A cell migration behavior prediction method based on physical information width learning

Abstract

The application discloses a cell migration behavior prediction method based on physical information width learning, which utilizes cell scratch experiment data to construct a base function matrix, combines a Fisher-KPP reaction diffusion equation to construct a nonlinear least square problem containing physical mechanism constraints, and adopts an enhanced nonlinear least square disturbance algorithm to iteratively solve output weights, so as to realize prediction of cell density space-time evolution. The application converts traditional deep network iterative training into a nonlinear least square problem, greatly reduces a search space through physical information initialization and linearization approximation, and realizes order-of-magnitude improvement of training speed. The application constructs a width learning architecture containing feature nodes and enhanced nodes, effectively avoids the gradient vanishing problem in deep learning, combines analytical derivative calculation, eliminates cumulative error of automatic differentiation on high-order derivatives, and can more accurately capture space-time evolution characteristics in cell migration.

Description

Technical Field

[0001] This invention relates to the technical field of the intersection of biomedical engineering and computational biology, and in particular to a method for predicting cell migration behavior based on physical information width learning. Background Technology

[0002] Cell migration and proliferation are not only fundamental to life activities such as embryonic development, tissue regeneration, and wound healing, but also core driving forces in pathological processes such as cancer invasion and metastasis. In modern cell biology research, in vitro cell scratch assays have become the standard paradigm for observing and quantifying the movement characteristics of cell populations. However, traditional experimental analyses mainly rely on morphological observations of microscopic images or simple migration rate calculations. This qualitative or semi-quantitative approach is insufficient to reveal the intrinsic dynamic mechanisms of cell populations in complex spatiotemporal environments, nor can it accurately predict the future evolutionary state of cells. To overcome this limitation, mathematical modeling methods based on partial differential equations have emerged. Among them, the Fisher-KPP reaction-diffusion equation is widely recognized as the standard physical model for describing the spatiotemporal evolution of cell density because it can simultaneously characterize the random diffusion movement and logistic proliferation behavior of cells.

[0003] While mathematical models provide a theoretical framework for understanding cell behavior, solving the Fisher-KPP reaction-diffusion equation and retrieving parameters face multiple technical bottlenecks in practical applications. Traditional numerical methods, such as the finite element method or the finite difference method, are theoretically mature, but their computational accuracy and stability are highly dependent on the quality of the mesh generation. When dealing with irregular wound boundaries or complex biological tissue geometries commonly seen in cell scratch experiments, generating high-quality body-fit meshes is not only extremely time-consuming, but the computational cost also increases exponentially with increasing spatiotemporal resolution, making it difficult to meet the high-throughput and rapid analysis requirements of biological experiments. Furthermore, traditional numerical methods typically require prior knowledge of precise physical parameters, such as the diffusion coefficient and proliferation rate, which are often unknown in real biological experiments and must be obtained through inverse problem solving, further increasing computational complexity and uncertainty.

[0004] With the rise of artificial intelligence, data-driven methods based on deep learning have provided new insights into cell behavior prediction. However, purely data-driven deep neural networks are essentially black-box models, requiring massive amounts of labeled experimental data for training. In the biomedical field, acquiring high spatiotemporal resolution cell sequence images is costly and time-consuming, often facing severe data scarcity. More importantly, purely data-driven models lack physical interpretability and are prone to poor generalization ability and violations of fundamental physical conservation laws outside the scope of the training data, leading to reduced reliability of prediction results.

