Cluster adaptive path planning method based on geometric PDE and PINN

Abstract

The application discloses a cluster adaptive path planning method based on geometric PDE and PINN, which firstly models a dynamic model of a fault cluster with a time-varying topology.Secondly, a parabolic PDE system based on Riemannian manifold and heat flow based gradient evolution is constructed to obtain a reference trajectory of the cluster system in an obstacle environment. The steady-state solution of the cluster parabolic PDE system is learned by using a physical information neural network (PINN), and a fault-tolerant optimal control law of the cluster system based on geometric PDE and PINN is obtained under the conditions of time-varying topology, actuator failure and obstacle operation. The application ensures that the intelligent agent navigates from the initial position to the target position along the dynamic reference trajectory, simultaneously realizes the endogenous compensation of the fault effect, and realizes the unified solution of fault compensation and motion planning.

Description

Technical Field

[0001] This patent addresses the fault-tolerant control problem of cluster systems with time-varying topology and actuator failures in obstacle environments, proposing an adaptive fault-tolerant path planning algorithm based on geometric partial differential equations (PDEs) and physical information networks (PINNs). Background Technology

[0002] Cluster systems consist of multiple autonomous intelligent agents working collaboratively in a collective environment, demonstrating remarkable potential for performing complex tasks through coordinated interaction in obstacle-rich scenarios. The core application of such systems lies in motion planning for finding feasible paths, such as in autonomous driving systems, warehousing and logistics, and industrial automation. Existing path planning methods mainly fall into three categories: heuristic, optimization, and geometric approaches.

[0003] In particular, geometric methods focus on the geometric properties and dynamic constraints of the workspace, which are typically dynamically represented by PDEs, revealing the essential characteristics of motion planning problems. However, existing geometric PDE methods mostly focus on single-agent systems, neglecting the crucial role of communication and cooperation between agents in swarm dynamics. This stems from the theoretical challenges of integrating internal interaction information into the global PDE model to generate feasible trajectories.

[0004] In swarm systems with time-varying interactive topologies, time-varying behavior influenced by agent mobility and energy constraints can introduce dynamic uncertainties, increasing coordination difficulties. Actuator failures further exacerbate these challenges, potentially leading to control performance degradation, motion deviations, or actuator jamming, ultimately causing cascading propagation of faults within the swarm and jeopardizing task completion. Therefore, fault-tolerant control (FTC) techniques designed to compensate for the effects of faults are crucial for ensuring system reliability. Current research on FTC for multi-agent systems with time-varying topologies and actuator failures includes methods such as replacing nominal controllers with fault-adaptive controllers and deploying virtual actuators to construct reconstructed systems. However, these methods focus on local fault compensation while neglecting global trajectory reconstruction, failing to effectively coordinate fault-induced motion distortions with swarm motion planning.

[0005] While geometric PDE methods can characterize path planning problems from a dynamic perspective, their application in swarm systems faces two major challenges: the numerical complexity of solving high-dimensional PDEs and the real-time adaptability to dynamically interacting topologies. Traditional numerical methods (such as the finite element method and spectral methods) require discretization of the state equations for each agent when dealing with multi-agent coupled PDE models, leading to the curse of dimensionality. Existing research has shown that PINN exhibits superior performance in complex PDE problems such as multiphysics coupling and time-varying boundary conditions. For example, in fluid mechanics, PINN has successfully solved the Navier-Stokes equations with dynamically changing Reynolds numbers; in materials science, it has been used for strongly nonlinear problems such as crystal growth interface evolution. These achievements have laid the theoretical foundation for transferring PINN to fault-tolerant control of swarm systems. However, existing PINN research focuses primarily on static or continuously varying PDE systems, and its potential in discrete-continuous hybrid dynamics (such as topology time-varying and actuator failures) has not been fully explored, which is precisely the core problem of fault-tolerant path planning in swarms. Summary of the Invention

