CN104063594A - Complexity reliability calculation method based on optimized learning machine - Google Patents

Abstract

The invention provides a complexity reliability calculation method based on an optimized learning machine. According to the method, firstly, a reliability important influence region is determined; then, new samples are selected to be calculated in a purposive way in the region according to a certain strategy, so the calculation times of a limit state function is reduced to the maximum degree, meanwhile, the limit state function is cyclically rebuilt at high efficiency and high precision, finally, the simulated reliability calculation is fast carried out on the basis of a rebuilt approximate limit state function model by using an important sampling method, and the goal of obtaining high-precision reliability calculation results by calculating the limit state function in fewer times is finally achieved. The defect that the calculation precision and the calculation efficiency cannot be simultaneously considered in the conventional reliability calculation method is overcome, so the practicability of the method in the engineering reliability analysis is improved.

Description

A kind of computing method of the complicated fiduciary level based on Optimization Learning machine

Technical field

The present invention relates to a kind of computing method of the complicated fiduciary level based on Optimization Learning machine, it is the fiduciary level of accurate computational engineering integrity problem efficiently, is applicable to reliability assessment, the association areas such as reliability demonstration.

Background technology

In Engineering Reliability problem, limit state function often do not resolve, non-linear.Limit state function assessment each time need to be carried out extensive numerical solution by methods such as finite elements.For solving fiduciary level, need repeatedly calculating limit function of state.According to the existing different reliability methods that calculate, very little, the error of calculation is larger for limit state function calculation times; Limit state function calculation times is too many, and calculated amount is too large.

Reliability degree calculation method can be divided into two classes at present:

First kind method comprises the computing method based on Monte Carlo sampling and improves algorithm (as selective sampling method, Subset sampling method, line sampling etc.).These class methods are calculated simple, and applicability is strong, and precision is high, but because choosing of sample in computation process has randomness, for the complicated reliability issues of engineering, need extensive double counting limit state function, and overall efficiency is lower, is difficult to accept.

Equations of The Second Kind method, the i.e. computing method based on gradient.The method needs the number of times of calculating limit function of state few compared with first kind method, and efficiency is higher.But when reliability issues more complicated, limit state function is non-linear when stronger, exists the error of calculation large, not even not accurately shortcoming.

In the complicated integrity problem of engineering, its limit state function is non-linear, do not resolve, and often needs massive values computation.First kind method counting yield is low, and Equations of The Second Kind method computational accuracy is not enough, and engineering practicability is not strong.Therefore, exploring the efficient Method for Accurate Calculation of fiduciary level is of great significance for Practical Project reliability analysis tool.

Summary of the invention

The object of the invention is, for less number of times ground calculating limit function of state, just can obtain high precision fiduciary level result of calculation, proposed a kind of computing method of the complicated fiduciary level based on Optimization Learning machine.

The computing method of a kind of complicated fiduciary level based on Optimization Learning machine of the present invention, comprise the steps:

1) determine that fiduciary level affects the position of important area:

Integrity problem for comprising any stochastic variable, is converted Non-normal Variable is transformed to standard normal random variable by Rosenblatt, supposes that the stochastic variable x of G in limit state function (x) all obeys standardized normal distribution here, and its dimension is n;

1. step-length L=0.2～1.5 of advancing in standard normal space are set;

2. from the initial point in standard normal space, along the negative gradient direction of origin position place limit state function the step-length of advancing L is apart from arriving a new some P _i, i is the number of times advancing, new some P of every arrival _i, judge the some P that this is new _ig (x=P _i) whether be less than 0, if so, represent the some P that this is new _iin failed areas, halt, otherwise along P _ithe negative gradient direction of some place limit state function the step-length of advancing L is apart from arriving P _i+1point, until new point is in failed areas;

2) fiduciary level affects important area delimitation:

