A hemodynamic calculation method based on infinitesimal neural operators

Abstract

This invention discloses a hemodynamics calculation method based on infinitesimal neural operators, comprising: predicting the time shift of the intravascular flow field based on the initial flow field, vascular geometry, and boundary conditions using a pre-trained prediction network; the prediction network, by designing infinitesimal neural operators with infinitesimal correlation and displacement invariance, achieves efficient calculation of hemodynamics simulations in different computational domains; the prediction network includes a lifting layer, an inner block layer, and a projection layer; the prediction network is trained using supervised learning, minimizing the mean square error of multi-step recursive prediction within a multi-step cyclic framework, and learning a general time shift operator by decomposing the hemodynamics problem into infinitesimally correlated subproblems, thereby achieving rapid two-dimensional or three-dimensional simulation. Compared with existing technologies, this invention can adapt to multiple computational domains without retraining, significantly improving computational efficiency and accuracy, and is applicable to the field of medical blood flow simulation.

Description

Technical Field

[0001] This invention belongs to the field of computer simulation algorithms based on neural operators, and specifically relates to a hemodynamic calculation method based on infinitesimal neural operators. Background Technology

[0002] With the development of deep learning technology, neural operator methods have emerged as a new tool for solving partial differential equations (PDEs). Unlike traditional numerical methods, neural operator methods exhibit higher computational efficiency in solving PDEs. By learning the mapping from input functions to output functions, neural operators can quickly predict the behavior of physical fields. Among the many neural operator methods, DeepONet and FNO are representative techniques currently being researched and applied. The following will describe the technical details of these two methods in detail and analyze their shortcomings, especially the limitations related to hemodynamic calculations.

[0003] DeepONet (Deep Operator Network):

[0004] DeepONet is a neural network-based operator learning framework designed to learn the mapping from input functions (e.g., initial or boundary conditions) to output functions (e.g., solutions to PDEs). Its design is inspired by the general approximation theorem for operators.

[0005] Although DeepONet is theoretically very versatile, it has the following shortcomings in practical applications, especially in hemodynamic calculations:

[0006] Limitations of fixed computational domain: DeepONet is typically trained on a specific computational domain. For hemodynamic problems, due to significant differences in vascular geometry among individuals, DeepONet needs to be retrained or fine-tuned for each new geometry, increasing computational cost and time and limiting its flexibility.

[0007] High computational complexity: The separate design of the branch network and the backbone network makes DeepONet computationally expensive when processing high-dimensional or complex input functions. This becomes a bottleneck in scenarios requiring real-time simulation or large-scale computation (such as blood flow analysis).

[0008] Limited adaptability to non-periodic problems: Hemodynamics involves non-periodic boundary conditions and complex vessel wall geometry. DeepONet's general architecture is not optimized for these characteristics, which may result in insufficient ability to capture flow field details.

[0009] FNO (Fourier Neural Operator):

[0010] FNO is a neural operator method that uses Fourier transform to process functions in the frequency domain, and is particularly suitable for problems with periodic boundary conditions. Its core idea is to parameterize the integrator kernel to Fourier space.

[0011] FNO performs well in periodic problems, but has the following limitations in hemodynamic calculation scenarios:

[0012] Limitations of the periodicity assumption: The Fourier transform basis of FNO makes it more suitable for problems with periodic boundary conditions. However, the vascular geometry in hemodynamics is non-periodic, with complex and irregular boundaries, which leads to a decrease in the performance of FNO or requires additional preprocessing (such as padding or domain expansion), increasing the difficulty of implementation.

[0013] Limitations of a fixed computational domain: Similar to DeepONet, FNO is typically trained on a fixed computational domain. For blood flow simulations where vascular geometry varies from person to person, FNO cannot directly adapt to new domains and must be retrained, reducing its efficiency.

[0014] Insufficient capture of high-frequency details: FNO effectively captures global features through low-frequency modes, but in areas with more high-frequency details, such as near the vessel wall or in flow separation regions, the prediction accuracy may decrease, affecting the description of key flow field features.