[0006] This patent proposes a distributed adaptive fault-tolerant optimal control algorithm based on geometric partial differential equations (PDEs) and physical information networks (PINNs) for cluster systems operating under time-varying topologies, actuator failures, and obstacle environments. To address the cooperative imbalance problem caused by faults, a geometric metric for the trajectory is introduced to coordinate the cooperative mechanism between the nominal and virtual controllers. Based on this, the fault-tolerant optimization control problem is transformed into a trajectory optimization problem, where heat flux-based partial differential equations provide a reference trajectory for the system. This geometric modeling method characterizes the cluster dynamics evolution by combining Riemann length and heat flux-based gradient representations, and transforms communication constraints and external obstacles into state constraints, ensuring the stability and robustness of the cluster system's reference trajectory. To quickly and accurately obtain the following reference trajectory, a PINN architecture with independent networks is adopted, employing a parameter sharing mechanism to adaptively obtain the numerical solution of the reference trajectory and the fault-tolerant optimal control law, which is easy to apply in engineering practice.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0008] S1. Model a dynamic model of a fault cluster with a time-varying topology.

[0009] Consider by When a cluster of intelligent agents operates in a fault-free and unobstructed ideal environment, each agent follows a dynamic equation:

[0010]

[0011] variable These represent the intelligent agent 𝑖 in The planar position and control input at time, where and Representing intelligent agents respectively In horizontal Axis coordinate direction and vertical Displacement in the axial coordinate direction It is a nonlinear mapping. It is the set of all intelligent agents in the system.

[0012] Inter-agent communication is based on an undirected time-varying topology. Description, where the edge set The dynamic evolution is as follows: This indicates that the intelligent agent and The communication edge between them is formed if and only if the distance condition at the current time is met. When satisfied, the distance is defined as Time-varying signals In time The right limit at a given point is defined as:

[0013]

[0014] in, and Representing intelligent agents respectively and Communication status At the time of transition The values ​​before and after represent the communication edge. At any moment The instantaneous change. Conditions Defined interval The hysteresis identification region is simulated by introducing the hysteresis effect of edge addition / deletion to simulate the negotiation delay in the actual communication connection establishment process.

[0015] For cluster systems in this type of constrained time-varying topology In this scenario, the system's evolution relies on interaction-based potential functions between individuals to ensure global stability. These potential functions are characterized as the control input of the cluster system when it is fault-free. Specifically:

[0016]

[0017] in, This represents the total potential function variable with the agent distance as the variable. Let A represent the auxiliary potential function that ensures continuity and the potential function that maintains connectivity, respectively. The design involves adding and deleting potential functions simultaneously.

[0018]

[0019] parameter The selection of the potential function ensures At the threshold point It possesses 𝐶² continuity and applies to all intelligent agents. , and at any time It satisfies Lipschitz continuity. For intelligent agents The set of neighbors of a given , defined as follows: And it satisfies the symmetry condition: if and only if hour, Furthermore, in the aforementioned potential function, Individuals in a cluster system Connectivity-preserving sets in collaborative tasks For an attraction set, it is defined as When the judgment condition is met ( When the hysteresis threshold is reached, Included The attraction set activates the edge addition behavior; For an exclusion set, it is defined as The judgment conditions are met. ( When the hysteresis threshold is reached, Included The exclusion set activates the edge deletion behavior. and This is a binary discrimination signal. It takes the value 1 when the discrimination condition is met, and 0 otherwise.

[0020] Considering an unknown actuator failure in an agent, disrupting the interaction between agents and preventing the completion of a collaborative task, the dynamics of the failed agent can be characterized as follows:

[0021]

[0022] in, Indicates the first The state of a faulty agent, function The mathematical model representing the fault, the set of faulty intelligent agents is as follows .

[0023] Define the state vector as Having an initial state .in, Indicates the global state of the faulty cluster system. To generate auxiliary variables for synergistic forces, a dynamic augmentation model of fault clusters is constructed based on vector augmentation properties:

[0024]

[0025] To simplify the subsequent formulas, let the augmented matrix be... , ,in, , It is a Laplace matrix; middle for A column vector of all zeros for A matrix of all 1s. Furthermore, variables... This represents the interaction force of all fault-free intelligent agents, i.e., the nominal control force, where the force of a single fault-free intelligent agent is... . For the fault-tolerant control input to be designed, This represents the total potential function variable with the state of the faulty agent as the variable.