P sets up an office _mfor step 1) in the point of trying to achieve, postulated point P _mcoordinate is (x _p1..., x _pn), then obtain some P _mnear other M ₁～M _2n, its coordinate is respectively (x _p1± k' σ _x1, x _p2..., x _pn) ..., (x _p1..., x _pi-1, x _pi± k' σ _xi, x _pi+1..., x _pn), here coefficient k ' value 0.5, σ _xifor stochastic variable x _istandard deviation, set up respectively initial point with some P _m, M ₁～M _2nstraight line l ₀～l _2n, by method of interpolation, try to achieve above-mentioned straight line l ₀～l _2nintersection point Q with limit state function G (x) ₁～Q _2n, obtain the failpoint Q on limit state surface important area ₁～Q _2n;

By failpoint Q ₁～Q _2n, some P _m, M ₁～M _2nconstitute coordinate point set X, limit state function value corresponding to coordinate point set X constituted to limit state function response collection Y simultaneously, based on Optimization Learning machine OLEM, the mapping of setting up X → Y, builds initial threshold function of state recycling formula (2), obtains corresponding initial MPP point

$\begin{array}{c}\min {\mathrm{\β}}^0={\left|\right|x\left|\right|}^{1/2}\\ s.t.G_{\mathrm{OELM}}^0\left(x\right)=0\\ {\mathrm{\β}}^0=\left|\right|x_{\mathrm{Mpp}}^0\left|\right|\end{array}.---\left(2\right)$

Then constructing curve wherein △ β value 2, will be similar to limit state surface by constructing curve B (x) on meet curved surface B (x) >0 regional assignment be the important area of fiduciary level impact;

3) based on Optimization Learning machine OELM loop restructuring fiduciary level, affect limit state function in important area, building on the basis of (i>=0), reconstruct limit state function time, need to find three kinds of new sample points, to improve the reconstruction accuracy of limit state function;

1. I type sample point find: in reconstruct each time after, according to formula (2), will replace to searching correspondence until neighborhood calculation with error is very little, in time, stops finding, ε ₂get 0.001, by this point be called I type sample point

2. II type sample point find: according to formula (3), find Equations of The Second Kind new sample point

$\begin{array}{c}\max \mathrm{Dis}\tan \mathrm{ce}=\min \left\{\underset{j=1...d}{\mathrm{norm}}\right||x-x_{\mathrm{wj}}|\left|\right\}\\ s.t.G_{\mathrm{OELM}}^i\left(x\right)=0\\ B\left(x\right)\≥0\end{array}.---\left(3\right)$

Formula (3) is illustrated in fiduciary level to be affected in important area, on curved surface, find the location point of sparse region as new sample point

3. III type sample point find: according to formula (4), find the 3rd class new sample point

$\begin{array}{c}\max {\left|\right|x\left|\right|}^{1/2}\\ s.t.G_{\mathrm{OELM}}^i\left(x\right)=0\\ B\left(x\right)\≥0\\ H\left(x\right)={\left|\right|x\left|\right|}^{1/2}-\left|\right|x_0\left|\right|\≥0\end{array}---\left(4\right)$

The point of picking out outermost in coordinate point set X forms set X ', each point belonging in set X ' has following characteristic: exist in the coordinate of certain one dimension, its coordinate figure is in point set X or be maximum or minimum, above-mentioned in space distribution the outermost in coordinate point set X, then will gather X ' each point as initial value x ₀, according to formula (4), find the point farther apart from initial point, finally choose point farthest as III type sample point

4. reconstruct limit state function

In reconstruct limit state curve surface basis on, using the above-mentioned I～III type sample point searching out as new sample point, calculate the true ultimate limit state response of its correspondence, and add in coordinate point set X and limit state function response collection Y, then continue the mapping based on Optimization Learning machine OELM reconstruct X → Y, obtaining fiduciary level affects the higher limit state function of approximation accuracy in important area

4) fiduciary level is calculated:

In the complete limit state function of above-mentioned every reconstruct after, by Importance Sampling Method, with P _mpoint, for sampling center, utilizes agent model calculate corresponding failure probability pf ⁱ, when in time, stops calculating, wherein ε ₃=0.02, represent pf ⁱ～pf ^i-Mstandard deviation, by failure probability pf ⁱ, obtaining fiduciary level value is 1-pf ⁱ.

