A dynamic constraint maximum entropy distribution estimation optimization method for a complex lithography process landscape

Abstract

This invention provides a dynamic constraint maximum entropy distribution estimation optimization method for complex landscapes in photolithography processes, comprising the following steps: Step 1, constructing a nonlinear photolithography process response model including threshold effects and depth-of-focus attenuation to verify the effectiveness of candidate parameter combinations; Step 2, during the iterative search process, collecting feedback from the current sampling population and extracting local state indicators to characterize the landscape features of the current parameter space; Step 3, performing corresponding operations for different distributions. This invention introduces a local landscape feature perception mechanism, actively abandoning unsuitable second-order constraints in non-smooth regions and selecting reliable constraints based on currently known information to generate an unbiased distribution. This mechanism breaks the structural limitations of a single probability model, enabling the sampling distribution to truly fit the topological features of different local landforms, effectively avoiding blind convergence or stagnation caused by incorrect prior assumptions, and improving the optimization success rate within the distorted process window.

Description

Technical Field

[0001] This invention belongs to the fields of artificial intelligence, semiconductor manufacturing process control and computational lithography, and particularly relates to a dynamic constraint maximum entropy distribution estimation optimization method for complex landscapes in lithography processes. Background Technology

[0002] Photolithography is the most critical and costly step in semiconductor manufacturing. Its parameters (such as exposure dose, focal length shift, and development time) directly determine the linewidth accuracy and final yield of the chip. As integrated circuit manufacturing processes advance to advanced nodes, the photolithography process window becomes extremely narrow, and the relationship between parameters and linewidth errors exhibits highly nonlinear and strong coupling characteristics.

[0003] In existing technologies, parameter optimization for semiconductor processes such as photolithography widely employs search strategies based on fixed Gaussian priors (such as Bayesian optimization based on Gaussian process regression or standard Gaussian evolution strategies). These methods presuppose, at their underlying logic, strong prior assumptions that the local landscape possesses symmetry and unimodality.

[0004] However, the actual photolithography process suffers from problems such as the "threshold effect" and "asymmetric decay." For example, the exposure and formation of photoresist requires crossing a specific energy threshold. Once the parameter combination deviates slightly from the critical point, the linewidth error will experience a precipitous drop. In the oversaturated region, however, changes in parameters have almost no impact on the result. This complex terrain causes traditional Gaussian optimization algorithms that rely on second-order statistical moments to suffer from severe "variance shrinkage" due to extreme error penalties when facing critical abrupt changes, leading to a deadlock in the search. In flat dead zones, they face gradient vanishing and search stagnation.

[0005] Existing methods generally lack the ability to perceive the topological structure of discontinuous semiconductor process landscapes, forcing the use of a single smooth probability model to fit complex physical cliffs, which makes the search in critical regions extremely prone to failure. Therefore, how to construct a structured intelligent optimization method that can achieve high-precision convergence and overcome process dead zones through directional transitions under the condition of "non-stationary and discontinuous" process abrupt changes has become a key technical problem that urgently needs to be solved in this field. Summary of the Invention

[0006] Purpose of the invention: The technical problem to be solved by the present invention is to provide a dynamic constraint maximum entropy distribution estimation optimization method for the complex landscape of photolithography processes, in order to solve the problems of local deadlock and variance shrinkage that traditional Gaussian algorithms get into when facing critical process changes.

[0007] Specifically, in real photolithography processes, due to the nonlinearity of optical interference and photoresist chemical reactions, the performance evaluation space naturally exhibits characteristics of being narrow, asymmetric, and fragmented across multiple modes. Existing methods struggle to adapt to asymmetric process windows when dealing with objective functions exhibiting drastic changes and large dead zones. Furthermore, the traditional Gaussian fixed-symmetric search structure cannot guarantee an effective balance between local utilization and global exploration.

[0008] Therefore, the core technical problem of this invention lies in how to construct a lithography proxy evaluation model with real physical properties to perceive the topological features of the local landscape in real time, and design a dynamic statistical constraint switching mechanism based on the maximum entropy principle. By generating maximum entropy distributions under different constraints, the distribution estimation algorithm is endowed with directional transition capabilities in the "cliff" region and entropy increase exploration capabilities in the "platform" region, thereby achieving efficient and robust optimization of lithography parameters.

[0009] The method includes the following steps:

[0010] Step 1: Construct a nonlinear lithography process response model that includes threshold effect and depth-of-focus attenuation to verify candidate parameter combinations. The effectiveness, where E is the exposure dose, F is the focal length shift, and T is the development time;

[0011] Step 2: During the iterative search process, collect feedback from the current sampled population and extract local state indicators to characterize the landscape features of the current parameter space;

[0012] Step 3: Perform the corresponding operations for different distributions.

[0013] Step 1 includes: defining the effective exposure energy decay function. for:

[0014] ,

[0015] Where E is the exposure dose, an input parameter characterizing the energy intensity irradiated onto the photoresist during photolithography, measured in units of... F is the input parameter, representing the deviation of the exposure position from the optimal focal plane, in units of... ; This is the depth-of-focus attenuation coefficient, a system sensitivity parameter used to adjust the intensity of energy suppression by the amount of defocus. A recommended value range is [insert range here]. ; exp represents the natural exponential function.

[0016] Step 1 also includes: introducing the critical threshold effect of the photoresist chemical reaction to construct... Mutation function :

[0017] ,

[0018] in, This is the response value of the photoresist, with a range of values. , characterizing the chemical reaction conversion rate under specific exposure conditions; k is the threshold steepness, controlling the speed of the photoresist transitioning from "unreacted" to "fully reacted," with a recommended value of . The larger the value, the more pronounced the simulated "cliff" effect; It is the effective exposure energy; It is the critical exposure threshold, and its value is related to the effective exposure energy. The dimensions are kept consistent and determined based on the photoresist material, exposure dose unit, and process calibration results;

[0019] Calculate the actual generated linewidth based on the development time. And minimize the linewidth error as the single objective optimization function. :

[0020] .

[0021] in, It is a linewidth calculation model, defined as follows: ;

[0022] T is the development time, which is the time variable of the development process, and its unit is seconds (s). It is a reference constant that characterizes the initial feature size when there is no developing reaction; It is the response weighting coefficient, which characterizes the physical contribution of the degree of reaction and development time to the final linewidth; The target linewidth represents the ideal feature size required for chip design. It is an optimization objective function, whose output is the absolute error value between the final line width and the target line width. The optimization objective is to find the optimal exposure dose E, focal length offset F, and development time T through a search algorithm, and to find the parameter combination that minimizes the objective function value within the feasible region.

[0023] Step 2 includes: Since the response landscape of photolithography process parameters typically exhibits plateau regions, threshold abrupt change regions, potential multi-peak regions, and smooth process window regions, using a fixed single Gaussian distribution can easily introduce excessively strong mean and second-order moment priors when current statistical information is insufficient. Based on the maximum entropy principle, this invention determines which statistical information is reliably instructive at the current stage based on the feedback from the sampled population, and only converts reliable statistical information into distribution constraints to generate a sampling distribution with a smaller bias under the condition of satisfying the currently known information.