[0015] In summary, these methods typically assume a fixed computational domain, making it difficult to directly apply pre-trained models to computational domains with varying shapes or sizes. In hemodynamic simulations, the geometry of human blood vessels varies due to individual differences. Traditional neural operator methods require retraining the model each time a new computational domain is encountered, resulting in significant waste of time and computational resources. Furthermore, they lack geometric generalization capabilities, limiting their practicality and efficiency. Therefore, addressing the reusability issue of existing neural operator methods across different computational domains, there is an urgent need to propose a hemodynamic calculation method that can flexibly adapt to multiple computational domains to improve the efficiency and applicability of simulations. Summary of the Invention

[0016] The purpose of this invention is to overcome the shortcomings of the prior art and to propose a hemodynamic calculation method based on micro-element neural operators.

[0017] In view of this, the present invention proposes a hemodynamic calculation method based on the infinitesimal neural operator (MNO), comprising:

[0018] Based on the initial flow field, blood vessel geometry, and boundary conditions, a pre-trained prediction network is used to predict the time shift of the flow field within the blood vessel.

[0019] The prediction network achieves efficient computation of hemodynamic simulations in different computational domains by designing infinitesimal neural operators with infinitesimal correlation and displacement invariance. The prediction network includes a lifting layer, an inner block layer, and a projection layer. The prediction network is trained using supervised learning. Under a multi-step recursive framework, it minimizes the mean square error of multi-step recursive prediction. By decomposing the hemodynamic problem into infinitesimal correlation subproblems and learning a general time-shifting operator, it achieves fast simulation in two or three dimensions.

[0020] Preferably, the initial flow field includes: the horizontal velocity U at time t. t Vertical velocity V t and pressure P t ;

[0021] The blood vessel geometry is tortuous or bifurcated;

[0022] The boundary conditions include: inlet flow velocity and outlet pressure;

[0023] The time-shift prediction of the intravascular flow field is achieved through a pre-trained prediction network, satisfying the following equation:

[0024]

[0025] in, U represents the time-progression operator. t+Δt V t+Δt ,P t+Δt These are the predicted fields for horizontal velocity, vertical velocity, and pressure at time t+Δt, respectively, where Δt is the time step.

[0026] Preferably, the booster layer of the prediction network is used to process the input field X. t (x,y) is mapped to a high-dimensional feature h (0) (x, y), satisfying the following equation:

[0027] h (0) (x,y)=GELU(W lift X t (x,y)+b lift )

[0028] X t (x,y)=[U t (x,y),V t (x,y),P t (x,y)] T

[0029] Among them, W lift Let b be the weight matrix of the pointwise linear mapping. lift The term is the bias term, GELU is the activation function, (x,y) represents the spatial coordinates, and the superscript T indicates transpose.

[0030] Preferably, the inner block layer of the prediction network includes multiple stacked operator layers. In each operator layer, the output features of the parallel physical path and spectral path are concatenated or added along the channel dimension before proceeding to the next operator layer; wherein,

[0031] The physical path includes several infinitesimal correlation convolutional layers and pointwise layers, which are used to capture direct physical spatial relationships within the neighborhood. Their operation conforms to the conditions of infinitesimal correlation and displacement invariance.

[0032] The spectral path performs orthogonal reflection spectrum transformation on each local sliding window with a 12th-order Legendre polynomial as the basis, extracting global information while ensuring the correlation of infinitesimal elements, and completing the frequency domain coupling of function values ​​by learning the spectral mapping layer.

[0033] After stacking multiple operator layers, the initial features are fused with the deep layer output through residuals or cross-layer skip connections to minimize gradient vanishing in deep networks and provide stable feature transfer for multi-step autoregression.

[0034] Preferably, the projection layer of the prediction network is used to map the features back to the three-channel physical field output. A complete one-step time-progression operator is formed, satisfying the following equation:

[0035]

[0036] Among them, W proj Let b be the weight matrix of the Lth layer. proj h is the bias term for the Lth layer. L (X) represents the output feature of the L operator layer, where X represents the feature quantity corresponding to horizontal velocity, vertical velocity, and pressure.

[0037] Preferably, the radius range r of the micro-element neural operator min Satisfy the following formula:

[0038]

[0039] Where WD is the size of the local sliding window, N is the repetition rate, and n is the number of inner block layers.

