Underwater dynamic modeling method and device based on physical information neural network

Abstract

The application discloses a kind of underwater dynamic modeling methods and devices based on physical information neural network, it is related to underwater vehicle dynamics modeling technical field, comprising: obtaining the historical operation data of underwater vehicle under different marine environment working conditions and multiple navigation maneuver state, pre-constructed hybrid dynamics model and weighted joint loss function;The historical operation data is preprocessed, and standard training sample set is constructed;With standard training sample set as input, with minimizing weighted joint loss function as target, the hybrid dynamics model is trained offline, and the target hybrid dynamics model is obtained;The target hybrid dynamics model is deployed to underwater vehicle, and the acceleration of underwater vehicle is predicted, the problems that pure data-driven model is poor in physical consistency and pure physical model parameter identification are solved, and the model after deployment can output high-precision acceleration prediction value in real time, improve the navigation stability and control precision of underwater vehicle in complex marine environment.

Description

Technical Field

[0001] This application relates to the field of underwater vehicle dynamics modeling technology, and in particular to underwater dynamic modeling methods and devices based on physical information neural networks. Background Technology

[0002] Unmanned Underwater Vehicles (UUVs) are playing an increasingly important role in fields such as marine resource exploration, underwater infrastructure monitoring, environmental monitoring, and even national defense and security. High-precision real-time dynamic models are the core foundation for achieving accurate trajectory planning, stable motion control, and successful mission execution for UUVs, and are also key to building their digital twins. This model needs to accurately predict changes in the vehicle's speed, attitude, and other states based on control inputs in complex environments.

[0003] Currently, dynamic modeling of underwater unmanned vehicles (UAVs) typically employs physical models based on hydrodynamic principles or purely data-driven neural network models. While physical models possess clear physical interpretability, they struggle to accurately characterize complex, unmodeled dynamics such as high angle-of-attack flow field separation, thruster interaction, and biofouling. Purely data-driven models, while capable of fitting nonlinear characteristics, lack physical constraints, are prone to generating non-physical interpretations that violate mechanical laws, and their generalization ability significantly decreases outside the training data coverage. In recent years, physically-informed neural networks have attempted to balance data fitting and physical consistency by embedding physical equations into loss functions; however, existing methods often employ static training, where model parameters are not updated after deployment. Due to the highly time-varying nature of the marine environment and the slow drift of UAV characteristics caused by load changes and surface fouling, a continuous distributional shift exists between training and actual operational data. The prediction error of static models accumulates and amplifies with increasing mission duration, making it difficult to meet the demands of long-term autonomous operations for high-precision, high-reliability dynamic predictions. Summary of the Invention

[0004] The main purpose of this application is to provide an underwater dynamic modeling method and device based on physical information neural networks, which aims to solve the technical problem of the decline in prediction accuracy of static physical information neural network models in time-varying marine environments in current dynamic modeling.

[0005] To achieve the above objectives, this application proposes an underwater dynamic modeling method based on a physical information neural network, wherein the underwater dynamic modeling method based on a physical information neural network includes: Acquire historical operational data of underwater vehicles under different marine environmental conditions and various navigation maneuvers, as well as pre-constructed hybrid dynamics models and weighted joint loss functions; The historical operational data is preprocessed to construct a standard training sample set; Using the standard training sample set as input and minimizing the weighted joint loss function as the objective, the hybrid power model is trained offline to obtain the target hybrid power model; The target hybrid dynamics model is deployed to the underwater vehicle to predict its acceleration.

[0006] In one embodiment, the step of obtaining a pre-built hybrid dynamics model includes: Construct a physical model for calculating nominal acceleration; Construct a neural network residual model for calculating residual acceleration; The hybrid dynamics model is obtained based on the physical model and the neural network residual model.

[0007] In one embodiment, the step of constructing a physical model for calculating nominal acceleration includes: Acquire sample data of physical parameters of underwater vehicles; A dynamic equation is constructed based on the physical parameter sample data; A physical model for calculating nominal acceleration is constructed based on the aforementioned dynamic equations.

[0008] In one embodiment, the step of constructing a neural network residual model for calculating residual acceleration includes: Acquire twelve-dimensional state data consisting of six-dimensional velocity data and three-dimensional attitude angle data of the underwater vehicle; Feature extraction is performed on the twelve-dimensional state data to obtain a twelve-dimensional state vector; The input layer is configured according to the twelve-dimensional state vector, and the input layer is used to receive the twelve-dimensional state vector. The hidden layer's neurons and activation functions are configured to process the twelve-dimensional state vector; Set a linear activation function for the output layer. The output layer is used to receive and process the twelve-dimensional state vector processed by the hidden layer and output a six-dimensional residual acceleration. A neural network residual model is constructed based on the input layer, the hidden layer, and the output layer.

[0009] In one embodiment, the step of obtaining the pre-constructed weighted joint loss function includes: Obtain the mean square error between historical measured acceleration and historical predicted acceleration; Set the data fitting loss term based on the mean square error; Substituting the historical predicted acceleration into the dynamic equation, we obtain the equation residuals; Performing a L2 norm operation on the residuals of the equations yields the kinetic physical residual loss term; Set the weight coefficients for the data fitting loss term and the kinetic physical residual loss term, and construct a weighted joint loss function based on the weight coefficients, the data fitting loss term, and the kinetic physical residual loss term.

[0010] In one embodiment, the step of offline training the hybrid power model using the standard training sample set as input and minimizing the weighted joint loss function to obtain the target hybrid power model includes: Configure the training hyperparameters of the hybrid dynamics model, including optimizer type, initial learning rate, batch size, and gradient clipping threshold; The standard training sample set is input into the hybrid power model, the optimizer calculates the gradient based on the weighted joint loss function, and the parameters of the hybrid power model are updated. During training, a learning rate scheduling strategy is implemented, which automatically reduces the learning rate when the validation set loss no longer decreases within a preset number of rounds. Gradient clipping is performed during training. When the gradient norm exceeds the gradient clipping threshold, the gradient is scaled. Monitor the changes in the validation set loss, and stop training when the validation set loss reaches the preset iteration termination condition to obtain the target hybrid dynamics model.

[0011] In one embodiment, the step of acquiring historical operational data of an underwater vehicle under different marine environmental conditions and various navigation maneuvers includes: Control the underwater vehicle to perform various navigation maneuver tests under different preset marine environmental conditions; During the execution of the various navigation maneuver state tests, the sensors mounted on the underwater vehicle collect in real time the historical six-dimensional velocity data, historical three-dimensional attitude angle data, historical control input data, and historical measured acceleration data measured by the inertial measurement unit. The collected six-dimensional velocity data, three-dimensional attitude angle data, control input data, and measured acceleration data are time-stamped to obtain historical operational data of the underwater vehicle under different marine environmental conditions and various navigation maneuvers.

[0012] In one embodiment, the step of preprocessing the historical running data to construct a standard training sample set includes: The historical operating data is smoothed and denoised to obtain the smoothed and denoised data. The smoothed and denoised data is then normalized to obtain a standard training sample set.

[0013] In one embodiment, the step of deploying the target hybrid dynamics model to an underwater vehicle and predicting the acceleration of the underwater vehicle includes: The target hybrid dynamics model is deployed to the underwater vehicle, and real-time six-dimensional velocity data and real-time three-dimensional attitude angle data are collected during the operation of the underwater vehicle. The real-time six-dimensional velocity data and the real-time three-dimensional attitude angle data are input into the target hybrid power dynamics model to obtain the nominal acceleration output by the physical model in the target hybrid power dynamics model and the residual acceleration output by the neural network residual model in the target hybrid power dynamics model. The nominal acceleration and the residual acceleration are linearly superimposed to obtain the real-time predicted acceleration of the underwater vehicle.

[0014] Furthermore, to achieve the above objectives, this application also proposes an underwater dynamic modeling device based on a physical information neural network, the underwater dynamic modeling device based on a physical information neural network comprising: The acquisition module is used to acquire historical operational data of underwater vehicles under different marine environmental conditions and various navigation maneuvers, as well as pre-built hybrid dynamics models and weighted joint loss functions; The preprocessing module is used to preprocess the historical running data and construct a standard training sample set; The training module is used to train the hybrid power model offline with the standard training sample set as input and the goal of minimizing the weighted joint loss function, so as to obtain the target hybrid power model. The prediction module is used to deploy the target hybrid dynamics model to the underwater vehicle and predict the acceleration of the underwater vehicle.

[0015] Furthermore, to achieve the above objectives, this application also proposes an underwater dynamic modeling device based on a physical information neural network. The device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. The computer program is configured to implement the steps of the underwater dynamic modeling method based on a physical information neural network as described above.

[0016] In addition, to achieve the above objectives, this application also proposes a storage medium, which is a computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, it implements the steps of the underwater dynamic modeling method based on physical information neural networks as described above.

[0017] In addition, to achieve the above objectives, this application also provides a computer program product, which includes a computer program that, when executed by a processor, implements the steps of the underwater dynamic modeling method based on physical information neural networks as described above.

