High-dimensional random system order reduction decoupling method and system based on stochastic near-conjugacy

Abstract

This invention discloses a method and system for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximation conjugacy, belonging to the field of complex system modeling and analysis technology. This method acquires multi-condition state data of the high-dimensional system, constructs a reduced-order analysis framework including embedding mapping, a low-dimensional decoupled dynamic model, and lifting mapping. Using stochastic approximation conjugacy as a constraint, it minimizes the total loss function to complete joint model training. After solving the system's stochastic response characteristics within a low-dimensional feature manifold, it maps back to the high-dimensional space to complete dynamic behavior analysis. The corresponding system includes modules for data acquisition, model construction, model training, low-dimensional solution, and mapping analysis. This invention solves the curse of dimensionality problem of traditional methods, ensures the reliability and interpretability of the reduced-order model, has strong versatility, and can be widely applied to intelligent operation and maintenance and control optimization of major engineering systems.

Description

Technical Field

[0001] This invention relates to the field of complex system modeling and analysis technology, specifically to a method and system for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy. Background Technology

[0002] In major engineering systems such as aero-engines and power systems, dynamic characteristics that are high-dimensional, strongly nonlinear, multi-parameter coupled, and influenced by stochastic excitations are common. Traditional mechanism-based modeling methods, such as directly performing Monte Carlo simulations of high-dimensional stochastic differential equations or solving the Fokker-Planck equations, face the curse of dimensionality problem when the system dimension is high, resulting in extremely high computational costs and failing to meet the needs of online analysis and real-time control.

[0003] Most existing dimensionality reduction methods are only applicable to linear systems or nonlinear systems with specific structures, and are difficult to handle the ubiquitous strong nonlinearity and stochastic coupling problems. In recent years, deep learning-based dimensionality reduction methods have been attempted to be applied to this field, but these methods usually lack rigorous theoretical guarantees and cannot ensure that the reduced model can retain the key stochastic dynamic characteristics of the original system, resulting in insufficient reliability and interpretability of their prediction results.

[0004] Therefore, there is an urgent need to develop a method that integrates data-driven approaches and physical mechanisms, and can provide mathematical guarantees for the stability, convergence, and error limits of the reduced-order analysis model, in order to overcome the technical bottleneck of online analysis of high-dimensional stochastic systems. Summary of the Invention

[0005] The purpose of this invention is to provide a method and system for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximation conjugacy, so as to solve the dual problems of low computational efficiency and lack of theoretical guarantees faced by existing technologies when dealing with high-dimensional coupled nonlinear stochastic systems.

[0006] The technical solution of the present invention to solve the above-mentioned technical problems is as follows: A method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy includes: S1: Obtain state data of a high-dimensional coupled nonlinear stochastic dynamic system under multiple operating conditions; S2: Construct a reduced-order analysis framework that includes an embedding mapping, a low-dimensional decoupled dynamic model, and a lifting mapping. The embedding mapping realizes the nonlinear mapping from the high-dimensional state space to the low-dimensional feature manifold, the lifting mapping realizes the reconstruction mapping from the low-dimensional feature manifold to the original high-dimensional state space, and the low-dimensional decoupled dynamic model is used to characterize the dynamic evolution law of the low-dimensional feature manifold. S3: Based on the state data, the embedding mapping, low-dimensional decoupled dynamics model and boosting mapping are jointly trained by minimizing a preset loss function. The training process satisfies the random approximate conjugate constraint. S4: Using the trained low-dimensional decoupled dynamic model, solve the stochastic response characteristics of the high-dimensional coupled nonlinear stochastic dynamic system within the low-dimensional characteristic manifold; S5: Through the lifting mapping, the random response characteristics within the low-dimensional feature manifold are mapped back to the original high-dimensional state space, thus completing the dynamic behavior analysis of the original high-dimensional system.

[0007] In a further embodiment, the multiple operating conditions in step S1 include different initial conditions, different random excitation intensities, and different system parameter configurations, and the state data is time series data.

[0008] According to claim 1, the method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximation conjugacy is characterized in that, in step S2, the dimension m of the low-dimensional feature manifold is much smaller than the dimension N of the original high-dimensional state space, and the low-dimensional decoupling dynamic model is an m-dimensional stochastic differential equation, the expression of which is: In the formula, z is the state variable of the low-dimensional feature manifold, g(z) is the drift function of the low-dimensional system, γ(z) is the diffusion function of the low-dimensional system, W(t) is the standard Wiener process, and g(z) and γ(z) are both parameterized by the neural network.

