A flow field reconstruction and deduction method based on K-A theorem
By using the KAN network based on the KA theorem to decouple and reduce the dimensions of irregular waves, an equivalent regular wave model is constructed, which solves the problems of low efficiency and insufficient physical interpretability in the existing technology, and realizes efficient and accurate flow field reconstruction and ship response modeling.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- QINGDAO INNOVATION & DEV CENT OF HARBIN ENG UNIV
- Filing Date
- 2025-10-20
- Publication Date
- 2026-06-23
AI Technical Summary
Existing technologies are inefficient, have limited accuracy, and lack physical interpretability when reconstructing irregular wave flow fields in real seas. They are also difficult to effectively utilize the physical prior knowledge of KAN networks for high-dimensional and highly coupled wave data processing.
By using a KAN network based on the KA theorem, irregular wave time series are decoupled and dimensionality reduced, transforming them into a structured, low-dimensional equivalent regular wave combination model. The parameters are optimized using B-spline curves to construct an equivalent wave superposition model, and the network is fine-tuned using a backpropagation algorithm to reduce errors.
It enables accurate and efficient modeling of wave flow fields under complex sea conditions, improves the robustness and generalization ability of the model, provides physical interpretability, simplifies the learning process, and reduces dependence on data.
Smart Images
Figure CN121302973B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of ship hydrodynamics, specifically to a flow field reconstruction and deduction method based on the KA theorem. Background Technology
[0002] In the field of shipbuilding and ocean engineering, irregular waves and flow field variations in real-world marine environments significantly impact ship attitude, drag, and structural response. Traditional physical model-based methods suffer from limited modeling accuracy and high computational costs when facing complex boundary conditions and high-dimensional nonlinear disturbances. In recent years, machine learning and deep learning technologies have shown great promise in solving partial differential equations and modeling physical fields. In particular, the Kolmogorov-Arnold Network (KAN), based on the Kolmogorov-Arnold representation theorem (KA theorem), replaces fixed activation functions with learnable parameterized spline functions, enhancing physical interpretability while maintaining model expressiveness, and exhibiting stronger generalization ability and higher solution accuracy. Addressing the high-dimensional nonlinear characteristics of irregular waves under real-world marine conditions, this invention proposes a wave decoupling and flow field reconstruction method based on the KA theorem. The KAN network is used to perform function decoupling and dimensionality reduction on measured wave time series, extracting a wave superposition model composed of equivalent regular wave groups, significantly reducing model input redundancy and improving data structure stability. This wave equivalent model can serve as an important input basis for digital flow field reconstruction and ship response simulation, providing key support for hydrodynamic analysis and multi-source coupled environment modeling under complex sea conditions.
[0003] Current technical approaches tend to treat KAN as an end-to-end regression model, directly inputting high-dimensional, coupled measured wave time series containing information such as direction, amplitude, and period, aiming to directly map to the final flow field parameters. The fundamental flaw of this approach is that it reduces a system with a clear physical structure (i.e., irregular waves can be considered as a superposition of multiple regular waves) to a purely high-dimensional mathematical fitting problem. The network is forced to explore a vast input space filled with redundancy and noise, significantly increasing the model's computational burden and data dependence. More importantly, due to the lack of effective constraints from prior physical knowledge, its learning process struggles to converge to a globally optimal solution with clear physical meaning, thus impairing the model's generalization ability and stability. Raw wave data from actual sea measurements is high-dimensional, strongly coupled, and contains a large amount of noise and redundant information. When this high-dimensional, raw wave data is directly input into the network, the spline function shapes learned by the network edges become extremely complex in order to fit the complex internal and external relationships, making it impossible for technicians to intuitively and effectively identify the dominant physical processes or key influencing parameters. Summary of the Invention
[0004] This invention aims to provide a KAN-based flow field reconstruction and extrapolation method. By decoupling and reducing the dimensionality of measured high-dimensional, nonlinear irregular wave time series, it transforms them into a structured, low-dimensional, and physically meaningful equivalent regular wave combination model. The goal is to overcome the technical difficulties faced by existing technologies in reconstructing irregular wave flow fields in real seas, such as low efficiency, limited accuracy, and insufficient physical interpretability. This model serves as the basic input for subsequent digital flow field reconstruction and ship attitude response modeling, thereby enabling accurate, efficient, and reliable modeling and extrapolation of wave flow fields under complex sea conditions.
