KR102561402B1 - surrogate model constructing method for extrapolation prediction and apparatus of thereof - Google Patents

Abstract

Embodiments are directed to methods and apparatus for generating surrogate models that predict extrapolation phenomena. A method for generating a surrogate model for predicting an extrapolation phenomenon according to an embodiment includes estimating PCE coefficients based on a polynomial order and determining parameters of an autocorrelation function; constructing a plurality of substitute models having different Gaussian variances according to the iteration order of an iterative algorithm through PCE coefficients and the parameters; and selecting an alternative model having a minimum deviation from an actual result from among a plurality of alternative models.

Description

A method for generating a surrogate model for predicting an extrapolation phenomenon and an apparatus therefor

Embodiments relate to a method and apparatus for generating a surrogate model predicting an extrapolation phenomenon.

In general, data-based models (including machine learning and deep learning-based models) learn and predict phenomena based on currently collected data. Such models have a disadvantage in that their predictive power is greatly reduced for data spaces (eg, extreme events such as floods) that deviate significantly from the trained data space.

In this regard, PCE (Polynomial Chaos Expansion) models are used to capture trend trends in prototypical models using a set of orthogonal polynomials. When constructing a PCE model, sufficient training data is generally required to prevent underfitting.

Ordinary kriging is also about creating an overall trend model and interpolating data using local deviations obtained using maximum likelihood.

In the invention according to the embodiment, an alternative model that reproduces the behavior of the original model by modeling the overall tendency or trend of the data using the PCE technique and supplementing the local deformation that the trend does not reflect with the Kriging technique, and We want to provide a way to create an alternative model.

A method of generating a surrogate model for predicting an extrapolation phenomenon, comprising: estimating PCE coefficients based on a polynomial order and determining parameters of an autocorrelation function; constructing a plurality of alternative models having different Gaussian variances according to an iteration order of an iterative algorithm through the PCE coefficient and the parameters; and selecting an alternative model having a minimum deviation from an actual result from among the plurality of alternative models.

The configuring of the plurality of substitute models may include deriving a variance by a trend function corresponding to the PCE coefficient and the parameter and the autocorrelation function according to a corresponding polynomial order.

The plurality of replacement models may be in a form in which a new term is added to the trend function for each iteration order by the iterative algorithm.

Selecting the surrogate model may include calculating a leave-one-out error for each of the plurality of surrogate models.

The selecting of the surrogate model may include selecting a surrogate model corresponding to a degree having a minimum deviation from the actual result.

It may further include quantifying uncertainty for the selected surrogate model based on a figure of merit.

The figure of merit may be defined based on a Euclidean distance between an ensemble set of the selected surrogate model and the original model and a relative runtime of the selected surrogate model and the original model.

The parameter x may be obtained through a maximum likelihood estimation method or a leave-one-out cross-validation.

The PCE coefficient ε _α may be calculated using at least one of least square regression, Bayesian compressive sensing, and least angular regression.

Through the invention according to the embodiment, an alternative model that reproduces the behavior of the original model by modeling the overall trend or trend of the data using the PCE technique and supplementing the local deformation that the trend does not reflect with the Kriging technique and a method for generating an alternative model.

1 is a flow chart of a method for generating a surrogate model for predicting an extrapolation phenomenon, in an embodiment.
2 is a diagram for explaining a method of generating a surrogate model and evaluating the performance of the surrogate model in an embodiment.
3 is a diagram illustrating performance related to error of an alternative model in an embodiment.
4 is a graph showing results of applying an alternative model to an event that actually occurred in an embodiment.
5 is a diagram illustrating the effect of acceptance thresholds on the performance of the original model and the surrogate model in an embodiment.

Hereinafter, embodiments will be described in detail with reference to the accompanying drawings. However, since various changes can be made to the embodiments, the scope of the patent application is not limited or limited by these embodiments. It should be understood that all changes, equivalents or substitutes to the embodiments are included within the scope of rights.

Terms used in the examples are used only for descriptive purposes and should not be construed as limiting. Singular expressions include plural expressions unless the context clearly dictates otherwise. In this specification, terms such as "include" or "have" are intended to designate that there is a feature, number, step, operation, component, part, or combination thereof described in the specification, but one or more other features It should be understood that the presence or addition of numbers, steps, operations, components, parts, or combinations thereof is not precluded.