The computing method of a kind of complicated fiduciary level based on Optimization Learning machine of the present invention, first determine fiduciary level material impact region, then in this region according to certain strategy, purposively select to calculate new samples, when reducing to greatest extent limit state function calculation times, high-efficiency high-accuracy loop restructuring limit state function, finally, on the basis of the approximate limit function of state model of this reconstruct, utilize Importance Sampling Method to carry out fast analog computation fiduciary level, finally realize less number of times ground calculating limit function of state, just can obtain high precision fiduciary level result of calculation, overcome the shortcoming that conventional reliability degree calculation method computational accuracy and counting yield are difficult to take into account, thereby improved the practicality of the present invention in engineering reliability is analyzed.

Accompanying drawing explanation

Fig. 1 is that fiduciary level of the present invention affects important area position definite schematic diagram fast;

Fig. 2 obtains the primary failure point schematic diagram on limit state surface important area under two-dimensional random variable situation of the present invention;

Fig. 3 is that fiduciary level of the present invention affects important area delimitation schematic diagram;

Fig. 4 is that fiduciary level of the present invention affects II type new sample point in important area searching schematic diagram;

Fig. 5 is that fiduciary level of the present invention affects III type new sample point in important area searching schematic diagram.

Below in conjunction with the drawings and specific embodiments, the present invention is further described.

Embodiment

As Fig. 1-5, the present invention proposes a kind of computing method of the complicated fiduciary level based on Optimization Learning machine, with the fiduciary level that comprises two-dimensional random variable situation, is calculated as example, and concrete steps are as follows:

1) determine that fast fiduciary level affects important area position;

For the integrity problem that comprises any stochastic variable, by Rosenblatt, convert and Non-normal Variable can be transformed to standard normal random variable, here suppose that the stochastic variable X in limit state function G (X) all obeys standardized normal distribution, its dimension is n;

1. step-length L=0.2～1.5 of advancing in standard normal space are set;

2. from the initial point in standard normal space, along the negative gradient direction of the limit state function G of origin position place (X) ( ) advance L apart from arriving new some P ₁if, judgement P ₁put in failed areas, i.e. G (X=P ₁) stop during <0, otherwise execution step is 3.;

3. along the negative gradient direction of the P1 point limit state function G of place (X), advance L apart from arriving P _ipoint, if P _iput in failed areas, i.e. G (X=P ₁) stop during <0, otherwise repeated execution of steps is 3., until new point is in inefficacy territory, supposes to advance m time, and last location point is P _m, with this point, represent that fiduciary level affects the approximate location of important area;

It should be noted that step 1) in the situation of not resolving for limit state function, can adopt difference method, approximate its gradient that obtains, shown in (1):

$\begin{array}{c}\&PartialD;G/\&PartialD;x_i=[G(x_1...x_i+\mathrm{k\&sigma;}{x}_i,...x_n)\\ -G(x_1,...x_n)]/\mathrm{k\&sigma;}{x}_i\end{array}.---\left(1\right)$

In formula: σ x _ifor standard normal random variable x _istandard deviation, be 1, conventionally coefficient k is less, gradient is approximate more accurate, in the present embodiment, k gets 0.001.

In Fig. 1, can find out, this limit state function is bunch non-linear very strong, and gradient from left to right alters a great deal.Here get step-length L=0.5, from the initial point in standard normal space, along the negative gradient direction of the limit state function G of origin position place (X) ( ) advance L apart from arriving new some P ₁, then at a P ₁place, along the negative gradient direction of its limit state function again advance L apart from arriving new some P ₂, so circulation, until limit state function G (X=P ₇) <0, judge P ₇point, in failed areas, halts and with P ₇point represents that fiduciary level affects the approximate location of important area;

2) fiduciary level affects important area delimitation;

Point P _mbe step 1) in the point of trying to achieve, here postulated point P _mcoordinate is (x _p1..., x _pn), obtain some P _mnear other M ₁～M _2n, its coordinate is respectively (x _p1± k' σ _x1, x _p2..., x _pn) ..., (x _p1..., x _pi-1, x _pi± k' σ _xi, x _pi+1..., x _pn), here coefficient k ' get 0.5, σ _xifor stochastic variable x _istandard deviation, then set up respectively initial point with some P _m, M ₁～M _2nstraight line l ₀～l _2n, by method of interpolation, can try to achieve the intersection point Q of above-mentioned straight line and limit state function G (X) ₁～Q _2n, be the failpoint on limit state surface important area;