[0026] S2. Construct a reference trajectory based on geometric heat flow for the dynamic augmented model of the fault cluster in step S1 using a Riemannian manifold.

[0027] Through positive definite symmetric Riemannian metric tensors Define the trajectory length of the faulty cluster system in step S1. :

[0028]

[0029] in, Indicates the agent's time The trajectory of motion, the Riemannian metric tensor The design aims to suppress underactuated dynamic effects caused by faults and satisfy the system constraints:

[0030]

[0031] in Pre-calculated using Gram-Schmidt orthogonalization, It is a supplementary matrix, representing the original control input matrix. Auxiliary input direction of complementary underactuated components, parameters to be designed , indicating a time-varying regulator, The time-varying regularization weight parameters have a structure The adjusted gain satisfies and Actuator failure leads to an imbalance in the cooperative potential energy among agents, therefore a time-varying gain is introduced. Adjusting the Riemannian metric tensor This allows for adaptive adjustment of the system's trajectory length under the given metric. Gain is then introduced. Through constraints The agent achieves obstacle avoidance by adjusting the trajectory length. It should be noted that when the parameters... When a larger value is taken, it means that the most influential fault and obstacle is taken as the underactuated direction of the actuator, and a penalty term is applied to force the agent's trajectory to converge to the control allowable motion subspace of the cluster system and to be robust.

[0032] Consider As an additional constraint for local communication (where To address the local communication neighbor number constraint for agent 𝑖, Represents time (The total number of other intelligent agents that have direct communication or perceptual interaction with intelligent agent A), constructing a system that satisfies... Scalar functions transform such inequalities into constraints of a uniform form (totaling...) item, Specifically, the definition ,like This indicates a violation of the maximum number of adjacent edges constraint. (This is based on the constraint condition of the edge with the maximum degree.) ,in The number of virtual neighbors of agent 𝑖 can be further designed as a time-varying function as follows:

[0033]

[0034] Among them, the Herveside step function is passed through a logic function. , Approximate representation, it is obvious that when hour, ;when hour, Based on this, variables are introduced. To characterize the cumulative error, it is defined as follows:

[0035]

[0036] Furthermore, define vectors Under this definition, the dynamic model of a fault cluster system can be further expressed as a Riemann augmented system:

[0037]

[0038] Define state vector ,in , and By introducing an augmented Riemannian metric tensor It integrates topological constraints and adjacency quantity constraints, and its specific form is given by the following formula:

[0039]

[0040] in, ,element Indicates the time-varying function The penalty coefficient for exceeding the limit (i.e., when the number of adjacent units of agent A exceeds the limit).

[0041] matrix The constraints characterizing time-varying topologies based on Riemannian metrics are essentially derived from the constraints of the time-varying edges of the topology. , ,in It is a dynamically adjusted penalty coefficient used to quantify opposite edges. Deviation from the preset target distance during the optimization process The severity of the punishment.

[0042] Using the fault cluster Riemann augmented model, the trajectory length is specified as:

[0043]

[0044] Among them, scalar It is the total constraint penalty item. The constraint violations of each agent are mapped into a vector form, where each component corresponds to a weighted violation amount of a constraint. Indicates the first The constraint function for each agent. It should be noted that the constraint conditions for any agent *x*... If not satisfied, it will lead to and And then through Apply a penalty to the trajectory length. Conversely, when At that time, Xiang It does not affect the trajectory length. This penalty mechanism also applies to constraints. Furthermore, as can be seen from the trajectory length formula of the Riemann augmented system, when seeking the minimum length trajectory, the Riemann metric... Adaptive adjustment parameters introduced in This will compensate for actuator uncertainties caused by faults, thereby optimizing the agent's trajectory.

