A method and system for spatiotemporal physical field reconstruction based on fractional Fokker-Planck regularization and implicit sampling

Abstract

This application provides a method and system for reconstructing spatiotemporal physical fields based on fractional Fokker-Planck regularization and implicit sampling. The method includes: inputting sparse sensor observation data into a trained deep neural network embedded with physical evolution laws, outputting a predicted fractional field, and iterating backward from pure Gaussian noise to obtain a noise-free physical field. The backward iteration includes: using the Tweedie formula to map the noise state of the current step to the estimated noise-free physical field, performing multi-physics coupling gradient calculation; calculating a deterministic drift term and a controlled random Gaussian perturbation term, superimposing a physical correction gradient, updating the noise state for the next time step, and cyclically decreasing until the step count is 0 to obtain the noise-free physical field. The advantages of this application are: significantly improved sampling speed; enhanced physical consistency; and stronger robustness under sparse observations.

Description

Technical Field

[0001] This application belongs to the interdisciplinary field of artificial intelligence and scientific computing, specifically involving a spatiotemporal physical field reconstruction method and system based on fractional Fokker-Planck regularization and implicit sampling. Background Technology

[0002] Partial differential equations form the fundamental mathematical framework describing the laws of physics such as acoustics and fluid dynamics. However, when dealing with the inverse problem of reconstructing high-fidelity spatiotemporal physical fields from sparse observations, traditional numerical methods often face challenges such as high computational costs and ill-conditioned problems. In recent years, generative artificial intelligence, especially fractional diffusion models, has provided a new paradigm for modeling physical fields. These models define the forward diffusion process through stochastic differential equations (SDEs), gradually transforming data into noise, and driving the inverse process to generate samples by learning a fractional function (the gradient of the logarithmic density of the data). Although fractional diffusion models have achieved significant success in fields such as image generation, significant technical bottlenecks remain when applying them to scientific computing, where high accuracy and physical consistency are required.

[0003] Existing training frameworks suffer from insufficient dynamic consistency. Current mainstream denoising score matching (DSM) methods primarily focus on minimizing the Euclidean distance between the estimated and true scores, but numerically they do not guarantee that the learned score function satisfies the Score Fokker-Planck Equation (ScoreFPE). Research shows that models trained solely on DSM often fail to produce score fields that satisfy the inverse martingale properties required for the reverse diffusion process, a bias particularly pronounced in the early stages of diffusion when noise levels are low. This inconsistency between theory and numerical values ​​results in a lack of rigorous dynamic stability guarantees in the generation process, making the generated physical field prone to non-physical artifacts or distortions, and failing to meet the stringent physical conservation requirements of scientific computing.

[0004] Low sampling efficiency is another major obstacle limiting the application of this technology in real-time applications. Fractional generative models typically require numerical solutions to inverse SDEs or probabilistic flow ordinary differential equations (ODEs) to generate samples. Standard numerical solvers rely on the assumption of infinitesimal time steps to control discretization errors, meaning that generating a single sample often requires thousands of iterative computations. In time-sensitive applications such as digital twins and real-time monitoring, this high computational latency is unacceptable. Although some accelerated sampling algorithms exist, most are designed for variance-preserving (VP) SDEs, and directly increasing the time step can lead to significant deviations in the discrete approximations of drift and diffusion terms, causing the generated trajectory to deviate from the true data manifold. For SDEs of the variance explosion (VE) form commonly used in physical modeling, there is currently a lack of efficient implicit sampling mechanisms that can maintain high accuracy and stability even with large time steps. Summary of the Invention

[0005] The purpose of this application is to overcome the shortcomings of existing fraction-based generative models in the field of scientific computing, such as low sampling efficiency and inconsistent dynamics.

[0006] To achieve the above objectives, this application proposes a spatiotemporal physics field reconstruction method based on fractional Fock-Planck regularization and implicit sampling, comprising: The sparse sensor observation data is input into a trained deep neural network with embedded physical evolution laws, and the output is a predicted fractional field. Starting from pure Gaussian noise, it is iterated in reverse to obtain a noise-free physical field. The reverse iteration includes: Using the Tweedie formula, the noise state of the current step is mapped to the estimated noise-free physical field, and multi-physics coupled gradient calculation is performed. Calculate the deterministic drift term and the controlled random Gaussian perturbation term, superimpose the physical correction gradient, update the noise state for the next time step, and iterate until the step count is 0 to obtain the noise-free physical field.

[0007] As an improvement to the above method, the deep neural network embedding the physical evolution law is a fractional network based on the 3D U-Net architecture.

[0008] As an improvement to the above method, the training process of the deep neural network embedding physical evolution laws includes: Obtain physical simulation data as initial noise-free physical field data. Set a discrete time step sequence , The total number of discretized noise scale sequences; the noise variance scheduling sequence of the variance explosion stochastic differential equation is defined. ;Towards Injecting standard Gaussian noise Generate the noisy physical field state corresponding to the time step. ; Noisy physical field state and noise scale Input a deep neural network that embeds physical evolution laws, and output a predicted score field.

[0009] As an improvement to the above method, the loss function of the deep neural network embedding physical evolution laws is: ; in, The denoising fractional matching loss is used to fit the statistical distribution characteristics of the physical signal; The regularization loss for the fractional Fock-Planck equation is constructed based on the partial differential equations that the true fractional function must satisfy.