[0040] Preferably, the method further includes a training step for the prediction network, comprising:

[0041] Step S1) Construct a three-channel dataset including horizontal velocity, vertical velocity, and pressure for a two-dimensional blood flow cross-section;

[0042] Step S2) The Bootstrap-style multi-step sequence sampling strategy is used to input the prediction network. Under the framework of multi-step loop, multi-step autoregressive training is carried out until the training requirements are met, and the trained prediction network is obtained.

[0043] Preferably, step S1) includes:

[0044] The computational domain of the two-dimensional blood flow section is defined as Ω = [-1,1] × [-1,1], and 128 × 128 grid cells are equally spaced within this computational domain, with the grid spacing set to [value missing].

[0045] A stochastic excitation strategy based on the superposition of multi-frequency sine and cosine basis functions and Gaussian perturbation is introduced to construct horizontal and vertical velocities respectively.

[0046] Global sampling of dimensionless numbers in a three-dimensional parameter space;

[0047] Independent variation in three dimensions—horizontal velocity U, vertical velocity V, and pressure P—results in multiple parameter combinations, and spatial augmentation is applied to the original sample.

[0048] Normalization was performed on the three dimensions of horizontal velocity U, vertical velocity V, and pressure P to obtain global time-domain data.

[0049] Preferably, step S2) includes:

[0050] In each training iteration, a starting time point is randomly selected from the global time series data, and a truncation of length r is performed. rounds The sequence is used, and in the forward computation of the network, the model's output of the previous prediction is used as the input for the next step to achieve multi-step autoregressive training.

[0051] Preferably, in the training of the prediction network, the optimizer is Adam, and the single-step prediction operator is... Recursively apply r on the same sequence rounds Next, define the comprehensive multi-step mean square error loss. for:

[0052]

[0053] Where i represents the i-th recursive application, X t+iΔt and X t These represent the characteristic quantities at time t+iΔt and time t, respectively; Δt is the time step.

[0054] By jointly monitoring the velocity field (U,V) and pressure P, the prediction network achieves a balance between short-term and medium-term prediction accuracy, thus suppressing error accumulation.

[0055] Compared with the prior art, the advantages of the present invention are:

[0056] 1. Ability to adapt to different computing domains

[0057] By learning infinitesimal-related time-shift operators, the infinitesimal neural operator can reuse pre-trained models in computational domains with different blood vessel geometries without retraining. This design of infinitesimal correlation and displacement invariance significantly improves the model's versatility and computational efficiency.

[0058] 2. Handling non-periodic problems

[0059] The infinitesimal neural operator uses the Legendre Transform instead of the Fourier Transform, which is more suitable for non-periodic boundary conditions and can effectively capture local and global features in blood flow, thus improving prediction capabilities.

[0060] 3. Efficiency of boundary handling

[0061] The infinitesimal neural operator integrates the Immersed Boundary Method (IBM), which efficiently handles complex geometries and significantly reduces computational overhead by directly modifying the physical field in the near-boundary region.

[0062] 4. Prediction accuracy

[0063] The micro-element neural operator improves prediction accuracy by about 20% compared to FNO and DeepONet by combining the parallel design of physical paths (convolutional layers) and spectral paths (Legend transform) with the ability to capture local and global features.

[0064] 5. Computational efficiency

[0065] The infinitesimal neural operator optimizes computational efficiency through local operations and parallel paths, and its prediction time is about 600 times faster than the traditional finite element method (FEM), and it outperforms FNO and DeepONet. Attached Figure Description

[0066] Figure 1 This is a diagram of the micro-element neural operator architecture;

[0067] Figure 2 This is a flowchart of the training process for a micro-element neural operator;

[0068] Figure 3 It is a graph of local invariance and time-shift invariance;

[0069] Figure 4 This is a flowchart of the prediction process for the pre-trained infinitesimal neural operator;

[0070] Figure 5 These are intelligent simulation results under different blood vessel geometries. Detailed Implementation

[0071] This invention proposes a hemodynamics calculation method based on infinitesimal neural operators (MNOs), addressing the shortcomings of DeepONet and FNO. While DeepONet and FNO, as the closest existing solutions, hold important positions in the field of neural operators, their fixed computational domain, insufficient adaptability to aperiodic problems, and limitations in computational efficiency make them unsuitable for directly meeting the diverse vascular geometries required in hemodynamics. This invention significantly improves the applicability and accuracy of the calculation method by introducing flexible, cross-domain reusable infinitesimal operators, spectral methods optimized for aperiodic problems (such as Legendre transform), and efficient boundary handling techniques (such as the immersion boundary method).