[0005] In recent years, Physical Information Neural Networks (PINs), as an emerging paradigm integrating physical mechanisms and data-driven approaches, have addressed the problems of small-sample learning and grid dependency to some extent by embedding the residuals of partial differential equations into the loss function. However, most existing PIN architectures still utilize the deep network structures of traditional deep learning, and their training process relies on gradient-based backpropagation algorithms. This iterative, non-convex optimization process often suffers from extremely slow convergence, high computational cost, and susceptibility to local minima. For cell biology research scenarios requiring real-time feedback or rapid processing of large numbers of experimental samples, the low training efficiency of existing PIN methods has become a key bottleneck restricting their practical application. Therefore, there is an urgent need to develop a novel method that can embed physical mechanisms to ensure the biological validity of predictions while also breaking free from the constraints of deep network gradient training to achieve second-level rapid modeling and prediction. Summary of the Invention

[0006] The purpose of this invention is to address the technical problems of existing deep learning-based partial differential equation solving methods, which employ deep network architectures and suffer from slow training convergence speeds, susceptibility to local minima, sensitivity to hyperparameters, and high computational costs. This invention provides a cell migration behavior prediction method based on width learning of physical information. This method constructs a width learning architecture containing feature nodes and enhancement nodes, utilizes feature mapping and enhancement mapping to replace deep stacking, and transforms the solution of the Fisher-KPP reaction-diffusion equation containing cell migration mechanisms into a nonlinear least squares problem, thereby achieving efficient and high-precision solutions without backpropagation.

[0007] To achieve the above objectives, the technical solution provided by this invention is: a method for predicting cell migration behavior based on physical information width learning, comprising the following steps:

[0008] Step S1: Obtain data from the cell scratch experiment and physical model parameters for subsequent construction of physical constraints; simultaneously, use spatiotemporal coordinate data as a separate input variable, and convert the input variable into multiple sets of feature nodes through nonlinear mapping, then combine these multiple sets of feature nodes to form a feature matrix; based on the feature matrix, further generate multiple sets of enhancement nodes, and combine these multiple sets of enhancement nodes to form an enhancement matrix; wherein, the weights and bias parameters inside the feature nodes and enhancement nodes remain fixed after initialization, and only the output weights are trainable parameters; finally, the feature matrix and enhancement matrix are horizontally concatenated to construct the basis function matrix;

[0009] Step S2: Calculate the derivative matrix of the basis function matrix obtained in Step S1 with respect to the input variables based on the chain rule; then, substitute the derivative matrix and the physical model parameters obtained in Step S1 into the Fisher-KPP reaction-diffusion equation, boundary conditions, and initial conditions that control cell migration and proliferation processes to construct a physical residual equation for measuring prediction bias; finally, construct a nonlinear least squares problem that includes constraints on cell migration mechanisms with the goal of minimizing the physical residual.

[0010] Step S3: Based on the nonlinear least squares problem obtained in step S2, the output weights are iteratively updated using an enhanced nonlinear least squares perturbation algorithm until convergence to obtain the optimal output weights. The enhanced nonlinear least squares perturbation algorithm first uses linearization approximation to obtain the initial weight estimate, and then uses this as the starting point to perform a local search using a nonlinear least squares optimization algorithm. If the preset error requirement is not met or the maximum number of attempts is not reached, random noise perturbation is applied to the current weights to generate a new starting point until the optimal output weights are found.

[0011] Step S4: The optimal output weights obtained in step S3 are used to perform a linear combination calculation with the basis function matrix constructed in step S1 to obtain an approximate solution for the cell density at any point in the spatiotemporal domain to be predicted, and this is used as the predicted cell density value for that point. Based on the data of each predicted spatiotemporal coordinate point, the output presents the overall spatiotemporal evolution distribution of cell density, thereby realizing the prediction of cell population migration and proliferation behavior.

[0012] Furthermore, in step S1, the data of the cell scratch experiment includes cell density distribution data at the initial moment; the physical model parameters include cell diffusion coefficient, intrinsic cell growth rate, and crowding effect coefficient; and the spatiotemporal coordinate data includes time variables and spatial location variables.