[0005] 1. A flow field reconstruction and derivation method based on the KA theorem, characterized by comprising the following steps:
[0006] S1. Obtain the direction of the irregular wave at a preset time. Amplitude ,cycle ,frequency The velocity data components u(z,t) and v(z,t) at depth z will be used for each time step. The collected wave parameters are integrated into a high-dimensional wave vector. ;
[0007] S2. Use the B-spline curve as a learnable function of the KAN network, and use the high-dimensional wave vector... The input is fed into the KAN network, and the output is a low-dimensional list of parameters, containing... Amplitude of the dominant equivalent regular wave angular frequency and initial phase The decoupling objective function for constructing an equivalent wave superposition model using the parameter list is described. and the real target Compare and calculate the loss function The loss function As a feedback signal, the control point parameters of all B-spline curves in the network are fine-tuned using the backpropagation algorithm until the error is corrected. Converging to the minimum value or reaching the preset number of training iterations;
[0008] S3. Parameter list output by the KAN network Construct the final wave superposition model Calculate the macroscopic statistical parameters; combine these macroscopic statistical parameters with the parameter list set, and encapsulate them into the final output vector. .
[0009] Preferably, in step S1, the high-dimensional wave vector Described in the following vector form Irregular wave state at time:
[0010] ;
[0011] In the formula, This represents the number of components required to describe the irregular wave state, corresponding to readings at multiple measurement points or the number of wavelets obtained through spectral analysis.
[0012] Preferably, in step S2, the parameter vector of the KAN network output is:
[0013] ;
[0014] In the parameter vector , and They represent The equivalent amplitude, equivalent angular frequency, and equivalent initial phase of a dominant equivalent regular wave;
[0015] This set of parameter vectors is used to construct, in real time, a decoupling objective function representing the equivalent wave superposition model. :
[0016] .
[0017] Preferably, a loss function is defined. Quantify the fitting error:
[0018] ;
[0019] In the formula, The loss function represents the ground truth values used as the training baseline. The physical meaning is within a time period Internally, the equivalent wave model constructed from network parameters. Compared with the actual observed wave process The root mean square error between the two values is used by the backpropagation algorithm to guide the optimization and adjustment of all B-spline function control points within the network, until the loss function is reached. It converges to the minimum value.
[0020] Preferably, in step S3, the equivalent amplitude output by the KAN network is... Equivalent angular frequency and equivalent initial phase The final wave superposition model is constructed as follows:
[0021] ;
[0022] In the formula, Represents any time and location The equivalent water surface rise and fall, which is caused by Wavenumber is formed by the linear superposition of regular waves; The equivalent angular frequency output by the network is obtained through the wave dispersion relation. The calculation shows that:
[0023] ;
[0024] In the formula, This is the acceleration due to gravity.
[0025] Preferably, in step S3, the average wave height :
[0026] ;
[0027] Effective period : ;
[0028] in, From the maximum equivalent amplitude Sure;
[0029] Dominant wave direction : ;
[0030] Output data vector as follows: .
[0031] Preferably, in step S3, the macroscopic statistical parameters include: significant wave height. Effective period With the dominant wave direction Output data vector : .
[0032] Preferably, in step S2, the model is constructed before calculating the loss function. and real data Perform Fourier transform or wavelet transform respectively to convert the time-domain signal into the frequency-domain spectrum or time-frequency spectrum; define a loss function in the frequency domain and quantize the error by calculating the difference between the two spectra.
[0033] Compared with the prior art, the present invention has the following beneficial effects:
[0034] I. A fundamental leap from functionally interpretable to physically interpretable
[0035] The conventional approach to KAN is to use it as an end-to-end regression tool. While its internal B-spline function is mathematically interpretable, when the entire network is used to fit a complex high-dimensional mapping, its overall behavior still tends to resemble a gray box that is difficult to understand intuitively. This invention changes the task objective of KAN from directly predicting results to inferring the parameters of physical models. This allows technicians not only to see the prediction results but also to deeply understand the dominant physical components constituting the current complex sea state. It achieves a transformation from simply fitting mathematical functions to deconstructing physical processes, providing a solid foundation for model validation, diagnosis, and trust.
[0036] II. The robustness and generalization ability of the model are significantly enhanced.
