Algorithm 1.1 ILLE algorithm

Input: $\left(\overleftarrow{x},\overrightarrow{x}\right)=\left\{\left({\overleftarrow{x}}_1,{\overleftarrow{x}}_1\right),\left({\overleftarrow{x}}_2,{\overleftarrow{x}}_2\right),\mathrm{...},\left({\overleftarrow{x}}_n,{\overleftarrow{x}}_n\right)\right\}\subset \left({\overleftarrow{R}}^{D\times n},{\overleftarrow{R}}^{D\times n}\right),$$\left({\overleftarrow{x}}_j,{\overrightarrow{x}}_i\right)\in \left({\overleftarrow{R}}^D,{\overleftarrow{R}}^D\right)\left(i=1,2,\mathrm{...},n\right)$

Output:$\left(\overleftarrow{Y},\overrightarrow{Y}\right)=\left\{\left({\overleftarrow{Y}}_1,{\overrightarrow{Y}}_1\right),\left({\overleftarrow{Y}}_2,{\overrightarrow{Y}}_2\right),\left({\overleftarrow{Y}}_n,{\overrightarrow{Y}}_n\right)\right\}\subset \left({\overleftarrow{R}}^{d\times n},{\overleftarrow{R}}^{d\times n}\right),$$\left({\overleftarrow{y}}_j,{\overrightarrow{y}}_j\right)\in \left({\overleftarrow{R}}^d,{\overleftarrow{R}}^d\right)\left(l=1,2,\mathrm{...},n\right)$

For each $\left({\overleftarrow{x}}_j,{\overleftarrow{x}}_j\right)$ do

1.Choose neighbourhood: The k neighbourhoods $\left\{\left({\overleftarrow{x}}_i^1,{\overleftarrow{x}}_i^1\right)|=1,2,\mathrm{...},k\right\}$ of each point $\left({\overleftarrow{x}}_i,{\overrightarrow{x}}_i\right)$ are determined according to the DGSOM method, where k may be different for each neighbourhood. A linear combination of $\left\{\left({\overleftarrow{x}}_j^1,{\overleftarrow{x}}_j^1\right)|l=1,2,\mathrm{...},k\right\}$ is used to denote $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right),$ that is, the hyperboloid passing through $\left\{\left({\overleftarrow{x}}_j^1,{\overleftarrow{x}}_j^1\right)|l=1,2,\mathrm{...},k\right\}$ is used to approximate the local block near $\left\{\left({\overleftarrow{x}}_j^i,{\overleftarrow{x}}_j^i\right)|l=1,2,\mathrm{...},k\right\}$ in the manifold.

2.Get weight: For each $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right),\text{find}\left({\overleftarrow{W}}_i,{\overleftarrow{W}}_i\right)=\left(\left({\overleftarrow{W}}_j^1,{\overleftarrow{W}}_j^1\right),\left({\overleftarrow{W}}_j^2,{\overleftarrow{W}}_j^2\right),\mathrm{...},\left({\overleftarrow{W}}_j^k,{\overleftarrow{W}}_j^k\right)\right)$ that minimizes $\left(\overleftarrow{Q},\overleftarrow{Q}\right)\left({\overleftarrow{w}}_i,{\overleftarrow{w}}_i\right)={\Vert \left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)-{\displaystyle \sum _{i=1}^k\left({\overleftarrow{w}}_i^l,{\overleftarrow{w}}_i^l\right)\left({\overleftarrow{x}}_i^1,{\overleftarrow{x}}_i^1\right)}\Vert }^2+\left(\overleftarrow{\eta },\overrightarrow{\eta }\right){\Vert {\displaystyle \sum _{l=1}^k\left({\overleftarrow{w}}_i^1,{\overleftarrow{w}}_i^1\right)}\Vert }^2.$ and finally obtain the dynamic fuzzy topology matrix (, ).

3. Reduce dimension: Minimize the loss function

where $\left(\overleftarrow{Y},\overrightarrow{Y}\right)=\left(\left({\overleftarrow{Y}}_1,{\overrightarrow{Y}}_2\right),\left({\overleftarrow{Y}}_2,{\overrightarrow{Y}}_2\right),\mathrm{...},\left({\overleftarrow{Y}}_n,{\overrightarrow{Y}}_n\right)\right)\in \left({\overleftarrow{R}}^{d\times n},{\overleftarrow{R}}^{d\times n}\right),\left({\overleftarrow{y}}_i,{\overleftarrow{y}}_i\right)\in \left({\overleftarrow{R}}^d,{\overleftarrow{R}}^d\right)\left(i=\right)$ 1,2,. . .n). To make the solution unique, we add the constraint to $\left({\overleftarrow{Y}}_i,{\overleftarrow{Y}}_i\right)\in \left({\overleftarrow{R}}^d,{\overleftarrow{R}}^d\right)(i=$ 1,2,. . .n):

where ${\left(\overleftarrow{I},\overrightarrow{I}\right)}_{d\times d}$ is the dynamic fuzzy unit matrix.