Unless defined otherwise, all terms used herein, including technical or scientific terms, have the same meaning as commonly understood by a person of ordinary skill in the art to which the embodiment belongs. Terms such as those defined in commonly used dictionaries should be interpreted as having a meaning consistent with the meaning in the context of the related art, and unless explicitly defined in the present application, they should not be interpreted in an ideal or excessively formal meaning. don't

In addition, in the description with reference to the accompanying drawings, the same reference numerals are given to the same components regardless of reference numerals, and overlapping descriptions thereof will be omitted. In describing the embodiment, if it is determined that a detailed description of a related known technology may unnecessarily obscure the gist of the embodiment, the detailed description will be omitted.

In addition, in describing the components of the embodiment, terms such as first, second, A, B, (a), and (b) may be used. These terms are only used to distinguish the component from other components, and the nature, order, or order of the corresponding component is not limited by the term. When an element is described as being “connected,” “coupled to,” or “connected” to another element, that element may be directly connected or connected to the other element, but there may be another element between the elements. It should be understood that may be "connected", "coupled" or "connected".

Components included in one embodiment and components including common functions will be described using the same names in other embodiments. Unless stated to the contrary, descriptions described in one embodiment may be applied to other embodiments, and detailed descriptions will be omitted to the extent of overlap.

In an embodiment, a hydrological replacement model may be provided using PCE (Polynomial Chaos Expansion) and OK (ordinary kriging) techniques. The replacement model according to the embodiment can be simply expressed as Equation 1 below.

[Equation 1]

where M denotes a hydrologic model with X representing model inputs including parameter θ, model state x, and physical force u. Y represents the corresponding model output (Quantity of Interest, QoI), which can be a scalar quantity or a vector with multiple QoIs. The number of inputs X is N _X determined by the sum of the number of parameters (N _P ), the number of model states (N _S ), and the number of forcings (N _I ).

In general, when the model of Equation 1 includes N _ED sets of input-output relationships (also referred to as a training data set including experimental design X and corresponding model response Y), the PCK to be proposed in the embodiment is model M You can configure an alternative model (M ^su ) that can replace .

An alternative model to be provided through the present invention according to an embodiment is hereinafter referred to as PCK. PCK models the overall tendency or trend of the data using the PCE technique, and models a method of supplementing the local deformation that the trend does not reflect with the Ordinary Kriging technique.

In an embodiment, PCK may consist of two terms for input X as in Equation 2 below.

[Equation 2]

[Equation 3]

In Equation 2, the first term ∑ is equivalent to PCE and can be used as a trend function in equations.

Ψ _α (X) is a multivariate polynomial orthogonal to a random input variable X determined using the Weiner-Askey method for uniform, Gaussian, beta and gamma distributions. a is a multiple index identifying the components of the multivariate polynomial, and ε _α is the unknown PCE coefficient corresponding to Ψ _α (X).

The second term of Equation 2 (i.e., ), where σ ² is the Gaussian process variance (or kriging variance). Z(X) is a stationary Gaussian process with zero mean and unit variance, Z(X) is the autocorrelation function between two random model input sets X and X', R(X,X')=R(|X- X′|;δ), where δ is the hyperparameter of the autocorrelation function and needs to be estimated.

P in Equation 3 represents the number of polynomial basis terms (or the number of PCE coefficients) depending on the polynomial order (p), such as N _X .

PCK according to the embodiment can be built in three processes. The method according to the embodiment will be described with reference to FIG. 1 below.

1 is a flowchart for explaining a method of building a surrogate model (PCK) in an embodiment.

In step 110, the device estimates the coefficients of the surrogate model based on the polynomial order and determines the parameters of the autocorrelation function.

First, the PCE coefficient can be estimated. The PCE coefficient ε _α can be calculated using several methods including projection, least squares regression, Bayesian compressive sensing and least angle regression.

In the embodiment, the minimum angle regression method, which is more efficient than the least squares regression method used as a method of estimating the PCE coefficient with a small experimental design, can be used, and the PCE coefficient to be estimated is reduced so that the higher the dimension, the more uncertain the input. You can avoid problems with curses.

A method of estimating a PCE coefficient according to an embodiment may refer to Equation 4.