[0024] Landscape features include fitness and distance correlation, elite dispersion ratio, mean migration amplitude, and mean directional stability. Furthermore, these landscape features can be combined with auxiliary statistics from classical fitness landscape analysis and conventional statistical analysis, such as population performance variation coefficient, boundary touch rate, skewness, and kurtosis, to supplement the characterization of the current local response landscape state.

[0025] The fitness-distance correlation, elite dispersion ratio, mean migration amplitude, and mean directional stability indices are all calculated from the candidate lithography process parameter combinations of the sampled population and their corresponding objective function evaluation results during the optimization process. Specifically, fitness-distance correlation is characterized using the fitness-distance correlation coefficient (FDC) from classic fitness landscape analysis (FLA); elite dispersion ratio is characterized by the statistical form of pairwise distances between samples; mean migration amplitude is characterized by the change in the distribution center of elite samples in adjacent iterations; and mean directional stability is calculated based on the idea of ​​dimension-wise sign consistency detection and continuous iteration direction accumulation, according to the change in the migration sign of the elite sample distribution center in each parameter dimension during continuous iterations.

[0026] Calculate the correlation between fitness and distance: Fitness and distance in generation t Used to evaluate the topology and optimization difficulty of the search landscape:

[0027] ,

[0028] in Let i be the objective function value corresponding to the i-th individual; It is a very small positive number, used to prevent the denominator from being zero and to enhance the stability of numerical calculations; Let be the Euclidean distance between the i-th individual and the optimal candidate combination of photolithography process parameters in the current population; The average search distance of the t-th generation population; The average objective function value is N; N is the total number of individuals.

[0029] The elite sample set of generation t Relative to the complete population of generation t Elite discrete ratio Defined as:

[0030] ,

[0031] Wherein, the set of arbitrary samples in generation t Dimensional normalized average pairwise distance Defined as:

[0032] ,

[0033] in, Size of the sample set; Represents a set The Middle candidate parameter sample points and the b-th candidate parameter sample point The Euclidean distance between them; D represents the total number of dimensions of the candidate lithography process parameters;

[0034] Mean migration magnitude in generation t for:

[0035] ,

[0036]

[0037] in, Let be the standard deviation of the r-th dimension in generation t; and let be the center of the elite sample distribution in generation t. The value in the r-th dimension is M represents the total number of samples in the current generation t elite sample set; This represents the parameter value of the i-th sample in the r-th dimension within the t-th generation elite sample set;

[0038] Mean direction stability This is used to evaluate the consistency of the population's evolutionary direction over the most recent L generations. To avoid small perturbations and search stagnation being misjudged as valid directions, the normalized mean migration of the r-th parameter dimension in the t-th generation is first defined. :

[0039] ,

[0040] in, This represents the sampling standard deviation of this dimension in generation t;

[0041] Based on the set migration threshold The normalized mean transport is transformed into a discrete directional sign. :

[0042] ,

[0043] Based on this, within an observation window of length L, the directional consistency of the r-th parameter dimension Defined as:

[0044] ,

[0045] in, The indicator function is considered to have maintained effective directional consistency within the interval if and only if the migration direction signs of two adjacent generations are the same and both are not 0 (i.e., non-stagnant state).

[0046] Finally, by averaging across all parameter dimensions, the overall mean directional stability index for generation t is obtained. :

[0047] ,

[0048] This indicator The range of values ​​is The larger the value, the more significant the overall directional advancement trend of the system in the current stage.

[0049] Step 2 further includes: based on the landscape features, constructing a general distribution adaptive generation mechanism. First, the probability distribution of the parameter search space is defined as follows: probability distribution Information entropy Defined as:

[0050] ,

[0051] Where d represents the differential symbol;

[0052] The search strategy for each stage is determined by solving the following constrained maximum entropy optimization problem:

[0053] ,

[0054] ,

[0055] ,

[0056] Where min represents the optimization objective as finding a probability distribution that minimizes the global minimum of the objective function. In this maximum entropy framework, this means minimizing the negative value of entropy by optimizing the distribution parameters; m represents the number of constraints in the current sampled distribution. It is the probability density function value of the sample point under the current policy distribution; Indicates that it is subject to, This represents the feature mapping function corresponding to the q-th statistical constraint, used to characterize the properties of candidate parameter combinations in the lithography process landscape; This represents the empirical statistical value of the q-th statistical constraint calculated from the sampled population;

[0057] The m statistical constraints employ a dynamic adaptive activation mechanism, rather than a fixed activation. The specific activation logic is determined by the current local landscape features. The corresponding constraint is incorporated into the maximum entropy distribution solution process only when the local landscape features indicate that the q-th statistical constraint has reliable search guidance. The specific correspondence between local feature indicators and constraint conditions will be explained later.

[0058] Step 2 also includes: using the Lagrange multiplier method, constructing the following Lagrange functional:

[0059] ,

[0060] in, Represents the Lagrange functional. This represents the Lagrange multiplier corresponding to the normalization constraint (i.e., the probability integral is 1); This represents the Lagrange multiplier corresponding to the q-th statistical constraint;

[0061] right By taking the variational derivative and setting it to zero, the general maximum entropy distribution model is derived:

[0062] .

[0063] Step 3 includes: the different distributions include stagnant platform state, directional migration state, multi-region candidate state, and smooth basin state.

[0064] In step 3, the stagnant platform state includes: the maximum entropy distribution estimation algorithm iterating to the... The threshold logic for determining whether the current local landscape is in a stagnant platform state is as follows:

[0065] ,

[0066] in It is a low threshold for the correlation coefficient between fitness and distance, used to determine the lack of correlation between the objective function and the distance of the parameters; It is a high threshold for the proportion of elites, used to determine whether elite individuals are still in a highly discrete state in space; It is a low threshold for mean migration amplitude, used to determine whether the current population center is in a stagnant state; It is a low threshold for directional stability, used to determine the direction of advancement in the current search path where there is a lack of certainty.

[0067] When the above criteria are simultaneously met, the maximum entropy distribution estimation algorithm determines that reliable prior information on the center, scale, and orientation is currently lacking. At this point, all moment constraints in the maximum entropy optimization problem (i.e., ...) All items are set to invalid. According to the maximum entropy principle, with only boundary constraints and normalization constraints retained, the optimal probability density function degenerates into a uniform distribution:

[0068] ,

[0069] in, Indicates the combination of candidate photolithography process parameters The uniform probability density function; Represents the physical search space (or feasible region) for candidate parameters. This represents the hypervolume (or measure size) of the search space.