[0072] This invention proposes a hemodynamics calculation method based on a microelement neural operator, aiming to achieve efficient calculation of hemodynamics simulations in different computational domains by designing a microelement-dependent and displacement-invariant microelement neural operator (MNO). The following is a detailed description of the technical solution:

[0073] 1. Overview of Technical Solution

[0074] This method utilizes MNO to learn time-shift operators in hemodynamics, enabling the pre-trained model to be reused in the computational domain for different vascular geometries.

[0075]

[0076] Where (U, V, P) represent the three-channel physical fields of horizontal velocity, vertical velocity, and pressure. The entire training process is conducted using supervised learning, inputting pre-generated samples into the network and minimizing the mean square error of multi-step recursive predictions within a multi-step recurrent (RNN-like) framework. By decomposing the hemodynamic problem into infinitesimally related subproblems, a general time-shifting operator is learned, thereby achieving rapid simulation. The MNO architecture and training method are optimized to meet the specific needs of hemodynamics (such as nonlinear Navier-Stokes equations and complex boundary conditions). It should be noted that the embodiments in this application use a three-channel physical field including horizontal velocity, vertical velocity, and pressure as an example to achieve two-dimensional field simulation, but this is not a limitation; this method is also applicable to three-dimensional fields.

[0077] Key techniques used in training include:

[0078] (1) Bootstrap-style multi-step sequence sampling strategy

[0079] In each training iteration, a starting time point t is randomly selected from the global time series data, and a truncation of length r is performed. rounds=10 sequence (where X = (U, V, P)), and in the network forward computation, the model is used to evaluate the previously predicted X. t+(i-1)Δt The output is used as the input for the next step to achieve multi-step autoregressive training. This method can effectively cover the flow field evolution characteristics with different initial phases and improve the stability and robustness of the model for long-sequence recursive predictions.

[0080] (2) Dynamic filling of the input domain based on the maximum receptive field

[0081] To address the maximum receptive field size of the MNO network, which is formed by the superposition of multiple layers of local operator modules, periodic boundary data is automatically filled around the original 128×128 grid. This mechanism ensures that the network can obtain complete contextual information at any spatial location, and the output is precisely aligned with the original domain without the need for special boundary engineering.

[0082] (3) Multi-step recursive supervised loss design

[0083] Single-step prediction operator Recursively apply r on the same sequence rounds Next, define the comprehensive multi-step mean square error loss. By jointly supervising the velocity field (U,V) and pressure P, the model can achieve a balance between short-term and medium-term prediction accuracy and suppress error accumulation.

[0084] (4) Training hyperparameters and optimization strategies

[0085] Optimizer: Adam;

[0086] Learning rate scheduling: Initial η = 1 × 10 -3 It decays by 0.7 every 20 epochs;

[0087] Training cycle: 200 epochs;

[0088] Steps per epoch: 500 iterations;

[0089] Weight initialization: An improved Xavier method is used for physical convolution and spectral mapping layers;

[0090] Gradient stabilization: When necessary, combine gradient clipping techniques to prevent gradient explosion.

[0091] (5) Implementing the platform and hardware environment

[0092] Training and testing are both implemented based on the PyTorch framework, using a single NVIDIA RTX 4090 card for parallel acceleration. The typical single-step forward inference time does not exceed tens of milliseconds, in order to meet the application requirements of real-time hemodynamic simulation.

[0093] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0094] Example

[0095] An embodiment of the present invention proposes a hemodynamic calculation method based on infinitesimal neural operators, which specifically includes the following steps:

[0096] Step 1: Data Preparation

[0097] To ensure the efficiency and generalization ability of the Micro-Element Neural Operator (MNO) in hemodynamic simulation, this invention features a systematic design in the data preparation stage. A systematic workflow suitable for three-channel (horizontal velocity U, vertical velocity V, and pressure P) inputs is constructed for two-dimensional blood flow cross-sections. The main steps are as follows:

[0098] First, the computational domain of the two-dimensional blood flow section is defined as Ω = [-1,1] × [-1,1], and this domain is divided into 128 × 128 grid cells at equal intervals, with the grid spacing set to [value missing]. In the initial field generation stage, to simulate the pulsating flow characteristics at the blood inlet, this invention introduces a stochastic excitation strategy based on multi-frequency sine-cosine basis functions superimposed with Gaussian perturbations. Specifically, horizontal and vertical velocities are constructed separately.