[0013] Furthermore, in step S1, the input variables are projected and generated using a first nonlinear activation function. A set of feature nodes is constructed, where the weights and biases of the feature nodes are randomly initialized and fixed; all feature nodes are concatenated to obtain a feature matrix, which is then generated using a second nonlinear activation function. The group of augmentation nodes has its weights and biases randomly initialized and fixed; the final basis function matrix is ​​formed by horizontally concatenating the feature matrix and the augmentation matrix.

[0014] Furthermore, the first nonlinear activation function and the second nonlinear activation function are selected from any one of the hyperbolic tangent function, sine function, sigmoid function, Gaussian function or Swish function.

[0015] Furthermore, in step S2, the specific form of the Fisher-KPP reaction-diffusion equation is as follows: ;in, Indicates cell density, Indicates time, Indicates spatial location, Indicates the cell diffusion coefficient. Indicates the intrinsic growth rate of cells. This represents the crowding effect coefficient. This represents the second derivative of cell density with respect to space. This represents the first derivative of cell density with respect to time.

[0016] Furthermore, in step S2, the derivative matrix is ​​calculated by directly differentiating the basis function matrix based on the chain rule; for feature nodes, the first and second derivatives of the first nonlinear activation function are calculated and combined with the weights of the feature nodes to obtain the feature derivative matrix; for enhancement nodes, the enhancement derivative matrix is ​​calculated by combining the feature derivative matrix, the weights of the enhancement nodes, and the derivative of the second nonlinear activation function; finally, the feature derivative matrix and the enhancement derivative matrix are horizontally concatenated to obtain the derivative matrix of the basis function matrix with respect to the input variables, which is used to construct the physical residual equation.

[0017] Furthermore, in step S3, the nonlinear least squares optimization algorithm is any one of the trust region reflection algorithm, the Levenberg-Marquardt algorithm, or the Gauss-Newton method.

[0018] Furthermore, in step S3, the random noise perturbation applied to the current weights adopts a scaling additive hybrid strategy, and the formula for calculating the new weights after perturbation is as follows: ;in, This represents the new weights generated after the perturbation. This represents the optimal weight found in the current search. For random scaling factor, The additive noise term follows a uniform distribution, and its boundary is dynamically determined by a preset maximum disturbance amplitude and a random scaling factor.

[0019] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0020] 1. Improved Solution Speed: This invention transforms the traditional iterative training of deep networks into a nonlinear least squares problem. By initializing with physical information and using linearization approximation, the search space is significantly reduced, resulting in an order-of-magnitude improvement in training speed compared to traditional gradient-based backpropagation algorithms.

[0021] 2. Enhanced expressive power and accuracy: By constructing a wide learning architecture that includes feature nodes and enhancement nodes, the gradient vanishing problem in deep learning is effectively avoided; combined with analytical derivative calculation, the cumulative error of automatic differentiation on higher-order derivatives is eliminated, enabling more accurate capture of the spatiotemporal evolution features in cell migration.

[0022] 3. The algorithm has strong stability and good robustness: The proposed enhanced nonlinear least squares perturbation algorithm effectively solves the problem of easily getting trapped in local minima in nonlinear optimization. Attached Figure Description

[0023] Figure 1 This is a flowchart illustrating the method of the present invention.

[0024] Figure 2 This is a schematic diagram of a width learning architecture including feature nodes and enhancement nodes according to an embodiment of the present invention.

[0025] Figure 3 This is a detailed execution flowchart of the enhanced nonlinear least squares perturbation algorithm.

[0026] Figure 4 This is a diagram showing the predicted results of the cell scratch experiment in an embodiment of the present invention. Detailed Implementation

[0027] The present invention will be further described in detail below with reference to the embodiments and accompanying drawings, but the embodiments of the present invention are not limited thereto.

[0028] like Figure 1 As shown, this embodiment discloses a cell migration behavior prediction method based on width learning of physical information. This method addresses the problems of data scarcity in cell biology research, the difficulty of traditional numerical simulation grid partitioning, and the low training efficiency of deep learning models. It proposes a fast prediction architecture combining the Fisher-KPP reaction-diffusion mechanism with width learning. This embodiment uses in vitro cell scratch experiments as an application scenario. By constructing a width learning architecture containing feature nodes and enhancement nodes, it achieves accurate prediction of the spatiotemporal evolution of cell population density.

