A method and system for implementing simulated optical link topology based on deep learning

Abstract

The present application belongs to the technical field of simulated optical link topology, and specifically relates to a method and system for realizing simulated optical link topology based on deep learning, S1. Abstracting different photonic devices or sub-links into sub-modules with specific transmission coefficients, and regarding the output response of the overall target as the superposition of the output responses of multiple optical paths between the input port and the output port, with different numbers of photonic devices randomly arranged on each optical path, to construct a simulated optical link topology; S2. Encoding the topology of the simulated optical link to obtain a unified parameterized matrix; S3. Using a genetic algorithm to search for the topology closest to the target response, the forward network model is used as a fast link topology output performance evaluation module to improve the search efficiency of the optimization algorithm. Its advantage is that, without considering the changes in the physical parameters of the photonic devices themselves, the automated design of the simulated optical link topology is realized for the first time.

Description

Technical Field

[0001] The present application belongs to the technical field of simulated optical link topology structure, and specifically relates to a method and system for realizing simulated optical link topology structure based on deep learning. Background Art

[0002] Current automated topology design methods for analog circuits mostly use graph encoding or structural encoding to abstract components or subcircuits into nodes or functional modules. These methods then employ optimization algorithms to design the topology. However, these methods are currently limited to simple circuit structures, such as low-order amplifiers, and are difficult to extend to more complex circuit designs. Generally speaking, research on topology generation for analog integrated circuits involves searching and selecting from simple topology libraries. After abstracting or encoding the structure, the system then performs global-to-partial separation or part-to-global integration according to connectivity rules. The correspondence between structure and output is not modeled, and the design process becomes increasingly complex as the design scale increases. Summary of the Invention

[0003] This application abstracts different photonic devices or sub-links into sub-modules with specific transmission coefficients, and the output response of the overall target can be regarded as the superposition of the output responses of multiple optical paths between the input port and the output port. Each optical path is a random arrangement of different numbers of photonic devices. This construction idea is applicable to optical links of different sizes. Therefore, we propose a topology encoding method based on this idea. When the maximum number of devices and the maximum number of optical paths are determined, they are unified into a fixed-size matrix expression. A network model is established to simulate the relationship between the optical link topology parameters and the output response, and a genetic optimization algorithm is further used to achieve the optimal topology structure generation under a given target response. The technical solution is as follows:

[0004] A method for implementing a simulated optical link topology structure based on deep learning, comprising the following steps:

[0005] S1. Different photonic devices or sub-links are abstracted into sub-modules with specific transmission coefficients. The overall target output response is considered to be the superposition of the output responses of multiple optical paths from the input port to the output port. Each optical path is a random arrangement of different numbers of photonic devices, constructing a simulated optical link topology.

[0006] S2. Encode the topology of the simulated optical link to obtain a unified parameterized matrix;

[0007] S3. Genetic algorithm is used to search for the topology closest to the target response. The forward network model is used as a fast link topology output performance evaluation module to improve the search efficiency of the optimization algorithm.

[0008] Preferably, the optical link topology includes multiple different photonic devices and multiple branch splitters and combiners. Different photonic devices are replaced by single micro-ring structures with different and fixed physical parameters. The different topological structures formed have two input and output ports as a whole, and the internal structure is a free combination of devices, thereby forming different device connection structures with different output responses.

[0009] Preferably, in step S2, the component components are treated as a single node according to the graph encoding method, and the parameterization matrix is ​​as follows:

[0010]

[0011] The value of each element in the parameterized matrix is ​​0 or 1, indicating whether the unit device represented by the column index exists on the optical path represented by the row; assuming that all branch devices are evenly divided, any connection structure can ultimately be equivalent to an M*N matrix.

[0012] Preferably, the total transmission coefficient of the simulated optical link topology can be expressed as:

[0013]

[0014] Where H n Represents the transmission coefficient of a unit device, where only a mn The corresponding matrix elements ≠0 participate in the formula operation; due to the equivalent expansion, multiple actual connection structures may be generated according to the parameter matrix, but these structures are equivalent and have a unique total transmission coefficient, which does not affect the correspondence of the parameter matrix to structures with different performances.