[0045] In the constrained Riemannian metric tensor Under the influence of heat flow gradient theory, an energy functional based on Riemann length is established:

[0046]

[0047] in , the Lagrange quantity of an instantaneous variable system.

[0048] The Euler-Lagrange equations are corrected using the variational principle, and a correction term is introduced. This yields the heat flux partial differential equation (PDE), which is the reference trajectory dynamics of the cluster system with actuator failure and time-varying topology in step S1, expressed in parabolic PDE as follows:

[0049]

[0050] in:

[0051]

[0052] in It is time homotopy variables, , Connect the initial positions of the faulty cluster under Riemannian metrics and finish line The trajectory. Representation and Measurement Related Levi-Civita contacts, and This reflects the gradient term after metric adjustment. The trajectory acceleration and nominal dynamics are described. The deviation between them. By reshaping the potential field, the curvature and trajectory length are minimized, and the gradient term is utilized. Adjust the trajectory to Time-varying measure In Introduced, generating additional acceleration terms This reflects the impact of dynamic adjustments to constraints on the trajectory deformation rate.

[0053] Based on the LaSalle invariant set principle, the non-increasing property of the Lyapunov function is obtained, and consequently, the state of the parabolic PDE system converges to the steady-state value. , where the state By decomposing the steady state We obtain this. According to isomorphism theory, This represents the ideal trajectory of the cluster system under actuator failure, time-varying topology, and obstacles, also known as the reference trajectory.

[0054] S3. Based on the reference trajectory of the cluster system based on the parabolic PDE system obtained in step S2, the steady-state value of the established parabolic PDE system is accurately approximated using a Physics-Informed Neural Network (PINN). This method overcomes the difficulties and large errors in solving PDEs based on methods such as the finite difference method, especially since the parabolic PDE system contains unknown variables. Furthermore, the fault-tolerant optimal control law based on geometric PDE and PINN is obtained for a cluster system with time-varying topology and actuator faults:

[0055] .

[0056] The details are as follows:

[0057] S3.1 Solving the parabolic PDE equation in S2 using PINN

[0058] First, define the input variable as the homotopy parameter. and time Output variables are divided into main outputs and auxiliary outputs, with the main output being the state. ,Right now A real vector of dimension, with an auxiliary output containing unknown parameters. .

[0059] Next, two parallel neural networks are designed. Network 1 learns the state. Network 2 computes parameterized time-varying parameters State network 1 uses a spatiotemporal bivariate architecture. As input, it dynamically represents the system's evolution trajectory; parameter network 2 uses only time... For input, focus on time-varying parameters Identification.

[0060] This decoupled structure can effectively improve model training efficiency and generalization performance: the state network is specifically designed to handle spatiotemporal bivariate variables. The complex mapping relationship, while the parameter network focuses on time-varying features. The network employs an innovative design that combines explicit decoupling with implicit coupling. This design avoids gradient conflicts in multiple tasks and improves training stability through independent branch structures. It also uses the PDE residual loss term to construct implicit parameter-state relationships under physical constraints and enhances cross-network information interaction by sharing the underlying temporal feature extractor. While maintaining the lightweight nature of the model, it achieves a balance between accuracy and efficiency in the collaborative identification of fast and slow variables in complex coupled systems.

[0061] Finally, the PDE residuals are calculated using automatic differentiation. The specific steps are as follows:

[0062] 1. Calculate the Riemann metric :

[0063]

[0064] in Pre-calculated using Gram-Schmidt orthogonalization, based on which the following is obtained .

[0065] 2. Calculate the gradient term :

[0066] First, use automatic differentiation calculation Then calculate the Levi-Civita connection.

[0067] 3. Calculate the correction term based on the Riemann metric and gradient term. :

[0068]

[0069] S3.2 Loss Function Design

[0070] The total loss function consists of four parts:

[0071] 1. PDE residual loss

[0072] To ensure the neural network output satisfies the PDE equation in S2:

[0073]

[0074] in This represents the number of configuration points.