[0070] Execute the maximum entropy increase strategy to force a global restart in order to escape the flat region;

[0071] The directional migration state includes the following: During the optimization of photolithography process parameters, when the maximum entropy distribution estimation algorithm detects that the population is in a directional migration state (i.e., the population center is moving significantly along a defined evolutionary direction and has high consistency), it indicates that the current main reliable guiding information is the migration direction, rather than local variance. At this time, the maximum entropy distribution estimation algorithm should utilize the maximum entropy principle, extract only the mean as a constraint, and use an exponential distribution for directional sampling.

[0072] The threshold logic for determining whether the current local landscape is in a directional migration state when iterating to generation t is as follows:

[0073] ,

[0074] in It is a high threshold for mean migration magnitude, used to determine that the search center is in a rapid migration phase; It is a high threshold for directional stability, used to determine that the current evolution path has a highly deterministic directional inertia; when the determination criterion is met, the maximum entropy distribution estimation algorithm determines that the current search has entered a valid directional guidance region;

[0075] First, calculate the orientation scale vector of the t-th generation. And construct the search boundary in the r-th dimension:

[0076] ,

[0077] ,

[0078] in, Represents the global orientation scale vector of the t-th generation; Let represent the mean vector of the elite samples sampled by the exponential distribution in generation t; Let represent the optimal parameter vector in the t-th generation elite sample; Represents the direction scale vector The component in the r-th parameter dimension; This represents the value of the optimal parameter vector in the r-th dimension; These represent the left and right boundaries of the exponential distribution sampling, respectively. and The parameter is the physical search range, and c is the orientation scale adjustment coefficient, which determines the degree of trust that the maximum entropy distribution estimation algorithm has in the "past migration trends".

[0079] According to the scale The sign of the sampling offset determines the direction and starting point:

[0080] Negative promotion strategy, scale : by the left boundary Starting from a high-density base, exponential sampling is performed to the right:

[0081] ,

[0082] Positive advancement strategy, scale : by the right boundary Starting from a high-density base, exponential sampling is performed to the left:

[0083] ,

[0084] in, Let represent the value of the i-th candidate sample generated using the exponential distribution in the t-th generation on the r-th dimension; Indicates in The interval follows a standard uniform distribution Random numbers; The rate parameter representing the exponential distribution of the t-th generation on the r-th dimension is used to control the degree of sampling divergence along the propagation direction;

[0085] The multi-region candidate state includes: when it is determined that the current state is in a local dominant but structurally uncertain state, a generalized long-tailed distribution is introduced as an auxiliary sampling distribution;

[0086] The state of local advantage but structural uncertainty refers to a situation where the current elite sample set shows a certain concentration trend relative to the overall sampling population, indicating that the current sampling area has high utilization value; however, the fitness-distance correlation is insufficient, and the mean directional stability is also insufficient, indicating that the current search has not yet formed a stable direction of advancement that can be used for directional exponential distribution and a deterministic local landscape of a single basin that can be used for Gaussian distribution. Therefore, this state is distinct from the stagnant plateau state, the directional migration state, and the smooth basin state.

[0087] The threshold logic for determining whether the current local landscape is in a potential multi-peak state when iterating to generation t is as follows:

[0088] ,

[0089] in The threshold value in the elite dispersion ratio indicates that the current elite samples have shown a certain concentration trend, and the search area has preliminary utilization value. It is the threshold in fitness-distance correlation, indicating that the landscape has not yet formed a definite smooth basin structure; It is the threshold of directional stability, indicating that there is currently a lack of stable directional advancement trend; The low threshold of mean migration indicates that the search process is still evolving and is not a completely stagnant plateau.

[0090] To prevent the maximum entropy distribution estimation algorithm from prematurely getting stuck in a single basin due to the second-order moment constraint, a logarithmic distance moment constraint is introduced. This constraint allows sampling points to retain a certain probability of being distributed outside the center, enhancing global exploration capabilities. The constraint equation is defined as:

[0091] ,

[0092] in It is a combination of candidate photolithography process parameters The sampling probability density function; It is the location parameter of the long-tail distribution, and its value is determined by the current distribution center of the elite samples. It is the scaling parameter of the long-tailed distribution, used to adjust the convergence width of the probability density near the center; It is the empirical statistical value corresponding to the logarithmic distance moment constraint, calculated from the distribution characteristics of the current population; It is a logarithmic distance feature mapping function, which weakens the penalty for distance by using a logarithmic form, so that distant parameter points still have the possibility of being sampled;

[0093] Substituting the logarithmic distance feature function into the general maximum entropy distribution expression:

[0094] ,

[0095] Using the properties of the exponential function, it can be simplified to a generalized long-tailed distribution model:

[0096] ,

[0097] in, Represents the center of the sample distribution (or the mean vector); Lagrange multipliers for controlling the thickness of the distribution tail; Z is a smoothing parameter used to control the scale range of the logarithmic distance feature; Z is a normalization constant (or partition function) used to ensure that the integral of the probability density function over the entire search space is 1.

[0098] The smooth basin state includes: when the system determines that the current local landscape is in a smooth basin state, the high-confidence mean and second moment are extracted as joint constraints, and substituted into the general model to derive that the system strictly degenerates into a Gaussian distribution, as shown in the formula:

[0099] ,

[0100] in, The probability density function represents a one-dimensional Gaussian distribution; This represents the mean of a local Gaussian distribution; This represents the standard deviation of a local Gaussian distribution;

[0101] The threshold logic for determining whether the current local landscape is in a smooth basin state when iterating to generation t is as follows:

[0102] ,

[0103] in A high threshold for the correlation coefficient between fitness and distance indicates that a stable negative correlation has been established between the objective function value and the parameter distance, with a clear descent guidance; The low threshold of the elite discrete proportion indicates that the elite samples have been highly clustered near a single optimal basin; the Gaussian distribution utilizes strong local continuity to achieve fine-tuning convergence within the process window.

[0104] The present invention also provides an electronic device, including a processor and a memory, the memory storing program code that, when executed by the processor, causes the processor to perform the steps of the method.

[0105] The present invention also provides a storage medium storing a computer program or instructions that, when the computer program or instructions are run on a computer, execute the steps of the method described.

[0106] Beneficial effects: Compared with the current traditional Gaussian evolutionary optimization strategy, this invention, by introducing local terrain perception and dynamic constraint maximum entropy mechanism, achieves the following technical effects in discontinuous and highly nonlinear semiconductor manufacturing scenarios:

[0107] First, this invention overcomes the inherent flaw of traditional Gaussian distribution models, which suffer from excessive local prior assumptions. Traditional distribution estimation algorithms rely heavily on second-order statistical moments, implicitly injecting a strong prior assumption of "symmetric and unimodal local terrain" into the system. This is severely inconsistent with the complex terrain encountered in actual photolithography processes, such as asymmetric narrow windows and critical cliffs. This invention introduces a local landscape feature perception mechanism, actively abandoning unsuitable second-order constraints in non-smooth regions and selecting reliable constraints based on currently known information to generate an unbiased distribution. This mechanism breaks the structural limitations of a single probability model, enabling the sampling distribution to truly fit the topological features of different local landforms. This effectively avoids blind convergence or stagnation caused by incorrect prior assumptions, improving the optimization success rate within distorted process windows.