[0015] Preferably, in step S3,

[0016] S31. Initialize the population size and treat the parameterized matrix p as an individual in the population;

[0017] S32. Set the number of iterations; if the number of iterations reaches the set threshold and no individual meeting the conditions appears, reinitialize the population and search for the global optimal solution again;

[0018] S33. Calculate individual fitness using absolute error;

[0019] S34. Determine the size of the absolute error and the error threshold. If the absolute error is less than the error threshold, output the individual parameters and end this iteration; if the error is not less than the error threshold, use the genetic operator to process, generate a new population, and repeat S32-S34.

[0020] Preferably, in step S32, a forward network model between the input structure parameters and the corresponding output responses is established based on topological coding, and the network model used is a fully connected network;

[0021] The loss function for network training is the mean squared error between the true target response and the response predicted by the forward network:

[0022]

[0023] Among them, y i is the target response to the input, is the predicted response of the output structure, and m is the number of samples. During network training, the network calculates the loss of the input data samples and then uses backpropagation to optimize and adjust the weights and biases of each layer in the neural network. Through multiple rounds of iteration, the network model with the minimum error is obtained.

[0024] A deep learning-based simulation optical link topology implementation system includes a data acquisition module, a data processing module, and an output module;

[0025] Data acquisition module: collects data information of optical link topology;

[0026] Data processing module: This module encodes the topology of the simulated optical link to obtain a unified parameterized matrix. It uses a genetic algorithm to search for the topology closest to the target response. The forward network model serves as a fast link topology output performance evaluation module to improve the search efficiency of the optimization algorithm.

[0027] Output module: outputs optimal topology parameters.

[0028] Compared with the prior art, this application has the following beneficial effects:

[0029] 1. The topology of the simulated optical link is encoded to obtain a unified parameterized expression. Compared with existing graph encoding methods, this reduces the possibility of multiple parameter encodings corresponding to the same topology and facilitates the conversion from parameter encoding to analytical formulas and actual topologies.

[0030] 2. After the topology structure of the simulated optical link is parameterized, a forward network model is established between different topologies and output responses, realizing the mapping expression between topology parameters and output responses. Compared with the process of structure construction and simulation calculation in simulation software, performance evaluation can be carried out more quickly.

[0031] 3. Use genetic algorithms to search for the topology closest to the target response, and use the forward network model as a fast topology performance estimation component to improve the search efficiency of the optimization algorithm;

[0032] 4. The applicability of this method under different numbers of devices was verified, and the forward network model still has a good response fitting effect for the topological structure with different numbers of optical devices. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 The topology of the optical link is simulated by two split paths.

[0034] Figure 2 It is a forward network model.

[0035] Figure 3 is the error change during the training process.

[0036] Figure 4 Comparison between the true response (original) and the model predicted response (ForwardNet).

[0037] Figure 5 For error.

[0038] Figure 6 It is a new topology structure; (a) four-way topology structure; (b) three-way topology structure.

[0039] Figure 7 For generalization test, (a) is the equivalent extension (Code1) and (b) is the zero-fill extension (Code2).

[0040] Figure 8 Implement a process for the genetic algorithm. DETAILED DESCRIPTION

[0041] Exemplary embodiments will be described in detail herein, examples of which are illustrated in the accompanying drawings. In the following description, when referring to the drawings, like numbers in different figures represent like or similar elements unless otherwise indicated. The embodiments described in the following exemplary embodiments are not intended to represent all embodiments consistent with the present invention. Instead, they are merely examples of apparatus and methods consistent with certain aspects of the present invention, as detailed in the appended claims.

[0042] A method for implementing a simulated optical link topology structure based on deep learning, comprising the following steps:

[0043] S1. Different photonic devices or sub-links are abstracted into sub-modules with specific transmission coefficients. The overall target output response is considered to be the superposition of the output responses of multiple optical paths from the input port to the output port. Each optical path is a random arrangement of different numbers of photonic devices, constructing a simulated optical link topology.

[0044] S2. Encode the topology of the simulated optical link to obtain a unified parameterized matrix;

[0045] In step S2, the component components are treated as single nodes according to the graph encoding method, and the parameterization matrix is ​​as follows:

[0046]

[0047] The value of each element in the parameterized matrix is ​​0 or 1, indicating whether the unit device represented by the column index exists on the optical path represented by the row; assuming that all branch devices are evenly divided, any connection structure can ultimately be equivalent to an M*N matrix.