[0018] One or more technical solutions proposed in this application have at least the following technical effects: 1) By collecting historical operational data covering multiple operating conditions and maneuver states, the dynamic characteristics of underwater vehicles in complex marine environments can be comprehensively reflected, providing a rich and representative data foundation for subsequent model training. The pre-built hybrid dynamics model combines the advantages of physical mechanisms and data-driven approaches, while the weighted joint loss function balances data fitting accuracy with physical constraints. Data preprocessing eliminates noise interference and dimensional differences in the original data, and the construction of a standard training sample set ensures the standardization and consistency of model training data, improving the stability of model training. Offline training avoids the occupation of real-time computing resources of the underwater vehicle by online training. The optimization objective of the weighted joint loss function enables the model to accurately fit the measured data while strictly adhering to the physical laws of underwater dynamics, effectively solving the problems of poor physical consistency in pure data-driven models and difficulty in parameter identification in pure physics models, thus improving the model's generalization ability and prediction accuracy. The deployed model can output high-precision acceleration prediction values ​​in real time, providing reliable dynamic parameter support for the attitude control, trajectory tracking, and fault diagnosis of the underwater vehicle, improving the navigation stability and control accuracy of the underwater vehicle in complex marine environments.

[0019] 2) A hybrid dynamics model is derived by constructing a physical model for calculating nominal acceleration and a neural network residual model for calculating residual acceleration. The physical model is based on mature underwater vehicle dynamics mechanisms, accurately describing the nominal motion characteristics of the vehicle and possessing clear physical meaning and good interpretability. The neural network residual model can adaptively learn residual terms such as nonlinear disturbances, unmodeled dynamics, and environmental disturbances that the physical model cannot cover. The hybrid dynamics model formed by combining the two retains the mechanistic advantages of the physical model while using the neural network to compensate for the accuracy deficiencies of the physical model. This achieves a deep fusion of mechanism and data, significantly improving the model's ability to represent the complex dynamic characteristics of underwater vehicles and solving the technical challenge of balancing accuracy and physical consistency with a single model.

[0020] 3) A data fitting loss term is set by obtaining the mean square error between historical measured acceleration and historical predicted acceleration. The historical predicted acceleration is substituted into the dynamic equation to obtain the equation residuals. The dynamic-physical residual loss term is obtained by L2 calculation, and then a weighted joint loss function is constructed by setting weight coefficients. The data fitting loss term uses the mean square error to measure the deviation between the model's predicted value and the measured value, ensuring the model's fitting accuracy to the measured data. The dynamic-physical residual loss term constrains the model output to satisfy the underwater dynamic equation, ensuring the physical consistency of the model and avoiding prediction results that violate physical laws. The weight coefficients can flexibly adjust the weight ratio of data fitting and physical constraints to adapt to the modeling needs under different working conditions. The construction of the weighted joint loss function enables the model to pursue both data fitting accuracy and physical mechanism constraints during training, effectively solving the problems of overfitting and poor physical consistency that are prone to occur in pure data-driven models, and improving the model's generalization ability and reliability. Attached Figure Description

[0021] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application.

[0022] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0023] Figure 1 This is a flowchart illustrating an embodiment of the underwater dynamic modeling method based on physical information neural networks in this application. Figure 2 This is a bar chart comparing the average RMSE performance of an embodiment of the underwater dynamic modeling method based on physical information neural networks in this application. Figure 3 This is yet another flowchart illustrating an embodiment of the underwater dynamic modeling method based on physical information neural networks in this application. Figure 4 This is yet another flowchart illustrating an embodiment of the underwater dynamic modeling method based on physical information neural networks in this application. Figure 5 This is yet another flowchart illustrating an embodiment of the underwater dynamic modeling method based on physical information neural networks in this application. Figure 6 This is a schematic diagram of the offline pre-training loss convergence curve provided in an embodiment of the underwater dynamic modeling method based on physical information neural network of this application; Figure 7A simplified flowchart is provided for an embodiment of the underwater dynamic modeling method based on physical information neural networks in this application; Figure 8 This is a schematic diagram of the module structure of the underwater dynamic modeling device based on physical information neural network according to an embodiment of this application; Figure 9 This is a schematic diagram of the equipment structure of the hardware operating environment involved in the underwater dynamic modeling method based on physical information neural network in the embodiments of this application.

[0024] The purpose, features, and advantages of this application will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0025] It should be understood that the specific embodiments described herein are merely illustrative of the technical solutions of this application and are not intended to limit this application.

[0026] To better understand the technical solution of this application, a detailed description will be provided below in conjunction with the accompanying drawings and specific implementation methods.

[0027] The main solution of this application embodiment is as follows: acquiring historical operating data of the underwater vehicle under different marine environmental conditions and various navigation maneuvers, a pre-constructed hybrid dynamics model, and a weighted joint loss function; preprocessing the historical operating data to construct a standard training sample set; using the standard training sample set as input and minimizing the weighted joint loss function as the objective, offline training of the hybrid dynamics model to obtain a target hybrid dynamics model; deploying the target hybrid dynamics model to the underwater vehicle to predict the acceleration of the underwater vehicle.

[0028] Because existing physical information neural network models still employ a static training mode, these models are deployed and used after training on historical data, without being updated. Due to the time-varying characteristics of the marine environment and the slow drift of UUV characteristics themselves, there is a continuous distribution shift between the training data and the actual operational data. This causes the prediction error of the static physical information neural network model to gradually accumulate and amplify as the mission duration increases.

[0029] It should be noted that the executing entity in this embodiment can be a computing service device with data processing, network communication, and program execution functions, such as a tablet computer, personal computer, or mobile phone, or an electronic device capable of performing the above functions, such as an underwater dynamic modeling device based on a physical information neural network. The following description uses an underwater dynamic modeling device based on a physical information neural network as an example to illustrate this embodiment and the following embodiments. All actions involving the acquisition of signals, information, or data in this application are performed in accordance with the relevant data protection regulations of the locality and with authorization from the owner of the corresponding device.

[0030] Based on this, embodiments of this application provide an underwater dynamic modeling method based on a physical information neural network, referring to... Figure 1 , Figure 1 This is a flowchart illustrating an embodiment of the underwater dynamic modeling method based on physical information neural networks according to this application.

[0031] In this embodiment, the underwater dynamic modeling method based on physical information neural networks includes steps S10~S40: Step S10: Obtain historical operational data of the underwater vehicle under different marine environmental conditions and various navigation maneuvers, as well as a pre-built hybrid dynamics model and a weighted joint loss function.

[0032] In this embodiment, underwater vehicles include not only unmanned underwater vehicles (UUVs), but also manned submersibles, underwater gliders, autonomous underwater robots, and other carriers that need to perform motion control or status monitoring underwater.

[0033] Different marine environmental conditions refer to various hydrological conditions that a vessel may encounter during actual operations, such as calm water environments, uniform ocean current environments, wave disturbance environments, and complex marine environments with drastic changes in temperature, salinity, and depth.

[0034] The various navigation maneuver states cover the typical motion modes of the aircraft when performing missions, such as constant depth straight navigation, horizontal plane turning, vertical plane depth change, Z-shaped maneuver, and hovering positioning.

[0035] Historical operational data refers to the collection and storage of time-series data acquired by various sensors mounted on the aircraft under the aforementioned operating conditions and states. This typically includes, but is not limited to, six-dimensional velocity (linear velocity and angular velocity), three-dimensional attitude angles, control input commands, and measured acceleration. It should be understood that the sources of historical operational data are not limited to actual sea trial data; they can also include high-fidelity numerical simulation data, tank towing test data, or hardware-in-the-loop simulation data, as long as these data can reflect the true dynamic response characteristics of the aircraft.

[0036] The pre-built hybrid dynamics model is the core architecture of this application. It is not a single physical equation or a black-box network, but rather a parallel combination of a physical model representing known mainstream dynamics and a neural network residual model representing unknown higher-order dynamics. The weighted joint loss function is the bridge connecting data and physics. It includes a data fitting loss term and a dynamic physics residual loss term, and adjusts the optimization weight of the two through weight coefficients.

[0037] In one feasible implementation, step S10, "acquiring historical operational data of the underwater vehicle under different marine environmental conditions and various navigation maneuvers," may include steps A11 to A13: Step A11: Control the underwater vehicle to perform various navigation maneuver tests under different preset marine environmental conditions; Understandably, controlling an underwater vehicle to perform various navigation maneuvers under preset marine environmental conditions is the data foundation for building a highly generalizable model. The different marine environmental conditions referred to in this application aim to cover typical hydrological conditions that underwater vehicles may encounter in actual operations. These include not only ideal calm water environments but also complex environments with uniform ocean currents, irregular wave disturbances, and drastic changes in temperature, salinity, and depth. Multiple navigation maneuvers refer to control modes that can fully stimulate the vehicle's six degrees of freedom dynamics, including but not limited to steady-state navigation at a constant depth, horizontal turning, vertical depth variation, Z-maneuvers, and hovering. Among these, Z-maneuvers and turning motions can effectively stimulate the coupled dynamics of sway and yaw, while depth variation motions are used to identify the hydrodynamic coefficients of heave and pitch. It should be understood that this requirement for completeness of operating conditions and maneuver states is to ensure that training data can traverse the vehicle's main operating envelope, avoiding models that are only effective under specific stable conditions but fail under complex maneuvers, thus distinguishing it from the traditional practice of obtaining data only in laboratory tanks or a single ideal environment.