[0108] Secondly, an adaptive constraint selection mechanism based on information theory is provided. Traditional optimization methods often impose fixed statistical constraints, which are prone to introducing erroneous biases in complex terrains. This invention transforms local landscape features into confidence assessments of statistical information, adaptively selecting reliable constraints to solve for the most unbiased distribution. This mechanism rigorously unifies the algorithm's exploration and utilization behaviors within the dynamic adjustment of information constraints, greatly enhancing the system's adaptability to unknown and harsh industrial environments from the underlying mathematical logic.

[0109] Furthermore, this invention replaces the extremely cumbersome full-process physical simulation with a parameterizable proxy model that includes threshold effects and interference characteristics. This not only accurately characterizes the dynamic and harsh features of the lithography parameter space, but also reduces the computational overhead of the entire evaluation and decision-making process by several orders of magnitude. It provides a lightweight and energy-efficient technical solution with great industrial application value for the automated parameter tuning of advanced semiconductor processes. Attached Figure Description

[0110] Figure 1 This is a flowchart of the method of the present invention.

[0111] Figure 2 This is a schematic diagram of the photolithographic landscape in this embodiment.

[0112] Figure 3 This is a diagram illustrating the optimized distribution ratio at each stage.

[0113] Figure 4 This is a diagram showing the comparison of optimization results. Detailed Implementation

[0114] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments, and the advantages of the present invention in the above and / or other aspects will become clearer.

[0115] This invention provides a dynamic constraint maximum entropy distribution estimation optimization method for complex landscapes in photolithography processes. This method constructs a hybrid distribution optimization framework based on dynamic constraint maximum entropy. This framework is based on a distribution estimation algorithm, provides environmental feedback through a physically inspired simplified process response model, identifies terrain using local statistical higher-order moments, and dynamically changes the constraints for maximum entropy solving using the Lagrange multiplier method, thereby adaptively generating an exploratory distribution.

[0116] First, a nonlinear lithography process response model incorporating threshold effect and depth-of-focus attenuation is constructed to validate candidate parameter combinations. The effectiveness of (where E is the exposure dose, F is the focal length shift, and T is the development time).

[0117] The effective exposure energy decay function is defined as follows:

[0118] ,

[0119] Introducing the critical threshold effect of photoresist chemical reactions, constructing Mutation function:

[0120] ,

[0121] Calculate the actual generated linewidth based on the development time. And the single objective optimization function is to minimize the linewidth error:

[0122] ,

[0123] The model mathematically reproduces the narrow process window, plateau period, and nonlinear critical cliff in real photolithography.

[0124] Based on this, during the iterative search process, the system collects feedback from the current sampled population and extracts local state indicators to characterize the landscape features of the current parameter space (fitness and distance relationship, population dispersion, population performance variation coefficient, and higher-order statistical moments, etc.).

[0125] Based on the aforementioned landscape features, this invention constructs a general distribution adaptive generation mechanism. First, the probability distribution of the parameter search space is defined as follows: Its information entropy is defined as:

[0126] .

[0127] To satisfy known process characteristics without introducing any unfounded prior biases, this invention determines the search strategy for each stage by solving the following constrained maximum entropy optimization problem:

[0128] ,

[0129] ,

[0130] ,

[0131] in, This represents the statistical feature function extracted by the system from the local landscape. These are the corresponding empirical statistical values.

[0132] Using the Lagrange multiplier method, the following Lagrange functional is constructed:

[0133] ,

[0134] right Taking the variational derivative and setting it to zero, we can derive the general maximum entropy distribution model:

[0135] ,

[0136] in, These are undetermined multipliers determined by known statistical information.

[0137] Specific different distributions:

[0138] (1) Stagnant platform status:

[0139] When the system determines that it is currently in an information-scarce dead zone, all statistical moment constraints are removed, leaving only normalization and boundary constraints. Substituting these into the general model, the system degenerates into a uniform distribution, as shown in the formula:

[0140] ,

[0141] This executes the maximum entropy increase strategy, forcibly triggering a global restart to escape the flat region.

[0142] (2) Critical cliff and abrupt change state:

[0143] When the system detects a drastic change in terrain that causes the traditional second-order variance to fail, it actively discards the variance constraint.

[0144] If only the mean is extracted as a constraint, the system solution is an exponential distribution:

[0145] ,

[0146] If the logarithmic moment is introduced as a constraint, the system will derive a Cauchy distribution or a long-tailed distribution with heavy-tailed characteristics:

[0147] ,

[0148] These two types of distributions, with their asymmetric or long-tail characteristics, endow the algorithm with the ability to leap across the cliff dead zone.

[0149] (3) Smooth basin condition:

[0150] When the system is determined to have good local continuity, the mean and second moment with high confidence are extracted as joint constraints. Substituting them into the general model, it can be derived that the system strictly degenerates into a Gaussian distribution, as shown in the formula:

[0151] ,

[0152] Gaussian distribution utilizes its strong local continuity to achieve high-precision fine-tuning convergence within the process window.

[0153] This invention provides a dynamic constraint maximum entropy distribution estimation optimization method for complex landscapes in photolithography processes. Many methods and approaches exist for implementing this technical solution; the above description is merely a preferred embodiment of the invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.

[0154] To further illustrate the feasibility of the method described in this invention, a specific embodiment based on three-dimensional lithography process parameters is given below. This embodiment is used to illustrate the execution process of this invention in a lithography proxy landscape with threshold effects, depth-of-focus attenuation, local perturbations, and potential multi-peak response characteristics, and does not limit the form of the lithography evaluation model used in actual production. In other embodiments, the objective function can be provided by lithography simulation software, actual process measurement data, machine learning proxy models, or multi-index weighted evaluation models.

[0155] In this embodiment, the candidate photolithography process parameter combination is as follows:

[0156] ,

[0157] Where E represents exposure dose, F represents focus shift, and T represents development time. The search ranges for the three parameters are as follows:

[0158] ,

[0159] The exposure dose E ranges from 20 to 80, the focal length offset F ranges from -2 to 2, and the development time T ranges from 30 to 90.

[0160] I. Construction of a photolithography proxy evaluation model;

[0161] This embodiment first constructs a physically-inspired lithography proxy evaluation model to simulate the effects of exposure dose, focal length offset, and development time on critical dimensions.

[0162] The formula for calculating effective exposure energy is:

[0163] ,

[0164] in, This represents the effective exposure energy under the combined effect of exposure dose E and focal length shift F. This represents the depth-of-focus attenuation coefficient.

[0165] The formula for calculating the degree of photoresist reaction is:

[0166] ,

[0167] in, This indicates the degree of photoresist reaction, and k represents the threshold steepness coefficient. This indicates the critical exposure threshold.