[0048] The total transmission coefficient of the simulated optical link topology can be expressed as:

[0049]

[0050] Where H n Represents the transmission coefficient of a unit device, where only a mn The corresponding matrix elements with ≠ 0 participate in the formula operation. Due to the equivalent expansion, multiple actual connection structures may be generated according to the parameter matrix, but these structures are all equivalent and have a unique total transmission coefficient, which does not affect the parameter matrix's correspondence with structures with different performance.

[0051] Based on the significant optimization effect of the forward network model, the inverse design of the simulated optical link structure under specific target response is further realized. Since the parameterization of the topology structure is mainly expressed by 0 and 1. This structural parameter is equivalent to the effect after binary encoding, it is natural to use genetic algorithms for global optimization search. The implementation process of this optimization algorithm can skip the encoding step. Its specific implementation steps are as follows: Figure 8 As shown:

[0052] S3. Genetic algorithm is used to search for the topology closest to the target response. The forward network model is used as a fast link topology output performance evaluation module to improve the search efficiency of the optimization algorithm.

[0053] S31. Initialize the population size and treat the parameterized matrix p as an individual in the population;

[0054] S32. Set the number of iterations; if the number of iterations reaches the set threshold and no individual meeting the conditions appears, reinitialize the population and search for the global optimal solution again;

[0055] S33. Calculate individual fitness using absolute error;

[0056] S34. Determine the size of the absolute error and the error threshold. If the absolute error is less than the error threshold, output the individual parameters and end this iteration; if the error is not less than the error threshold, use the genetic operator to process, generate a new population, and repeat S32-S34.

[0057] In step S33, based on topological coding, a forward network model between the input structure parameters and the corresponding output responses is established, and the network model used is a fully connected network;

[0058] The loss function for network training is the mean squared error between the true target response and the response predicted by the forward network:

[0059]

[0060] Among them, y i is the target response to the input, is the predicted response of the output structure.

[0061] During the network training process, the network will calculate the loss size of the input data samples, and then use the backpropagation method to optimize and adjust the weights and biases of each layer in the neural network, and obtain a network model with minimum error through multiple rounds of iterations.

[0062] Here, the trained forward network model replaces the original electromagnetic simulation process, enabling rapid individual fitness evaluation and accelerating the algorithm's computational speed. As a global optimization algorithm, the genetic algorithm has a relatively slow search speed, especially when the error threshold is low. To minimize the differences in iteration rates caused by different initialization populations, we implemented a population reinitialization operation. If no individuals meeting the conditions appear after every 500 genetic generations, the population is reinitialized and the search for the global optimal solution is repeated. While the initialization value generally affects the global search, periodically adjusting it can expand the search range and accelerate the search for the global optimal solution. Some hyperparameters of the genetic algorithm must be set in advance; the setting values ​​are shown in Table 3.

[0063] Table 3. Genetic algorithm hyperparameter settings

[0064]

[0065] Based on the above optimization algorithm, we randomly selected 100 target responses in the test data set and brought them into the genetic algorithm for optimization search. The final results are shown in Table 4.

[0066] Table 4. Genetic algorithm operation results

[0067]

[0068] Here, the response spectra of 100 test samples were selected as optimization targets, and the genetic algorithm optimization process was repeated 100 times, taking a total of approximately two hours. Of the 100 optimized design structural parameters, 82 were identical to the actual structural parameters. Of the different outputs, only two had misalignments, which did not affect the similarity between the output response and the optimization target.

[0069] The optical link topology includes multiple different photonic devices and multiple branch splitters and combiners. Different photonic devices are replaced by single micro-ring structures with different and fixed physical parameters. The different topological structures formed have two input and output ports as a whole, and the internal components are freely combined to form different device connection structures with different output responses.

[0070] A deep learning-based simulation optical link topology implementation system includes a data acquisition module, a data processing module, and an output module;

[0071] Data acquisition module: collects data information of optical link topology;

[0072] Data processing module: This module encodes the topology of the simulated optical link to obtain a unified parameterized matrix. It uses a genetic algorithm to search for the topology closest to the target response. The forward network model serves as a fast link topology output performance evaluation module to improve the search efficiency of the optimization algorithm.

[0073] Output module: outputs optimal topology parameters.