[0038] Step A12: During the execution of the various navigation maneuver state tests, the historical six-dimensional velocity data, historical three-dimensional attitude angle data, historical control input data, and historical measured acceleration data measured by the inertial measurement unit of the underwater vehicle are collected in real time by the sensors mounted on the underwater vehicle. During the execution of various navigation maneuver tests, real-time data acquisition via sensors mounted on the underwater vehicle is a crucial step in obtaining the true physical values. The acquired data must constitute a complete dynamic closed loop, specifically including: historical six-dimensional velocity data (three-axis velocity and three-axis angular velocity), typically provided by a Doppler velocimeter (DVL) or acoustic Doppler current profiler; historical three-dimensional attitude angle data (roll, pitch, and yaw), typically provided by attitude sensors or a compass; historical control input data (such as thruster speed, rudder angle, thrust commands, etc.), directly recorded from the vehicle's control system; and historical measured acceleration data obtained by the inertial measurement unit (IMU), serving as the labeled true values ​​for model training.

[0039] Step A13: Timestamp the collected six-dimensional velocity data, three-dimensional attitude angle data, control input data, and measured acceleration data to obtain historical operating data of the underwater vehicle under different marine environmental conditions and various navigation maneuvers.

[0040] The specific implementation method of timestamp alignment can be flexibly selected according to the characteristics of the sensor. For example, when aligning low-frequency state data (such as DVL velocity) to a high-frequency IMU time reference, linear interpolation, cubic spline interpolation, or predictive interpolation based on kinematic models can be used; for discrete control commands, zero-order hold or nearest neighbor matching strategies can be used.

[0041] Since underwater dynamics modeling essentially establishes an instantaneous mapping between state, control, and acceleration, if the data is not strictly aligned—for example, incorrectly associating acceleration at time t with velocity at time t+Δt—the model will learn spurious correlations rather than true physical laws. Especially during highly dynamic maneuvers, time deviations of tens of milliseconds can lead to a complete mismatch between peak acceleration and velocity trends, causing the neural network to fit erroneous residuals that violate causality. Therefore, only data that has undergone strict timestamp alignment can be considered physically consistent historical operational data, providing reliable factual basis for subsequently constructing standard training sample sets and offline training of hybrid dynamics models.

[0042] Step S20: Preprocess the historical operation data to construct a standard training sample set.

[0043] Understandably, raw historical operating data often contains problems such as sensor noise, dimensional differences, and timestamp misalignment. Directly using it for training can lead to difficulties in model convergence or the learning of incorrect physical laws.

[0044] Therefore, historical operational data needs to be preprocessed, including smoothing, noise reduction, and normalization, to filter out high-frequency measurement noise while preserving the low-frequency trend characteristics of the signal, and to eliminate the order-of-magnitude differences between different physical quantities (such as velocity, angle, and acceleration), making the data distribution more consistent with the input assumptions of the neural network. The dataset obtained after the above processing is the standard training sample set. This standard training sample set not only contains the correspondence between input state variables and output acceleration variables, but also implicitly contains the dynamic evolution law of the vehicle under specific operating conditions.

[0045] In one feasible implementation, step S20 may include steps A21-A22: Step A21: Perform smoothing and noise reduction on the historical running data to obtain smoothed and noise-reduced data; In the step of smoothing and denoising historical operational data, considering that the data collected by sensors such as inertial measurement units and Doppler velocimeters on underwater vehicles in complex marine environments are often superimposed with high-frequency vibration noise, electromagnetic interference and quantization errors, if noisy data is directly input into the model, the neural network is very likely to mistakenly identify random noise as unmodeled high-order dynamics and fit it, causing the model to lose its physical generalization ability.

[0046] Therefore, this embodiment preferably uses a Savitzky-Golay filter to smooth and reduce noise in historical operational data. Compared to traditional moving average filtering or low-pass filtering, the Savitzky-Golay filter, based on the principle of local polynomial least squares fitting, can effectively filter out high-frequency noise while preserving the higher-order moment characteristics of the signal to the greatest extent, such as peak height, width, and rate of change. This is particularly crucial for underwater dynamics modeling, because the acceleration spikes generated by the vehicle during transient operations such as Z-maneuvers and emergency braking are the core characteristics characterizing its hydrodynamic coefficients. If these spikes are smoothed out by traditional filters, the model will lose crucial dynamic response information.

[0047] As a preferred example, for inertial measurement unit data with a sampling rate of 100Hz, the filter window size can be set to 21 and the polynomial order to 3. In practical applications, the window size and polynomial order should be adaptively adjusted according to the sensor sampling rate, signal bandwidth, and maneuver frequency. The smoothed and denoised data obtained through this step eliminates the interference of measurement noise on gradient calculation while fully preserving the dynamic trends that reflect the true physical laws, providing a high signal-to-noise ratio data foundation for subsequent training.

[0048] Step A22: Normalize the smoothed and denoised data to obtain a standard training sample set.

[0049] Because the state variables of underwater vehicles have significant heterogeneous dimensional characteristics—for example, linear velocity is measured in meters per second (m / s), attitude angle in radians (rad), and acceleration in meters per second squared (m / s²)—the numerical ranges of these physical quantities can differ by several orders of magnitude. If the raw dimensional data is used directly for training, the gradient update of the neural network will be dominated by large numerical features, causing the learning of small numerical features (such as attitude angle changes) to be submerged. More seriously, in the weighted joint loss function, the calculation results of the data fitting loss term and the dynamic physical residual loss term will be on completely different orders of magnitude due to the difference in dimensions. This causes the carefully designed weight coefficients in the aforementioned embodiments to lose their balancing and adjusting effect, and the physical constraints to degenerate into invalid terms. Therefore, this embodiment adopts a zero-mean unit variance normalization method, that is, calculating the mean and standard deviation of each feature dimension separately, and transforming it according to the following formula: in, This represents the normalized i-th eigenvalue. This represents the data after smoothing and noise reduction. This represents the mean of the feature on the training set. This represents the standard deviation of the feature on the training set.

[0050] Through this transformation, all input features are mapped to a standard normal distribution with a mean of 0 and a standard deviation of 1. This not only eliminates the negative impact of dimensional differences on network optimization and accelerates the convergence speed of gradient descent, but more importantly, it brings the data loss term and the physical loss term back to the same numerical scale. This ensures that the weighted joint loss function can truly drive the model to evolve towards a dual optimal solution of "data fit" and "physical consistency." The dataset obtained after the above smoothing, denoising, and normalization processes constitutes a high-quality standard training sample set. It provides a standardized and physically reliable input foundation for the offline training of the hybrid dynamics model, fundamentally guaranteeing the prediction accuracy and robustness of the final target model.

[0051] Step S30: Using the standard training sample set as input and minimizing the weighted joint loss function as the objective, the hybrid power model is trained offline to obtain the target hybrid power model.

[0052] In practice, unlike traditional pure data-driven training, the optimization objective of this embodiment is to minimize the weighted joint loss function. This means that in each iteration, the model must not only learn how to fit historical measured data (guided by the data fitting loss term), but also learn how to satisfy the constraints of the underwater dynamics equations (guided by the dynamic physics residual loss term).

[0053] This dual constraint mechanism forces the model to search for the optimal solution in the solution space that conforms to both observed facts and physical laws, thus effectively avoiding the non-physical problems that are prone to occur in pure neural network models. After offline training, the physical parameters and neural network weights in the hybrid power dynamics model are optimized to a stable state, and the resulting model is the target hybrid power dynamics model. This model combines the interpretability of a physical model with the strong nonlinear fitting capability of a neural network, and can maintain good generalization performance under unseen operating conditions.

[0054] Step S40: Deploy the target hybrid dynamics model to the underwater vehicle and predict the acceleration of the underwater vehicle.

[0055] During actual operation, the airborne system collects real-time six-dimensional velocity data and three-dimensional attitude angle data, and inputs them into the target hybrid dynamics model. The physical model and the neural network residual model within the model are calculated in parallel, outputting nominal acceleration and residual acceleration respectively, and the final real-time predicted acceleration is obtained by linear superposition.

[0056] This predicted acceleration can be used for trajectory tracking control, attitude stabilization control, or state synchronization of digital twin systems. It should be understood that although this embodiment uses acceleration prediction as an example, based on the same modeling approach, by deploying the target hybrid dynamics model onto the underwater vehicle, a complete closed loop from offline modeling to online application is achieved. This ensures that theoretical research results can be effectively transformed into engineering practice capabilities, significantly improving the perception and control level of underwater vehicles in complex dynamic environments.

[0057] In one feasible implementation, step S40 may include steps A31 to A33: Step A31: Deploy the target hybrid dynamics model onto the underwater vehicle and collect real-time six-dimensional velocity data and real-time three-dimensional attitude angle data during the operation of the underwater vehicle; Deploying the target hybrid dynamics model to an underwater vehicle refers to storing the parameter files of the offline-trained and validated target hybrid dynamics model, including the hydrodynamic parameters of the physical model and the weight matrix and bias vector of the neural network residual model, in the onboard embedded computing unit of the underwater vehicle. At this stage, the model no longer undergoes parameter updates but operates as a deterministic inference engine. Simultaneously, sensors mounted on the underwater vehicle collect real-time state data during operation, namely real-time six-dimensional velocity data and real-time three-dimensional attitude angle data.