[0009] In a further embodiment, the preset loss function in step S3 is the total loss function, expressed as: In the formula, This is the total loss function; For reconstruction loss; For stochastic dynamic conjugate loss; Loss induced by dimensional independence; , , These are the weighting coefficients for each loss, and >0、 >0、 >0.

[0010] In a further embodiment, the formal definition of the stochastic approximate conjugate constraint is: (1) Drift conjugate constraint: (2) Diffusion conjugate constraint: (3) Reconstructing approximate constraints: In the formula, φ(x) is the embedding map, ψ(z) is the lifting map, and f(x) and σ(x) are the drift function and diffusion function of the original high-dimensional system, respectively. To embed the Jacobian matrix of the mapping with respect to the high-dimensional state x, This represents the upper limit of the drift conjugate error. This represents the upper limit of diffusion conjugation error. This represents the upper limit of the reconstruction error.

[0011] In a further embodiment, the random response characteristics in step S4 include time-series response, steady-state probability density distribution, first-turn time, extreme value distribution, and system failure probability.

[0012] In a further embodiment, the dynamic behavior analysis in step S5 includes safety analysis and stability analysis; S51: Security Analysis: Define a security function V(z) within a low-dimensional characteristic manifold. If V(z) satisfies Then, a composite security function V(φ(x)) is constructed to provide a security lower bound for the original high-dimensional system; S52: Stability Analysis: Define a stability function U(z) within a low-dimensional characteristic manifold. If U(z) satisfies Then, construct the composite stability function U(φ(x)) and prove that the trajectory of the original high-dimensional system is attracted to the path of the system. , , Within the defined bounded set of attraction.

[0013] In a further embodiment, the joint training in step S3 employs a gradient descent-type optimization algorithm, which includes any one of the following: stochastic gradient descent, Adam algorithm, Adagrad algorithm, and L-BFGS algorithm.

[0014] A high-dimensional stochastic system reduction and decoupling system based on stochastic approximation conjugacy for implementing the above method includes: a data acquisition module for acquiring state-time series data of a high-dimensional coupled nonlinear stochastic dynamic system under multiple operating conditions; a model building module for constructing a reduction analysis framework including embedding mapping, a low-dimensional decoupling dynamic model, and lifting mapping; a model training module for jointly training the parameters of the reduction analysis framework based on the state-time series data by minimizing the total loss function, ensuring that the training process satisfies the stochastic approximation conjugacy constraint; a low-dimensional solution module for solving the stochastic response characteristics of the system using the trained low-dimensional decoupling dynamic model; and a mapping analysis module for mapping the stochastic response characteristics of the low-dimensional characteristic manifold back to the original high-dimensional state space, completing the dynamic behavior analysis of the original high-dimensional system.

[0015] In a further embodiment, the system is integrated into the health management unit, real-time control unit, or fault early warning unit of a complex engineering system, which includes an aero-engine, a gas turbine, a launch vehicle, and a power system.

[0016] The present invention has the following beneficial effects: This invention transforms the analysis problem of high-dimensional coupled nonlinear stochastic systems into the solution problem of low-dimensional decoupled systems by constructing a reduced-order analysis framework based on stochastic approximate conjugacy. This effectively solves the curse of dimensionality problem faced by traditional methods, significantly improves computational efficiency, and ensures the reliability and interpretability of the reduced-order model through strict theoretical constraints. Meanwhile, this method does not rely on the dynamic mechanism of a specific system and has strong versatility. It can be applied to various complex systems that exhibit collective behavior and can provide core analysis tools for health monitoring, fault early warning and real-time control optimization of major engineering systems, thus possessing outstanding engineering application value. Attached Figure Description

[0017] Figure 1 This is a system block diagram of the present invention; Figure 2 The diagram illustrates the specific steps of the method of this invention. Figure 3 Comparison of covariance between high-dimensional systems and reconstructed systems. Detailed Implementation

[0018] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0019] This implementation method revolves around the overall process of acquiring state data of high-dimensional systems, constructing a reduction analysis framework, jointly training models, solving low-dimensional responses, and performing high-dimensional dynamic analysis, thus completing the entire process of reducing and decoupling high-dimensional systems. At the same time, it builds a system that includes a data acquisition module, a model construction module, a model training module, a low-dimensional solution module, and a mapping analysis module, realizing the engineering implementation of the reduction and decoupling method for high-dimensional stochastic systems based on stochastic approximation conjugacy.

