Hemodynamics calculation method based on infinitesimal neural operator

Abstract

The invention discloses a hemodynamics calculation method based on an infinitesimal neural operator, which comprises the following steps: realizing time shift prediction of an intravascular flow field through a pre-trained prediction network based on an initial flow field, vascular geometry and boundary conditions; the prediction network realizes efficient calculation of hemodynamic analogue simulation of different computational domains by designing infinitesimal neural operators with infinitesimal correlation and invariable displacement; the prediction network comprises a lifting layer, an inner block layer and a projection layer; the prediction network is trained in a supervised learning mode, a mean square error of multi-step recursive prediction is minimized under a multi-step circulation framework, and a universal time shift operator is learned by decomposing a hemodynamics problem into subproblems related to infinitesimal elements, so that two-dimensional or three-dimensional rapid simulation is realized. Compared with the prior art, the method can adapt to various computational domains without retraining, the computational efficiency and precision are remarkably improved, and the method is suitable for the field of medical blood flow simulation.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of computer simulation algorithms based on neural operators, and particularly relates to a blood flow dynamics calculation method based on micro-element neural operators. BACKGROUND

[0002] With the development of deep learning technology, neural operator methods have become a new tool for solving partial differential equations (PDEs). Unlike traditional numerical methods, neural operator methods exhibit high computational efficiency in solving PDEs. Neural operators learn the mapping from input functions to output functions and can quickly predict the behavior of physical fields. Among numerous neural operator methods, DeepONet and FNO are representative technologies currently being researched and applied. The technical details of both will be described in detail below, and their limitations, particularly in relation to blood flow dynamics calculations, will be analyzed.

[0003] DeepONet (Deep Operator Network):

[0004] DeepONet is a neural network-based operator learning framework designed to learn the mapping from input functions (such as initial conditions or boundary conditions) to output functions (such as solutions to PDEs). Its design inspiration comes from the universal approximation theorem of operators.

[0005] Although DeepONet has strong universality in theory, in practical applications, especially in blood flow dynamics calculations, there are the following shortcomings:

[0006] Limitation of fixed computational domain: DeepONet is usually trained on a specific computational domain. For blood flow dynamics problems, due to significant differences in blood vessel geometry among different 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 separation of branch network and trunk network makes DeepONet have large computational overhead when dealing with high-dimensional or complex input functions. This becomes a bottleneck in scenarios that require real-time simulation or large-scale calculations, such as blood flow analysis.

[0008] Limited adaptability to non-periodic problems: Blood flow dynamics involves non-periodic boundary conditions and complex blood vessel wall geometry. The universal architecture of DeepONet is not optimized for such characteristics, which may result in insufficient ability to capture the details of the flow field.

[0009] FNO (Fourier Neural Operator):

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

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

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

[0013] Limitations of fixed calculation domain: Similar to DeepONet, FNO is usually trained on a fixed calculation domain. For blood flow simulation with individualized blood vessel geometry, 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 high-frequency details such as near the vessel wall or flow separation regions, the prediction accuracy may decrease, affecting the description of key flow field characteristics.

[0015] In summary, these methods usually assume that the calculation domain is fixed, and pre-trained models cannot be directly applied to calculation domains with different shapes or sizes. In blood flow dynamics simulation, due to individual differences in the geometry of human blood vessels, traditional neural operator methods require retraining of the model each time a new calculation domain is encountered, resulting in a significant waste of time and computational resources. They do not have the ability to generalize to different geometries, limiting their practicality and efficiency. Therefore, to address the problem of reusability of existing neural operator methods in different calculation domains, there is an urgent need to develop a blood flow dynamics calculation method that can adapt to multiple calculation domains, to improve the efficiency and applicability of simulation and simulation. SUMMARY

[0016] The purpose of the present application is to overcome the defects of the prior art and provide a blood flow dynamics calculation method based on micro-element neural operator.

[0017] Therefore, the present application provides a blood flow dynamics calculation method based on micro-element neural operator (MNO), which comprises:

[0018] Based on the initial flow field, blood vessel geometry, and boundary conditions, the time shift prediction of the flow field in the blood vessel is realized through a pre-trained prediction network.