[0058] Step A32: Input the real-time six-dimensional velocity data and the real-time three-dimensional attitude angle data into the target hybrid electric vehicle model to obtain the nominal acceleration output by the physical model in the target hybrid electric vehicle model and the residual acceleration output by the neural network residual model in the target hybrid electric vehicle model; After real-time six-dimensional velocity data and real-time three-dimensional attitude angle data are input into the target hybrid dynamics model, a parallel computing mechanism is triggered within the model. Specifically, the physical model in the target hybrid dynamics model calculates the nominal acceleration conforming to classical mechanics laws in real time based on the built-in dynamic equations and physical parameters. Simultaneously, the neural network residual model in the target hybrid dynamics model calculates the residual acceleration used to compensate for unmodeled dynamics through forward propagation, based on the same input state or the twelve-dimensional state vector after feature extraction. This parallel computing architecture not only fully utilizes the multi-core concurrency capabilities of modern processors and shortens the time consumed in a single inference, but more importantly, it reflects the physical essence of hierarchical modeling: the basic dynamic response and the higher-order nonlinear correction are two independent but complementary physical processes that occur synchronously on the time axis, rather than being sequentially dependent cascaded. Therefore, the parallel output ensures precise phase alignment of the two components, avoiding fusion errors caused by time lag introduced by serial processing.

[0059] Step A33: Linearly superimpose the nominal acceleration and the residual acceleration to obtain the real-time predicted acceleration of the underwater vehicle.

[0060] It is understandable that the nominal acceleration vector output by the physical model module is... The residual module of the neural network outputs the residual acceleration vector as follows: By linearly adding the corresponding dimensions of the two, the final acceleration prediction vector of the hybrid model is obtained. The calculation is as follows: In the formula All are 6-dimensional vectors, corresponding to the six degrees of freedom of UUV sway, roll, heave, pitch, yaw, and bow accelerations, respectively. The linear superposition and fusion result serves as the final acceleration prediction value of the hybrid model.

[0061] This application adopts a hierarchical hybrid modeling approach, dividing the model into two independent units: a physical model module and a neural network residual module. During the training phase, a weighted joint optimization strategy of data loss plus physical loss is used, and the underwater dynamic equations are embedded into the network training as soft constraints. With the complete data preprocessing and offline pre-training process, the hybrid model is built, and the final real-time predicted acceleration has both physical consistency and data fitting accuracy, which can accurately reflect the real dynamic response of the underwater vehicle in the current state.

[0062] like Figure 2 As shown in the figure, this intuitively demonstrates the performance advantages of the underwater dynamic modeling method based on physical information neural networks proposed in this embodiment in actual sea trials. In a full-duration sea trial lasting 4200 seconds, the average root mean square error (RMSE) of the proposed method was only 0.0036, a 90.7% reduction compared to the traditional static PINN model (RMSE of 0.0386) and a 79.4% reduction compared to the purely online PINN model without physical constraints (RMSE of 0.0175). This significant improvement in accuracy fully verifies the effectiveness of the hierarchical hybrid architecture and linear superposition fusion mechanism. Static PINN lacks adaptability to time-varying environmental characteristics, leading to rapid error accumulation as the mission duration increases; while purely online PINN can track some changes, it is prone to generating non-physical interpretations that violate physical laws under conditions of sparse data or insufficient excitation, resulting in a higher overall error. This application provides stable baseline constraints through a physical model, absorbs environmental drift and unmodeled dynamics in real time through a neural network residual model, and achieves seamless integration of the two through linear superposition. Thus, it maintains high-precision and high-reliability dynamic prediction capabilities throughout the entire mission cycle, effectively supporting the long-term autonomous operation requirements of underwater vehicles in complex marine environments.

[0063] This embodiment provides an underwater dynamic modeling method based on a physical information neural network. By collecting historical operational data covering multiple operating conditions and maneuver states, it can comprehensively reflect the dynamic characteristics of underwater vehicles in complex marine environments, providing a rich and representative data foundation for subsequent model training. The pre-constructed hybrid dynamics model combines the advantages of physical mechanisms and data-driven approaches, while the weighted joint loss function balances data fitting accuracy with physical constraints. Data preprocessing eliminates noise interference and dimensional differences in the original data, and the construction of a standard training sample set ensures the standardization and consistency of model training data, improving the stability of model training. Offline training avoids the real-time computational resources of the underwater vehicle occupied by online training. The optimization objective of the weighted joint loss function enables the model to accurately fit the measured data while strictly adhering to the physical laws of underwater dynamics, effectively solving the problems of poor physical consistency in pure data-driven models and difficulty in parameter identification in pure physical models, thus improving the model's generalization ability and prediction accuracy. Once deployed, the model can output high-precision acceleration predictions in real time, providing reliable dynamic parameter support for the attitude control, trajectory tracking, and fault diagnosis of underwater vehicles, thereby improving the navigation stability and control accuracy of underwater vehicles in complex marine environments.

[0064] Based on the above embodiments of this application, in another embodiment of this application, the same or similar content as the above embodiments can be referred to the above description, and will not be repeated hereafter. Based on this, please refer to... Figure 3The step of "obtaining the pre-built hybrid dynamics model" in step S10 includes steps S101 to S103: Step S101: Construct a physical model for calculating nominal acceleration.

[0065] This embodiment employs a layered decoupled modeling strategy, breaking down the complex dynamic response of an underwater vehicle into a deterministic foundational component and an uncertain correction component, each handled by models of different natures. Specifically, a physical model is constructed to calculate the nominal acceleration, aiming to establish a physical benchmark for the vehicle's motion based on classical mechanics principles. This physical model is typically built upon Fossen's six-degree-of-freedom rigid body dynamics equations, incorporating known physical quantities such as the vehicle's mass, moment of inertia, added mass coefficient, damping coefficient, and hydrostatic parameters. Through these parameterized mathematical equations, the physical model can calculate the theoretically expected acceleration response of the vehicle under given conditions and control inputs—the nominal acceleration. This process fully complies with Newton's second law and fundamental laws of fluid mechanics, possessing clear physical interpretability and stability, ensuring that the model output does not deviate too far from actual physical behavior, and providing a reliable "anchor point" for the entire hybrid model.

[0066] In one feasible implementation, step S101 may include steps B11 to B13: Step B11: Obtain sample data of the physical parameters of the underwater vehicle; Physical parameter sample data refers to a set of various parameters that can characterize the rigid body motion and fluid interaction characteristics of an underwater vehicle. In this embodiment, physical parameter sample data is not limited to parameters from a specific source or of a specific type. It may include, but is not limited to, inertial parameters such as vehicle mass, three-axis rotational inertia, center of gravity coordinates, hydrostatic parameters such as displacement volume, buoyancy magnitude, center of buoyancy coordinates, and hydrodynamic coefficients such as added mass matrix, linear and nonlinear damping coefficients, and lift coefficients. In terms of data sources, physical parameter sample data can be obtained through actual ship testing, computational fluid dynamics (CFD) simulation calculations, or by consulting design manuals or estimating using empirical formulas. It can even be a fusion of data from multiple sources.

[0067] For example, for a certain type of torpedo-type underwater vehicle, the measured mass is 89.165 kg, the displacement volume is 0.089 m³, and the external surface area is 0.85 m². The three-axis rotational inertia is obtained by integral solution through a three-dimensional CAD model. The coordinates of the center of gravity are measured and calibrated as (0.02 m, 0, 0), and the coordinates of the center of buoyancy are (0, 0, 0.01 m). The core added mass coefficient and damping coefficient are selected by combining the medium and low speed conditions (speed less than 2 m / s and angular rate less than 0.3 rad / s).

[0068] In practical applications, the specific content and accuracy of physical parameter sample data depend on the type of aircraft, the operating environment, and the modeling requirements. By acquiring a wide range of multi-source physical parameter sample data, a solid material foundation can be provided for subsequent equation construction, avoiding the failure of the physical model due to missing or distorted parameters.

[0069] Step B12: Construct the dynamic equations based on the physical parameter sample data; In this embodiment, the dynamic equations preferably adopt the Fossen six-degree-of-freedom rigid body dynamic equation framework. This framework comprehensively considers factors such as rigid body inertial forces, Coriolis forces, centripetal forces, hydrostatic forces, and hydrodynamics, and can comprehensively describe the motion laws of underwater vehicles in three-dimensional space. The construction process is not a simple application of fixed formulas, but rather an instantiation and adaptive simplification of the general equations based on physical parameter sample data. For example, for underwater vehicles with a slender body configuration, some small cross-damping coefficients of coupling terms can be ignored; for ROVs with symmetrical structures, the symmetry can be used to simplify the additional mass matrix. This parameterized construction method makes the method of this application highly versatile. It can be adapted to underwater vehicles of different shapes and sizes simply by replacing the corresponding physical parameter sample data and adjusting the equation structure. By constructing dynamic equations that conform to Newton's second law and the basic principles of fluid mechanics, it is ensured that the subsequently calculated nominal acceleration strictly follows the physical conservation laws, providing an insurmountable physical boundary for the entire hybrid dynamics model and effectively preventing prediction results that violate the common sense of mechanics that may be produced by a purely data-driven model.