[Equation 4]

An embodiment is an input model X consisting of Nx samples and It relates to a method of estimating an optimal PCE coefficient estimated in relation to an output model Y corresponding to .

where k is the index of N _x , λ is a non-negative constant, ε ₁ corresponds to the regularization term for minimization to follow the Spare solution, can be calculated as Equation 4 can be expressed as Equation 5 below.

[Equation 5]

Here, F corresponds to an information matrix of N××P, and the general term can be expressed as follows.

[Equation 6]

In the second procedure, the parameter δ of the autocorrelation function and the Gaussian process variance this can be determined. Parameter δ can be obtained through maximum likelihood estimation method or leave-one-out cross-validation.

Among these, the leave-one-out cross-validation method can be expressed as in Equation 7 below.

[Equation 7]

here, denotes the optimal δ, Is of x, two samples and means the redemption matrix of

In step 120, the apparatus constructs a plurality of alternative models having different Gaussian variances according to the iteration order of the iterative algorithm through the PCE coefficients and the above parameters.

In an embodiment, x, y, R(X, X'), , ε _α Ψ _α (X), an iterative algorithm can be applied after optimizing the Gaussian variance σ ² . ε _α Ψ _α (X) is considered as a candidate for the trend part of kriging, and the number of candidates P for each iteration algorithm can be represented by (a=1, ... , P).

In an embodiment, the initial value of the trend part (a=1) is PCK, which can be expressed in terms of ε ₁ Ψ ₁ (X). Polynomials can be iteratively added to the trend part one by one by the iterative algorithm, and the Gaussian variance in each iteration can be estimated as shown in Equation 8 below.

[Equation 8]

denotes the optimal correlation matrix. F is an information matrix calculated by Equation 6, and an alternative model PCK _a with a different variance can be constructed for each iteration of the algorithm.

In step 130, the device selects a surrogate model having the smallest deviation from the actual result from among a plurality of surrogate models.

Among the number of P of each PCK model, an optimal PCK in which the deviation of the result of the prototype model is minimized may be selected. In order to quantitatively express this deviation, leave-one-out error() may be used as shown in Equation 9.

[Equation 9]

In Equation 9, PCK _x(-k) is size experimental design Corresponds to the PCK model built using

In an embodiment, the GLUE technique may be used to quantify uncertainty for an alternative model provided through the present invention. In an embodiment, two criteria of accuracy-based threshold and efficiency-based threshold may be set, and pros and cons and results thereof may be presented.

The parameter inference method can be applied to infer the probability that the value of the uncertain parameter θ can be applied to the predictive model that will match the actual observation result. From the inference results, the observed streamflow can generate the posterior distribution of the parameters given, and prove possible ensemble outcomes for that distribution. The posterior distribution of a parameter conditional on the observed value y _obs can generally be expressed using the Bayes' Rule as

[Equation 10]

where ρ(θ) is the prior distribution of θ generated based on prior knowledge. denotes the conditional probability of a model outcome given a set of parameters, is the posterior distribution of the parameter θ.

In an embodiment, the GLUE technique may be applied to the embodiment. It is simple to implement and allows for a flexible choice of informal probability function and corresponding cut-off threshold. The technique avoids over-tuning and can exclude some model parameter sets that do not guarantee predictive performance.

In the context of GLUE, various likelihood functions can be applied. In an embodiment, a combination of the Nash-Sutcliffe efficiency (NSE), the peak error (PE), and the volume error (VE) can be used to simultaneously characterize the deviation of the shape, peak, and volume of the flood hydrograph. can also be expressed as a function (L).

[Equation 11]

where y _t ^Obs and y _t are the observed and simulated stream flow at time t, respectively. T is the total number of time steps for the flood event, and y _max ^Obs and y _max are the observed and simulated river flow at the peak, respectively. and denote the total volume of the observed and simulated hydrographs, respectively.

The partial equations in the three parentheses of Equation 11 represent complementary NSE (1-NSE), PE and VE, respectively. Equation 11 has a value between 0 and 1, and the closer the value is to the minimum, the smaller the error. In an embodiment, the cutoff threshold may be designated as an allowable deviation of the likelihood function (hereinafter, expressed as an accuracy-based threshold) or a fixed ratio of the total number of simulations (hereinafter, expressed as an efficiency-based threshold). The overall simulation performed can be divided into runs that meet or do not meet this threshold condition, which is utilized to track the value of Equation 10.