Algorithm 1.2 Dynamic fuzzy machine learning

Input: $\left(\overleftarrow{x},\overrightarrow{x}\right)$

Output: $\left({\overleftarrow{y}}_k,{\overleftarrow{y}}_k\right)$

$//\left({\overleftarrow{u}}_{{\overleftarrow{A}}_l^j},{\overleftarrow{u}}_{{\overleftarrow{A}}_l^j}\right),\left({\overleftarrow{u}}_{{\overleftarrow{b}}_l^j},{\overleftarrow{u}}_{{\overrightarrow{b}}_l^j}\right)$ is the corresponding dynamic fuzzy membership, $\left({\overleftarrow{y}}_d,{\overleftarrow{y}}_d\right)$ is the desired output //

(1)Use the ILLE method to reduce the dimension of the DFD;

(2)

(3)${\left(\overrightarrow{v},\overleftarrow{v}\right)}^l={\displaystyle \prod _{j=1}^n\left({\overleftarrow{u}}_{{\overleftarrow{A}}_j^1},{\overleftarrow{u}}_{{\overleftarrow{A}}_j^1}\right)\left[\left(\overleftarrow{x},\overrightarrow{x}\right)\left(k-l+1\right)\right]}\left({\overleftarrow{u}}_{{\overrightarrow{b}}_j^1},{\overleftarrow{u}}_{{\overleftarrow{b}}_j^1}\right)\left[\left(\overleftarrow{u},\overrightarrow{u}\right)\left(k\right)\right];$

(4)$\left({\overleftarrow{y}}_k,{\overrightarrow{y}}_k\right)=\frac{{\displaystyle \sum _{j-1}^k\left({\overleftarrow{h}}_j,{\overleftarrow{h}}_j\right)\left(\overleftarrow{x},\overrightarrow{x}\right)}\left(i\right)}{{\displaystyle \sum _{j-1}^k\left({\overleftarrow{h}}_j,{\overleftarrow{h}}_j\right)}};$

(5)$\left({\overleftarrow{e}}_k,{\overleftarrow{e}}_k\right)=\left({\overleftarrow{y}}_d,{\overleftarrow{y}}_d\right)-\left({\overleftarrow{y}}_k,{\overleftarrow{y}}_k\right);$

(6)$\left(\overleftarrow{\varphi },\overrightarrow{\varphi }\right)\left(\left({\overleftarrow{e}}_k,{\overleftarrow{e}}_k\right)\left({\overleftarrow{c}}_k,{\overleftarrow{c}}_k\right)\right)=2.\exp \left(-\left({\overleftarrow{\lambda }}_k,{\overleftarrow{\lambda }}_k\right)\left|\left({\overleftarrow{e}}_k,{\overleftarrow{e}}_k\right)-\left({\overleftarrow{c}}_k,{\overleftarrow{c}}_k\right)\right|\right);$$//\left({\overleftarrow{\lambda }}_k,{\overleftarrow{\lambda }}_k\right)\mathrm{and}\left({\overleftarrow{c}}_k,{\overleftarrow{c}}_k\right)$ are determined as in [17] //

(7)$\left(\overleftarrow{p},\overrightarrow{p}\right)\left(k\right)=\frac{1}{2}{\displaystyle \sum _{j=1}^k{\left|\left({\overleftarrow{e}}_i,{\overleftarrow{e}}_i\right)\right|}^2;//\text{performance//}}$

(8)$\text{if}\left|\left({\overleftarrow{e}}_k,{\overleftarrow{e}}_k\right)>\left(\overleftarrow{\epsilon },\overrightarrow{\epsilon }\right)\right|.$

//$\left(\overleftarrow{\epsilon },\overrightarrow{\epsilon }\right)$ is the maximum tolerance error, its value can be based on system requirements or experience. If the learning error is greater than the maximum tolerance, then the learning is invalid//

then Discard $\left({\overleftarrow{y}}_k,{\overleftarrow{y}}_k\right),$ feedback to the rule base, notify the adjustment rules of the parameters, perform a parametric learning algorithm.

$\left(\overleftarrow{u},\overrightarrow{u}\right)\left(k\right)=\left(\overleftarrow{u},\overrightarrow{u}\right)\left(k\right)+\alpha \left({\overleftarrow{e}}_k,{\overleftarrow{e}}_k\right)$ //α is the gain learning coefficient

Continue from scratch

else if | $\left({\overleftarrow{e}}_k,{\overleftarrow{e}}_k\right)\le \left(\overleftarrow{\delta },\overrightarrow{\delta }\right)|$

//$\left(\overleftarrow{\delta },\overrightarrow{\delta }\right)$ is an acceptable error, much smaller than $\left(\overrightarrow{\epsilon },\overleftarrow{\epsilon }\right),$, its value can be given according to system requirements or experience. If the error after learning is less than (or equal to) this acceptable level, it is considered to have reached the accuracy of the learning requirements //

then The learning process is finished, corrected, and output

$\left({\overleftarrow{y}}_k,{\overleftarrow{y}}_k\right)=\left({\overleftarrow{y}}_k,{\overleftarrow{y}}_k\right)+\beta \left(\overleftarrow{\varphi },\overrightarrow{\varphi }\right)\left(\left({\overleftarrow{e}}_k,{\overleftarrow{e}}_k\right)\left({\overleftarrow{c}}_k,{\overleftarrow{c}}_k\right)\right)=\left({\overleftarrow{y}}_k,{\overleftarrow{y}}_k\right)//\beta$ is the correction factor

else Feedback to the rule base, notify the adjustment rules in the parameters, and implement the parameter learning algorithm.

Continue from scratch;

end if

end if