[0070] For a torpedo-type UUV platform, the basic physical parameters were measured and calibrated as follows: cruising mass 89.165 kg, displacement volume 0.089 m³, and external surface area 0.85 m². The three-axis rotational inertia was solved by integration from a 3D CAD model, with the center of gravity coordinates (0.02 m, 0, 0) and the center of buoyancy coordinates (0, 0, 0.01 m). The dynamic equations were simplified based on low-to-medium speed conditions (speed < 2 m / s, angular rate < 0.3 rad / s), and core added mass and damping coefficients were selected for calculation. This module characterizes the basic and mainstream dynamic characteristics of the UUV, ensuring the overall physical interpretability of the model and avoiding non-physical outputs.

[0071] Step B13: Construct a physical model for calculating nominal acceleration based on the dynamic equations.

[0072] This physical model takes the underwater vehicle's real-time state variables, such as six-dimensional velocity, three-dimensional attitude, and control input variables, as inputs. It performs forward solving using dynamic equations with pre-filled parameters and outputs the theoretically expected six-dimensional acceleration response, i.e., the nominal acceleration, under the current state. This nominal acceleration represents the vehicle's basic dynamic response to control commands and environmental stimuli under ideal fluid assumptions and known parameters.

[0073] In the overall architecture of the hybrid dynamics model, the physical model is responsible for capturing the mainstream, deterministic dynamics, and its output constitutes the main part of the prediction results. Since the physical model is entirely based on analytical equations, its computational process is transparent and highly interpretable, and it maintains a reasonable trend extrapolation capability even outside the training data coverage. Meanwhile, complex nonlinear effects that the physical model cannot accurately describe due to simplification assumptions and parameter errors are naturally extracted and left to the neural network residual model to learn and compensate for. This division of labor makes the physical model not only an independent computational unit, but also guides the neural network to focus on learning the structured priors of the true residual distribution, thereby significantly reducing the learning difficulty of the neural network and improving the overall model's convergence speed and generalization performance.

[0074] Step S102: Construct a neural network residual model for calculating residual acceleration.

[0075] In practical implementation, a neural network residual model is constructed to calculate residual acceleration, primarily to characterize unmodeled dynamics and higher-order nonlinear effects that cannot be precisely described by physical models. In actual marine environments, underwater vehicles are affected by a variety of complex factors, such as high angle-of-attack flow field separation, propeller-rudder effect interaction, surface biofouling, and time-varying ocean current disturbances. These factors are difficult to express precisely using analytical formulas. The neural network residual model, in a data-driven manner, learns the deviation patterns between real observation data and physical model predictions. Its output residual acceleration is essentially a dynamic compensation for errors in the physical model. It should be understood that the "residual" here is not a simple numerical difference, but a structured correction quantity that includes rich environmental information and unmodeled characteristics of the system. By introducing the neural network residual model, the shortcomings of the pure physical model in terms of accuracy under complex operating conditions are compensated, and the overall model is endowed with the ability to adaptively fit unknown dynamics.

[0076] It is understandable that the neural network residual model uses a fully connected feedforward neural network as the residual correction network. In one feasible implementation, step S102 may include steps B11 to B16: Step B11: Acquire twelve-dimensional state data consisting of six-dimensional velocity data and three-dimensional attitude angle data of the underwater vehicle; Six-dimensional velocity data typically includes linear velocities along three axes of the carrier coordinate system and angular velocities around three axes. The linear velocities along the three axes include sway, roll, and heave, while the angular velocities around the three axes include roll, pitch, and yaw. The three-dimensional attitude angle data includes roll angle, pitch angle, and yaw angle.

[0077] In practical applications, to fully characterize the dynamic response of the aircraft, the remaining three dimensions can preferably be three-axis control input commands, such as thrust and rudder angle, because the control input directly excites the system to generate an acceleration response, which is an indispensable source of information for predicting residuals. Alternatively, the remaining dimensions can also be derived features obtained from the original velocity and attitude through kinematic transformations, such as angle of attack, drift angle, or dynamic pressure coefficient. This design, which integrates state observation with control excitation or derived features into a unified twelve-dimensional input, ensures that the residual model can capture the complex unmodeled dynamics caused by the coupling of external manipulation and environment, avoiding the information loss caused by relying solely on state feedback.

[0078] Step B12: Extract features from the twelve-dimensional state data to obtain a twelve-dimensional state vector; Feature extraction involves more than just simple numerical concatenation; it also implicitly includes preprocessing operations such as normalization, detrending, or sliding temporal windows, aiming to eliminate differences between different physical dimensions and highlight dynamic trends. The resulting twelve-dimensional state vector serves as the unified input interface for the network, preserving the integrity of the original physical information while adapting to the computational needs of subsequent fully connected layers.

[0079] Step B13: Configure the input layer according to the twelve-dimensional state vector, wherein the input layer is used to receive the twelve-dimensional state vector; The input layer is configured based on a twelve-dimensional state vector. The input layer is used to receive the twelve-dimensional state vector. This means that the number of nodes in the input layer strictly corresponds to the dimension of the state vector, that is, 12 input nodes. Each node carries a state component with a clear physical meaning, thus establishing the correspondence between data and physical objects at the network entry point and preventing information confusion.

[0080] Step B14: Set the neurons and activation functions of the hidden layer to process the twelve-dimensional state vector; In this embodiment, the hidden layers preferably employ a four-layer fully connected structure, with 128 neurons per layer. This combination of depth and width is an optimal configuration that balances model capacity and the risk of overfitting, validated through extensive experiments. More importantly, the activation function for the hidden layers is the hyperbolic tangent function (tanh). The choice of tanh, rather than the ReLU function commonly used in deep learning, is a targeted design based on the specific properties of underwater dynamic systems.

[0081] Step B15: Set the linear activation function of the output layer, which is used to receive and process the twelve-dimensional state vector processed by the hidden layer and output a six-dimensional residual acceleration. As a continuous physical vector, the residual acceleration should not be subject to artificial nonlinear truncation or compression. Linear output ensures the network can express corrections of any magnitude without distortion. Since the final prediction of the hybrid dynamics model is a linear superposition of nominal and residual acceleration, introducing nonlinearity into the output layer would destroy the physical interpretability of this superposition relationship, rendering the hierarchical architecture of "nominal + residual" meaningless. Furthermore, the linear output layer ensures the existence and continuity of the second derivative of the entire network with respect to the input variables, providing a mathematical basis for calculating loss functions involving higher-order physical constraints.

[0082] Step B16: Construct a neural network residual model based on the input layer, the hidden layer, and the output layer as set.

[0083] Based on the set input layer, hidden layer, and output layer, a neural network residual model is constructed, ultimately forming a dedicated feedforward neural network with an input dimension of 12, an output dimension of 6, and four layers of tanh nonlinear transformation.

[0084] In practical implementation, the network weights can be initialized using the Xavier uniform initialization strategy, with the bias initialized to 0, resulting in a total number of parameters on the order of approximately 50,000. This model is compact and computationally efficient, meeting the real-time requirements of airborne deployment while achieving high-precision, physically consistent representation of the complex, unmodeled dynamics of underwater vehicles through specific activation functions and hierarchical design. It forms a complementary and structurally synergistic organic whole with the aforementioned physical model.

[0085] Step S103: Obtain the hybrid dynamics model based on the physical model and the neural network residual model.

[0086] In this embodiment, a linear superposition mechanism after parallel computation is used to fuse the two models, thereby obtaining a hybrid dynamics model. Specifically, when the same state data is input, the physical model and the neural network residual model run independently and simultaneously, outputting the nominal acceleration vector and the residual acceleration vector, respectively. Since the outputs of the two models are designed to be six-dimensional acceleration vectors with the same physical dimensions (containing three linear acceleration components of sway, roll, and heave, and three angular acceleration components of pitch, yaw, and yaw), they can be directly algebraically added to obtain the final predicted acceleration.

[0087] This embodiment constructs a physical model for calculating nominal acceleration; constructs a neural network residual model for calculating residual acceleration; and obtains a hybrid dynamics model based on the physical model and the neural network residual model. Through a hierarchical decoupling design, the basic dynamic response is represented by the physical model, while the unmodeled dynamics are compensated by the neural network residual model. This preserves the interpretability and stability of the physical model while endowing it with the ability to learn complex nonlinear effects, avoiding the trade-off between accuracy and reliability inherent in a single model.

[0088] Based on the above embodiments of this application, in another embodiment of this application, the same or similar content as the above embodiments can be referred to the above description, and will not be repeated hereafter. Based on this, please refer to... Figure 4 The step of "obtaining the pre-constructed weighted joint loss function" in step S10 includes steps S104 to S108: Step S104: Obtain the mean square error between historical measured acceleration and historical predicted acceleration.

[0089] Obtaining the mean square error between historical measured acceleration and historical predicted acceleration is a fundamental step in quantifying the deviation between model predictions and actual physical world observations. Historical measured acceleration originates from sensors such as the inertial measurement unit onboard the underwater vehicle and represents a comprehensive response incorporating environmental noise and real dynamics; historical predicted acceleration is the output estimate of the hybrid dynamics model for the state at the same moment under current parameters.