In embodiments, a framework for quantifying uncertainty based on alternative models may be proposed.

Figure 2 shows how to construct a surrogate model in an embodiment, and combine the surrogate model and GLUE to infer model parameters in an efficient manner.

Step 201 relates to a method of obtaining a collection of experimental designs x and model responses y.

In general, as in step 201, the set of inputs and outputs of the hydrologic model (or design of experiments and model responses) can be used to construct a surrogate model 202. The surrogate model 202 can enable fast computation of inferences for uncertain parameters of the hydrological model 203.

Step 202 describes how to construct a surrogate model. The method may refer to the description of FIG. 1 .

Step 203 is for the inference step and describes how to use the surrogate model created in step 202 to perform the GLUE. Embodiments may obtain ensemble streamflows and posterior parameters based on likelihood functions, two types of thresholds (accuracy-based thresholds or efficiency-based thresholds) and observed streamflows.

Specifically, according to step 201, the experimental design x consists of Nx sets of θ, x and u, where the θ values are uniform (prior) distributed ρ(θ) using Latin Hypercube sampling (LHS). can be selected randomly through State x is initialized with a 0 vector and the value of physical force u can be collected using past climate data (eg rainfall, etc.). The corresponding response y value can be obtained by applying x to the hydrologic model M.

Step 202 shows the procedure for constructing a surrogate model for a given x and y.

First, the PCE coefficient (ε _α ) can be estimated by Equation 4 given x, y and the polynomial order (p), and the parameter δ of the autocorrelation function R (X, X′) is as shown in Equation 7 can be optimized. Once the PCE coefficients and parameters are all determined, the Gaussian variance can be optimized by an iterative algorithm, and starting to construct an alternative model, which can continue up to the number of iterations P. Among the configured surrogate models, a surrogate model with the smallest error calculated by Equation 9 may be selected as an optimal surrogate model.

Once a surrogate model is constructed, it can be used for computationally inexpensive parameter inference. Step 203 provides the posterior distribution of the parameters and the uncertain interval of the simulated streamflow. Among them, the prediction of the stream flow can match the observed stream flow and meet the acceptance threshold. In the embodiment, two types of threshold values, the acceptable accuracy threshold and the acceptable efficiency threshold, may be unified. In GLUE, θ and x are initialized similarly to step 201 while the force u value can be used in potential future events including possible extreme events.

In order to investigate how accurately the aforementioned three surrogate models (PCE, OK and PCK) mimic the prototypical model for design of experiments during learning, the one-time error in Equation 9 ( ) can be used. A smaller value indicates that the emulator is more accurate.

Ensemble prediction requires evaluation through deterministic and probabilistic measurements. The likelihood function L adopted by GLUE can also be utilized as a deterministic measure. Continuous Rank Probability Score (CRPS) and Spread can be selected for probabilistic measures. CRPS can measure the closeness between the distribution of ensemble predictions and observations in a single time step. In the embodiment, the average of CRPS for flood events in Equation 12 ( ) is important, and the ideal value is zero. The second probability metric can represent the variance of the ensemble prediction compared to the observed value. Smaller values indicate more reliable predictions. Spread equals the square root of the average ensemble variance over the evaluation period formulated in Equation 13.

[Equation 12]

[Equation 13]

where F(y) and F(y ^Obs ) are the cumulative distributions of the ensemble streamflow predictions and observations at time t, respectively, and l is the index for the ensemble member of the NE-sized model prediction.

In addition to the above metrics, a performance score (PS) can be proposed to evaluate the overall performance of the surrogate model (M ^su ) by weighting the accuracy and efficiency of the surrogate model compared to the original model.