To obtain failpoint Q ₁～Q _2n, some P _mand M ₁～M _2nconstitute coordinate point set X, limit state function value corresponding to coordinate point set X constituted to limit state function response collection Y simultaneously, based on Optimization Learning machine OLEM, the mapping of setting up X → Y, can build initial threshold function of state, is made as recycling formula (2), can obtain corresponding initial MPP point

$\begin{array}{c}\min {\mathrm{\&beta;}}^0={\left|\right|x\left|\right|}^{1/2}\\ s.t.G_{\mathrm{OELM}}^0\left(x\right)=0\end{array}.---\left(2\right)$ Wherein ${\mathrm{\&beta;}}^0=\left|\right|x_{\mathrm{Mpp}}^0\left|\right|;$

Then, constructing curve wherein △ β gets 2, by constructing curve B (x), will be similar to limit state surface on meet curved surface B (x) >0 regional assignment be important area;

In Fig. 2, some P _mbe step 1) in the point of trying to achieve, here postulated point P _mcoordinate is (x _p1, x _p2), then can obtain a P _mnear other M ₁～M ₄, its coordinate is respectively (x _p1-k' σ _x1, x _p2), (x _p1, x _p2-k' σ _x2), (x _p1+ k' σ _x1, x _p2), (x _p1, x _p2+ k' σ _x2), here coefficient k ' get 0.5, σ _xifor stochastic variable x _istandard deviation, then set up respectively initial point with some P _m, M ₁～M ₄straight line l ₀～l ₄, by method of interpolation, can try to achieve the intersection point Q of above-mentioned straight line and limit state function G (X) ₀～Q _4m, be the failpoint on limit state surface important area;

Here in the hope of Q ₁point is example, provides Fast Interpolation derivation algorithm:

1. make t=0 line correspondence l ₁on initial point, t=1 line correspondence l ₁on some M1;

2. make T=[01], Y=[G (x=0), G (x=x _m1)], in the time of can obtaining G=0 by three Hermite interpolation methods of segmentation, corresponding t _newvalue;

3. obtain thus t _newline correspondence l ₁on some x _new=0+t _new* (x _m1-0);

4. assess x _newthe limit state function value G (x=x of position _new), if | G (x=x _new) | < ε ₁, ε ₁represent convergency value, when meeting | G (x=x _new) | < ε ₁, ε ₁get 0.001 o'clock, think an x _newapproach the intersection point of straight line and limit state function G curved surface, so x _newas the intersection point of straight line and limit state function G curved surface, stop calculating; Otherwise make T=[T (2) t _new], Y=[Y (2) G (x=x _new)], and obtain t by three Hermite interpolation methods of segmentation _newvalue, then repeating step 3..

After the primary failure point getting on above-mentioned limit state surface important area, by failpoint Q ₀～Q4 _m, and some P _m, M ₁～M ₄constitute coordinate point set X and corresponding initial threshold function of state response collection Y thereof, then based on Optimization Learning machine OLEM, the mapping of setting up X → Y, can build initial threshold function of state, is made as recycling formula (2), can obtain corresponding initial MPP point

Then constructing curve in Fig. 3 $B\left(x\right)=\left|\right|x\left|\right|-{\mathrm{\&beta;}}_{*}^0$ Wherein ${\mathrm{\&beta;}}_{*}^0={\mathrm{\&beta;}}^0+\mathrm{\&Delta;\&beta;},$ ${\mathrm{\&beta;}}^0=\left|\right|x_{\mathrm{Mpp}}^0\left|\right|,$ △ β gets 2.By constructing curve B (x), will be similar to limit state surface on meet curved surface B (x) >0 regional assignment be important area (Fig. 3 shadow region).