[0168] The formula for predicting critical dimensions is:

[0169] ,

[0170] in, Indicates the predicted critical dimensions. Indicates the base line width parameter. This represents the coefficient indicating the influence of the photoresist reaction degree on critical dimensions. This represents the coefficient indicating the influence of development time on critical dimensions. This indicates a process disturbance term.

[0171] In this embodiment, in order to simulate the local non-smooth response caused by standing wave effect, focal depth window variation, and development loading fluctuation in actual photolithography process, the process perturbation term... By standing wave disturbance term Depth of focus window perturbation item and developing loading perturbation terms composition:

[0172] ,

[0173] Among them, the threshold sensitivity factor is defined. for:

[0174] ,

[0175] when When the value is close to 0 or 1, it indicates underexposure or a saturated reaction area; this is the threshold sensitivity factor. Smaller; when A value close to 0.5 indicates the area is in the critical exposure region, which is the threshold sensitivity factor. Larger.

[0176] The standing wave disturbance term is:

[0177] ,

[0178] The depth-of-focus window perturbation term is:

[0179] ,

[0180] The development loading perturbation term is:

[0181] ,

[0182] in, This represents the standing wave disturbance term. This represents the depth-of-focus window perturbation term. This indicates the development loading disturbance term; These represent the amplitude coefficients of the three types of disturbance terms, respectively; These represent the perturbation period parameters in the exposure dose direction, focus offset direction, and development time direction, respectively; and These represent the window width parameters in the focal length offset direction and the exposure dose direction, respectively; This indicates the center parameter of the exposure dose.

[0183] The objective function is defined as:

[0184] ,

[0185] in, Indicates the target's critical dimensions. This represents the critical dimension error corresponding to the candidate lithography process parameter combination x. The objective is to minimize... .

[0186] In one specific embodiment, the model parameters are taken as follows: focal depth attenuation coefficient Threshold steepness coefficient Critical exposure threshold Basic line width parameters Photoresist reaction influence coefficient Development time influence coefficient Target critical dimensions ;

[0187] The values ​​for the disturbance term parameters are as follows:

[0188] Standing wave disturbance amplitude coefficient Exposure dose perturbation period Focal length offset window width Focal depth window perturbation amplitude coefficient Focal length shift disturbance period Exposure dose center parameters Exposure dose window width The amplitude coefficient of the loading disturbance during development Development time disturbance period ;

[0189] The above parameters are used to construct a physically-inspired lithography proxy evaluation model. It should be noted that this embodiment does not arbitrarily construct the objective function, but rather simplifies the modeling based on common physical response characteristics in lithography processes. Specifically, the exposure dose E and focal length offset F jointly determine the effective exposure energy; focal length offset leads to an attenuation of effective energy. Photoresist reactions typically have a critical exposure threshold; therefore, when the effective exposure energy approaches the threshold... At certain times, the response of critical dimensions changes rapidly; however, in areas of underexposure or reaction saturation, different combinations of candidate parameters may produce similar objective function values, thus exhibiting plateau characteristics. Furthermore, this embodiment simulates the local periodic fluctuations, depth-of-focus window changes, and development loading fluctuations that may occur during actual photolithography by using standing wave perturbation terms, depth-of-focus window perturbation terms, and development loading perturbation terms, giving the proxy response landscape local non-smoothness and potential multi-peak characteristics.

[0190] Therefore, the surrogate evaluation model in this embodiment is used to approximately characterize common features in real photolithography parameter tuning processes, such as narrow process windows, threshold response, depth-of-focus sensitivity, plateau regions, local non-smoothness, and potential multi-peak characteristics. This surrogate model is only used as a specific evaluation module to illustrate the implementability of the present invention; in practical applications, the objective function can also be provided by commercial photolithography simulation software, wafer experimental measurement data, process databases, or machine learning surrogate models.

[0191] The objective function landscape obtained from the above model can be displayed through the 2D objective function heatmap in the attached figure, for example. Figure 2 As shown. In Figure 2 In the figure, the fixed development time T is the development time corresponding to the final optimal candidate combination of lithography process parameters. The distribution of the objective function value on the plane of exposure dose E and focal length offset F is plotted, and the position of the final optimal candidate combination of lithography process parameters is marked. This figure is used to illustrate that the evaluation model in this embodiment can present response characteristics such as local plateaus, threshold transitions, and local excellent regions consistent with the lithography process parameter tuning.

[0192] II. Set optimization parameters;

[0193] This embodiment uses two methods for comparison:

[0194] The first method is the fixed Gaussian cross-entropy method (CEM), which uses a Gaussian distribution for sampling, elite selection, and distribution parameter updates in each generation.

[0195] The second method is the dynamic hybrid maximum entropy distribution estimation optimization method described in this invention. This method calculates local landscape features for each generation based on the sampled population and dynamically adjusts the sampling weights among uniform distribution, directional exponential distribution, generalized long-tailed distribution, and Gaussian distribution.

[0196] The main optimized parameter settings in this embodiment are as follows:

[0197] ,

[0198] Where N represents the size of the sampled population in each generation, and G represents the maximum number of iterations. Indicates the proportion of elites.

[0199] The number of elite samples in the exponential distribution is: ;

[0200] The length of the observation window for directional consistency is: ;

[0201] The initial mean was set to the center of the search range, and the initial search standard deviation was set to one-fifth of the search range length. The random seed was set to 42 to ensure the reproducibility of the results in this embodiment.

[0202] III. Local landscape features obtained from each generation of calculations;

[0203] In each iteration, the system evaluates the current candidate combinations of lithography process parameters and calculates the following four local landscape features:

[0204] Fitness and distance correlation in generation t This is used to characterize the correlation between the objective function value and the distance from the candidate parameters to the current optimal candidate parameters;

[0205] Elite dispersion ratio of generation t It is used to characterize the degree of concentration of the elite sample set relative to the overall sampled population;

[0206] Mean migration magnitude in generation t It is used to characterize the magnitude of the shift of the elite sample distribution center relative to the current search scale between adjacent iterations;

[0207] Stability of the overall mean direction in generation t It is used to characterize the stability of the dimensional migration direction of the elite sample distribution center in consecutive iterations.

[0208] IV. Switch the sampling distribution according to the local landscape condition;

[0209] In this embodiment, the system classifies the current response landscape into four states based on four local landscape indicators:

[0210] (1) Stagnant platform status

[0211] When the fitness of generation t is correlated with distance Lower, elite dispersion ratio Higher mean migration Low and mean-directed stability When the value is low, the system determines that it is currently in a stagnant plateau state. This state indicates that the objective function values ​​in the current sampling region are not sufficiently distinguishable, and a reliable local center or stable direction of propagation has not yet been formed. The first-order and second-order moment constraints are unreliable, corresponding to only a uniform distribution of boundary and normalized constraints.