[0020] High-dimensional system state data acquisition A fully coupled neural network consisting of N units is selected as an example of a high-dimensional coupled nonlinear stochastic dynamical system. The stochastic differential equation of this system is: in, For the state variables of the neuron, This refers to the neuronal membrane potential. To restore the variables, the overall state of the system ; k is the coupling strength coefficient, and ϵ, a, and b are system parameters; , Noise intensity coefficient; , It is an independent Wiener process that characterizes external stochastic stimuli.

[0021] For this system, multiple operating conditions are set, including different initial potentials, different noise intensities, and different coupling strengths. High-dimensional state time series data under each operating condition are generated through numerical simulation, which serves as the dataset for model training and validation.

[0022] Construction of a reduced-order analysis framework Embedding Mapping and Lifting Mapping: Embedding Mapping and lifting mapping Both employ a multilayer perceptron structure, where m ≪ 2N. The input to the embedding map is a high-dimensional state variable x, and the output is a low-dimensional feature manifold variable z; the input to the lifting map is a low-dimensional feature manifold variable z, and the output is a reconstructed high-dimensional state variable. .

[0023] Low-dimensional decoupling dynamics model: The low-dimensional decoupling dynamics model is defined as an m-dimensional stochastic differential equation: Where z represents the state variables of the low-dimensional feature manifold; g(z) is the drift function of the low-dimensional system, parameterized using a multilayer perceptron; and γ(z) is the diffusion function of the low-dimensional system, parameterized using a multilayer perceptron with an output lower triangular matrix to ensure its corresponding covariance matrix. Positive definiteness; W(t) is an m-dimensional standard Wiener process.

[0024] Joint training of models Total loss function setting: The goal of model training is to minimize the total loss function, which is expressed as: In the formula, This is the total loss function; For reconstruction loss; For stochastic dynamic conjugate loss; Loss induced by dimensional independence; >0、 >0、 >0 represents the weighting coefficient for each loss, which can be flexibly adjusted according to the characteristics of the actual system.

[0025] The reconstruction loss is used to constrain the consistency between the high-dimensional state and the reconstructed state, and its expression is: In the formula, M is the number of samples in the dataset; This represents the high-dimensional state of the i-th sample. It is a 2-norm.

[0026] The stochastic dynamical conjugate loss is used to constrain the dynamical consistency between the low-dimensional system and the original high-dimensional system, and its expression is: In the formula, Let be the state-time derivative of the i-th sample after mapping to the low-dimensional feature manifold; Let be the derivative of the Wiener process corresponding to the i-th sample.

[0027] The inter-dimensional independence induced loss is used to constrain the independence of each dimension of a low-dimensional system, achieving dimensional decoupling. Its expression is: In the formula, Let g(z) be the Jacobian matrix of the drift function g(z) with respect to the low-dimensional state z; The covariance matrix corresponding to the diffusion function; An operator for extracting off-diagonal elements of a matrix; It is the F-norm.

[0028] Random approximate conjugate constraint The model training process must satisfy the stochastic approximate conjugate constraint, specifically including: Drift conjugate constraint: Diffusion conjugate constraint: Reconstructing approximate constraints: In the above formula, f(x) and σ(x) are the drift function and diffusion function of the original high-dimensional system, respectively; The Jacobian matrix of the embedding mapping φ(x) with respect to the high-dimensional state x; , , These are the upper limits for drift conjugate error, diffusion conjugate error, and reconstruction error, respectively.

[0029] Optimization algorithm selection The stochastic gradient descent algorithm is used to jointly optimize the parameters of the embedding mapping, the low-dimensional decoupled dynamic model, and the lifting mapping. An appropriate learning rate and number of training rounds are set until the total loss function converges to a stable value.

[0030] Response solution within low-dimensional characteristic manifolds Monte Carlo simulation: Based on the trained low-dimensional decoupled dynamics model, Monte Carlo simulation is performed to generate a large number of low-dimensional state trajectories. The simulation formula is as follows: In the formula, This represents the low-dimensional state at time k. For time step; For time step The Wiener process increment within the simulation. Based on the low-dimensional trajectory obtained from the simulation, the response characteristics such as the frequency of synchronous firing events in the neural network are statistically analyzed.