[0019] The prediction network realizes efficient calculation of hemodynamic simulation of different calculation domains by designing micro-element related and displacement invariant micro-element neural operators; 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 the mean square error of multi-step recursive prediction in a multi-step cycle framework, decomposes the hemodynamic problem into micro-element related sub-problems, learns a general time shift operator, and thus realizes fast simulation in two or three dimensions.

[0020] Preferably, the initial flow field comprises horizontal velocity U t , vertical velocity V t and pressure P t at time t.

[0021] The blood vessel geometry is curved or bifurcated.

[0022] The boundary conditions include inlet flow rate and outlet pressure.

[0023] The time shift prediction of the flow field in the blood vessel is realized by the pre-trained prediction network; the following formula is satisfied:

[0024]

[0025] Wherein, represents a time advancing operator, U t+Δt , V t+Δt and 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.

[0026] Preferably, the lifting layer of the prediction network is used to map the input field X t (x,y) to high-dimensional features h (0) (x,y), satisfying the following formula:

[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] Wherein, W lift is a weight matrix of point-by-point linear mapping, b lift is a bias term, GELU is an activation function, (x,y) represents spatial position coordinates, and the superscript T represents transposition.

[0030] Preferably, the inner block layer of the prediction network comprises a plurality of stacked operator layers, and after the output features of the parallel physical path and spectral path are spliced or added in the channel dimension in each operator layer, they continue to enter the next operator layer; wherein,

[0031] The physical path comprises a plurality of micro-element related convolution layers and pointwise layers for capturing direct physical space correlation in the neighborhood, and the operation conforms to the conditions of micro-element correlation and displacement invariance;

[0032] The spectral path performs forward and backward spectral transformation on each local sliding window based on 12-order Legendre polynomials as the basis to extract global information while ensuring micro-element correlation, and completes the frequency domain coupling of function values through a spectral mapping layer.

[0033] After stacking a plurality of operator layers, 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.

[0034] Preferably, the projection layer of the prediction network is used to map the features back to the three-channel physical field output to form a complete one-step time propagation operator, which satisfies the following formula:

[0035]

[0036] wherein, 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 L operator layer, and X represents the characteristic quantity corresponding to the horizontal velocity, vertical velocity and pressure.

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

[0038]

[0039] wherein, 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 comprises a training step of the prediction network, comprising:

[0041] Step S1) constructing a three-channel data set comprising horizontal velocity, vertical velocity and pressure for a two-dimensional blood flow section;

[0042] Step S2) uses a Bootstrap multi-step sequence sampling strategy to input the prediction network, realizes multi-step autoregressive training under the framework of multi-step cycles, until the training requirements are met, and obtains the trained prediction network.

[0043] Preferably, the step S1) comprises:

[0044] The calculation domain of the two-dimensional blood flow section is defined as Ω = [-1, 1] x [-1, 1], and 128 x 128 grid cells are equally divided on the calculation domain, and the grid spacing is set as

[0045] A random excitation strategy based on superimposed Gaussian disturbance of multi-frequency sine-cosine basis function is introduced, and the horizontal velocity and vertical velocity are constructed respectively;

[0046] The dimensionless number is globally sampled in the three-dimensional parameter space;

[0047] The horizontal velocity U, the vertical velocity V and the pressure P are independently varied in three dimensions to obtain a variety of parameter combinations, and the original sample is spatially enhanced;

[0048] The horizontal velocity U, the vertical velocity V and the pressure P are normalized respectively to obtain global time domain data.

[0049] Preferably, the step S2) comprises:

[0050] In each training iteration, the starting time point is selected in a random manner from the global time series data, a sequence with a length of r rounds is intercepted, and in the network forward calculation, the model is used to take the previously predicted output as the next step input, realizing multi-step autoregressive training.