[0212] In this state, the system increases the uniformly distributed sampling weights to expand the exploration range and avoid premature contraction in areas with insufficient information.

[0213] (2) Directional migration state

[0214] when higher and When the value is high, the system determines that it is currently in a directional migration state. This state indicates that the center of the elite sample distribution has maintained stable migration along several parameter dimensions over multiple consecutive iterations, meaning that only the first-order moment constraint is reliable, while the second-order moment constraint is weakened in the current local state. Correspondingly, in this state, the system increases the sampling weight of the directional exponential distribution. The directional exponential distribution does not directly impose a one-dimensional exponential distribution on the entire candidate lithography process parameter combination, but rather calculates the directional scale based on the elite sample set of the exponential distribution.

[0215] ,

[0216] in, Let c represent the orientation scale vector of generation t, and let c represent the orientation scale adjustment coefficient. This represents the center of the elite sample distribution in the exponential distribution of generation t. This represents the current optimal candidate combination of lithography process parameters for the t-th generation exponential distribution.

[0217] when When the system has a high-density exponential distribution starting point on the left boundary in the r-th parameter dimension, the tail extends to the right; when At this point, the system uses the right boundary as the starting point for an exponential high-density distribution, with the tail extending to the left. This mechanism ensures that sampling points are distributed more extensively outside the current optimal candidate point, while retaining the possibility of retreating to the elite sample distribution area.

[0218] (3) Potential multi-peak state

[0219] when The data shows that the elite sample already exhibits a certain concentration trend, but and If it is not sufficient to indicate that the current region has formed a smooth basin or a stable direction, the system determines that it is currently in a potential multimodal state. The second moment is still not completely reliable. Therefore, a logarithmic distance constraint that penalizes the slower growth of distant samples is adopted, corresponding to the generalized long-tailed distribution of the first moment and logarithmic moment constraints.

[0220] This state indicates that the current location has shown some advantages, but the possibility of potentially better process windows in other areas cannot be ruled out. Compared to a uniform distribution, this state does not allow for complete abandonment of the current area; compared to a directional exponential distribution, this state has not yet formed a definite direction; and compared to a Gaussian distribution, this state has not yet formed a reliable local basin. Therefore, the system introduces a generalized long-tailed distribution to enhance long-distance exploration capabilities while retaining the sampling probability of the current better area.

[0221] The generalized long-tail distribution can be represented as:

[0222] ,

[0223] in, This represents the generalized long-tail sampling probability density function. This represents the location parameters of the long-tail distribution. This represents the scale parameter of the long-tail distribution. This represents the tail decay parameter.

[0224] (4) Smooth basin state

[0225] when higher lower and When the value is low or slightly below medium, the system determines that it is currently in a smooth basin state. This state indicates that the elite samples have formed a stable concentrated region, there is a relatively clear structural relationship between the objective function value and the search distance, the migration amplitude of the elite sample distribution center is reduced, and the first moment (mean) and second moment (variance) of the local state information are reliable, corresponding to a Gaussian distribution constrained by the first moment and second moment.

[0226] In this state, the system increases the sampling weights of the Gaussian distribution and performs local fine-tuning convergence.

[0227] The process of the four types of states changing with iteration can be shown on the state-time axis in the attached figure, such as... Figure 3 As shown. Figure 3 The horizontal axis represents the iteration number, and the vertical axis represents the state number or state name, used to indicate the main sampling state used in each generation.

[0228] V. Specific operation process and intermediate results;

[0229] The flowchart of this method is as follows: Figure 1 As shown. In one run of this embodiment, the system is optimized with a sampling budget of 40 generations and 100 candidate samples per generation. The Dynamic Mixture Maximum Entropy Distribution Estimation Optimization Method and the Fixed Gaussian Cross-Entropy Method (CEM) use the same initial mean, initial standard deviation, search range, and evaluation budget.

[0230] In generation 0, the system generates 100 candidate combinations of lithography process parameters and calculates the predicted critical dimension and target error corresponding to each candidate parameter using a lithography proxy evaluation model. This generation calculates the current optimal target error. for:

[0231] ,

[0232] According to the indicator rules, the system determines that the local structural information in the initial sampling area is still insufficient, and a reliable local center, stable advancement direction, or smooth basin structure has not yet formed. Therefore, the system retains a high exploration ratio in subsequent sampling to avoid prematurely shrinking to a single local area. The output of this step includes the set of candidate lithography process parameters for generation 0, the objective function value corresponding to each candidate parameter, the current optimal combination of candidate lithography process parameters, four local landscape indicators, and the state determination result.

[0233] Based on the average migration amplitude of the third generation and overall mean direction stability index The system determines that the current search process has shown a certain directional migration trend. Therefore, the system increases the sampling ratio of the directional exponential distribution and calculates the directional scale based on the elite sample set of the exponential distribution to generate the next generation of directional exponential sampling candidate parameters. This step reflects that when directional information begins to emerge, the present invention no longer relies entirely on symmetric Gaussian sampling, but increases the sampling probability near the effective migration direction.

[0234] In subsequent iterations, the system continues to use the fitness-distance correlation coefficient of generation t. Elite dispersion ratio Mean migration magnitude and overall mean direction stability index The system adapts to the changes in the state, switching between stagnant platform, directional migration, potential multi-peak, and smooth basin states. Specifically, a uniform distribution is used to maintain search coverage during the information-scarce phase; a directional exponential distribution is used to enhance directional sampling during the directional migration phase; a generalized long-tail distribution is used to maintain long-distance exploration capabilities in the potential multi-peak state; and a Gaussian distribution is used for local refinement in the smooth basin state.

[0235] In the 31st generation, the system obtains candidate lithography process parameter combinations through Gaussian distribution sampling under smooth basin conditions:

[0236] ,

[0237] The candidate parameter corresponds to the predicted key size: ;

[0238] The corresponding target error is: ;

[0239] This result represents the best outcome in this 40-generation iteration. Therefore, it can be seen that in this embodiment, the system, through a dynamic distribution switching process of "initial exploration—direction migration—potential multi-peak exploration—smoothing basin refinement," ultimately obtained a result close to the target critical size. The candidate lithography process parameter combinations.

[0240] The results indicate that, in the lithography proxy response landscape constructed in this embodiment, the method of the present invention can output a candidate combination of lithography process parameters close to the target critical size of 52.0.

[0241] The final convergence curve can be shown in the attached figure, such as... Figure 4 As shown (blue represents the maximum entropy method, which is superior to Gaussian alone). Figure 4 The horizontal axis represents the iterative algebra, and the vertical axis represents the historical best target error. The historical best error curves of the dynamic hybrid method and the fixed Gaussian cross-entropy method CEM are plotted respectively.

[0242] VI. Comparison with the fixed Gaussian cross-entropy method;

[0243] To illustrate the effectiveness of the method under a limited evaluation budget, this embodiment compares the dynamic mixed maximum entropy distribution estimation optimization method with the fixed Gaussian cross-entropy method (CEM). Both methods employ the same lithographic surrogate evaluation model, the same search range, the same initial distribution, the same number of samples per generation, and the same number of iterations.