[0074] Example 1:

[0075] The entire link consists of ten different photonic devices and two Y-branch beam splitters and combiners. Different photonic devices are replaced by single micro-ring structures with different and fixed physical parameters. The different topological structures formed have two input and output ports. The internal components are freely combined to form different device connection structures with different output responses. The Y-branch beam splitters and combiners are 1:1 equally divided, and each photonic device is represented by a different transmission coefficient: H1, H2, H3, H4, H5, H6, H7, H8, H9, H 10 The form of the analog link it constitutes is as follows Figure 1 shown. Figure 1 The overall transmission coefficient H of the link structure is:

[0076]

[0077] For the coding of this topological structure, the component devices are regarded as single nodes according to the graph coding method, which can be parameterized into the form of adjacency matrix. Figure 1 The undirected graph adjacency matrix of the structure can be expressed as:

[0078]

[0079] The matrix size of formula (2) is 10×10, such as Figure 1 In the example, H1 and H4 are connected, so the element at the 4th column and row 1 in the matrix is ​​1, and 0 indicates no connection. The size of this parameter matrix is ​​related to the number of devices, but it is easy to cause multiple different parameter matrices to correspond to the same topology, such as Figure 1 The swapping of H1 and H4, H3 and H5 does not affect the overall structure and output response, but it manifests itself in a different adjacency matrix form. Therefore, we propose a more concise parameterization method that does not affect the swapping of the positions of the series devices. Figure 1 Taking the structure as an example, its parameterization results can be as follows:

[0080]

[0081] The encoding form in formula (3) is related to the form of formula (1). The two row vectors correspond to the two terms in formula (1) respectively. Each row vector has 10 elements, corresponding to 10 unit devices. The element 1 in the first row indicates that the first item contains the device H corresponding to the column index index. index , 0 indicates that the device is not included. Similarly, in actual topological structures, if both rows of elements in the same column are 1, the device is on the main path; if only one row is 1, the device is on a branch path. This parameterization method can represent the devices on every path between the input and output ports, regardless of the order of the input and output ports. This solves the problem of position swaps in the adjacency matrix affecting the parameterization results and reduces the occurrence of multiple values ​​during the parameterization process.

[0082] Based on Figure 1 The link structure is shaped like the topological encoding of formula (3), and a forward network model is established between the input structure parameters and the corresponding output response. The input parameters are the 20 element values ​​after flattening formula (3), and the output is a spectrum discretized into 500 values. The network model used is a commonly used fully connected network, and the model structure is as follows: Figure 2 As shown, it is equivalent to a forward simulation operation, which is used to compare the performance of the newly designed structure with the target performance.

[0083] The settings of some hyperparameters of the network model are shown in Table 1:

[0084] Table 1. Model hyperparameter settings

[0085]

[0086] In this supervised learning network model, the dataset needs to obtain data samples of features and labels composed of different optical link topology parameters and corresponding output responses. The topology parameters, as shown in formula (3), are randomly generated by the program. The structural parameters are then converted into an expression for the total transmission coefficient of the corresponding link structure, and the corresponding output response spectrum is calculated. The total calculated dataset size is 15,000, which is divided into training set, validation set, and test set in a ratio of 3:1:1. The training set is used to train the network model, the validation set is used to test whether the model is overfitting during training and to determine the model with the lowest validation error. The test set is used to test the performance of the final network model.

[0087] During the network training process, the network will calculate the loss size of the input data sample, and then use the backpropagation method to optimize and adjust the weights and biases of each layer in the neural network. After multiple rounds of iterations, the network model with the minimum error is obtained, such as Figure 3 As shown in the figure, the error in the training process continues to decrease, and the lowest test error is about 4.1e-6.

[0088] After obtaining the network model with the lowest verification error, the topological structure parameters in part of the test set are input into the model for forward testing. The response of the designed structure is compared with the response of the real structure, and the magnitude of the absolute error (MAE) between the two is used to represent the test results of part of the test set sample. Figure 4 shown.

[0089] Figure 4 The figure shows four target responses (original) randomly selected from the test set. After inputting their corresponding topological structure parameters into the forward network model, the corresponding predicted performance results are output. These four target responses are then compared with the model prediction results. They show extremely high similarity in shape and error. The model can effectively obtain the mapping relationship between the input topological structure parameters and the output response, realizing rapid testing of structural performance.