[0099]

[0100]

[0101] Here, Λ1 and Λ2 are independent random matrices whose elements follow N(0,1), and the coefficient 0.6 is used to limit the amplitude of the velocity field.

[0102] To improve the model's adaptability to different blood flow states, this invention performs global sampling of dimensionless numbers in the three-dimensional parameter space: centered on the reference point (Re,Me,Δt)=(100,0.2,0.05), sampling is performed at...

[0103] Re∈{20,100,500},Ma∈{0.1,0.2,0.4},Δt∈{0.03,0.05,0.07}s

[0104] Independent mutations were performed across three dimensions, resulting in 27 parameter combinations. Each configuration generated 225 samples of length 500, and the data was partitioned into a training set:validation set ratio of 200:25. Spatial augmentation was then applied to all original samples, including seven methods: rotation around the origin (90° / 180° / 270°) and flipping (x=0, y=0, y=x, y=-x) to improve the model's robustness to multi-directional flow. Finally, normalization was performed on the U, V, and P channels to eliminate differences in dimensions and magnitudes and accelerate network convergence. To meet the multi-step time-progression training requirements of the MNO model, this invention employs a random sequence pruning method during training: in each augmented sample, a sequence of length r is randomly selected at a starting time t. rounds =10 sequence Where X = (U, V, P), and organized as input-target pairs for model updates. This method not only covers more initial states but also improves the stability of multi-step recursive predictions.

[0105] Finally, to ensure consistent data management and reproducibility, this invention organizes all preprocessed samples into a three-level directory structure: "Parameter Configuration → Enhancement Method → ​​Time Series," and stores them in HDF5 format. The files simultaneously save the original physical quantities, normalization coefficients, and random seed information, facilitating subsequent experimental comparisons and result reproduction. Through these steps, this invention constructs a high-quality, scalable data preparation workflow for three-channel hemodynamic simulation, laying a solid foundation for the subsequent training and application of micro-element neural operator models.

[0106] Step 2: Model Training

[0107] The infinitesimal neural operator (MNO) network proposed in this invention designs the following detailed architecture for the three-channel physical field (U,V,P) in blood flow simulation: First, the physical field X is input... t (x,y)=[U t (x,y),V t (x,y),P t (x,y)] T In the lifting layer, features are mapped to a feature space of dimension C. For each spatial location (x, y), the output features are defined as follows:

[0108] h (0) (x,y)=GELU(W lift X t (x,y)+b lift ),

[0109] Among them, W lift : Weight matrix of pointwise linear mapping; b liftBias term; GELU: Activation function. Each layer is internally divided into two parallel paths, such as... Figure 1 :

[0110] • Physical path: It consists of several local convolutional layers and pointwise layers, which are used to capture direct physical spatial relationships within the neighborhood. Its operation conforms to the conditions of "local correlation" and "displacement invariance" (coupling only occurs when ||x1-x2||≤r).

[0111] • Spectral path: On each local sliding window, a normal reflection spectrum transformation (Legendre Transform T and its inverse T-1) is performed based on the 12th order Legendre polynomial to extract global information while ensuring the correlation of infinitesimal elements. Frequency domain coupling of function values ​​is completed through a learnable spectrum mapping layer.

[0112] The output features of the two paths are concatenated or added along the channel dimension before proceeding to the next operator layer. After all operator layers are stacked, a projection layer maps the features back to the three-channel physical field output. Forming a complete one-step time-progression operator:

[0113]

[0114] W proj : The weight matrix of the Lth layer; b proj : Level L bias term; h L (x) represents the output feature of the L operator layer. After stacking L operator layers, the initial features are fused with the deep layer output through residuals or skip connections, which minimizes gradient vanishing in deep networks and provides stable feature propagation for multi-step autoregression. The overall operator architecture is illustrated below. Figure 2 As shown.