[0051] Preferably, in the training of the prediction network, the optimizer is Adam, the single-step prediction operator is recursively applied r rounds times on the same sequence, and the comprehensive multi-step mean square error loss is defined as:

[0052]

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

[0054] Through joint supervision of the velocity field (U, V) and the pressure P, the prediction network balances between short-term and medium-term prediction accuracy, and suppresses error accumulation.

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

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

[0057] The micro-element neural operator can reuse the pre-trained model on different blood vessel geometry computational domains without retraining by learning the time shift operator related to the micro-element. This micro-element correlation and displacement invariance design significantly improves the generality and computational efficiency of the model.

[0058] 2. Processing of non-periodic problems

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

[0060] 3. Efficiency of boundary processing

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

[0062] 4. Prediction accuracy

[0063] The micro-element neural operator improves the prediction accuracy by about 20% compared with FNO and DeepONet through the parallel design of physical path (convolution layer) and spectral path (Legendre Transform), combining local and global feature capture ability.

[0064] 5. Computational efficiency

[0065] The micro-element neural operator optimizes computational efficiency through local operation and parallel path, with prediction time about 600 times faster than traditional finite element method (FEM) and better than FNO and DeepONet. BRIEF DESCRIPTION OF DRAWINGS

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

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

[0068] Figure 3 is the local invariance and time shift invariance diagram;

[0069] Figure 4 is the micro-element neural operator prediction flowchart after pre-training;

[0070] Figure 5 is the intelligent simulation result under different blood vessel geometries. DETAILED DESCRIPTION

[0071] The present application proposes a microelement neural operator (MNO) based hemodynamic calculation method, which improves the deficiencies of DeepONet and FNO. As the closest existing solution, DeepONet and FNO have an important position in the field of neural operators, but their fixed calculation domain, insufficient adaptability to non-periodic problems, and limitations in computational efficiency make it difficult to directly meet the needs of diversified blood vessel geometry in hemodynamics. The present application introduces flexible and cross-domain reusable microelement operators, as well as spectral methods (such as Legendre transform) optimized for non-periodic problems and high-efficiency boundary processing techniques (such as immersed boundary method), which significantly improve the applicability and accuracy of the calculation method.

[0072] The embodiment of the present application proposes a hemodynamic calculation method based on a microelement neural operator, which aims to realize efficient calculation of hemodynamic simulation of different calculation domains by designing a microelement neural operator (MNO) related to microelements and invariant to displacement. The following is a detailed description of the technical solution:

[0073] 1. Technical solution overview

[0074] This method uses MNO to learn the time shift operator in hemodynamics, allowing the pre-trained model to be reused in different blood vessel geometry calculation domains.

[0075]

[0076] where (U, V, P) are the horizontal velocity, vertical velocity and pressure three-channel physical fields. The entire training is performed in a supervised learning manner, with pre-generated samples input into the network, and under the framework of multi-step recursion (similar to RNN), the mean square error of multi-step recursive prediction is minimized. By decomposing the hemodynamic problem into a microelement-related subproblem, a general time shift operator is learned, enabling fast simulation. MNO combines the specific needs of hemodynamics (such as the nonlinear Navier-Stokes equation, complex boundary conditions), and optimizes its architecture and training method. It should be noted that the embodiments of the present application use a three-channel physical field including horizontal velocity, vertical velocity and pressure as an example to realize two-dimensional field simulation, but this is not a limitation, and the method is also applicable to three-dimensional fields.

[0077] Key technologies in training include:

[0078] (1) Bootstrap 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 length of r roundssequence of length 10 (where X = (U, V, P)), and in the network forward computation, the model takes the previous predicted X t+(i-1)Δt as the next step input, realizing multi-step autoregressive training. This method can effectively cover the evolution characteristics of flow fields with different initial phases, improving the stability and robustness of the model for long sequence recursive prediction.

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

[0081] The maximum receptive field size formed by the superposition of multiple local operator modules in the MNO network is automatically filled with periodic boundary data outside the original 128x128 grid. This mechanism ensures that the network can obtain complete context information at any spatial location, and the output is accurately aligned with the original domain without special boundary engineering processing.