[0029] Step S1: Obtain data from the cell scratch experiment and physical model parameters for subsequent construction of physical constraints; simultaneously, use spatiotemporal coordinate data as input variables. In this embodiment, the classic in vitro cell scratch experiment is used as the research object. The migration and proliferation dynamics of the cell population are described by the Fisher-KPP reaction-diffusion equation:

[0030] ;

[0031] in, Indicates cell density, Indicates time, Indicates spatial location, Indicates the cell diffusion coefficient. Indicates the intrinsic growth rate of cells. This represents the crowding effect coefficient. This represents the second derivative of cell density with respect to space. This represents the first derivative of cell density with respect to time.

[0032] In this embodiment, the physical parameters in the above equations are known constants. Cell diffusion coefficient. Set as Intrinsic growth rate of cells Set as Crowding effect coefficient Set as Spatial domain length for Upper limit of the time domain for The cell density distribution observed at the initial time (time 0) was used as the initial condition data. The boundaries of the computational domain were set to zero-flux von Neumann boundary conditions to simulate the barrier effect of the culture dish walls. Randomly sampled or uniformly generated collocation points within the spatiotemporal domain served as constraint points for the physical equations. To eliminate the influence of dimensional differences on the algorithm, the input spatiotemporal coordinate data was normalized.

[0033] Secondly, a width learning architecture (also known as a width learning network) containing feature nodes and augmentation nodes is constructed to approximate the cell density function. For example... Figure 2 As shown, this architecture abandons the deep stacking structure of traditional deep neural networks and adopts a horizontal scaling strategy.

[0034] To construct a basis function space with strong nonlinear approximation capability, this embodiment employs a strategy combining feature mapping and enhancement mapping. Specifically, the input variables... Projection and generation via a first nonlinear activation function Group of feature nodes, the first Group feature nodes The calculation formula is:

[0035] ;

[0036] in, and They represent the first The weight matrix and bias vector of the feature nodes are randomly generated through a preset probability distribution during the initialization phase and remain fixed in subsequent calculations. The activation function is a nonlinear activation function. To ensure the feasibility of analytical differentiation, the activation function should be a continuous and differentiable nonlinear function, such as the hyperbolic tangent function, sine function, sigmoid function, Gaussian function, or Swish function. In this embodiment, the first nonlinear activation function... Second nonlinear activation function Specifically, the hyperbolic tangent function is selected.

[0037] Furthermore, to enhance the ability to represent complex physical fields, a feature matrix is ​​formed by concatenating all feature nodes. As input, it is further generated through a second nonlinear activation function. Group enhancement nodes. (Number) Group Enhancement Nodes The calculation formula is:

[0038] ;

[0039] in, and They represent the first The weight matrix and bias vector of the group enhancement nodes also adopt a strategy of random initialization and fixation; This is the second nonlinear activation function. Consistent with the feature mapping stage, this activation function can also be a smooth nonlinear function with similar properties, such as the hyperbolic tangent function, the sine function, the sigmoid function, the Gaussian function, or the Swish function.

[0040] Finally, the feature matrix The enhancement matrix composed of enhancement nodes By performing horizontal concatenation, a basis function matrix is ​​constructed to represent the approximate solution of the partial differential equation to be solved. :

[0041] ;

[0042] At this point, the approximate solution to the partial differential equation to be solved... Defined as a basis function matrix With the output weight to be determined A linear combination, i.e. Since the internal weights are fixed, only the output weights need to be trained. This greatly reduces the complexity of optimization.