[0090] Although Figure 4 The results are good, but there are some quirks. The network training only considers the link structure that occurs when all 10 devices and two Y-branch splitters are used. It lacks verification of the generalizability of the network model when the number of devices in the link structure varies. Therefore, a new dataset containing link structures with different numbers of devices has been added. The data distribution is shown in Table 2.

[0091] Table 2. Distribution of different types of data in the dataset

[0092]

[0093] The data of groups 4, 5, 7, 9, and 10 are mixed and then divided into training and test sets. The data of groups 6 and 8 are not involved in training and serve as new test data to verify the generalization of the model. The structure and hyperparameter settings of the network remain unchanged. The error changes during the training process and the test results of the test set data are as follows: Figure 5 shown.

[0094] Figure 5 (a) The optimal test error after network training is 4.6e-6, which is Figure 3 The test error of is of the same order of magnitude. After the network training is completed, the test set of mixed data and two new sets of data, 6 and 8, are input into the network model to verify its performance generalization, as shown in the following example: Figure 5 As shown in (b), loss_devices_mix represents the test set from the mixed dataset. loss_devices_6 and loss_devices_8 are two sets of data that were not used in training. Each set contains 2500 data points. After being input into the network model, the overall error distribution is obtained. The solid horizontal lines in the boxes represent the medians, which are 5.3e-4, 4.9e-4, and 6.1e-4, respectively. The dashed horizontal lines represent the means, which are 6.8e-4, 5.9e-4, and 7.5e-4, respectively. The error distributions of these three test data sets are very similar, demonstrating that the network model is universally applicable to simulated optical links with different numbers of components.

[0095] At the same time, considering the combination design of more complex link topology structures, the effectiveness of the forward model in a variety of complex structures is verified. Therefore, two other types of connection structures are designed here, such as Figure 6 shown.

[0096] The two structures have three and four light paths respectively to synthesize the final output light field. The overall transmission coefficients of the two structures are:

[0097]

[0098] According to the new structural encoding method, the obtained structural parameterization matrix is ​​in the form of:

[0099]

[0100] The two parameter matrices P a , P b The number of rows and the transmission coefficient H a , H bThe number of monomials is the same as the number of 1s in each row. The column index of the position of the 1 in each row indicates the transmission coefficient of the corresponding device included in the corresponding monomial. For the actual connection structure, the number of rows determines the total number of optical paths from input to output. The number and position of 1s in each column determine whether the device is shared by all paths (i.e., all 1s), or shared by several paths (i.e., the device is distributed across the total path of these paths), or has only one 1 (i.e., the path passes through only this one device from input to output). Although the matrix parameters may generate more than one structure that meets the requirements, the matrix parameters correspond one-to-one to the overall transmission coefficient of the structure, so these structures must have the same output response and are functionally equivalent.

[0101] Here, the two structural parameter matrices have different sizes, which does not meet the requirement of uniform network input parameter size. Therefore, it is necessary to expand the smaller parameter matrix to the maximum matrix size, and to increase it from small to large so as to fully preserve the encoded combination structure information. Matrix expansion usually uses the zero padding method, but it is obvious that a full zero row has no practical meaning. Therefore, an equivalent expansion method is proposed here, such as Figure 6 In (b), the three devices in the original single branch can be replaced by the content pointed by the arrow: the original three devices are copied into two identical branches. This method expands the branch without affecting the overall output effect. The new transmission coefficient is

[0102] Compared with the transmission coefficient in the original formula (6), the 1 / 2 coefficient term is equivalently decomposed into two identical 1 / 4 coefficient terms, which does not change the overall output response. Its parameterized matrix becomes

[0103]

[0104] This is to copy the last row of the matrix to a new row. This extension has a practical structural equivalence and is conducive to the generalization verification of the model on different structures. In order to verify the effectiveness of this method, Figure 1 and Figure 6 The three topological structures are connected and the data sampling of different outputs is carried out. Figure 6 The two structures of the data set are mixed and used to train a new forward network model. Figure 1 After the structure of is equivalently expanded, it is used as new test data to verify the generalization performance of the network model. At the same time, we compare the zero-fill expansion method with the equivalent expansion method. The test structure is as follows:

[0105] like Figure 7 As shown, under the equivalent extension method, Figure 6 The error performance of the model trained with data of two mixed structures in its own test set (Code1_mixed) is the same as Figure 1The test errors for the structure (Code1_new) in this model are relatively similar, demonstrating good generalization. However, the model with zero-padding encoding exhibits poor generalization. Therefore, we can conclude that, given any number N of devices and the maximum number M of possible branch paths in the connection structure, any connection structure under these conditions can be represented by an M*N matrix.