[0115] Using Adam as the optimizer, the single-step prediction operator... Recursively apply r on the same sequence rounds Next, define the comprehensive multi-step mean square error loss. for:

[0116]

[0117] The infinitesimal neural operator (MNO) network proposed in this invention fully utilizes two major properties in its design: shift-invariance in time and local-related invariance. Figure 3This ensures the MNO's efficient and universal solution capability for the three-channel physical field (U, V, P) in blood flow simulation. These characteristics not only enhance the MNO's generalization ability under complex vascular geometry and dynamic boundary conditions but also significantly reduce computational complexity, making it suitable for real-time blood flow simulation and long-term prediction. Time-shift invariance refers to the translational invariance of the time-progression operator G learned by the MNO network in the time dimension. Specifically, the MNO solves the physical field (U, V, P) by... t V t ,P t Mapping at time t to the next time step (U) t+Δt V t+Δt ,P t+Δt This implements a mapping process that is independent of a specific time point t. The form of this operator depends only on the time step Δt and the spatial distribution of the input physical field, and its behavior does not change as time progresses.

[0118] In blood flow simulation, time-shift invariance enables the Model-Non-Automatic (MNO) to accurately predict the long-term dynamic evolution of the intravascular flow field by iteratively applying operator G, starting from arbitrary initial conditions (U0, V0, P0), without retraining the model for different time periods. Neither the convolutional nor spectral transform layers of the MNO introduce time-dependent parameters, and combined with skip connections, this ensures the stability of the deep network in multi-step autoregressive predictions. Time-shift invariance also allows the MNO to efficiently reuse a single operator when simulating periodic dynamics such as pulsatile blood flow or cardiac cycles, significantly reducing computational costs.

[0119] Local invariance means that the operations of the MNO network in the spatial dimension depend only on the physical field information within the local neighborhood, rather than the global state of the entire computational domain. This characteristic mimics the local differential property of partial differential equations, ensuring the flexibility and efficiency of the operator when dealing with complex vascular geometry. Specifically, for spatial location (x, y), the prediction of the MNO is coupled only with neighborhood information within a distance ‖X1-X2‖≤r, where r is the range of the infinitesimal element correlation.

[0120] MNO achieves local invariance through the following architectural design:

[0121] Physical Path: The infinitesimal correlation convolutional layer uses a finite-size convolutional kernel (e.g., 3x3) to limit the information propagation range to r = HΔd, where H is the kernel size and Δd is the grid spacing. This design captures the direct physical correlation between the local velocity field (U,V) and pressure field P in blood flow.

[0122] Spectral Path: By applying a normal reflection spectrum transform based on a 12th-order Legendre polynomial basis within a local sliding window, the MNO extracts frequency domain features in a local subdomain. Simultaneously, a learned spectral mapping layer limits the coupling range to IΔd. This ensures the localization of global information, balancing efficiency and accuracy. Here, I represents the width of the infinitesimal operator's spectral transform.

[0123] Erosion Width Management: Due to local operations, the output domain has an erosion width compared to the input domain. MNO (Mean Novel) compensates for this by using numerical methods (such as the finite element method) or the Infiltrated Boundary Method (IBM) to ensure the accuracy of complex vessel wall boundary conditions. The advantage of local invariance is that MNO is insensitive to vessel geometry (such as tortuosity, branching, or stenosis), allowing direct application to unseen domains and suitability for personalized blood flow simulations. By limiting the information propagation range, MNO significantly reduces computational complexity, maintaining high efficiency even at high-resolution meshes. Combined with IBM, MNO can handle complex vessel wall boundary conditions (such as no-slip conditions), seamlessly integrating local prediction with boundary correction, such as... Figure 4 As shown.

[0124] The following is the operational formula for MNO, introducing the operators A, P, C, T, W, and F. The operator consists of three parts: the ascending / descending layer as the head operator P, the projection layer as the tail operator A, and n modules as the main body. The convolution operator is C, the Legendre transform operator is F, and the linear layer operator is W.

[0125] The paths in the spectral channels are constructed using the Legendre transform, i.e. We have in a 2D model:

[0126]

[0127] here z is the input, z' is the output. For spatial domain. This is the product of matrix elements. Therefore, the architecture of MONO can be derived: In the above formulas, subscripts are used to identify operations with independent trainable weights in different layers.