[0043] Specifically, the architecture parameters configured in this embodiment are as follows: number of feature node groups. The system is configured with 20 groups, each containing 10 nodes, for a total of 200 feature nodes; the number of enhanced node groups is [not specified]. The system is configured with 20 groups, each containing 10 nodes, for a total of 200 enhancement nodes. The architecture comprises a total of 400 nodes, representing the output weights. The dimension is 400. The first nonlinear activation function... Second nonlinear activation function Hyperbolic tangent function is used for all features. Feature node weights. and enhance node weight Obey during the initialization phase The even distribution of the particles is maintained during training.

[0044] Step S2: Construct a nonlinear least squares problem constrained by cell migration mechanisms. Based on the governing equations, boundary conditions, and initial conditions of the Fisher-KPP reaction-diffusion equation, construct the physical residual equation. The core of this step lies in utilizing the analytical form of a width-based learning architecture containing feature nodes and augmentation nodes, and deriving the analytical derivative formula with respect to the input variables based on the chain rule, thus replacing automatic differentiation.

[0045] The specific analytical derivative calculation is as follows: For a feature node, its derivative with respect to the input variable is... Each component of The formula for calculating the partial derivative is:

[0046] ;

[0047] in, Represents the activation function First derivative, It represents the Hadamah accumulation. This represents the elements of the corresponding row in the weight matrix. The power of 1. It should be noted that the specific expression for calculating the analytical derivative depends on the chosen activation function. In this embodiment, since the activation function is the hyperbolic tangent function, its first derivative is... The second derivative is For the enhancement nodes, nested derivative calculations are performed using the derivative results of the aforementioned feature nodes. Using the aforementioned analytical derivatives, the basis function matrices are calculated with respect to time. The first derivative matrix and about space The second derivative matrix is ​​then obtained. Subsequently, the first-order temporal derivative matrix and the second-order spatial derivative matrix are substituted into the control operator, boundary operator, and initial condition operator of the Fisher-KPP reaction-diffusion equation to construct a nonlinear least squares problem constrained by cell migration mechanisms.

[0048] Step S3: Solve for the optimal output weights using an enhanced nonlinear least squares perturbation algorithm. The objective is a nonlinear least squares optimization problem. To solve this nonlinear optimization problem efficiently and stably, this embodiment employs the following... Figure 3 An enhanced nonlinear least squares perturbation algorithm is shown. This algorithm combines deterministic search with random noise perturbation mechanism, and the specific process is as follows:

[0049] First, physical information is initialized. Specifically, the nonlinear quadratic terms in the Fisher-KPP reaction-diffusion equation are ignored, simplifying the original equation into a linear equation. This simplified equation is then solved using the linear least squares method, and the solution is used as the initial weight guess. This step provides a high-quality starting point for nonlinear optimization.

[0050] Secondly, with Starting with the nonlinear least squares optimization algorithm, a deterministic local search is performed. During this process, the analytical derivative derived in step S2 is used to accurately calculate the Jacobian matrix of the residuals with respect to the weights. Among them, the physical residual vector Defined as the deviation after substituting the architecture prediction solution into the Fisher-KPP equation, i.e. The formula for calculating the Jacobian matrix is:

[0051] ;

[0052] in, This represents the variation of the differential operator with respect to the solution. Using analytic Jacobian matrices instead of numerical differentiation can significantly improve computational efficiency and convergence stability.

[0053] Finally, a random noise perturbation mechanism is introduced to escape local minima. If the preset error requirement is not met or the maximum number of attempts is not reached during the local search process, random noise perturbation is applied to the current weights.

[0054] In this embodiment, the random noise perturbation employs a scaling-additive hybrid strategy. Specifically, the new weights after perturbation... The calculation formula is as follows:

[0055] ;

[0056] in, This represents the new weights generated after the perturbation. This represents the optimal weight found in the current search. For random scaling factor, The noise term is additive, and it follows a uniform distribution. Its boundary is dynamically determined by a preset maximum disturbance amplitude and a random scaling factor. In this embodiment, the parameter is set as: random scaling factor. Obedience range Uniform distribution; additive noise term Obedience range The distribution is uniform, the maximum number of attempts is 5, and the preset error is [missing information]. With this configuration, the model can quickly converge to the optimal output weights while ensuring the accuracy of physical constraints.