[0075] 2. Boundary condition loss

[0076] Initial and terminal conditions are satisfied:

[0077]

[0078] 3. Parameter regularization loss

[0079] constraint parameters :

[0080]

[0081] in This represents the total number of time sampling points after time discretization. Time step For safety thresholds (e.g.) ).

[0082] 4. Data-driven loss

[0083] Reference data is generated through the offline training phase. Based on a fault-free scenario and a consistent topology, a reference solution for the PDE is generated. At this point, the PDE model, i.e., the parabolic partial differential equation (PDE) in step S2, is in a time-invariant Riemannian metric-free state. Penalty items The expression for time. Using the reference solution to train PINN, the spatiotemporal evolution of geometric PDEs is learned.

[0084] Based on reference data The data-driven loss is constructed as follows:

[0085]

[0086] in This represents the total number of observation data points used for supervised training.

[0087] Beneficial effects of this invention: This invention aims to propose a fault-tolerant control framework for geometric partial differential equations in cluster systems operating under time-varying topology and actuator failure environments. This method ensures that the agent navigates from its initial position to its target position along a dynamic reference trajectory, while simultaneously achieving endogenous compensation for failure effects. The main innovative contributions include:

[0088] 1. Problem Modeling Level: The fault-tolerant control problem is transformed into a distributed trajectory optimization problem using geometric PDEs, unifying the mathematical descriptions of faults, topology, and motion constraints. Coordination forces are autonomously generated through dynamic adjustment of Riemannian metric weights. Geometric heat flow theory is used to decouple kinematic constraints and communication limitations, establishing an intrinsic connection between geometric constraints and swarm dynamics. An improved Riemannian metric-coordinated nominal controller and virtual controller are constructed to address the imbalance of coordination forces caused by actuator failures. Reference trajectories satisfying swarm interaction constraints are derived through isomorphic parabolic PDE equations, achieving a unified solution for fault compensation and motion planning.

[0089] 2. Solution method level: By utilizing the parameter sharing mechanism of PINN, efficient adaptive optimal solution finding for complex PDEs is achieved, breaking through the bottlenecks of traditional numerical methods in terms of real-time performance and scalability; through independent network division of labor and lightweight fine-tuning, theoretical algorithms can be directly embedded into practical systems. Attached Figure Description

[0090] Figure 1 This is a schematic diagram of the PINN architecture;

[0091] Figure 2 A graph showing the number of training iterations versus the loss in PINN;

[0092] Figure 3 This is the optimal state solution based on PINN;

[0093] Figure 4 For solutions based on PINN and the finite difference method Norm;

[0094] Figure 5 This is a trajectory space diagram for multiple agents. Detailed Implementation

[0095] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0096] A cluster adaptive path planning method based on geometric PDE and PINN includes the following steps:

[0097] S1. Model a dynamic model of a fault cluster with a time-varying topology.

[0098] S2. By using geometrical thermal flux and Riemannian manifolds, a fault cluster with time-varying topology is constructed to generate a reference trajectory in an obstacle environment based on a parabolic system PDE model.

[0099] S3. Based on the reference trajectory of the cluster-based parabolic PDE system, the steady-state value of the established cluster parabolic PDE system is quickly solved using the physical information neural network PINN, and the fault-tolerant optimal control law of the cluster system with time-varying topology and actuator faults based on geometric PDE and PINN is obtained.

[0100] Example 1:

[0101] This patent embodiment provides an adaptive distributed fault-tolerant optimal control algorithm based on geometric partial differential equations (PDEs) for a cluster system with time-varying topology and actuator failures in an obstacle environment. The algorithm solves the parabolic partial differential equation (PDE) in S2 using a Physical Information Neural Network (PINN). Its core idea is to fuse the physical equation constraints and boundary conditions into a loss function through the neural network, achieving end-to-end learning of the equation solution. Specifically, for a formation system consisting of seven agents, the algorithm considers the agents' initial time... Having a connected topology, in Despite an actuator bias fault occurring in the intelligent agent, the PDE-PINN-based control method can ultimately ensure that the center position of the formation deployment reaches [the target value]. The target position is at coordinates (-2, 25). The implementation process includes the following steps:

[0102] Step S100: In the augmented dynamic model Below, compensation amount is introduced. Obtain the parabolic PDE system .