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

[0083] The single-step prediction operator is applied recursively r rounds times on the same sequence, and the comprehensive multi-step mean square error loss is defined Through joint supervision of the velocity field (U, V) and pressure P, the model can balance the 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 , decay by 0.7 every 20 epochs;

[0087] Training period: 200 epochs;

[0088] Number of steps per epoch: 500 iterations;

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

[0090] Gradient stabilization: if necessary, use gradient clipping technology to prevent gradient explosion.

[0091] (5) Implementation platform and hardware environment

[0092] Training and testing are 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, meeting the application requirements of real-time hemodynamic simulation.

[0093] The technical solutions of the present application will be described in detail below in combination with the drawings and examples.

[0094] Embodiment

[0095] The embodiment of the present application proposes a blood flow dynamics calculation method based on micro-element neural operator, specifically including the following steps:

[0096] Step 1: Data preparation:

[0097] In order to ensure the efficiency and generalization ability of micro-element neural operator (MNO) in blood flow dynamics simulation, the present application designs a systematic process in the data preparation stage, which is suitable for "three-channel" (horizontal velocity U, vertical velocity V, pressure P) input for two-dimensional blood flow cross section. The main steps are as follows:

[0098] Firstly, define the calculation domain of two-dimensional blood flow cross section as Ω = [-1, 1] × [-1, 1], and divide it into 128 × 128 grid cells with equal spacing, and the grid spacing is set as In the initial field generation link, in order to simulate the pulsatile flow characteristics at the blood inlet, the present application introduces a random excitation strategy based on multi-frequency sine-cosine basis function superimposed Gaussian disturbance. Specifically, the horizontal velocity and vertical velocity are constructed as

[0099]

[0100]

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

[0102] In order to improve the adaptability of the model to different blood flow states, the present application globally samples the dimensionless numbers in three-dimensional parameter space: taking the reference point (Re, Me, Δt) = (100, 0.2, 0.05) as the center, respectively in

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

[0104] The three dimensions are independently varied, and 27 parameter combinations are formed, 225 samples of 500 lengths are generated under each configuration, and the data is divided into training set: validation set = 200:25. Subsequently, spatial enhancement is performed on all original samples, including 7 ways of data expansion such as rotating around the origin by 90° / 180° / 270° and flipping x = 0, y = 0, y = x, y = -x, to improve the robustness of the model to multi-directional flow; finally, normalization processing is performed on the U, V, P three channels respectively to eliminate the dimension and order of magnitude difference and accelerate the network convergence; in order to meet the multi-step time advancement training needs of the MNO model, the present application adopts the method of random cropping sequence during training: in each enhanced sample, a sequence of length r rounds = 10 is extracted at a random starting time t Where X = (U, V, P), and is organized as an input and target pair for model updating. This method not only covers more initial states, but also improves the stability of multi-step recursive prediction.

[0105] Finally, in order to ensure consistent management and reproducibility of data, the present application organizes all preprocessed samples according to the three-level directory of 'parameter configuration → enhancement method → time sequence', and stores them in HDF5 format. The original physical quantity, normalization coefficient and random seed information are saved in the file at the same time, which is convenient for subsequent experiments and result reproduction. Through the above links, the present application constructs a set of high-quality and expandable data preparation process for three-channel hemodynamic simulation, which lays a solid foundation for the training and application of the subsequent micro-element neural operator model.

[0106] Step 2 model training

[0107] The micro-element neural operator (MNO) network proposed in the present application is designed as follows for the three-channel physical field (U, V, P) in blood flow simulation: first, the input physical field X t (x, y) = [U t (x, y), V t (x, y), P t (x, y)] T The lifting layer (Lifting Layer) is mapped to a feature space with dimension C. For each spatial position (x, y), the output feature is defined as:

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

[0109] Where W lift : point-by-point linear mapping weight matrix; b lift: bias term; GELU: activation function. Each layer is divided into two parallel paths as Figure 1 :

[0110] • Physical path: composed of several local convolutional layers and pointwise layers, which are used to capture the direct physical space correlation within the neighborhood, and the operation is consistent with the conditions of "local correlation" and "displacement invariance" (coupling only when ||x1-x2||≤r).