[0090] Mean Squared Error (MSE), as a classic regression loss metric, can effectively penalize large prediction biases.

[0091] Specifically, the mean square error (MSE) between historical measured acceleration and historical predicted acceleration can be calculated as follows: In the above formula, For data fitting loss term, For the sample size, For historical measured acceleration, Accelerating historical predictions.

[0092] Step S105: Set the data fitting loss term according to the mean square error.

[0093] Understandably, setting the data fitting loss term based on the mean squared error means directly using this statistical indicator as a component of the optimization function. The role of the data fitting loss term is to provide a realistic anchor for model training, ensuring that the model output is always pulled towards real observation data, and preventing the model from deviating from actual operating conditions due to over-reliance on theoretical equations. For example, when an underwater vehicle encounters unknown ocean current disturbances or surface attachments that increase drag, the prediction of a purely physical model will be underestimated. In this case, the data fitting loss term will generate a large gradient signal, driving the neural network residual model to learn this bias, thereby correcting the prediction results.

[0094] Step S106: Substitute the historical predicted acceleration into the dynamic equation to obtain the equation residual.

[0095] The dynamic equations are the six-degree-of-freedom rigid body motion equations upon which the physical model constructed in the aforementioned embodiments is based. Unlike traditional methods that only use equations to calculate nominal values, this embodiment substitutes the predicted output of the neural network into the left side of the equations to calculate whether it satisfies Newton's second law and the fluid dynamic equilibrium relationship under the current state.

[0096] If the predicted acceleration perfectly conforms to the laws of physics, the equations are balanced and the residuals are zero. Conversely, if the predicted value violates the laws of mass conservation or momentum theorem, non-zero residuals will occur. Performing L2 norm operations on the equation residuals yields the dynamic physical residual loss term.

[0097] Step S107: Perform L2 norm operation on the residuals of the equation to obtain the dynamic physical residual loss term.

[0098] The L2 norm is essentially a measure of the energy of the residual vector. Compared to the absolute value norm, the L2 norm is differentiable everywhere, which is more conducive to the stable convergence of gradient-based backpropagation algorithms. The dynamic physics residual loss term acts as a soft constraint. It does not force the network output to be strictly equal to the solution of the equation (i.e., a hard constraint), but allows for deviations within a certain range to accommodate unmodeled dynamics. However, it regulates the solution space by penalizing excessive physical violations. This soft constraint mechanism effectively avoids "illusionary" predictions that violate the common sense of mechanics from purely data-driven models in sparse data regions, ensuring the physical consistency of the model across the entire envelope.

[0099] Step S108: Set the weight coefficients of the data fitting loss term and the kinetic physical residual loss term, and construct a weighted joint loss function based on the weight coefficients, the data fitting loss term, and the kinetic physical residual loss term.

[0100] The mathematical expression of the weighted joint loss function is as follows: In the above formula, The sum of the weighted joint loss function. The data loss weight is set, with a default value of 1.0. This is the weight for physical loss, with a default value of 0.1. The loss term is a dynamic physical residual. A soft constraint approach is adopted, which is different from the hard constraint approach. This approach ensures physical consistency and avoids pathological optimization of the problem, thereby improving training stability.

[0101] This application sets a non-zero and appropriate proportion This design allows the physical loss term to provide regularization guidance when data noise is high, provide trend constraints in data-scarce regions, and allow the model to deviate moderately from the equation to fit the true residuals in data-rich regions. This weighted design fundamentally solves the gradient competition problem between physical constraints and data fitting, ensuring that the hybrid dynamics model can accurately reproduce historical operating data and strictly adhere to the basic laws of fluid dynamics, achieving a dual improvement in accuracy and reliability.

[0102] This embodiment obtains the mean square error between historical measured acceleration and historical predicted acceleration; sets a data fitting loss term based on the mean square error; substitutes the historical predicted acceleration into the dynamic equation to obtain the equation residual; performs a L2 operation on the equation residual to obtain a kinetic physical residual loss term; sets weight coefficients for the data fitting loss term and the kinetic physical residual loss term, and constructs a weighted joint loss function based on the weight coefficients, the data fitting loss term, and the kinetic physical residual loss term. By weighting and combining data-driven and physical constraint losses, the model training is simultaneously guided by measured data and governed by physical laws. The introduction of weight coefficients can adjust the optimization priority of the two types of objectives, avoiding convergence instability caused by gradient competition, and ensuring that the model does not violate basic mechanical principles while fitting the data.

[0103] Based on the above embodiments of this application, in another embodiment of this application, the same or similar content as the above embodiments can be referred to the above description, and will not be repeated hereafter. Based on this, please refer to... Figure 5 Step S30 includes steps S301 to S305: Step S301: Configure the training hyperparameters of the hybrid dynamics model, including optimizer type, initial learning rate, batch size, and gradient pruning threshold.

[0104] The offline training process in this embodiment is not a general neural network training process, but a special training strategy designed for the gradient competition and non-convex optimization problems unique to the Underwater Dynamics Physical Information Neural Network (PINN).

[0105] Configuring the training hyperparameters of the hybrid dynamics model is fundamental to initiating efficient optimization. The preferred optimizer type is the Adam optimizer. Compared to traditional stochastic gradient descent (SGD), Adam can adaptively adjust the learning rate of each parameter to address the problem of large differences in the magnitude and sparse distribution of gradients between the physical loss term and the data loss term, thus accelerating convergence. For example, the initial learning rate is set to 0.003, a balance achieved between ensuring rapid exploration of the solution space in the early stages of training and avoiding oscillations in the later stages. The batch size is set to 64, ensuring both the statistical stability of gradient estimation and training efficiency before deployment on an airborne embedded platform. The gradient pruning threshold is set to 1.0, a key safety boundary to prevent abnormal physical residual gradients.

[0106] Step S302: Input the standard training sample set into the hybrid power model, use the optimizer to calculate the gradient according to the weighted joint loss function, and update the parameters of the hybrid power model.

[0107] In practical implementation, gradients can be calculated using the backpropagation algorithm. This algorithm calculates not only the gradient of the data fitting error with respect to the network weights, but also the gradient of the residuals of the dynamic equations with respect to the network weights using automatic differentiation. The two gradients are weighted and fused according to the weight coefficients determined in the aforementioned embodiment to form a total gradient vector, which then drives the optimizer to update the neural network weights and learnable physical parameters in the hybrid dynamics model. This joint update mechanism ensures that every adjustment to the model parameters simultaneously moves in two directions: "fitting the measured data" and "satisfying the physical equations," avoiding the model's unbalanced performance caused by single-objective optimization.

[0108] Step S303: Execute a learning rate scheduling strategy during training. When the validation set loss no longer decreases within a preset number of rounds, automatically reduce the learning rate.

[0109] Implementing a learning rate scheduling strategy during training—automatically reducing the learning rate when the validation set loss no longer decreases within a preset number of rounds—is a key technique for addressing the highly nonconvex nature of the PINN loss surface. Due to the complex nonlinear physical constraints introduced by the underwater dynamics equations, the loss function surface of hybrid dynamics models often exhibits numerous saddle points and local minima. If a fixed learning rate is used, the model is highly susceptible to getting trapped in suboptimal solutions or oscillating violently around minima, thus failing to converge.

[0110] This embodiment employs the ReduceLROnPlateau strategy, a plateau detection mechanism based on validation set loss. For example, the patience epoch is set to 100, meaning the preset epochs are 100, and the decay factor is 0.5. This means that when the validation set loss does not show a significant decrease for 100 consecutive training epochs, the system determines that the current learning rate is too high or close to a local optimum, and thus halves the learning rate. This adaptive adjustment mechanism gives the training process the ability to dynamically switch between "coarse search" and "fine tuning," helping the model escape shallow local optima and find a flat minimum region with better generalization performance, thereby significantly improving the prediction robustness of the target hybrid dynamics model under unseen conditions.

[0111] Step S304: Perform gradient clipping during training. When the gradient norm exceeds the gradient clipping threshold, scale the gradient.

[0112] Scaling the gradient when its norm exceeds the gradient clipping threshold is a necessary safeguard to ensure the numerical stability of PINN training. In training PINNs, the dynamic physical residual loss term typically involves higher-order derivatives of the network output with respect to the input, making gradient calculation extremely sensitive to network parameters. During certain training phases, especially during initialization or when encountering anomalous samples, the physical residual gradient may exhibit instantaneous spikes with amplitudes far exceeding the normal range. Without restriction, this can lead to erroneous and significant updates to network weights, or even cause NaN (NaN) errors, resulting in a complete training failure. This embodiment sets the gradient clipping threshold to 1.0, forcibly limiting the L2 norm of the total gradient within a safe range. When the gradient norm is detected to exceed the threshold, the gradient vector is scaled proportionally to bring its norm back below the threshold. This operation suppresses the destructive power of anomalous gradients without changing the gradient direction, ensuring that the optimization trajectory of minimizing the weighted joint loss function remains within a controllable range. This is an indispensable technical guarantee for achieving long-term stable training.