[0244] In this embodiment, the sampled population size is 100 per generation, and the maximum number of iterations is 40. Therefore, the objective function evaluation budget for each independent run is:

[0245] (Second-rate),

[0246] To avoid the results of a single run being affected by random initialization or early sampling fluctuations, this embodiment uses continuous random seeds for repeated verification. The set of random seeds is as follows:

[0247] ,

[0248] Under the same evaluation budget, the dynamic hybrid maximum entropy distribution estimation optimization method of this invention achieved a lower final target error than the fixed Gaussian cross-entropy method (CEM) in 8 out of 10 independent runs, with a winning count of 8 / 10 and a win rate of 0.80. The statistical results of the 10 independent runs are shown in Table 1 below.

[0249] Table 1

[0250]

[0251] The above results show that, in the lithographic proxy response landscape constructed in this embodiment with threshold response, depth-of-focus attenuation, local perturbation and potential multi-peak characteristics, the method of the present invention achieves a lower average target error and a higher win rate under continuous random seed testing, indicating that it has good optimization stability under a limited evaluation budget.

[0252] Therefore, this invention enables adaptive sampling of different local landscape states in complex, non-smooth lithographic response landscapes, matching the sampling distribution with current reliable statistical information, thereby reducing the impact of erroneous prior assumptions on the search process and improving sampling efficiency, cross-random seed stability, and optimization reliability of target process parameters under limited evaluation budgets.

[0253] VII. The phased contributions of each sampling distribution;

[0254] In one run of this embodiment, the statistics of the best sample sources for each generation are as follows:

[0255] ,

[0256] Where gaussian represents Gaussian distribution sampling, exp represents directional exponential distribution sampling, tail represents generalized long-tailed distribution sampling, and uniform represents uniform distribution sampling.

[0257] The sampling sources that broke the historical global optimum include:

[0258] Generation 0: Gaussian distribution sampling;

[0259] Generation 1: Uniform distribution sampling;

[0260] 4th generation: Directional index distribution sampling;

[0261] Generation 33: Gaussian distribution sampling.

[0262] The results indicate that the uniform distribution plays an extended search role in the early stage when information is insufficient, the directional exponential distribution generates effective candidate parameters in the direction migration stage, the generalized long-tailed distribution maintains the cross-regional exploration capability in the potential multi-peak state, and the Gaussian distribution completes local refinement in the later smooth basin state.

[0263] VIII. Technical problems solved and technical effects achieved;

[0264] The practical technical problem addressed in this embodiment is that in the optimization of photolithography process parameters, parameters such as exposure dose, focal length offset, and development time jointly affect critical dimensions. The target response landscape may simultaneously possess plateau regions, critical threshold regions, potential multi-peak regions, and smooth window regions. If a fixed Gaussian distribution is used for sampling in all stages, it is easy to prematurely shrink in the information-insufficient stage, generate more reverse or low-contribution sampling in the direction migration stage, over-concentrate on the current local area in the potential multi-peak stage, or prematurely converge to local convergence before the smooth window has been formed.

[0265] The method of this invention calculates local landscape features in real time using sampled populations and dynamically selects the maximum entropy sampling distribution based on the local response landscape state, so that the sampling distribution matches the lithographic response landscape state. Specifically, a uniform distribution is used in the stagnant plateau state, a directional exponential distribution is used in the directional migration state, a generalized long-tailed distribution is used in the potentially multimodal state, and a Gaussian distribution is used in the smooth basin state.

[0266] Compared to the fixed Gaussian cross-entropy method, the superiority of this embodiment lies in the fact that it does not force the same symmetric distribution to be used in all search stages. Instead, it judges the current reliable constraints based on local landscape indicators and obtains the probability distribution that minimizes the bias of the unknown region under the current local state for next-generation sampling. Therefore, this invention can achieve adaptive sampling for different local landscape states in complex, non-smooth lithography response scenarios, matching the sampling distribution with the current reliable statistical information. This reduces the impact of erroneous prior assumptions on the search process and improves sampling efficiency, convergence stability, and the reliability of optimizing target process parameters under limited evaluation budgets.

[0267] This invention provides a dynamic constraint maximum entropy distribution estimation optimization method for complex landscapes in photolithography processes. Many methods and approaches exist for implementing this technical solution; the above description is merely a preferred embodiment of the invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.

Claims

1. A dynamic constraint maximum entropy distribution estimation and optimization method for complex landscapes in photolithography processes, characterized in that, Includes the following steps: Step 1: Construct a nonlinear lithography process response model that includes threshold effect and depth-of-focus attenuation to verify candidate parameter combinations. The effectiveness, where E is the exposure dose, F is the focal length shift, and T is the development time; Step 2: During the iterative search process, collect feedback from the current sampled population and extract local state indicators to characterize the landscape features of the current parameter space; Step 3: Perform the corresponding operations for different distributions.

2. The method according to claim 1, characterized in that, Step 1 includes: defining the effective exposure energy decay function. for: ， Where E is the exposure dose; is the depth-of-focus attenuation coefficient; exp represents the natural exponential function.

3. The method according to claim 2, characterized in that, Step 1 also includes: introducing the critical threshold effect of the photoresist chemical reaction to construct... Mutation function : ， in, This is the response value of the photoresist, with a range of values. k is the threshold steepness; It is the effective exposure energy; It is the critical exposure threshold; Calculate the actual generated linewidth based on the development time. And minimize the linewidth error as the single objective optimization function. : ， in, It is a linewidth calculation model, defined as follows: ; T is the development time; It is a reference constant; It is the response weighting coefficient; It is the target line width; The objective function is to find the optimal exposure dose E, focal length offset F, and development time T through a search algorithm, and to find the parameter combination that minimizes the objective function value within the feasible region.