[0103] Step S110: Define the isomorphic parameter solution domain Boundary conditions include the lower boundary. hour Upper boundary hour initial conditions hour Configuration point sampling is performed within the domain. Inner uniform sampling Configuration points Samples were taken at the boundary locations. The sampling density is dynamically adjusted based on the residual distribution at each point.

[0104] Step S120: Employ a four-layer fully connected network (2D input layer, 32D x 3 hidden layers, 1D output layer) with Tanh activation function. Input To Network 1, output the predicted values. Value. Input Go to Network 2, and get To ensure Furthermore, it changes smoothly over time, taking an exponential function on the output of Network 2. The network approximates the spatial distribution of PDE solutions through nonlinear mapping.

[0105] Step S130: Automatic differential calculation and This forces the network to satisfy the PDE equations in S2. The mean square error of the residuals constitutes the PDE residual loss. The key improvement lies in directly calculating using PyTorch's higher-order automatic differential functions. Meanwhile, the automatic mixed-precision training module is used to accelerate computation and reduce GPU memory usage.

[0106] Step S140: Boundary Condition Loss constraint and hour The consistency between the predicted values ​​and the fixed values ​​of 2 and 0.3 indicates data-driven loss. make sure hour Established. The total loss is the weighted sum of four items:

[0107]

[0108] For acquiring real-world data, sensors are deployed in the actual physical system to record spatiotemporal data. High-precision numerical simulations are then achieved using reference solutions generated by the finite element method (FEM) or the finite difference method (FDM). If no data is available, this step can be omitted. Relying solely on the physical equation residuals and boundary condition loss .

[0109] Step S150: Minimize the total loss using the Adam optimizer (learning rate 0.001), and simultaneously update the weight learning rate decay strategies for Network 1 and Network 2 via backpropagation: initial values The loss is reduced by 0.5 times every 1000 steps. The training run is repeated for 20,000 epochs, with the loss output every 1000 epochs to monitor convergence, and uniform grid points are generated.

[0110] To address potential convergence difficulties during training, this patent implements a multi-dimensional diagnostic and correction mechanism: First, it introduces gradient clipping technology, using PyTorch's gradient clipping function to constrain parameter update magnitudes, effectively suppressing gradient explosion; second, it constructs an adaptive balancing mechanism for the loss function, based on... (Physical equation residuals) and The weight coefficients are dynamically adjusted based on the real-time convergence state of the boundary conditions to eliminate the dominant bias in multi-objective optimization. Furthermore, considering the physical constraints of the parameter network, a uniformly distributed initialization function is used to perform asymmetric initialization of the weights in the final layer of the network, ensuring the time-varying parameters... The initial output strictly satisfies the positive definiteness requirement, laying the foundation for physically compliant optimized trajectories.

[0111] The trained network predicts the overall solution and generates the solution. Save the loss curve as it changes with training epochs to analyze training stability, and calculate the PDE residuals. The norm changes with the number of training rounds. Figure 1 Two independent networks are shown, but they share common parameters. Figure 2 This demonstrates that as the number of training iterations increases, the value of the loss function approaches zero. Figure 3 The state evolution of the parabolic PDE based on PINN is shown, which eventually reaches a stable value, thus obtaining the reference trajectory of the cluster under fault and time-varying topology. Figure 4 The solutions based on the finite difference method and the PINN method are shown. As can be seen from the figure, the evolution of the norm over time shows that the PINN-based method obtains a stable solution faster than the finite difference method. Figure 5 The diagram shows the trajectories of multiple agents reaching the target position from the initial moment. It can be seen that actuator failure causes the agents' trajectories to deviate from their intended path. Fluctuations then occurred, and under the action of the PDE-PINN controller, at Reach the target location and avoid obstacles during operation.