[0113] Step S305: Monitor the change in the validation set loss. When the validation set loss reaches the preset iteration termination condition, stop training and obtain the target hybrid dynamics model.

[0114] Historical operational data of underwater vehicles inevitably contains sensor measurement noise and random environmental disturbances. If too many training epochs are performed, the model may shift from learning underlying physical laws to memorizing high-frequency noise in the data, resulting in excellent performance on the training set but a sharp decline in performance in real-world applications. This embodiment configures an early stopping mechanism, using the validation set loss as a real-time indicator of the model's generalization ability. For example, the preset iteration termination condition is set as follows: early stopping duration is 200 epochs, and the maximum number of training epochs is 1000. When the validation set loss does not improve for 200 consecutive epochs, the system automatically terminates training and rolls back to the model parameters at the point of lowest validation loss. This not only saves computational resources but, more importantly, ensures that the obtained target hybrid dynamics model is at the optimal balance between "underfitting" and "overfitting," possessing optimal engineering practical value.

[0115] like Figure 6 As shown in the figure, this figure illustrates the convergence curve of the offline pre-training loss obtained using the training strategy described in this embodiment. It can be seen from the figure that, under the synergistic effect of the ReduceLROnPlateau strategy, gradient pruning, and early stopping mechanism, the weighted joint loss function exhibits a smooth exponential decay characteristic, without obvious gradient explosion or oscillating divergence. The early stopping mechanism is triggered after approximately 1600 training iterations, and the loss eventually converges stably to 10. -8 The convergence behavior demonstrates that the targeted training strategy proposed in this embodiment can effectively overcome the inherent difficulties of underwater dynamics PINN training, successfully achieving simultaneous optimization of data fitting accuracy and physical consistency, and laying a solid model foundation for subsequent deployment and application.

[0116] This embodiment configures the training hyperparameters of the hybrid power model, including optimizer type, initial learning rate, batch size, and gradient clipping threshold. The standard training sample set is input into the hybrid power model, and the optimizer calculates the gradient based on the weighted joint loss function, updating the parameters of the hybrid power model. During training, a learning rate scheduling strategy is implemented; when the validation set loss no longer decreases within a preset number of iterations, the learning rate is automatically reduced. Gradient clipping is performed during training; when the gradient norm exceeds the gradient clipping threshold, the gradient is scaled. Changes in the validation set loss are monitored; when the validation set loss reaches a preset iteration termination condition, training is stopped, resulting in the target hybrid power model. Through adaptive learning rate adjustment, gradient clipping, and early stopping mechanisms, the risks of gradient explosion and overfitting during training are effectively suppressed, improving the model's convergence stability and generalization performance, ensuring that the obtained target hybrid power model still possesses reliable predictive capabilities under different operating conditions.

[0117] For example, to help understand the implementation process of the underwater dynamic modeling method based on physical information neural networks obtained by combining this embodiment with the above embodiment one, please refer to... Figure 7 , Figure 7 A simplified flowchart of an underwater dynamic modeling method based on physical information neural networks is provided. Specifically, the method involves: firstly, collecting and preprocessing historical operational data; then, designing a joint loss function and building a hybrid power dynamics model architecture; finally, training the hybrid power dynamics model offline based on the preprocessed data and the joint loss function; and finally, deploying and applying the hybrid power dynamics model to complete the overall modeling process.

[0118] It should be noted that the above examples are only for understanding this application and do not constitute a limitation on the underwater dynamic modeling method based on physical information neural networks in this application. Any simple modifications based on this technical concept are within the protection scope of this application.

[0119] This application also provides an underwater dynamic modeling device based on a physical information neural network. Please refer to [link / reference]. Figure 8 The underwater dynamic modeling device based on physical information neural networks includes: The acquisition module 10 is used to acquire historical operating data of the underwater vehicle under different marine environmental conditions and various navigation maneuvers, as well as a pre-built hybrid dynamics model and a weighted joint loss function.

[0120] The preprocessing module 20 is used to preprocess the historical running data and construct a standard training sample set.

[0121] Training module 30 is used to train the hybrid power model offline with the standard training sample set as input and the goal of minimizing the weighted joint loss function, so as to obtain the target hybrid power model.

[0122] The prediction module 40 is used to deploy the target hybrid dynamics model to the underwater vehicle and predict the acceleration of the underwater vehicle.

[0123] The underwater dynamic modeling device based on physical information neural networks provided in this application, employing the underwater dynamic modeling method based on physical information neural networks in the above embodiments, can solve the technical problem of the prediction accuracy decay of static physical information neural network models in time-varying ocean environments in current dynamic modeling. Compared with the prior art, the beneficial effects of the underwater dynamic modeling device based on physical information neural networks provided in this application are the same as those of the underwater dynamic modeling method based on physical information neural networks provided in the above embodiments, and other technical features in the underwater dynamic modeling device based on physical information neural networks are the same as those disclosed in the methods of the above embodiments, and will not be repeated here.

[0124] This application provides an underwater dynamic modeling device based on a physical information neural network. The underwater dynamic modeling device based on a physical information neural network includes: at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores instructions that can be executed by the at least one processor, and the instructions are executed by the at least one processor to enable the at least one processor to execute the underwater dynamic modeling method based on a physical information neural network in the above embodiment 1.

[0125] The following is for reference. Figure 9 This document illustrates a structural schematic diagram of an underwater dynamic modeling device based on a physical information neural network, suitable for implementing embodiments of this application. The underwater dynamic modeling device based on a physical information neural network in these embodiments may include, but is not limited to, mobile terminals such as mobile phones, laptops, digital broadcast receivers, PDAs (Personal Digital Assistants), tablets, PMPs (Portable Media Players), and vehicle-mounted terminals (e.g., vehicle navigation terminals), as well as fixed terminals such as digital TVs and desktop computers. Figure 9 The underwater dynamic modeling device based on physical information neural networks shown is merely an example and should not impose any limitations on the functionality and scope of use of the embodiments of this application.

[0126] like Figure 9As shown, the underwater dynamic modeling device based on physical information neural networks may include a processing unit 1001 (e.g., a central processing unit, a graphics processing unit, etc.), which can perform various appropriate actions and processes according to programs stored in ROM (Read Only Memory) 1002 or programs loaded from storage device 1003 into RAM (Random Access Memory) 1004. RAM 1004 also stores various programs and data required for the operation of the underwater dynamic modeling device based on physical information neural networks. The processing unit 1001, ROM 1002, and RAM 1004 are interconnected via bus 1005. Input / output (I / O) interface 1006 is also connected to the bus. Typically, the following systems can be connected to I / O interface 1006: input devices 1007 including, for example, touchscreens, touchpads, keyboards, mice, image sensors, microphones, accelerometers, gyroscopes, etc.; output devices 1008 including, for example, LCDs (Liquid Crystal Displays), speakers, vibrators, etc.; storage devices 1003 including, for example, magnetic tapes, hard disks, etc.; and communication devices 1009. Communication device 1009 allows the underwater dynamic modeling equipment based on physical information neural networks to exchange data wirelessly or via wired communication with other devices. Although the figure shows an underwater dynamic modeling equipment based on physical information neural networks with various systems, it should be understood that it is not required to implement or possess all the systems shown. More or fewer systems can be implemented alternatively.

[0127] Specifically, according to the embodiments disclosed in this application, the processes described above with reference to the flowcharts can be implemented as computer software programs. For example, embodiments disclosed in this application include a computer program product comprising a computer program carried on a computer-readable medium, the computer program containing program code for performing the methods shown in the flowcharts. In such embodiments, the computer program can be downloaded and installed from a network via a communication device, or installed from storage device 1003, or installed from ROM 1002. When the computer program is executed by processing device 1001, it performs the functions defined in the methods of the embodiments disclosed in this application.

[0128] The underwater dynamic modeling device based on physical information neural networks provided in this application, employing the underwater dynamic modeling method based on physical information neural networks in the above embodiments, can solve the technical problem of the prediction accuracy decay of static physical information neural network models in time-varying ocean environments in current dynamic modeling. Compared with the prior art, the beneficial effects of the underwater dynamic modeling device based on physical information neural networks provided in this application are the same as those of the underwater dynamic modeling method based on physical information neural networks provided in the above embodiments, and other technical features in this underwater dynamic modeling device based on physical information neural networks are the same as those disclosed in the previous embodiment method, and will not be repeated here.

[0129] It should be understood that the various parts disclosed in this application can be implemented using hardware, software, firmware, or a combination thereof. In the description of the above embodiments, specific features, structures, materials, or characteristics can be combined in any suitable manner in one or more embodiments or examples.

[0130] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

[0131] This application provides a computer-readable storage medium having computer-readable program instructions (i.e., a computer program) stored thereon, the computer-readable program instructions being used to execute the underwater dynamic modeling method based on a physical information neural network in the above embodiments.