4. The method according to claim 3, characterized in that, Step 2 includes: landscape features such as fitness and distance correlation, elite dispersion ratio, mean migration magnitude, and mean directional stability; Calculate the correlation between fitness and distance: Fitness and distance in generation t Used to evaluate the topology and optimization difficulty of the search landscape: ， in Let i be the objective function value corresponding to the i-th individual; It is a very small positive number; Let be the Euclidean distance between the i-th individual and the optimal candidate combination of photolithography process parameters in the current population; The average search distance of the t-th generation population; The average objective function value is N; N is the total number of individuals. The elite sample set of generation t Relative to the complete population of generation t Elite discrete ratio Defined as: ， Wherein, the set of arbitrary samples in generation t Dimensional normalized average pairwise distance Defined as: ， in, Size of the sample set; Represents a set The Middle candidate parameter sample points and the b-th candidate parameter sample point The Euclidean distance between them; D represents the total number of dimensions of the candidate lithography process parameters; Mean migration magnitude in generation t for: ， in, Let be the standard deviation of the r-th dimension in generation t; and let be the center of the elite sample distribution in generation t. The value in the r-th dimension is M represents the total number of samples in the current generation t elite sample set; This represents the parameter value of the i-th sample in the r-th dimension within the t-th generation elite sample set; Mean direction stability This is used to assess the consistency of the evolutionary direction of a population over the most recent L generations; firstly, the normalized mean migration of the r-th parameter dimension in the t-th generation is defined. : ， in, This represents the sampling standard deviation of this dimension in generation t; Based on the set migration threshold The normalized mean transport is transformed into a discrete directional sign. : ， Within an observation window of length L, the directional consistency of the r-th parameter dimension. Defined as: ， in, For indicator functions; Finally, by averaging across all parameter dimensions, we obtain the overall mean directional stability index for generation t. : 。 5. The method according to claim 4, characterized in that, Step 2 further includes: based on the landscape features, constructing a general distribution adaptive generation mechanism. First, the probability distribution of the parameter search space is defined as follows: probability distribution Information entropy Defined as: ， Where d represents the differential symbol; The search strategy for each stage is determined by solving the following constrained maximum entropy optimization problem: ， ， ， Where min represents the optimization objective as finding a probability distribution that makes the objective function reach its global minimum; m represents the number of constraints in the current sampling distribution; It is the probability density function value of the sample point under the current policy distribution; Indicates that it is subject to, This represents the feature mapping function corresponding to the q-th statistical constraint; This represents the empirical statistical value of the q-th statistical constraint calculated from the sampled population.

6. The method according to claim 5, characterized in that, Step 2 also includes: using the Lagrange multiplier method, constructing the following Lagrange functional: ， in, Represents the Lagrange functional. This represents the Lagrange multiplier corresponding to the normalization constraint; This represents the Lagrange multiplier corresponding to the q-th statistical constraint; right By taking the variational derivative and setting it to zero, the general maximum entropy distribution model is derived: 。 7. The method according to claim 6, characterized in that, Step 3 includes: the different distributions include stagnant platform state, directional migration state, multi-region candidate state, and smooth basin state.

8. The method according to claim 7, characterized in that, In step 3, the stagnant platform state includes: the maximum entropy distribution estimation algorithm iterating to the... The threshold logic for determining whether the current local landscape is in a stagnant platform state is as follows: ， in It is the low threshold of the correlation coefficient between fitness and distance; It is a high threshold for the proportion of elites who are scattered. It is a low threshold for mean shift. It is a low threshold for directional stability; According to the maximum entropy principle, when only boundary constraints and normalization constraints are retained, the optimal probability density function degenerates into a uniform distribution: ， in, Indicates the combination of candidate photolithography process parameters The uniform probability density function; Represents the physical search space for candidate parameters; This represents the hypervolume of the search space; Execute the maximum entropy increase strategy to force a global restart in order to escape the flat region; The directional migration state includes: during the optimization of photolithography process parameters, when the maximum entropy distribution estimation algorithm detects that the population is in a directional migration state, it indicates that the current reliable guidance information is the migration direction. At this time, the maximum entropy distribution estimation algorithm should use the maximum entropy principle, extract only the mean as a constraint, and use an exponential distribution for directional sampling. The threshold logic for determining whether the current local landscape is in a directional migration state when iterating to generation t is as follows: ， in It is a high threshold for mean shift. It is a high threshold for directional stability; when the judgment criterion is met, the maximum entropy distribution estimation algorithm determines that the current search has entered the effective directional guidance area; First, calculate the orientation scale vector of the t-th generation. And construct the search boundary in the r-th dimension: ， ， in, Represents the global orientation scale vector of the t-th generation; Let represent the mean vector of the elite samples sampled by the exponential distribution in generation t; Let represent the optimal parameter vector in the t-th generation elite sample; Represents the direction scale vector The component in the r-th parameter dimension; This represents the value of the optimal parameter vector in the r-th dimension; These represent the left and right boundaries of the exponential distribution sampling, respectively. and Where c is the physical search range for the parameters, and c is the orientation scale adjustment coefficient; According to the scale The sign of the sample offset determines the direction and starting point: Negative promotion strategy, scale : by the left boundary Starting from a high-density base, exponential sampling is performed to the right: ， Positive advancement strategy, scale : by the right boundary Starting from a high-density base, exponential sampling is performed to the left: ， in, Let represent the value of the i-th candidate sample generated using the exponential distribution in the t-th generation on the r-th dimension; Indicates in The interval follows a standard uniform distribution Random numbers; This represents the rate parameter of the exponential distribution in the t-th generation along the r-th dimension; The multi-region candidate state includes: when it is determined that the current state is in a local dominant but structurally uncertain state, a generalized long-tailed distribution is introduced as an auxiliary sampling distribution; The aforementioned local advantage but structural uncertainty state refers to the following: the current elite sample set has shown a concentration trend relative to the overall sampling population, and the current search has not yet formed a stable direction of advancement that can be used for directional exponential distribution and a deterministic local landscape of a single basin that can be used for Gaussian distribution. The threshold logic for determining whether the current local landscape is in a potential multi-peak state when iterating to generation t is as follows: ， in It is the threshold in the elite discrete proportion; It is the threshold in fitness-distance correlation; It is the threshold in directional stability; It is a low threshold for mean shift. The constraint equations are defined as follows: ， in It is a combination of candidate photolithography process parameters The sampling probability density function; These are the location parameters of the long-tailed distribution; It is the scale parameter of the long-tailed distribution; These are empirical statistical values ​​corresponding to the logarithmic distance moment constraints; It is a logarithmic distance feature mapping function; Substituting the logarithmic distance feature function into the general maximum entropy distribution expression: ， Using the properties of the exponential function, it can be simplified to a generalized long-tailed distribution model: ， in, Indicates the center of the sample distribution; Lagrange multipliers for controlling the thickness of the distribution tail; Z is a smoothing parameter used to control the range of logarithmic distance feature scales; Z is a normalization constant. The smooth basin state includes: when the system determines that the current local landscape is in a smooth basin state, the high-confidence mean and second moment are extracted as joint constraints, and substituted into the general model to derive that the system strictly degenerates into a Gaussian distribution, as shown in the formula: ， in, The probability density function represents a one-dimensional Gaussian distribution; This represents the mean of a local Gaussian distribution; This represents the standard deviation of a local Gaussian distribution; The threshold logic for determining whether the current local landscape is in a smooth basin state when iterating to generation t is as follows: ， in It is the high threshold of the correlation coefficient between fitness and distance; It is a low threshold for the proportion of elites.

9. An electronic device, characterized in that, It includes a processor and a memory, the memory storing program code that, when executed by the processor, causes the processor to perform the steps of the method as described in any one of claims 1 to 8.

10. A storage medium, characterized in that, It stores a computer program or instructions that, when run on a computer, perform the steps of the method as described in any one of claims 1 to 8.

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