[0111] • Spectral path: on each local sliding window, the forward and inverse Legendre transform (Legendre Transform T and its inverse T-1) is performed based on the 12-order Legendre polynomial, which extracts global information while ensuring local correlation, and the frequency domain coupling of function values is completed through a learnable spectral mapping layer.

[0112] The output features of the two paths are spliced or added in the channel dimension, and then continue to enter the next operator layer. After all the operator layers are stacked, the projection layer is used to map the features back to the three-channel physical field output to form a complete one-step time propagation operator:

[0113]

[0114] W proj : the L-th layer weight matrix; b proj : the L-th layer bias term; h L (x) is the output feature of the L operator layer. After stacking L operator layers, the initial features and deep layer outputs are fused through residual or cross-layer skip connection to maximize the gradient vanishing of deep network and provide stable feature transmission for multi-step autoregression. The overall operator architecture diagram is shown in Figure 2 .

[0115] The optimizer is Adam, and the single-step prediction operator is recursively applied r rounds times on the same sequence, and the comprehensive multi-step mean square error loss is defined as:

[0116]

[0117] The micro-element neural operator (MNO) network proposed in the application fully utilizes the two characteristics of shift-invariance in time and local-related invariance in design as Figure 3To ensure its efficient and universal solution capability for three-channel physical fields (U, V, P) in blood flow simulation. These features not only enhance the generalization ability of MNO under complex vascular geometry and dynamic boundary conditions, but also significantly reduce the computational complexity, making it suitable for real-time blood flow simulation and long-term prediction. Time-shift invariance means that the time-advancing operator G learned by the MNO network has the property of translation invariance in the time dimension. Specifically, MNO transforms the physical field (U t ,V t ,P t ) at time t is mapped to the next time step (U t+Δt ,V t+Δt ,P t+Δt ), realizing a mapping process that is independent of the 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 advances.

[0118] In blood flow simulation, time-shift invariance enables MNO to accurately predict the long-term dynamic evolution of the intravascular flow field by iteratively applying the operator G, starting from arbitrary initial conditions (U0, V0, P0), without having to retrain the model for different time periods. MNO's convolutional and spectral transformation layers do not introduce time-dependent parameters, and combined with residual connections (skip connections), they ensure the stability of the deep network in multi-step autoregressive prediction. Time-shift invariance enables MNO to efficiently reuse a single operator when simulating periodic dynamics such as pulsatile blood flow or the cardiac cycle, significantly reducing computational costs.

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

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

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

[0122] Spectral Path: By applying the forward reflection spectral transform of 12th order Legendre polynomial basis in a local sliding window, MNO extracts the frequency domain features in a local subdomain, while limiting the coupling range to IAd by learning the spectral mapping layer. This ensures the localization of global information, balancing efficiency and accuracy. Where I is the micro-element operator spectral transform width.

[0123] Erosion width management: Due to local operation, there is a corrosion width in the output domain compared with the input domain, and MNO supplements the processing through numerical methods (such as finite element method) or immersed boundary method (IBM) of boundary area, to ensure the accuracy of complex blood vessel wall boundary conditions. The advantage of local invariance is that MNO is not sensitive to the geometry of blood vessels (such as bending, branching or stenosis), and can be directly applied on unseen domains, suitable for personalized blood flow simulation. By limiting the range of information propagation, MNO significantly reduces the computational complexity, especially under high-resolution grid, still maintains high efficiency. Combined with IBM, MNO can handle complex boundary conditions of blood vessel wall (such as no slip condition), seamlessly connect local prediction and boundary correction, as shown in Figure 4 .

[0124] The following is the operation formula of MNO, A, P, C, T, W, F operators are introduced into the operator, the operator is composed of three parts, the head operator P, the tail operator A, and n modules as the main part. The convolution operator is C, the Legendre transform operator is F, and the linear layer operator is W.