[0031] Solving the Fokker-Planck equations: Numerical solution of the Fokker-Planck equations corresponding to the low-dimensional decoupled dynamic model to obtain the steady-state probability density distribution of the system. The equation expression is as follows: In the formula, Let z be the probability density function of the low-dimensional state z at time t; It is a divergence operator; This is the Laplace operator. By solving this equation, the steady-state probability density of the low-dimensional system can be obtained. The peak position and distribution can directly reflect the collective behavior pattern of the system.

[0032] Analysis of dynamic behavior of high-dimensional systems High-dimensional state reconstruction: The typical state trajectory z(t) within the low-dimensional feature manifold is input into the lifting map to obtain the reconstructed high-dimensional state trajectory. The reconstructed trajectory is compared with the real trajectory of the original high-dimensional system to verify the accuracy of the reduced-order model.

[0033] Security Analysis: Defining a Security Function within a Low-Dimensional Characteristic Manifold ,satisfy If the time derivative of the function satisfies Then, a composite security function V(φ(x)) is constructed, which can directly provide a security lower bound for the original high-dimensional system.

[0034] Stability Analysis: Define a stability function U(z) within a low-dimensional characteristic manifold, satisfying U(z) > 0. If the time derivative of this function satisfies... Then, we construct a composite stability function U(φ(x)). Through this composite stability function, we can prove that the trajectory of the original high-dimensional system will be attracted to a point where... , , Within the defined bounded set of attraction.

[0035] System modules work together The data acquisition module is responsible for collecting high-dimensional state time series data of the aforementioned neural network under multiple operating conditions; the model building module is responsible for building a reduced-order analysis framework that includes embedding mapping, low-dimensional decoupled dynamic model, and lifting mapping; the model training module is responsible for jointly optimizing the model parameters based on the dataset and the total loss function; the low-dimensional solution module is responsible for performing Monte Carlo simulation and solving the Fokker-Planck equation to obtain the stochastic response characteristics of the low-dimensional system; and the mapping analysis module is responsible for mapping the low-dimensional solution results back to the high-dimensional space to complete the safety and stability analysis of the original high-dimensional system.

[0036] This system can be directly integrated into the health management unit, real-time control unit, or fault early warning unit of complex engineering systems such as aero-engines and power systems, providing a basis for decision-making for online performance analysis of the system.

[0037] The principle of this invention: This invention uses stochastic approximate conjugacy as its theoretical anchor, integrating data-driven approaches and physical mechanisms to construct a closed-loop analysis framework of "high-dimensional embedding - low-dimensional decoupling - high-dimensional reconstruction," enabling efficient and reliable order reduction analysis of high-dimensional coupled nonlinear stochastic systems. This principle progresses layer by layer from three core levels: By building a two-way bridge between high-dimensional and low-dimensional spaces through embedding mapping and lifting mapping, the complex high-dimensional dynamic problem is transformed into a problem to be solved within a low-dimensional feature manifold. At the same time, the evolution law of the feature manifold is characterized by a low-dimensional decoupled dynamic model, and the model is forced to tend towards a diagonalized structure by dimensional independence-induced loss, thus achieving dimensional decoupling. Using stochastic approximate conjugacy as a rigid constraint, we establish a dynamic consistency relationship between the low-dimensional system and the original high-dimensional system from three dimensions: drift term matching, diffusion term matching, and state reconstruction accuracy. By combining the weighted optimization of reconstruction loss and stochastic dynamic conjugacy loss, we ensure the high fidelity of the low-dimensional analysis results while satisfying the conjugacy error bound. By leveraging efficient low-dimensional space solution tools to calculate stochastic response characteristics, and based on the theoretical guarantee of conjugacy, the safety and stability conclusions obtained in low-dimensional space are quantitatively transferred to high-dimensional systems, forming a complete logical chain of "dimensionality reduction solution - conclusion transfer". This entire principle not only overcomes the dimensionality curse bottleneck of traditional mechanistic modeling but also compensates for the lack of theoretical guarantees in existing deep learning dimensionality reduction methods. Ultimately, it forms a general analytical paradigm that combines computational efficiency and reliability, providing core theoretical support for the intelligent operation and maintenance and control optimization of various complex engineering systems.