3) based on limit state function in Optimization Learning machine OELM loop restructuring important area;

In step 2) in, initial due to what construct sample point very little, larger with true limit state function error.Next the strategy according to certain is purposively found to new samples, so that limit state function in reconstruct important area then on the limit state function basis of this new reconstruct, continue to find limit state function in the further loop restructuring important area of new samples until meet the requirement of regulation;

Building on the basis of (i>=0), reconstruct limit state function time, need to find three kinds of new sample points here, to improve to greatest extent the reconstruction accuracy of limit state function, be respectively described below:

1. I type sample point find.

Because MPP point is the nearest point of distance initial point in limit state surface, find this point, for in important area, whole approaching to reality curved mask is significant.Therefore, in reconstruct each time after, according to formula (2), will replace to searching correspondence (this point is called to I type point, ), until neighborhood calculation with error is very little ( here ε ₂get 0.001, mean a little with point each coordinate figure basic identical, the point of adjacent iterative computation is described converged near real MPP point, can be similar to and think a little so be MPP point, explanation near true MPP point position, approaching to reality limit state surface precision is very high, stops finding this type sample point.

2. II type sample point find.

Suppose reconstruct limit state function the i time after, in important area, searched out the known failure point in limit state surface, be assumed to be a W ₁, W ₂w _d(for as shown in Figure 4).Due to the approximation ratio of true limit state surface is depended on to these failpoints, therefore, on curved surface, the place far away apart from these points, the error of approaching to reality limit state surface is larger.Therefore, here to find point on curved surface on the most sparse region is as new sample point according to formula (3), find Equations of The Second Kind new sample point

$\begin{array}{c}\max \mathrm{Dis}\tan \mathrm{ce}=\min \left\{\underset{j=1...d}{\mathrm{norm}}\right||x-x_{\mathrm{wj}}|\left|\right\}\\ s.t.G_{\mathrm{OELM}}^i\left(x\right)=0\\ B\left(x\right)\&GreaterEqual;0\end{array}.---\left(3\right)$

Formula (3) is illustrated in fiduciary level to be affected in important area, on curved surface, find the location point of sparse region as new sample point this point with on curved surface, other unknown failpoints are compared and are had following characteristic: i.e. this point and all known failure point W ₁, W ₂w _dbee-line be also maximum, shown in (3).X in formula (3) _wjin important area the point having found on curved surface is (as the some W in Fig. 4 ₁, W ₂w _d).Formula (3) is illustrated in important area on curved surface, find the location point of sparse region, and using this point as new sample point conventionally curved surface exists the predicted value precision of point is lower, finds the point of this sparse region, and calculates its ultimate limit state response, contributes to improve the reconstruction accuracy of curved surface.

It should be noted that, in solving the process of formula (3), optimization routine algorithm or Intelligent cluster algorithm not necessarily always find globally optimal solution, are absorbed in unavoidably local solution.Therefore, x when given initial solution ₀, make respectively x ₀=x _wj(j=1 ... d), can obtain different local solutions thus, then the local solution of chosen distance maximum is as this type sample point can obtain like this result relatively preferably.

3. III type sample point find, according to formula (4), find the 3rd class new sample point

$\begin{array}{c}\max {\left|\right|x\left|\right|}^{1/2}\\ s.t.G_{\mathrm{OELM}}^i\left(x\right)=0\\ B\left(x\right)\&GreaterEqual;0\\ H\left(x\right)={\left|\right|x\left|\right|}^{1/2}-\left|\right|x_0\left|\right|\&GreaterEqual;0\end{array}---\left(4\right)$

The point of picking out outermost in point set X forms set X ', each point belonging in set X ' has following characteristic: exist in the coordinate of certain one dimension, its coordinate figure is in coordinate point set X or be maximum or minimum, above-mentioned in space distribution the outermost in point set X, will gather X ' each point as initial value x ₀, then according to formula (4), find the point farther apart from initial point, finally choose point farthest as III type sample point find sample point object be in order to improve limit state function at the approximation accuracy at fiduciary level material impact zone boundary place.