[0037] Applying KAN directly to regression on high-dimensional, noisy measured data results in the network searching within a vast function space, making it highly susceptible to learning spurious correlations and noise in the data. This leads to the model's sensitivity to input perturbations and limited generalization ability. This invention embeds the prior physical knowledge that irregular waves can be superimposed from regular waves into the training objective, imposing a strong physical constraint on the network's learning process. This guides the network to find physically real solutions rather than mathematically optimal fits, effectively avoiding overfitting. Therefore, the model trained using this method is insensitive to data noise and exhibits stronger robustness and generalization ability when facing unfamiliar sea conditions because it grasps the physical essence of the problem.
[0038] Third, it significantly improves learning efficiency and reduces dependence on data.
[0039] Conventional end-to-end learning paradigms require networks to explore the complex relationships between the entire high-dimensional input space and the output from scratch, which typically requires massive amounts of data and lengthy training times. This invention transforms a complex function fitting problem into a parameter optimization problem with a more defined objective and a smaller search space. The network no longer needs to discover that fluctuations are sinusoidal, but only needs to determine the key parameters of these sinusoids. This simplification of the task allows the network to converge faster and potentially learn high-quality models even with relatively limited data.
[0040] Fourth, it created highly modular and reusable engineering application modules.
[0041] Conventional end-to-end models are strongly coupled to specific tasks; if the downstream application's objective changes, the entire model may need to be redesigned and retrained. The structured data vectors output by this invention... As an independent and highly condensed information module, it can be flexibly used as a unified input for various downstream applications, greatly enhancing the integration capability and reuse value of this invention in complex engineering systems. Attached Figure Description
[0042] Figure 1 This is a network structure diagram of the present invention;
[0043] Figure 2 For technical flowcharts;
[0044] Figure 3 To decouple irregular waves into regular wave diagrams. Detailed Implementation
[0045] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the scope of protection of the present invention.
[0046] I. Data Preparation and Input Standardization
[0047] To achieve the objectives of this invention, the technical solution first defines and standardizes the raw measurement data from the real marine environment, thus establishing an accurate and standardized input foundation for subsequent function decoupling and dimensionality reduction based on KAN.
[0048] The data source for this scheme is real-time navigation survey data, which refers to data continuously collected during the actual navigation of a ship using shipboard or deployed professional marine environmental monitoring equipment. The aim is to simultaneously acquire two types of core environmental information: firstly, a description of the overall characteristics of the wave field around the ship, including the direction of irregular waves at a specific moment. Amplitude ,cycle With frequency Secondly, it is the flow field information recorded at specific measurement points, that is, the flow velocity components of the water body in the X and Y directions at different depths z are u(z,t) and v(z,t), respectively.
[0049] Furthermore, in order to transform these dynamically changing multidimensional measurement information into a structured format suitable for network model processing, this invention continuously records the evolution of key physical quantities describing the irregular wave state over time, and records each discrete moment... The wave measurement data are integrated into a unified high-dimensional wave vector. It can be represented in the following vector form:
[0050] ;
[0051] In the formula, vector A comprehensive description The irregular wave state at any given moment, and This represents the number of components required to describe the state, which can correspond to readings from multiple measurement points or the number of wavelets obtained through spectral analysis.
[0052] To eliminate the potential adverse effects of dimensional differences between different physical quantities on the model training process and to significantly improve the convergence speed and stability of subsequent network model training, this scheme performs strict normalization processing on physical quantities of the same category before inputting data into the network. All input features are transformed to a similar numerical scale, thereby providing a high-quality, unbiased data foundation for the subsequent decoupling of the KAN execution function.
[0053] II. Function Decoupling and Dimensionality Reduction Based on KAN
[0054] The task at this stage is to process the high-dimensional, normalized irregular wave vectors generated in the previous stage. By using the KAN network for deep processing, a set of low-dimensional core parameters describing the main physical characteristics of waves can be extracted. In other words, a complex, multivariate physical mapping problem is decomposed into a series of simpler, interpretable function combinations.
[0055] In this invention, the functions on the KAN edges are composed of parameterized B-spline curves. During training, the network can adaptively learn and iteratively optimize the control points to fit the optimal functional relationship implied by the local data transformation it processes, thereby giving the network higher approximation accuracy and flexibility.