[Equation 14]

[Equation 15]

[Equation 16]

where L(M ^su ) and L(M) denote the likelihood function values for motion executions of the replacement and original models, respectively. GLUE(M ^su ) and GLUE(M) refer to the total runtime to achieve replacement and operation execution of the original model by GLUE. RT(M ^su ) and RT(M) represent the runtimes required for a single run for the replacement and original models, respectively, and Nruns(M ^su ) and Nruns(M), respectively, a predefined number of times that meet the cutoff threshold condition. Indicates the number of previous runs required to acquire an action run.

in Equation 14 Is to represent the accuracy performance, which can be defined as the Euclidean distance between two ensemble sets L(M ^su ) and L(M), referring to Equation 15. may indicate that the replacement model has the same accuracy as the original model and effectively replaces the original model. represents the efficiency performance and can be defined as the relative difference in the total runtime required to quantify the uncertainty. The smaller the corresponding value, the better the efficiency of the alternative model.

The range of possible values of PS combining the two terms of Equation 14 is from 0 to infinity. When PS is close to 0, it means that the selected surrogate model has similar accuracy to the original model and the uncertainty quantification is completed in a very short time, whereas when PS approaches infinity, it can indicate that both the accuracy and efficiency of the surrogate model are very low. .

3 is a diagram illustrating performance related to error of an alternative model in an embodiment.

The embodiment shows the error comparison result according to the alternative model when learning with a small number of data. The PCK surrogate model accurately reproduces the behavior of the original model, but the PCE and OK surrogate models show relatively low performance. For example, it can be confirmed that the leave one out error is about 2 to 7 times different.

4 is a graph illustrating results of applying a prototype model and replacement models to actual events in an embodiment.

4 (a) shows the prediction results of the NAM model for events 1 to 8 that are exemplarily applied, FIG. 4 (b) is the PCE model, and FIG. 4 (c) is the OK model. , and FIG. 4(d) shows the prediction result of the PCK model of the embodiment.

The embodiment shows the observed and predicted streamflow in the 95% confidence band of a certain number of ensemble elements obtained through each model for 8 events based on the GLUE technique using an accuracy-based threshold of 0.1 there is.

As shown, it can be confirmed that the prediction result of the PCK model proposed through the embodiment is more accurate than other models for an extreme event.

5 is a diagram illustrating the effect of acceptance thresholds on the performance of the original model and the surrogate model in an embodiment.

In an embodiment, two types of acceptance thresholds (accuracy-based thresholds or efficiency-based thresholds) affect accuracy and efficiency with the performance of the prototype model and each of the three types of surrogate models for the eight events associated with FIG. shows the influence.

The shaded area in FIGS. 5(a) and 5(c) represents the 95% confidence band of 2000 values of L corresponding to the threshold for each model, and FIGS. 5(b) and 5(d) show each model. Indicates the total number of model executions required to implement the GLUE technique for .

In performing uncertainty quantification using the GLUE technique, the pros and cons and results of these two criteria, accuracy-based threshold and efficiency-based threshold, can be presented.

In the embodiment, the prototype model NAM and the replacement model PCK can obtain more accurate results by lowering the accuracy standard, but the replacement model PCE and OK cannot obtain accuracy higher than a certain level if the accuracy standard is lowered.

In the examples, when uncertainty is quantified based on efficiency, it can be seen that the accuracy of the original model NAM and the surrogate model PCK is better than that of the surrogate model PCE and OK.

The method according to the embodiment may be implemented in the form of program instructions that can be executed through various computer means and recorded on a computer readable medium. The computer readable medium may include program instructions, data files, data structures, etc. alone or in combination. Program commands recorded on the medium may be specially designed and configured for the embodiment or may be known and usable to those skilled in computer software. Examples of computer-readable recording media include magnetic media such as hard disks, floppy disks and magnetic tapes, optical media such as CD-ROMs and DVDs, and magnetic media such as floptical disks. - includes hardware devices specially configured to store and execute program instructions, such as magneto-optical media, and ROM, RAM, flash memory, and the like. Examples of program instructions include high-level language codes that can be executed by a computer using an interpreter, as well as machine language codes such as those produced by a compiler. The hardware devices described above may be configured to operate as one or more software modules to perform the operations of the embodiments, and vice versa.

Software may include a computer program, code, instructions, or a combination of one or more of the foregoing, which configures a processing device to operate as desired or processes independently or collectively. The device can be commanded. Software and/or data may be any tangible machine, component, physical device, virtual equipment, computer storage medium or device, intended to be interpreted by or provide instructions or data to a processing device. , or may be permanently or temporarily embodied in a transmitted signal wave. Software may be distributed on networked computer systems and stored or executed in a distributed manner. Software and data may be stored on one or more computer readable media.