[0125] The path in the spectral channel is composed of Legendre transform, that is We have in a 2D model:

[0126]

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

[0128] Step 3 model calling process:

[0129] The micro-element neural operator (MNO) network proposed in the application provides dynamic solution for three-channel physical fields (U, V, P) in blood flow simulation through an efficient cyclic prediction framework, based on initial flow field, blood vessel geometry and boundary conditions (such as inlet flow rate, outlet pressure), to quickly generate the spatio-temporal evolution of flow field in blood vessels.

[0130] The calling process begins with input preparation: initial physical fields (U0, V0, P0), domain shape defined by vessel geometry reconstructed from medical images (e.g., bends or bifurcations), and a mask matrix M(x, y) to distinguish fluid and solid regions; boundary conditions include parabolic flow rate at the inlet (e.g., maximum 0.5 m / s) and constant pressure at the outlet (e.g., 0 mmHg). The MNO utilizes pre-trained operators to map the input fields to high-dimensional features (spatial channel number C = 40) through lifting layers, and combines local convolution (3x3 kernel, coupling range ‖X1-X2‖≤Δd) and 12th-order Legendre spectral transform (local window range NΔd) in 4-layer operators to process physical and spectral paths, with residual connections ensuring deep stability, and projection layers outputting predicted fields U t+Δt ,V t+Δt ,P t+Δt . Local invariance enables the MNO to efficiently predict flow fields far from the boundary region Ω1, while the near-boundary region Ω2 is handled by the immersed boundary method (IBM) to enforce complex vessel wall no-slip conditions, combined with inlet flow rate and outlet pressure to generate a complete flow field. Time-shift invariance supports recurrent prediction: starting from u0, the MNO is iteratively called with Δt = 0.001 seconds to generate a time series suitable for long-term blood flow dynamics (e.g., shear stress analysis of coronary artery bifurcations). The efficiency (orders of magnitude faster than FEM), geometry independence, and robustness of the MNO make it a significant advantage in complex vessel simulations, providing real-time analysis for clinical diagnosis. Figure 5 The results of fast simulation of different geometries under different vessel geometries are shown.

[0131] Core invention points:

[0132] 1. Concept and architecture of micro-element neural operator (MNO): a micro-element related and shift-invariant neural operator that can learn a general blood flow dynamics time-shift operator suitable for different computational domains.

[0133] 2. Combination of physical and spectral paths: parallel processing of features through convolution layers and Legendre transform to improve model expressiveness and prediction accuracy.

[0134] 3. Application of Legendre transform: use of 12th-order Legendre polynomials in the spectral path to adapt to the non-periodic characteristics of blood flow dynamics.

[0135] 4. Integration of immersed boundary method (IBM): handling of complex vessel boundary wall conditions through IBM to improve computational efficiency and applicability.

[0136] 5. Training and application process: including 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 examples are merely used to illustrate the technical solutions of the present application but not to limit. Although the present application is explained in detail with reference to the examples, those skilled in the art should understand that the technical solutions of the present application can be modified or replaced equivalently without departing from the spirit and scope of the present application, and all of them should be covered in the scope of the claims of the present application.

Claims

1. A hemodynamic calculation method based on a microelement neural operator, comprising: Based on the initial flow field, vascular geometry and boundary conditions, the time-shift prediction of the intravascular flow field is achieved through a pre-trained prediction network; The prediction network achieves efficient calculation of hemodynamic simulation in different computational domains by designing microelement-related and displacement-invariant microelement neural operators; the prediction network includes a lifting layer, an inner block layer, and a projection layer; the prediction network is trained using supervised learning, and minimizes the mean square error of multi-step recursive prediction under a multi-step loop framework. By decomposing the hemodynamic problem into microelement-related subproblems, a universal time-shift operator is learned, thereby achieving rapid two-dimensional or three-dimensional simulation.