[0128] Step 3: Model invocation process:

[0129] The proposed Micro-Element Neural Operator (MNO) network provides a dynamic solution for the three-channel physical field (U,V,P) in blood flow simulation through an efficient cyclic prediction framework. Based on the initial flow field, blood vessel geometry, and boundary conditions (such as inlet velocity and outlet pressure), it rapidly generates the spatiotemporal evolution of the flow field within the blood vessel.

[0130] The invocation process begins with input preparation: an initial physical field (U0, V0, P0), defining the domain shape from the reconstructed vascular geometry (e.g., tortuous or bifurcated) of medical images, and using a mask matrix M(x, y) to distinguish between fluid and solid regions; boundary conditions include an inlet parabolic velocity (e.g., maximum 0.5 m / s) and an outlet constant pressure (e.g., 0 mmHg). The MNO utilizes pre-trained operators to map the input field to high-dimensional features (spatial channel number C = 40) through a lifting layer. In four layers of operators, it combines local convolution (3x3 kernel, coupling range ‖X1-X2‖ ≤ Δd) and a 12th-order Legendre spectral transform (local window range NΔd) to process the physical and spectral paths. Residual connections ensure deep stability, and the projection layer outputs the predicted field U. t+Δt V t+Δt ,P t+Δt Local invariance enables the MNO to efficiently predict the flow field far from the boundary region Ω1, while the near-boundary region Ω2 is handled by the Infiltrated Boundary Method (IBM) to handle the complex no-slip conditions of the vessel wall. Combining inlet velocity and outlet pressure, a complete flow field is generated. Time-shift invariance supports cyclic prediction: starting from u0, the MNO is iteratively called at Δt = 0.001 seconds to generate a time series. It is suitable for long-term blood flow dynamics (such as shear stress analysis of coronary artery bifurcation). The high efficiency (several times faster than FEM), geometry independence, and robustness of MNO make it significantly advantageous in complex vascular simulation, providing real-time analysis for clinical diagnosis. Figure 5 The results of rapid simulations are shown for different geometries of blood vessels.

[0131] Key Invention Points:

[0132] 1. Concept and architecture of infinitesimal neural operators (MNO): A neural operator with infinitesimal correlation and displacement invariance, capable of learning general hemodynamic time-shifting operators, applicable to different computational domains.

[0133] 2. Combination of physical and spectral paths: Features are processed in parallel through convolutional layers and Legendre transform to improve the model's expressive power and prediction accuracy.

[0134] 3. Application of Legendre Transform: Using the 12th-order Legendre polynomial in the spectral path to adapt to the non-periodic characteristics of hemodynamics.

[0135] 4. Integration of Immersion Boundary Method (IBM): By using IBM to handle the solid wall conditions of complex blood vessel boundaries, computational efficiency and applicability are improved.

[0136] 5. Training and application process: This includes data generation, preprocessing, model training, and time-shift prediction and boundary handling during application to ensure the practicality of the method.

[0137] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for hemodynamic calculation based on micro-element neural operator, comprising: time shift prediction of the flow field in the blood vessel is realized by a pre-trained prediction network based on an initial flow field, blood vessel geometry and boundary conditions; the prediction network realizes efficient calculation of hemodynamic simulation of different calculation domains by designing micro-element related and displacement invariant micro-element neural operator; the prediction network comprises a lifting layer, an inner block layer and a projection layer; the prediction network is trained in a supervised learning manner, minimizes mean square error of multi-step recursive prediction in a multi-step loop framework, decomposes a hemodynamic problem into micro-element related sub-problems, learns a general time shift operator, and thus realizes fast simulation in two or three dimensions; the inner block layer of the prediction network comprises a plurality of stacked operator layers; the output features of parallel physical paths and spectral paths in each operator layer are spliced or added in the channel dimension and then continue to enter the next operator layer; wherein the physical path comprises a plurality of micro-element related convolution layers and a pointwise layer, which are used to capture direct physical space correlation in the neighborhood and operate in accordance with the conditions of micro-element correlation and displacement invariance; the spectral path performs forward and backward reflection spectrum transformation on each local sliding window based on 12-order Legendre polynomials as a basis to extract global information while ensuring micro-element correlation, and completes frequency domain coupling of function values through a spectral mapping layer; after the plurality of operator layers are stacked, the initial features and deep layer outputs are fused through residual or cross-layer jumping to relieve the gradient vanishing of the deep network to the greatest extent and provide stable feature transmission for multi-step autoregression.