As described above, although the embodiments have been described with limited drawings, those skilled in the art can apply various technical modifications and variations based on the above. For example, the described techniques may be performed in an order different from the method described, and/or components of the described system, structure, device, circuit, etc. may be combined or combined in a different form than the method described, or other components may be used. Or even if it is replaced or substituted by equivalents, appropriate results can be achieved.

Therefore, other implementations, other embodiments, and equivalents of the claims are within the scope of the following claims.

Claims (19)

A method for generating a surrogate model in an apparatus for generating a surrogate model predicting an extrapolation phenomenon,
estimating polynomial chaos expansion (PCE) coefficients based on polynomial order and determining parameters of an autocorrelation function;
constructing a plurality of alternative models having different Gaussian variances according to an iteration order of an iterative algorithm through the PCE coefficient and the parameter; and
Selecting an alternative model having a minimum deviation from an actual result from among the plurality of alternative models
including,
How to create an alternative model.
According to claim 1,
The step of constructing the plurality of substitute models,
Deriving a variance by a trend function corresponding to the PCE coefficient and the parameter and the autocorrelation function according to a corresponding polynomial order
including,
How to create an alternative model.
According to claim 2,
The plurality of alternative models,
A form in which a new term is added to the trend function for each iteration order by the iterative algorithm,
How to create an alternative model.
According to claim 1,
The step of selecting the alternative model,
Calculating a leave-one-out error for each of the plurality of substitute models
including,
How to create an alternative model.
According to claim 1,
The step of selecting the alternative model,
Selecting an alternative model corresponding to the order with the minimum deviation from the actual result
including,
How to create an alternative model.
According to claim 1,
quantifying uncertainty for the selected surrogate model based on a figure of merit;
Including more,
How to create an alternative model.
According to claim 6,
The performance index is,
Accuracy performance represented by the Euclidean distance between the selected surrogate model and the ensemble set of the original model and a weight given to each efficiency performance defined based on the relative runtimes of the selected surrogate model and the original model.
How to create an alternative model.
According to claim 1,
The parameter can be obtained through a maximum likelihood estimation method or a leave-one-out cross-validation,
How to create an alternative model.
According to claim 1,
The PCE coefficient ε _α is calculated by at least one method of least squares regression, Bayesian compressive sensing, and least angular regression.
How to create an alternative model.
A computer program stored in a computer readable medium to be combined with hardware to execute the method of any one of claims 1 to 9.
An apparatus for generating an alternative model for predicting an extrapolation phenomenon,
one or more processors;
Memory; and
one or more programs stored in the memory and configured to be executed by the one or more processors;
said program,
estimating polynomial chaos expansion (PCE) coefficients based on polynomial order and determining parameters of an autocorrelation function;
constructing a plurality of alternative models having different Gaussian variances according to an iteration order of an iterative algorithm through the PCE coefficient and the parameters; and
Selecting an alternative model having a minimum deviation from an actual result from among the plurality of alternative models
including,
Device.
According to claim 11,
The step of constructing the plurality of substitute models,
Deriving a variance by a trend function corresponding to the PCE coefficient and the parameter and the autocorrelation function according to a corresponding polynomial order
including,
Device.
According to claim 12,
The plurality of alternative models,
A form in which a new term is added to the trend function for each iteration order by the iterative algorithm,
Device.
According to claim 11,
The step of selecting the alternative model,
Calculating a leave-one-out error for each of the plurality of substitute models
including,
Device.
According to claim 11,
The step of selecting the alternative model,
Selecting an alternative model corresponding to the order with the minimum deviation from the actual result
including,
Device.
According to claim 11,
quantifying uncertainty for the selected surrogate model based on a figure of merit;
Including more,
Device.
According to claim 16,
The performance index is,
Defined based on the Euclidean distance between the selected surrogate model and the ensemble set of the original model and the relative runtime of the selected surrogate model and the original model,
Device.
According to claim 11,
The parameter can be obtained through a maximum likelihood estimation method or a leave-one-out cross-validation,
Device.
According to claim 11,
The PCE coefficient ε _α is calculated by at least one method of least squares regression, Bayesian compressive sensing, and least angular regression.
Device.