[0038] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy, characterized in that, include: S1: Obtain state data of a high-dimensional coupled nonlinear stochastic dynamic system under multiple operating conditions; S2: Construct a reduced-order analysis framework that includes an embedding mapping, a low-dimensional decoupled dynamic model, and a lifting mapping. The embedding mapping realizes the nonlinear mapping from the high-dimensional state space to the low-dimensional feature manifold, the lifting mapping realizes the reconstruction mapping from the low-dimensional feature manifold to the original high-dimensional state space, and the low-dimensional decoupled dynamic model is used to characterize the dynamic evolution law of the low-dimensional feature manifold. S3: Based on the state data, the embedding mapping, low-dimensional decoupled dynamics model and boosting mapping are jointly trained by minimizing a preset loss function. The training process satisfies the random approximate conjugate constraint. S4: Using the trained low-dimensional decoupled dynamic model, solve the stochastic response characteristics of the high-dimensional coupled nonlinear stochastic dynamic system within the low-dimensional characteristic manifold; S5: Through the lifting mapping, the random response characteristics within the low-dimensional feature manifold are mapped back to the original high-dimensional state space, thus completing the dynamic behavior analysis of the original high-dimensional system.

2. The method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy as described in claim 1, characterized in that, The multiple operating conditions in step S1 include different initial conditions, different random excitation intensities, and different system parameter configurations, and the state data is time series data.

3. The method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy as described in claim 1, characterized in that, In step S2, the dimension m of the low-dimensional feature manifold is smaller than the dimension N of the original high-dimensional state space, and the low-dimensional decoupled dynamic model is an m-dimensional stochastic differential equation.

4. The method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy as described in claim 1, characterized in that, The preset loss function in step S3 is the total loss function, and its expression is: In the formula, This is the total loss function; For reconstruction loss; For stochastic dynamic conjugate loss; Loss induced by dimensional independence; , , These are the weighting coefficients for each loss, and >0、 >0、 >

0.

5. The method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy according to claim 4, characterized in that, The formal definition of the stochastic approximate conjugate constraint is: Drift conjugate constraint: Diffusion conjugate constraint: Reconstructing approximate constraints: In the formula, φ(x) is the embedding map, ψ(z) is the lifting map, and f(x) and σ(x) are the drift function and diffusion function of the original high-dimensional system, respectively. To embed the Jacobian matrix of the mapping with respect to the high-dimensional state x, This represents the upper limit of the drift conjugate error. This represents the upper limit of diffusion conjugation error. This represents the upper limit of the reconstruction error.

6. The method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy according to claim 1, characterized in that, The random response characteristics in step S4 include time-series response, steady-state probability density distribution, first-through time, extreme value distribution, and system failure probability.

7. The method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy according to claim 1, characterized in that, The dynamic behavior analysis in step S5 includes safety analysis and stability analysis. S51: Security Analysis: Define a security function V(z) within a low-dimensional characteristic manifold. If V(z) satisfies Then, a composite security function V(φ(x)) is constructed to provide a security lower bound for the original high-dimensional system; S52: Stability Analysis: Define a stability function U(z) within a low-dimensional characteristic manifold. If U(z) satisfies Then, construct the composite stability function U(φ(x)) and prove that the trajectory of the original high-dimensional system is attracted to the path of the system. , , Within the defined bounded set of attraction.

8. The method for order reduction and decoupling of high-dimensional stochastic systems based on stochastic approximate conjugacy according to claim 1, characterized in that, The joint training in step S3 employs a gradient descent-type optimization algorithm, which includes any one of the following: stochastic gradient descent, Adam algorithm, Adagrad algorithm, and L-BFGS algorithm.

9. A reduced-order decoupling system for a high-dimensional stochastic system based on stochastic approximation conjugacy for implementing the method of any one of claims 1-8, characterized in that, include: Data acquisition module: used to acquire state-time series data of high-dimensional coupled nonlinear stochastic dynamic systems under multiple operating conditions; Model building module: used to build a reduced-order analysis framework that includes embedding mapping, low-dimensional decoupled dynamic models, and lifting mapping; Model training module: used to jointly train the parameters of the reduced-order analysis framework based on the state time series data by minimizing the total loss function, ensuring that the training process satisfies the random approximate conjugate constraint; Low-dimensional solution module: used to solve the stochastic response characteristics of the system using the trained low-dimensional decoupled dynamic model; Mapping Analysis Module: Used to map the stochastic response characteristics of low-dimensional feature manifolds back to the original high-dimensional state space, thereby completing the dynamic behavior analysis of the original high-dimensional system.

10. The order-reduced decoupling system for high-dimensional stochastic systems based on stochastic approximate conjugacy according to claim 9, characterized in that, The system is integrated into the health management unit, real-time control unit, or fault early warning unit of a complex engineering system, which includes aero-engines, gas turbines, launch vehicles, and power systems.

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