[0057] This perturbation strategy not only introduces randomness but also preserves the directional characteristics of the current optimal solution through random scaling, thus maintaining search stability while ensuring escape from local extrema. After generating new guesses, the local search is performed again from these guesses until the optimal output weights that meet the error requirements are found.

[0058] Step S4: Output the spatiotemporal evolution distribution prediction results of cell density. Utilize the calculated output weights... The basis function matrix constructed in step S1 Through matrix multiplication The predicted value of any point in the spatiotemporal domain to be solved is calculated, thereby obtaining the cell density value at any time and location in the spatiotemporal domain. This enables quantitative prediction of cell population migration trends and proliferation density, solving the technical problem that traditional methods cannot simultaneously capture random diffusion and proliferation behavior.

[0059] To verify the effectiveness of the method of the present invention, the model constructed above was used to predict real in vitro cell migration experimental data. The experimental data came from scratch assays of PC-3 prostate cancer cells cultured in vitro. Cell migration images were acquired at preset time intervals using a real-time imaging system, and cell density distribution data were extracted using image meshing segmentation and cell counting techniques. The experimental setup was as follows: scratch assay data with an initial cell count of 10,000 cells were used. The model used only the cell density distribution at time 0 as training data, i.e., the initial condition, and the task was to predict the cell density evolution over the next 48 hours.

[0060] Figure 4 The figure illustrates the prediction results of cell migration behavior using the method of this embodiment. The horizontal axis represents the position along the scratch width direction, and the vertical axis represents cell density. The solid lines represent the cell density curves predicted by the method of this invention at various time points of 12, 24, 36, and 48 hours, while the scatter plots represent the actual observed experimental data.

[0061] The results show that although the model was trained using only the initial state data at time 0, its prediction curves highly overlapped with the experimental measurement points at various future time points. The model not only accurately captured the process of cell migration to the wound center dominated by the diffusion term, but also precisely reproduced the process of cell density logistic growth dominated by the response term.

[0062] Compared with traditional deep learning methods, the method in this embodiment maintains extremely high prediction accuracy while reducing training time from several hours to seconds and runtime to less than 1 second. It also does not require complex hyperparameter tuning, demonstrating great potential in rapid biomedical modeling applications.

[0063] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for predicting cell migration behavior based on physical information width learning, characterized in that, Includes the following steps: Step S1: Obtain data from the cell scratch experiment and physical model parameters for subsequent construction of physical constraints; simultaneously, use spatiotemporal coordinate data as a separate input variable, convert the input variable into multiple sets of feature nodes through nonlinear mapping, and combine these multiple sets of feature nodes to form a feature matrix. Based on the feature matrix, multiple sets of enhancement nodes are further generated, and these multiple sets of enhancement nodes are combined to form an enhancement matrix; wherein, the weights and bias parameters inside the feature nodes and enhancement nodes remain fixed after initialization, and only the output weights are trainable parameters; finally, the feature matrix and enhancement matrix are horizontally concatenated to construct a basis function matrix; Step S2: Calculate the derivative matrix of the basis function matrix obtained in Step S1 with respect to the input variables based on the chain rule; then, substitute the derivative matrix and the physical model parameters obtained in Step S1 into the Fisher-KPP reaction-diffusion equation, boundary conditions, and initial conditions that control cell migration and proliferation processes to construct a physical residual equation for measuring prediction bias; finally, construct a nonlinear least squares problem that includes constraints on cell migration mechanisms with the goal of minimizing the physical residual. Step S3: Based on the nonlinear least squares problem obtained in step S2, the output weights are iteratively updated using an enhanced nonlinear least squares perturbation algorithm until convergence to obtain the optimal output weights. The enhanced nonlinear least squares perturbation algorithm first uses linearization approximation to obtain the initial weight estimate, and then uses this as the starting point to perform a local search using a nonlinear least squares optimization algorithm. If the preset error requirement is not met or the maximum number of attempts is not reached, random noise perturbation is applied to the current weights to generate a new starting point until the optimal output weights are found. Step S4: The optimal output weights obtained in step S3 are used to perform a linear combination calculation with the basis function matrix constructed in step S1 to obtain an approximate solution for the cell density at any point in the spatiotemporal domain to be predicted, and this is used as the predicted cell density value for that point. Based on the data of each predicted spatiotemporal coordinate point, the output presents the overall spatiotemporal evolution distribution of cell density, thereby realizing the prediction of cell population migration and proliferation behavior.