2. The hemodynamic calculation method based on microelement neural operator according to claim 1 is characterized in that: The initial flow field includes: the horizontal velocity U at time t t , vertical speed V t and pressure P t ; The blood vessel geometry is curved or bifurcated; The boundary conditions include: inlet flow rate and outlet pressure; The time-shift prediction of the intravascular flow field is achieved by using a pre-trained prediction network, satisfying the following formula: in, represents 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 hemodynamic calculation method based on microelement neural operator according to claim 2 is characterized in that: The lifting layer of the prediction network is used to transform the input field X t (x,y) is mapped to high-dimensional feature h (0) (x,y), satisfying the following formula: h (0) (x,y)=GROUND(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 Among them, W lift is the weight matrix of the point-by-point linear mapping, b lift is the bias term, GELU is the activation function, (x, y) represents the spatial position coordinates, and the superscript T represents the transpose.

4. The hemodynamic calculation method based on microelement neural operator according to claim 1 is characterized in that: 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 spectrum path are spliced ​​or added in the channel dimension before continuing to the next operator layer; wherein, The physical path includes: several microelement-related convolutional layers and point-wise layers, which are used to capture direct physical spatial correlation within the neighborhood, and their operations meet the conditions of microelement-related and displacement-invariant; The spectrum path performs regular reflection spectrum transformation on each local sliding window based on the 12th-order Legendre polynomial, extracting global information while ensuring the correlation of microelements, and completing the frequency domain coupling of function values ​​by learning the spectrum mapping layer; After stacking multiple operator layers, the initial features are fused with the deep layer output through residuals or cross-layer skip connections to minimize the gradient vanishing of the deep network and provide stable feature transfer for multi-step autoregression.

5. The hemodynamic calculation method based on microelement neural operator according to claim 1 is characterized in that: The projection layer of the prediction network is used to map the features back to the three-channel physical field output This forms a complete one-step time advancement operator that satisfies the following formula: Among them, W proj is the L-th layer weight matrix, b proj is the L-th layer bias term, h L (X) is the output feature of the L operator layer, and X represents the feature quantity corresponding to horizontal velocity, vertical velocity and pressure.

6. The hemodynamic calculation method based on microelement neural operator according to claim 4 is characterized in that: The radius range of the microelement neural operator is r min Satisfy the following formula: Among them, WD is the local sliding window size, N is the repetition rate, and n is the number of inner block layers.

7. The hemodynamic calculation method based on microelement neural operator according to claim 1 is characterized in that: The method further comprises a step of training the prediction network, comprising: Step S1) constructing a three-channel data set including horizontal velocity, vertical velocity and pressure for a two-dimensional blood flow cross section; Step S2) adopts the Bootstrap multi-step sequence sampling strategy, inputs the prediction network, and implements multi-step autoregressive training in the framework of a multi-step cycle until the training requirements are met, thereby obtaining a trained prediction network.

8. The hemodynamic calculation method based on microelement neural operator according to claim 7 is characterized in that: The step S1) comprises: The computational domain of the two-dimensional blood flow section is defined as Ω = [-1, 1] × [-1, 1], and the computational domain is divided into 128 × 128 grid cells at equal intervals. The grid spacing is set to A random excitation strategy based on multi-frequency sine-cosine basis functions superimposed with Gaussian perturbations is introduced to construct horizontal and vertical velocities respectively. Global sampling of dimensionless numbers in three-dimensional parameter space; Independently mutate the horizontal velocity U, vertical velocity V, and pressure P in three dimensions to obtain multiple parameter combinations and perform spatial enhancement on the original sample; Normalization processing is performed on the three dimensions of horizontal velocity U, vertical velocity V and pressure P respectively to obtain global time domain data.

9. The hemodynamic calculation method based on microelement neural operator according to claim 8, characterized in that: The step S2) comprises: In each training iteration, the starting time point is randomly selected from the global time series data, and the intercept length is r rounds In the forward calculation of the network, the model's output of the previous prediction is used as the input for the next step to implement multi-step autoregressive training.

10. The hemodynamic calculation method based on microelement neural operator according to claim 9 is characterized in that: 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 times, and define the comprehensive multi-step mean square error loss for: Where i represents the i-th recursive application, X t+iΔt and X t They represent the characteristic quantities 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 can achieve a balance between short-term and medium-term prediction accuracy and suppress error accumulation.

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