[0106] Other embodiments of the present invention will readily occur to those skilled in the art after considering the specification and practicing the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the present invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not invented herein. The description and examples are to be considered as exemplary only, with the true scope and spirit of the invention being indicated by the following claims.

[0107] It should be understood that the present invention is not limited to the exact construction described above and shown in the drawings, and that various modifications and changes may be made without departing from the scope thereof. The scope of the present invention is limited only by the appended claims.

Claims

1. A method for implementing a simulated optical link topology structure based on deep learning, characterized in that: The following steps are involved: S1. Different photonic devices or sub-links are abstracted into sub-modules with specific transmission coefficients. The overall target output response is considered to be the superposition of the output responses of multiple optical paths from the input port to the output port. Each optical path is a random arrangement of different numbers of photonic devices, constructing a simulated optical link topology. S2. Encode the topology of the simulated optical link to obtain a unified parameterized matrix; In step S2, the component components are treated as single nodes according to the graph encoding method, and the parameterization matrix is ​​as follows: ; Each element in the parameterized matrix is ​​0 or 1, indicating whether the unit device represented by the column index exists in the optical path represented by the row. When all branch devices are evenly distributed, any connection structure is ultimately equivalent to an M*N matrix, where N is the number of devices and M is the maximum number of branches in the connection structure. The total transmission coefficient of the simulated optical link topology can be expressed as: ; Where, represents the transmission coefficient of a unit device, where only The corresponding matrix elements of participate in the formula operation; S3. Genetic algorithm is used to search for the topology closest to the target response. The forward network model is used as a fast link topology output performance evaluation module to improve the search efficiency of the optimization algorithm.

2. The method for implementing a simulated optical link topology structure based on deep learning according to claim 1, wherein: The optical link topology includes multiple different photonic devices and multiple branch splitters and combiners. Different photonic devices are replaced by single micro-ring structures with different and fixed physical parameters. The different topological structures formed have two input and output ports as a whole, and the internal components are freely combined to form different device connection structures with different output responses.

3. The method for implementing a simulated optical link topology structure based on deep learning according to claim 1, wherein: In step S3, S31. Initialize the population size and treat the parameterized matrix p as an individual in the population; S32. Set the number of iterations; if the number of iterations reaches the set threshold and no individual meeting the conditions appears, reinitialize the population and search for the global optimal solution again; S33. Calculate individual fitness using absolute error; S34. Determine the size of the absolute error and the error threshold. If the absolute error is less than the error threshold, output the individual parameters and end this iteration; if the error is not less than the error threshold, use the genetic operator to process, generate a new population, and repeat S32-S34.

4. The method for implementing a simulated optical link topology structure based on deep learning according to claim 3, wherein: In step S32, based on topological coding, a forward network model between the input structure parameters and the corresponding output responses is established, and the network model used is a fully connected network; The loss function for network training is the mean squared error between the true target response and the response predicted by the forward network: ; in, is the target response to the input, is the predicted response of the output structure, m is the number of samples; During the network training process, the network will calculate the loss size of the input data samples, and then use the backpropagation method to optimize and adjust the weights and biases of each layer in the neural network, and obtain a network model with minimum error through multiple rounds of iterations.

5. A system for implementing a simulated optical link topology structure based on deep learning, utilizing the method for implementing a simulated optical link topology structure based on deep learning according to any one of claims 1 to 4, characterized in that: It includes data acquisition module, data processing module and output module; Data acquisition module: collects data information of optical link topology; Data processing module: This module encodes the topology of the simulated optical link to obtain a unified parameterized matrix. It uses a genetic algorithm to search for the topology closest to the target response. The forward network model serves as a fast link topology output performance evaluation module to improve the search efficiency of the optimization algorithm. Output module: outputs optimal topology parameters.

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