[0132] The computer-readable storage medium provided in this application may be, for example, a USB flash drive, but is not limited to, electrical, magnetic, optical, electromagnetic, infrared, or semiconductor systems, devices, or any combination thereof. More specific examples of computer-readable storage media may include, but are not limited to: electrical connections having one or more wires, portable computer disks, hard disks, RAM (Random Access Memory), ROM (Read Only Memory), EPROM (Erasable Programmable Read Only Memory or Flash Memory), optical fibers, CD-ROM (CD-Read Only Memory), optical storage devices, magnetic storage devices, or any suitable combination thereof. In this embodiment, the computer-readable storage medium may be any tangible medium containing or storing a program that can be used by or in conjunction with an instruction execution system, system, or device. The program code contained on the computer-readable storage medium may be transmitted using any suitable medium, including but not limited to: wires, optical cables, RF (Radio Frequency), etc., or any suitable combination thereof.

[0133] The aforementioned computer-readable storage medium may be included in an underwater dynamic modeling device based on a physical information neural network; or it may exist independently and not be assembled into an underwater dynamic modeling device based on a physical information neural network.

[0134] The aforementioned computer-readable storage medium carries one or more programs. When these programs are executed by the underwater dynamic modeling device based on a physical information neural network, the underwater dynamic modeling device based on a physical information neural network performs the following actions: acquires historical operational data of the underwater vehicle under different marine environmental conditions and various navigation maneuvers, a pre-constructed hybrid dynamics model, and a weighted joint loss function; preprocesses the historical operational data to construct a standard training sample set; uses the standard training sample set as input and minimizes the weighted joint loss function as the objective to perform offline training on the hybrid dynamics model to obtain a target hybrid dynamics model; and deploys the target hybrid dynamics model to the underwater vehicle to predict the acceleration of the underwater vehicle.

[0135] Computer program code for performing the operations of this application can be written in one or more programming languages ​​or a combination thereof, including object-oriented programming languages ​​such as Java, Smalltalk, and C++, as well as conventional procedural programming languages ​​such as the "C" language or similar programming languages. The program code can be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving remote computers, the remote computer can be connected to the user's computer via any type of network—including LAN (Local Area Network) or WAN (Wide Area Network)—or can be connected to an external computer (e.g., via the Internet using an Internet service provider).

[0136] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of this application. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions indicated in the blocks may occur in a different order than those indicated in the drawings. For example, two consecutively indicated blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should also be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, can be implemented using a dedicated hardware-based system that performs the specified function or operation, or using a combination of dedicated hardware and computer instructions.

[0137] The modules described in the embodiments of this application can be implemented in software or hardware. The names of the modules do not necessarily limit the functionality of the unit itself.

[0138] The readable storage medium provided in this application is a computer-readable storage medium that stores computer-readable program instructions (i.e., computer programs) for executing the above-described underwater dynamic modeling method based on physical information neural networks. This solves the technical problem of the prediction accuracy degradation of static physical information neural network models in time-varying ocean environments in current dynamic modeling. Compared with the prior art, the beneficial effects of the computer-readable storage medium provided in this application are the same as those of the underwater dynamic modeling method based on physical information neural networks provided in the above embodiments, and will not be repeated here.

[0139] This application also provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the underwater dynamic modeling method based on physical information neural networks as described above.

[0140] The computer program product provided in this application can solve the technical problem of the decline in prediction accuracy of static physical information neural network models in time-varying marine environments in current dynamic modeling. Compared with the prior art, the beneficial effects of the computer program product provided in this application are the same as those of the underwater dynamic modeling method based on physical information neural networks provided in the above embodiments, and will not be repeated here.

[0141] The above description is only a part of the embodiments of this application and does not limit the patent scope of this application. All equivalent structural transformations made under the technical concept of this application and using the contents of the specification and drawings of this application, or direct / indirect applications in other related technical fields, are included in the patent protection scope of this application.

Claims

1. A method for underwater dynamic modeling based on physical information neural networks, characterized in that, The underwater dynamic modeling method based on physical information neural networks includes: Acquire historical operational data of underwater vehicles under different marine environmental conditions and various navigation maneuvers, as well as pre-constructed hybrid dynamics models and weighted joint loss functions; The historical operational data is preprocessed to construct a standard training sample set; Using the standard training sample set as input and minimizing the weighted joint loss function as the objective, the hybrid power model is trained offline to obtain the target hybrid power model; The target hybrid dynamics model is deployed to the underwater vehicle to predict its acceleration.

2. The method as described in claim 1, characterized in that, The steps to obtain a pre-built hybrid dynamics model include: Construct a physical model for calculating nominal acceleration; Construct a neural network residual model for calculating residual acceleration; The hybrid dynamics model is obtained based on the physical model and the neural network residual model.

3. The method as described in claim 2, characterized in that, The steps for constructing a physical model for calculating nominal acceleration include: Acquire sample data of physical parameters of underwater vehicles; A dynamic equation is constructed based on the physical parameter sample data; A physical model for calculating nominal acceleration is constructed based on the aforementioned dynamic equations.

4. The method as described in claim 2, characterized in that, The steps for constructing a neural network residual model for calculating residual acceleration include: Acquire twelve-dimensional state data consisting of six-dimensional velocity data and three-dimensional attitude angle data of the underwater vehicle; Feature extraction is performed on the twelve-dimensional state data to obtain a twelve-dimensional state vector; The input layer is configured according to the twelve-dimensional state vector, and the input layer is used to receive the twelve-dimensional state vector. The hidden layer's neurons and activation functions are configured to process the twelve-dimensional state vector; Set a linear activation function for the output layer. The output layer is used to receive and process the twelve-dimensional state vector processed by the hidden layer and output a six-dimensional residual acceleration. A neural network residual model is constructed based on the input layer, the hidden layer, and the output layer.

5. The method as described in claim 1, characterized in that, The steps to obtain a pre-constructed weighted joint loss function include: Obtain the mean square error between historical measured acceleration and historical predicted acceleration; Set the data fitting loss term based on the mean square error; Substituting the historical predicted acceleration into the dynamic equation, we obtain the equation residuals; Performing a L2 norm operation on the residuals of the equations yields the kinetic physical residual loss term; Set the weight coefficients for the data fitting loss term and the kinetic physical residual loss term, and construct a weighted joint loss function based on the weight coefficients, the data fitting loss term, and the kinetic physical residual loss term.

6. The method as described in claim 1, characterized in that, The steps of offline training the hybrid power model to obtain the target hybrid power model, using the standard training sample set as input and minimizing the weighted joint loss function as the objective, include: Configure the training hyperparameters of the hybrid dynamics model, including optimizer type, initial learning rate, batch size, and gradient clipping threshold; The standard training sample set is input into the hybrid power model, the optimizer calculates the gradient based on the weighted joint loss function, and the parameters of the hybrid power model are updated. During training, a learning rate scheduling strategy is implemented, which automatically reduces the learning rate when the validation set loss no longer decreases within a preset number of rounds. Gradient clipping is performed during training. When the gradient norm exceeds the gradient clipping threshold, the gradient is scaled. Monitor the changes in the validation set loss, and stop training when the validation set loss reaches the preset iteration termination condition to obtain the target hybrid dynamics model.

7. The method according to any one of claims 1 to 6, characterized in that, The steps for acquiring historical operational data of underwater vehicles under different marine environmental conditions and various navigation maneuvers include: Control the underwater vehicle to perform various navigation maneuver tests under different preset marine environmental conditions; During the execution of the various navigation maneuver state tests, the sensors mounted on the underwater vehicle collect in real time the historical six-dimensional velocity data, historical three-dimensional attitude angle data, historical control input data, and historical measured acceleration data measured by the inertial measurement unit. The collected six-dimensional velocity data, three-dimensional attitude angle data, control input data, and measured acceleration data are time-stamped to obtain historical operational data of the underwater vehicle under different marine environmental conditions and various navigation maneuvers.

8. The method according to any one of claims 1 to 6, characterized in that, The steps of preprocessing the historical operational data and constructing a standard training sample set include: The historical operating data is smoothed and denoised to obtain the smoothed and denoised data. The smoothed and denoised data is then normalized to obtain a standard training sample set.

9. The method according to any one of claims 1 to 6, characterized in that, The step of deploying the target hybrid dynamics model to the underwater vehicle and predicting the acceleration of the underwater vehicle includes: The target hybrid dynamics model was deployed to the underwater vehicle, and real-time six-dimensional velocity data and real-time three-dimensional attitude angle data were collected during the operation of the underwater vehicle. The real-time six-dimensional velocity data and the real-time three-dimensional attitude angle data are input into the target hybrid power dynamics model to obtain the nominal acceleration output by the physical model in the target hybrid power dynamics model and the residual acceleration output by the neural network residual model in the target hybrid power dynamics model. The nominal acceleration and the residual acceleration are linearly superimposed to obtain the real-time predicted acceleration of the underwater vehicle.

10. An underwater dynamic modeling device based on a physical information neural network, characterized in that, The device includes: The acquisition module is used to acquire historical operational data of underwater vehicles under different marine environmental conditions and various navigation maneuvers, as well as pre-built hybrid dynamics models and weighted joint loss functions; The preprocessing module is used to preprocess the historical running data and construct a standard training sample set; The training module is used to train the hybrid power model offline with the standard training sample set as input and the goal of minimizing the weighted joint loss function, so as to obtain the target hybrid power model. The prediction module is used to deploy the target hybrid dynamics model to the underwater vehicle and predict the acceleration of the underwater vehicle.

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