number of samplesnMuch greater than dimensionDThe time is accurate, but the condition may not be satisfied in practical application, which results in the reliability of the original correlation model being reduced. Therefore, the model needs to be modified, and the specific scheme is to carry out matrix modification on the correlation coefficient
Figure 71116DEST_PATH_IMAGE008
Performing selective shrinking, wherein the shrunk correlation coefficient matrix has the following form:
Figure DEST_PATH_IMAGE009
wherein
Figure 145382DEST_PATH_IMAGE008
=[
Figure 876578DEST_PATH_IMAGE010
ij ] D D×,
Figure DEST_PATH_IMAGE011
i, j = 1, …, D.
In the step 3a, the step of the method,clustering the dimension with larger correlation (measured by the absolute value of the correlation coefficient) in the original model obtained in the step 2, wherein the dimension belonging to the same cluster does not exceed a given valuemaxSize 0(this value is an algorithm parameter). The clustering method may use normalized spectral clustering [1 ]]Where the optimal number of clusters can be found in literature [2 ]]The method of (1), determining the similarity matrix W is given by:
W = [|
Figure 56893DEST_PATH_IMAGE010
ij |] D D×, i, j = 1, …, D.
after clustering, if there is dimension greater thanmaxSize 0Is greater than the dimensionmaxSize 0Respectively carrying out the same clustering operation again in the clusters until the dimension of all the clusters does not exceed the dimensionmaxSize 0Until now.
And 3b, constructing a target correlation coefficient matrix. According to the clustering result obtained in the step 3a, for the dimensionality belonging to the same cluster, the correlation coefficient between every two dimensionalities is kept; and setting the correlation coefficient of the dimensionality belonging to different clusters to be 0. Thereby obtaining a target correlation coefficient matrix RtarI.e. by
Rtar = [ρ ij ] D D×,
Figure 188053DEST_PATH_IMAGE012
i, j = 1, …, D,
Wherein blkiIs shown asiThe cluster to which each dimension belongs.
And 3c, calculating the optimal shrinkage coefficient. The shrinkage factor should be such that the desired square error of the resulting correlation coefficient matrix is minimized. Document [3] shows that the optimum shrinkage factor satisfying this condition is
Figure DEST_PATH_IMAGE013
Wherein
Figure 174332DEST_PATH_IMAGE014
Step 3d, calculating a corrected correlation coefficient matrix R according to the target correlation coefficient matrix obtained in the step 3b and the contraction coefficient obtained in the step 3c*And corresponding covariance matrix sigma*：
Figure DEST_PATH_IMAGE015
Figure 573084DEST_PATH_IMAGE016
And 4, outputting the correlation model. The model is a multivariate normal distribution N D (
Figure 65245DEST_PATH_IMAGE001
,Σ*) Using the mean vector obtained in step 2
Figure 621385DEST_PATH_IMAGE001
And the covariance matrix sigma obtained in step 3d*And (4) uniquely determining.
Table 1.1 shows the accuracy and efficiency of the parametric yield analysis;
table 1.1: accuracy and efficiency of rapid parametric yield analysis using various correlation models
Method Optimization ofnValue of Yield estimate Relative error Total time (hours) Acceleration ratio
Monte Carlo simulation --- 0.8830 --- 261 1.0
Using original correlation models 80 0.8730 1.1% 36.1 7.2
Scaling with covariance [4 ]] 60 0.8820 0.1% 89.7 2.9
Scaling with covariance [5 ]] 60 0.8810 0.2% 86.3 3.0
By using the invention 50 0.8820 0.1% 28.2 9.3
The result shows that although the original correlation model has a certain acceleration ratio, the model error is larger because the simulation data dimension is larger than the sample number, the scaling result weakens the correlation of strong correlation dimension by using the existing covariance scaling method, so that the efficiency of parameter yield analysis is reduced, and the parameter yield analysis result performed by using the method has smaller error compared with the traditional Monte Carlo method, and the analysis efficiency is greatly improved.

Claims (3)

1. A modeling method for correlation of simulation data of an integrated circuit is characterized in that correlation of performance fluctuation caused by process parameter fluctuation under different working conditions is described through correlation clustering and covariance contraction technologies based on simulation data of a small number of integrated circuit samples to obtain a correlation model; the model is used for proposing or improving the related algorithm of the computer aided design of the integrated circuit, and the efficiency of the circuit design and verification is improved;
the different working conditions comprise temperature, power supply voltage and circuit load;
the method comprises the following steps:
step 1, inputting simulation data: according to the circuit network table and the technological parameters of each circuit sample, carrying out complete circuit simulation on the circuit samples to obtain simulation data;
step 2, establishing an original correlation model: establishing an original multivariate normal distribution by taking the circuit sample in the simulation data obtained in the step 1 as a sample and taking the circuit expression as the dimensionality of the sample;
step 3, correcting the correlation model: for the actual application, the reliability of the correlation model is reduced due to the fact that the dimension is smaller than the number of samples, the original correlation model is corrected, and the selective contraction of the correlation coefficient matrix comprises the following sub-steps:
step 3a, clustering correlation dimensions in the original model obtained in the step 2, wherein the dimension belonging to the same cluster does not exceed a given value;
step 3b, constructing a target correlation coefficient matrix: according to the clustering result obtained in the step 3a, for the dimensionality belonging to the same cluster, the correlation coefficient between every two dimensionalities is kept; setting the correlation coefficient of the dimensionality belonging to different clusters to be 0, and thus obtaining a target correlation coefficient matrix;
step 3c, calculating an optimal shrinkage coefficient: the shrinkage factor should be such that the desired square error of the resulting correlation coefficient matrix is minimized;
step 3d, calculating a corrected correlation coefficient matrix and a corresponding covariance matrix according to the target correlation coefficient matrix obtained in the step 3b and the shrinkage coefficient obtained in the step 3 c;
and 4, outputting a correlation model: the model is a multivariate normal distribution and is uniquely determined by the mean vector obtained in step 2 and the covariance matrix obtained in step 3 d.
2. The method of claim 1, wherein in step 3a, the correlation in the original model is measured by the absolute value of the correlation coefficient.
3. The method of claim 1, wherein the correlation of performance fluctuations caused by fluctuations in circuit process parameters under different operating conditions is characterized by a multivariate normal distribution, wherein correlation clustering and covariance shrinkage techniques are used to ensure the reliability and accuracy of the model in high-dimensional, small-sample situations.
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