[0056] The theoretical basis of this architecture stems from the KA representation theorem, which mathematically proves that any complex multivariable continuous function... All of these can be represented as nested summations of a finite number of one-dimensional functions, and their general form is as follows:
[0057] ;
[0058] In the formula, the inner function For the j-th input variable The transformation is performed, with a total of n input variables and an external function. The results of the transformation of the internal functions are combined. q represents the order of the B-spline basis functions.
[0059] Unlike traditional neural networks, the activation functions on the edges of a KAN network are learnable functions composed of parameterized B-spline curves, i.e. ,in This is a trainable B-spline function. By optimizing the control point parameters of the B-spline during training, the network can adaptively learn the function shape that best describes the intrinsic relationships of the data.
[0060] When high-dimensional wave vector When passing through the network, the structure of KAN enables it to decouple and reduce the highly coupled nonlinear relationships in irregular waves at the function level along the network path. That is, it extracts the dominant wave components from the original multidimensional wave vector sequence, and equates the complex random wave process to a linear superposition of a finite set of regular waves.
[0061] Therefore, the output of the KAN network is a dimensionality-reduced parameter vector that contains the physical properties of each dominant equivalent wave core, and its specific form is as follows:
[0062] ;
[0063] In the parameter vector , and They represent The equivalent amplitude, equivalent angular frequency, and equivalent initial phase of a dominant equivalent regular wave.
[0064] Next, using the parameter vectors output by the KAN network, an analytical decoupling objective function representing the equivalent wave superposition model is constructed in real time. :
[0065] ;
[0066] This formula represents a mathematical reconstruction and simplified expression of the original irregular wave. To drive the network to learn the optimal parameter combination, the following loss function is defined. Quantify the fitting error:
[0067] ;
[0068] In the formula, The ground truth values representing the training baseline are typically key scalar time series extracted from pre-prepared raw measurement data, hence the loss function... The physical meaning is that within a time period Internally, the equivalent wave model constructed from network parameters. Compared with the actual observed wave process The root mean square error between them. This error value will guide the optimization and adjustment of all B-spline function control points within the network through the backpropagation algorithm, until the loss function... The training process converges to a minimum. This closed-loop training process ensures that the extracted low-dimensional parameters can most accurately reproduce the dominant physical characteristics of the original wave field.
[0069] III. Equivalent Model Construction and Structured Output
[0070] First, based on the equivalent amplitude output of the KAN network Equivalent angular frequency and equivalent initial phase The final wave superposition model is constructed as follows:
[0071] ;
[0072] In the formula, Represents any time and location The equivalent water surface rise and fall, which is caused by It is formed by the linear superposition of several regular waves; the wave number of the i-th regular wave. The equivalent angular frequency output by the network is obtained through the wave dispersion relation. The calculation shows that:
[0073] ;
[0074] In the formula, It is the acceleration due to gravity. Let be the wavelength of the i-th regular wave. In this way, this scheme combines data-driven network learning results with explicit fluid dynamics physics laws, ensuring the physical consistency of the constructed model.
[0075] After constructing the aforementioned microscopic superposition model, this scheme further performs the key macroscopic parameter extraction steps to calculate sea state statistical characteristics that are universally applicable in the field of marine engineering.
[0076] average wave height :
[0077] ;
[0078] Effective period :
[0079] ;
[0080] in, From the maximum equivalent amplitude It is confirmed that m0 represents the zeroth spectral moment, which is the integral of the wave energy spectrum over the entire frequency range.
[0081] Dominant wave direction :
[0082] ;
[0083] Through the above steps, this scheme further refines the microscopic physical parameters output by the KAN network into macroscopic, standardized sea state descriptors.
[0084] Finally, a comprehensive, structured data vector is output. as follows:
[0085] ;
[0086] This structured data vector Its sole purpose is to serve as a high-quality base wave vector for subsequent flow field reconstruction or ship attitude response modeling.
[0087] This invention cleans and compresses complex and redundant raw data into low-dimensional, robust data with clear physical meaning. When used as input for the next stage of intelligent modeling tasks, it can significantly improve the computational efficiency, prediction accuracy, and stability of subsequent models, thus providing key technical support for hydrodynamic analysis and multi-source coupled environment modeling under complex sea conditions.