Although through research find to adopt of overall importance can better Intelligent cluster algorithm, as population is calculated, the II type sample point of searching globally optimal solution not necessarily.As shown in Figure 4, W ₁～W _dbetween the region that covers be F, region A and C are W ₁～W _doutside region.Suppose be globally optimal solution, but the point that actual optimization algorithm is found really the sparse region of finding is always between W ₁and W _dwithin, probably reconstruct next time after, find still on the one hand, if the failpoint in this time domain F is very intensive, if continue to increase new failpoint in this region, for reconstruct next time precision do not help much; On the other hand, always in the F of region but not find failpoint in region A or C, show the II type sample point of finding by formula (3) for opening up W outward ₁～W _doutside the failpoint scarce capacity in region (as A or C).Above-mentioned two aspects have affected reconstruction accuracy and the reconstruct number of times of limit state function, reduce counting yield.

For above-mentioned outer problem of opening up scarce capacity, propose to calculate III type sample point here to improve the outer ability of opening up.

First at a W ₁, W ₂w _din group, the point of picking out outermost forms set X ', and each point belonging in X ' has following characteristic, has in the coordinate of certain one dimension its coordinate figure or be W ₁, W ₂w _dmiddle maximum or minimum, two-dimensional case as shown in Figure 4, W ₁and W _dfor a W ₁, W ₂w _doutermost point in group.

Then, each point of choosing respectively set X ' is as initial value x ₀, then according to formula (4), find the point farther apart from initial point, finally choose point farthest as III type sample point as shown in Figure 5, P ₁and P ₂corresponding x ₀choose respectively W ₁and W _dduring as initial point, solve the point obtaining.Due to OP ₁>OP ₂therefore, selected element P ₁as III type sample point

4. reconstruct limit state function

In reconstruct limit state surface basis on, using the above-mentioned I～III type sample point searching out as new sample point, calculate the true ultimate limit state response of its correspondence, and add coordinate point set X and limit state function response to collect Y, then continue the mapping based on OELM method reconstruct X → Y, can obtain fiduciary level affects the higher limit state function of approximation accuracy in important area repeat said process, along with reconstruct number of times i increases, can improve to greatest extent fiduciary level affects the limit state function approximation accuracy in important area.

4) fiduciary level is calculated:

In the complete limit state function of every reconstruct after, by Importance Sampling Method, with P _mpoint, for sampling center, utilizes agent model calculate fast corresponding failure probability pf ⁱ, when time, wherein represent pf ⁱ～pf ^i-Mstandard deviation, agent model is described precision is enough high, and the convergence of CALCULATION OF FAILURE PROBABILITY result stops calculating, and gets M=4 here, ε ₃=0.02.By failure probability pf ⁱ, obtaining fiduciary level value is 1-pf ⁱ.

The above, it is only preferred embodiment of the present invention, not technical scope of the present invention is imposed any restrictions, therefore any trickle modification, equivalent variations and modification that every foundation technical spirit of the present invention is done above embodiment all still belong in the scope of technical solution of the present invention.

Claims (1)

1. computing method for the complicated fiduciary level based on Optimization Learning machine, is characterized in that comprising the steps:

1) determine that fiduciary level affects the position of important area:

Integrity problem for comprising any stochastic variable, is converted Non-normal Variable is transformed to standard normal random variable by Rosenblatt, supposes that the stochastic variable x of G in limit state function (x) all obeys standardized normal distribution here, and its dimension is n;

1. step-length L=0.2～1.5 of advancing in standard normal space are set;

2. from the initial point in standard normal space, along the negative gradient direction of origin position place limit state function the step-length of advancing L is apart from arriving a new some P _i, i is the number of times advancing, new some P of every arrival _i, judge the some P that this is new _ig (x=P _i) whether be less than 0, if so, represent the some P that this is new _iin failed areas, halt, otherwise along P _ithe negative gradient direction of some place limit state function the step-length of advancing L is apart from arriving P _i+1point, until new point is in failed areas;

2) fiduciary level affects important area delimitation:

P sets up an office _mfor step 1) in the point of trying to achieve, postulated point P _mcoordinate is (x _p1..., x _pn), then obtain some P _mnear other M ₁～M _2n, its coordinate is respectively (x _p1± k' σ _x1, x _p2..., x _pn) ..., (x _p1..., x _pi-1, x _pi± k' σ _xi, x _pi+1..., x _pn), here coefficient k ' value 0.5, σ _xifor stochastic variable x _istandard deviation, set up respectively initial point with some P _m, M ₁～M _2nstraight line l ₀～l _2n, by method of interpolation, try to achieve above-mentioned straight line l ₀～l _2nintersection point Q with limit state function G (x) ₁～Q _2n, obtain the failpoint Q on limit state surface important area ₁～Q _2n;

By failpoint Q ₁～Q _2n, some P _m, M ₁～M _2nconstitute coordinate point set X, limit state function value corresponding to coordinate point set X constituted to limit state function response collection Y simultaneously, based on Optimization Learning machine OLEM, the mapping of setting up X → Y, builds initial threshold function of state recycling formula (2), obtains corresponding initial MPP point

$\begin{array}{c}\min {\mathrm{\&beta;}}^0={\left|\right|x\left|\right|}^{1/2}\\ s.t.G_{\mathrm{OELM}}^0\left(x\right)=0\\ {\mathrm{\&beta;}}^0=\left|\right|x_{\mathrm{Mpp}}^0\left|\right|\end{array}.---\left(2\right)$

Then constructing curve wherein △ β value 2, will be similar to limit state surface by constructing curve B (x) on meet curved surface B (x) >0 regional assignment be the important area of fiduciary level impact;

3) based on Optimization Learning machine OELM loop restructuring fiduciary level, affect limit state function in important area, building on the basis of (i>=0), reconstruct limit state function time, need to find three kinds of new sample points, to improve the reconstruction accuracy of limit state function;

1. I type sample point find: in reconstruct each time after, according to formula (2), will replace to searching correspondence until neighborhood calculation with error is very little, in time, stops finding, ε ₂get 0.001, by this point be called I type sample point

2. II type sample point find: according to formula (3), find Equations of The Second Kind new sample point

$\begin{array}{c}\max \mathrm{Dis}\tan \mathrm{ce}=\min \left\{\underset{j=1...d}{\mathrm{norm}}\right||x-x_{\mathrm{wj}}|\left|\right\}\\ s.t.G_{\mathrm{OELM}}^i\left(x\right)=0\\ B\left(x\right)\&GreaterEqual;0\end{array}.---\left(3\right)$

Formula (3) is illustrated in fiduciary level to be affected in important area, on curved surface, find the location point of sparse region as new sample point

3. III type sample point find: according to formula (4), find the 3rd class new sample point

$\begin{array}{c}\max {\left|\right|x\left|\right|}^{1/2}\\ s.t.G_{\mathrm{OELM}}^i\left(x\right)=0\\ B\left(x\right)\&GreaterEqual;0\\ H\left(x\right)={\left|\right|x\left|\right|}^{1/2}-\left|\right|x_0\left|\right|\&GreaterEqual;0\end{array}---\left(4\right)$

The point of picking out outermost in coordinate point set X forms set X ', each point belonging in set X ' has following characteristic: exist in the coordinate of certain one dimension, its coordinate figure is in point set X or be maximum or minimum, above-mentioned in space distribution the outermost in coordinate point set X, then will gather X ' each point as initial value x ₀, according to formula (4), find the point farther apart from initial point, finally choose point farthest as III type sample point

4. reconstruct limit state function

In reconstruct limit state curve surface basis on, using the above-mentioned I～III type sample point searching out as new sample point, calculate the true ultimate limit state response of its correspondence, and add in coordinate point set X and limit state function response collection Y, then continue the mapping based on Optimization Learning machine OELM reconstruct X → Y, obtaining fiduciary level affects the higher limit state function of approximation accuracy in important area

4) fiduciary level is calculated:

In the complete limit state function of above-mentioned every reconstruct after, by Importance Sampling Method, with P _mpoint, for sampling center, utilizes agent model calculate corresponding failure probability pf ⁱ, when in time, stops calculating, wherein ε ₃=0.02, represent pf ⁱ～pf ^i-Mstandard deviation, by failure probability pf ⁱ, obtaining fiduciary level value is 1-pf ⁱ.