Claims

1. A swarm adaptive path planning method based on geometric PDE and PINN, characterized in that, Includes the following steps: S1. Model a dynamic model of a fault cluster with time-varying topology, as follows: A cluster system composed of In the ideal environment without failure and obstacle, the dynamics of each agent is described as ； Variables denote the planar position and control input of agent at time and denote the displacement of agent in the horizontal axis coordinate direction and the vertical axis coordinate direction, is a nonlinear mapping, is the set of all agents in the system; Inter-agent communication is based on an undirected time-varying topology. Description, where the edge set The dynamic evolution of This indicates that the intelligent agent and The communication edge between them is formed if and only if the distance condition at the current time is met. When satisfied, among them Time-varying signals In time The right limit at a given point is defined as: ； in, and Representing intelligent agents respectively and Communication status At the time of transition The values ​​before and after represent the communication edge. At any moment Instantaneous changes; conditions Defined interval For the hysteresis identification region, the hysteresis effect of edge addition / deletion is introduced to simulate the negotiation delay in the actual communication connection establishment process. Cluster systems in time-varying topologies In this context, the evolution of a cluster system relies on the potential energy generated by interactions between individuals, specifically referring to the control force of a fault-free cluster system. ; The dynamics of the intelligent agent due to actuator failure are characterized as ,in, Indicates the first The state of a faulty agent, function Mathematical models representing faults It is a collection of faulty intelligent agents; Define the state vector as Having an initial state ;in, Indicates the global state of the faulty cluster system. To generate auxiliary variables for synergistic forces; based on vector augmentation, the dynamic augmentation model of the fault cluster is described as follows: Let the augmented matrix be , ,in, , It is a Laplace matrix; middle for A column vector of all zeros for A matrix of all 1s; in addition, variables This represents the interaction force of all fault-free intelligent agents, i.e., the nominal control force, where the force of a single fault-free intelligent agent is... ; For the fault-tolerant control input to be designed, The total potential function variable represents the state of the faulty agent. S2. Using geometric heat flow and Riemannian manifolds, a fault cluster with time-varying topology is constructed to generate a reference trajectory in an obstacle environment based on a parabolic system PDE model. S3. Based on the reference trajectory of the cluster based on the parabolic PDE system, the steady-state value of the established cluster parabolic PDE system is quickly solved using the physical information neural network PINN. The fault-tolerant optimal control law of the cluster system based on geometric PDE and PINN with time-varying topology and actuator faults is obtained, and the path planning is completed.

2. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 1, characterized in that, Step S2, based on the parabolic system PDE model, generates a reference trajectory under obstacle conditions. The specific implementation process is as follows: Riemannian metric tensor with positive definite symmetry Define the trajectory length of the faulty cluster system ,in, Indicates the agent's time The trajectory of the movement; ,in Pre-calculated using Gram-Schmidt orthogonalization, It is a supplementary matrix, representing the original control input matrix. Auxiliary input direction of complementary underactuated components, parameters to be designed , indicating a time-varying regulator, The time-varying regularization weight parameters satisfy The adjusted gain is and For actuator failures, time-varying gain Adjusting the Riemannian metric tensor This leads to adaptive adjustment of the trajectory length of the cluster system under the metric; and the introduction of gain. Through constraints Adjusting the trajectory length enables the agent to avoid obstacles; when the parameters This means taking the most impactful faults and obstacles as the underactuated direction of the actuator and applying a penalty term to force the agent's trajectory to converge to the control-allowed motion subspace of the swarm system and to be robust. Will As an additional constraint for local communication. Represents time The total number of other intelligent agents that have direct communication or perceptual interaction with intelligent agent A is defined as follows: ,like This indicates a violation of the maximum number of adjacent edges constraint, based on the constraint condition of the edge with the maximum degree. , To represent the number of virtual neighbors of agent 𝑖, design a time-varying function: ； , It is the Herveside step function, introducing variables. Characterize the cumulative error and define a vector. The dynamic model of the fault cluster system is represented as a Riemann augmented system: ； Define state vector ,in , and Using Riemann length theory, an augmented Riemannian metric tensor is introduced. Transform the topology constraints of the faulty cluster into constraints: ； in, ,element Indicates the time-varying function At that time, the penalty coefficient for exceeding the limit; matrix The constraints characterizing time-varying topologies based on Riemannian metrics are essentially derived from the constraints of the time-varying edges of the topology. , ,in It is a dynamically adjusted penalty coefficient used to quantify opposite edges. Deviation from the preset target distance during the optimization process The severity of the punishment; According to the time-varying Riemannian metric Based on the gradient evolution of heat flow, the energy functional considering the Riemann length is: ； in The Lagrangian of an instantaneous system; the Euler-Lagrange equations are corrected using variational principles, and a correction term is introduced. This yields the partial differential equation for heat flow (PDE), which is the reference trajectory dynamics of the cluster system with actuator failure and time-varying topology in step S2: ； ； in It is time homotopy variables, , Connect the initial positions of the faulty cluster under Riemannian metrics and finish line The trajectory; Representation and Measurement Related Levi-Civita contacts, and This reflects the gradient term after metric adjustment; the term Describing trajectory acceleration and nominal dynamics The deviation between them; minimizing curvature and trajectory length by reshaping the potential field, gradient term Adjust the trajectory to Time-varying measurement In Introducing and generating acceleration terms This reflects the impact of dynamic adjustments to constraints on the trajectory deformation rate; Based on the LaSalle invariant set principle, the non-increasing property of the Lyapunov function is obtained, and consequently, the state of the parabolic PDE system converges to the steady-state value. , where the state By decomposing the steady state According to isomorphism theory, This represents the ideal trajectory of the cluster system under actuator failure, time-varying topology, and obstacles, also known as the reference trajectory.

3. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 2, characterized in that, The specific implementation process of step S3, based on the fault-tolerant optimal control law of geometric PDE and PINN, is as follows: Based on the established steady-state value of the parabolic PDE system To overcome the presence of unknown variables in this parabolic PDE system And obtain the fault-tolerant optimal control law based on geometric PDE and PINN for a cluster system with time-varying topology and actuator faults: ； S3.1, Learning the solution of the parabolic PDE equation in S2 through PINN First, define the input variable as the homotopy parameter. and time ; Output variables are divided into main outputs and auxiliary outputs, with the main output being the state. ,Right now A real vector of dimension, with an auxiliary output containing unknown parameters. ; Next, two parallel neural networks are designed, with network 1 approximating the state. Network 2 computes parameterized time-varying parameters State network 1 uses a spatiotemporal bivariate architecture. As input, it dynamically represents the system's evolution trajectory; parameter network 2 uses only time... For input, focus on time-varying parameters Identification; Finally, the PDE residuals are calculated using automatic differentiation; S3.2 Design a loss function to train the physical information neural network PINN; The total loss function consists of four parts: PDE residual loss To ensure the neural network output satisfies the PDE equation in S2, boundary condition loss is applied. Satisfying initial and terminal conditions; parameter regularization loss constraint parameters Data-driven loss Reference data is generated through the offline training phase. .

4. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 3, characterized in that, The specific process for calculating the PDE residuals through automatic differentiation is as follows: First, calculate the Riemann metric. : ; in Pre-calculated using Gram-Schmidt orthogonalization, based on which... ; Secondly, calculate the gradient term. : First, use automatic differentiation calculation Then calculate the Levi-Civita connection. ; Finally, the correction term is calculated based on the Riemann metric and the gradient term. : 。 5. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 4, characterized in that, The specific implementation of S3.2 is as follows: PDE residual loss To ensure that the neural network output satisfies the PDE equation in S2, it is designed as follows: ； in Number of configuration points; Boundary condition loss The initial and terminal conditions are satisfied: ； Parameter regularization loss constraint parameters ; ； This represents the total number of time sampling points after time discretization. Time step This is a safety threshold; Data-driven loss Reference data is generated through the offline training phase. Based on a fault-free scenario and a consistent topology, a reference solution for the PDE is generated. Based on the reference data, the data-driven loss is constructed as follows: ； in This represents the total number of observation data points used for supervised training.

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