[0088] In a preferred embodiment, the learnable activation functions within the KAN network are parameterized by B-spline curves. As a finer-grained alternative, B-spline basis functions can be replaced by other types of function bases, such as radial basis functions (RBFs) or different types of orthogonal polynomials (e.g., Chebyshev polynomials). In this alternative, the learnable activation functions on the network edges are represented by linear combinations of these alternative basis functions, and the goal of network training shifts from optimizing the control points of the B-splines to optimizing the combined weights of these alternative basis functions or their own parameters. Since these function bases also possess good function approximation capabilities, using them as building blocks for learnable activation functions can also achieve adaptive learning of complex functional relationships.
[0089] In a preferred embodiment, it is used to measure the equivalent model. With real data Loss function of the error between This is based on the root mean square error in the time domain. As an alternative, the error in the frequency domain or the time-frequency joint error can be used as the optimization objective.
[0090] In this alternative, the model needs to be constructed before calculating the loss function. and real data The signal is converted from the time domain signal to the frequency domain spectrum or time-spectrum by performing Fourier transform or wavelet transform respectively. Then, a loss function is defined in the frequency domain, for example, by calculating the difference between the two spectra (such as the difference in spectral energy, the difference in spectral peak frequency, etc.) to quantify the error. This frequency-domain-based loss function is insensitive to the phase drift of the signal and may be more robust in some application scenarios, better guiding the network to learn an equivalent wave model with the same energy distribution as the real sea state.
[0091] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.
Claims
1. A flow field reconstruction and deduction method based on the KA theorem, characterized in that, Includes the following steps: S1. Obtain the direction of the irregular wave at a preset time. Amplitude ,cycle angular frequency The velocity data components u(z,t) and v(z,t) at depth z will be used for each time step. The collected wave parameters are integrated into a high-dimensional wave vector. ; S2. Use the B-spline curve as a learnable function of the KAN network, and use the high-dimensional wave vector... The input is fed into the KAN network, and the output is a low-dimensional list of parameters, containing... Amplitude of the dominant equivalent regular wave angular frequency and initial phase ; The decoupling objective function for constructing an equivalent wave superposition model using the aforementioned parameter list is as follows. and the real target Compare and calculate the loss function The loss function As a feedback signal, the control point parameters of all B-spline curves in the network are fine-tuned using the backpropagation algorithm until the loss function is reached. The KAN network converges to the minimum value or reaches the preset number of training iterations; the output parameter vector of the KAN network is: Using this set of parameter vectors, a decoupling objective function representing the equivalent wave superposition model is constructed in real time. : ； S3. Construct the final wave stacking model using the parameter list output by the KAN network. Calculate the macroeconomic statistical parameters, including: significant wave height. Effective period With the dominant wave direction ; The macroscopic statistical parameters and parameter list are combined and encapsulated into a final output vector. Based on the equivalent amplitude, equivalent angular frequency, and equivalent initial phase output by the KAN network, the final wave superposition model is constructed as follows: In the formula, Represents at any time and location The equivalent water surface rise and fall, which is caused by The wave number is formed by the linear superposition of several dominant equivalent regular waves. It is obtained through the dispersion relation of waves.
2. The flow field reconstruction and derivation method based on the KA theorem according to claim 1, characterized in that, In step S1, the high-dimensional wave vector Described in the following vector form Irregular wave state at time: ； In the formula, This represents the number of components required to describe the irregular wave state, corresponding to readings at multiple measurement points or the number of wavelets obtained through spectral analysis.
3. The flow field reconstruction and derivation method based on the KA theorem according to claim 1, characterized in that, Define loss function Quantify the fitting error: ； In the formula, Representing the true objective, it guides the optimization and adjustment of all B-spline function control points within the network, up to the loss function. It converges to the minimum value.
4. The flow field reconstruction and derivation method based on the KA theorem according to claim 1, characterized in that, In step S3, the effective wave height : ； ； Effective period : Where m0 represents the zeroth spectral moment, From the maximum equivalent amplitude Determined; Dominant Wave Direction : ； Final output vector for: 。 5. The flow field reconstruction and derivation method based on the KA theorem according to claim 1, characterized in that, In step S2, the model is constructed before calculating the loss function. and the real goal Perform Fourier transform or wavelet transform respectively to convert the time-domain signal into the frequency-domain spectrum or time-frequency spectrum; define a loss function in the frequency domain and quantize the error by calculating the difference between the two spectra.