2. The microperitter-based neuro- operator based hemodynamic calculation method of claim 1, wherein, The initial flow field includes: horizontal velocity U at time t t , vertical velocity V t , and pressure P t ; the blood vessel geometry is curved or bifurcated; the boundary conditions include inlet flow velocity and outlet pressure; the time shift prediction of the flow field in the blood vessel is realized by the pre-trained prediction network, which satisfies the following formula: where denotes the time advancement operator, U t+Δt , V t+Δt , P t+Δt are the predicted fields of horizontal velocity, vertical velocity and pressure at time t + Δt, respectively, and Δt is the time step.

3. The microperitter-based neuro- operator based hemodynamic calculation method of claim 2, wherein, a boosting layer of the prediction network, configured to map the input field X t (x,y) to high-dimensional features h (0) (x,y) that satisfy the following equation: h (0) (x,y) = GELU(W lift X t (x,y) + b lift ) X t (x,y) = [U t (x,y), V t (x,y), P t (x,y) T where W lift is the weight matrix of the point-wise linear mapping, b lift is the bias term, GELU is the activation function, and (x, y) represents the spatial position coordinates.

4. The microperitter-based neuro- operator based hemodynamic calculation method of claim 1, wherein, a projection layer of the prediction network, configured to map the features back to a three-channel physical field output forming a complete one-step time-propagating operator satisfying the following equation: where W proj is the weight matrix of the Lth layer, b proj is the bias term of the Lth layer, h L (X) is the output feature of the Lth operator layer, and X represents the feature quantities corresponding to the horizontal velocity, vertical velocity, and pressure.

5. The microperitteron-based neural operator based hemodynamic computation method of claim 1, wherein, The microelement neural operator radius range r min satisfies the following equation: where WD is the size of the local sliding window, N is the repetition rate, and n is the number of inner block layers.

6. The microperitter-based neural operator based hemodynamic computation method of claim 1, wherein, The method further comprises a training step of the prediction network, comprising: step S1) constructing a three-channel data set comprising horizontal velocity, vertical velocity and pressure for a two-dimensional blood flow section; step S2) inputting the prediction network using a Bootstrap multi-step sequence sampling strategy, realizing multi-step autoregressive training in a multi-step loop framework until the training requirements are met, and obtaining the trained prediction network.

7. The microperitter-based neuro- operator based hemodynamic calculation method of claim 6, wherein, The step S1) comprises: The calculation domain of the two-dimensional blood flow cross-section is defined as Ω = [-1, 1] x [-1, 1], and 128 x 128 grid units are equally divided on the calculation domain, and the grid spacing is set to a random excitation strategy based on superposition of multi-frequency sine-cosine basis functions and Gaussian disturbance is introduced to construct horizontal velocity and vertical velocity respectively; dimensionless numbers are globally sampled in a three-dimensional parameter space; a plurality of parameter combinations are obtained by independently varying horizontal velocity U, vertical velocity V and pressure P, and the original samples are spatially enhanced; horizontal velocity U, vertical velocity V and pressure P are normalized respectively to obtain global time domain data.

8. The microperitteron-based neural operator based hemodynamic computation method of claim 7, wherein, The step S2) comprises: In each training iteration, a starting time point is selected in a random manner from the global time series data, a sequence of length r is cut off rounds , and the multi-step autoregressive training is realized in the network forward calculation, taking the model's previously predicted output as the next step input.

9. The microperitteron-based neural operator based hemodynamic computation method of claim 8, wherein, In the training of the prediction network, the optimizer is Adam, the single-step prediction operator is recursively applied on the same sequence r rounds and the integrated multi-step mean square error loss is defined as ​ where i denotes the i-th recursive application, X t+iΔt and X t denote the feature quantity at time t+iΔt and t, respectively; Δt is the time step. by jointly supervising the velocity field (U, V) and pressure P, the prediction network balances between short-term and medium-term prediction accuracy and suppresses error accumulation.

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