2. The cell migration behavior prediction method based on physical information width learning according to claim 1, characterized in that, In step S1, the data from the cell scratch experiment includes cell density distribution data at the initial moment; the physical model parameters include cell diffusion coefficient, intrinsic cell growth rate, and crowding effect coefficient; and the spatiotemporal coordinate data includes time variables and spatial location variables.

3. The cell migration behavior prediction method based on physical information width learning according to claim 1, characterized in that, In step S1, the input variables are projected and generated using the first nonlinear activation function. A set of feature nodes is constructed, where the weights and biases of the feature nodes are randomly initialized and fixed; all feature nodes are concatenated to obtain a feature matrix, which is then generated using a second nonlinear activation function. The group of augmentation nodes has its weights and biases randomly initialized and fixed; the final basis function matrix is ​​formed by horizontally concatenating the feature matrix and the augmentation matrix.

4. The cell migration behavior prediction method based on physical information width learning according to claim 3, characterized in that, The first nonlinear activation function and the second nonlinear activation function are selected from any one of the hyperbolic tangent function, sine function, sigmoid function, Gaussian function or Swish function.

5. The cell migration behavior prediction method based on physical information width learning according to claim 1, characterized in that, In step S2, the specific form of the Fisher-KPP reaction-diffusion equation is as follows: ;in, Indicates cell density, Indicates time, Indicates spatial location, Indicates the cell diffusion coefficient. Indicates the intrinsic growth rate of cells. This represents the crowding effect coefficient. This represents the second derivative of cell density with respect to space. This represents the first derivative of cell density with respect to time.

6. The cell migration behavior prediction method based on physical information width learning according to claim 1, characterized in that, In step S2, the derivative matrix is ​​calculated by directly differentiating the basis function matrix based on the chain rule; For a feature node, the first and second derivatives of the first nonlinear activation function are calculated and combined with the weights of the feature node to obtain the feature derivative matrix; For the augmentation node, the augmentation derivative matrix is ​​calculated by combining the feature derivative matrix, the weight of the augmentation node, and the derivative of the second nonlinear activation function; Finally, the eigenderm matrix and the augmented derivative matrix are concatenated laterally to obtain the derivative matrix of the basis function matrix with respect to the input variables, which can be used to construct the physical residual equation.

7. The cell migration behavior prediction method based on physical information width learning according to claim 1, characterized in that, In step S3, the nonlinear least squares optimization algorithm is any one of the trust region reflection algorithm, the Levenberg-Marquardt algorithm, or the Gauss-Newton method.

8. The cell migration behavior prediction method based on physical information width learning according to claim 1, characterized in that, In step S3, the random noise perturbation applied to the current weights adopts a scaling additive hybrid strategy, and the formula for calculating the new weights after perturbation is as follows: ;in, This represents the new weights generated after the perturbation. This represents the optimal weight found in the current search. For random scaling factor, The additive noise term follows a uniform distribution, and its boundary is dynamically determined by a preset maximum disturbance amplitude and a random scaling factor.

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