[0010] As an improvement to the above method, the Tweedie formula is used to map the noise state of the current step to the estimated noise-free physical field, expressed as: ; in, Represents the estimated noiseless physical field; Indicates the first i The noisy physical field state of the step; Indicates the first i The noise scale of the step; The estimated score represents the output of a deep neural network that incorporates the laws of physical evolution.

[0011] As an improvement to the above method, the multiphysics coupled gradient calculation includes: Calculation of physical residual loss based on the partial differential equations of mass and momentum conservation discretized by the finite difference method. ; Calculate the data consistency loss on sparse observation points : ; in, This represents sparse sensor observation data; Represents the physical space observation mask matrix; Represents the 2-norm of a matrix; Differentiating the noisy physical field state yields the comprehensive gradient guiding term used for correction.

[0012] As an improvement to the above method, the update of the noise state at the next moment is expressed as: ; in, Injected standard Gaussian random noise, It is the identity matrix; It is an adjustable random hyperparameter; Represents the estimated noiseless physical field; Indicates the first i The noisy physical field state of the step; = ; = ; = , and They represent the first Step and the first The noise scale of the step.

[0013] This application also provides a spatiotemporal physics field reconstruction system based on fractional Fock-Planck regularization and implicit sampling, implemented using the above method. The system includes: The module for obtaining the predicted fractional field is used to input sparse sensor observation data into a trained deep neural network embedded with physical evolution laws and output the predicted fractional field. The module for obtaining noiseless physics is used to iterate backward from pure Gaussian noise to obtain noiseless physics.

[0014] Compared with existing technologies, the advantages of this application are: 1. Significantly Improved Sampling Speed: The proposed Implicit Fractional Sampler (ISS) breaks through the time step limitation of traditional SDE solvers, allowing for larger time steps while ensuring numerical stability. Experiments show that, at the same reconstruction accuracy, this method can reduce the number of iterations required for inference by an order of magnitude, greatly reducing computational costs and meeting the needs of near real-time applications.

[0015] 2. Enhanced Physical Consistency: By explicitly incorporating Score FPE regularization into the training objective, the bias in score matching in low-density regions is theoretically corrected. This dynamic constraint ensures that the physical field generated by the model not only fits the real data in statistical distribution but also strictly follows the law of physical diffusion in its evolution trajectory, effectively suppressing non-physical artifacts in the generated results.

[0016] 3. Robustness under sparse observations: Combining Tweedie estimation and implicit sampling mechanism, this method shows stronger robustness when dealing with sparse sensor data (such as only a few observation points), and can accurately recover the complete spatiotemporal structure from incomplete information, which has a significant advantage in scientific computing inverse problems with extremely scarce data. Attached Figure Description

[0017] Figure 1The diagram shows a physical constraint-based spatiotemporal sound field generation framework, illustrating the two core stages of this invention: the unconditional pre-training stage, which includes a Score FPE regularization term, and the conditional generation stage, which integrates temporal, observational, and physical consistency gradient guidance mechanisms. Figure 2 The diagram shows the system composition during the model training phase, illustrating the 3D U-Net-based score network architecture (including spatial convolution and spatiotemporal attention mechanisms), the denoising score matching (DSM) loss calculation module, and the Score FPE regularization loss calculation module constructed using Hutchinson trace estimation and finite difference method. Figure 3 The diagram shows the iterative update logic of the Implicit Fractional Sampler (ISS): it illustrates how, in the single-step iteration of reverse generation, the noise-free state is estimated from the noisy state using the Tweedie formula, and the process of achieving large-step state updates is achieved by combining the deterministic drift term and the random perturbation term derived from VE-SDE. Figure 4(a) shows the reconstruction performance evaluation of high Reynolds number Kolmogorov turbulence, comparing the vorticity field reconstruction under different methods, demonstrating that... , The reconstruction performance of four methods, SDE, ISS, FNO and LNO, is compared on the test samples. Here, the baseline model was trained on a 6x super-resolution task. Figure 4(b) shows the reconstruction performance evaluation of high Reynolds number Kolmogorov turbulence and the effect of FPE regularization fine-tuning on the kinetic energy spectrum. The first row is the energy spectrum result without FPE regularization fine-tuning, and the second row is the result after FPE regularization fine-tuning. Figure 4(c) shows the reconstruction performance evaluation of high Reynolds number Kolmogorov turbulence, comparing the errors of generated samples, and comparing the normalized root mean square error (nRMSE) of ISS (N=100), SDE (N=100, 200, 500, 1000) and FNO method. Figure 4(d) shows the reconstruction performance evaluation of high Reynolds number Kolmogorov turbulence, and the error evaluation under different sparsity conditions. It shows the trend of reconstruction error of each method as the spatial downsampling factor increases. The error bars represent the mean ± standard deviation, which are calculated based on the prediction results of the method on five solutions with different initial conditions. Detailed Implementation

[0018] The technical solution of this application will be described in detail below with reference to the accompanying drawings.

[0019] This invention provides a spatiotemporal physical field reconstruction method and system based on fractional Fokker-Planck regularization and implicit sampling. This method deeply integrates physical laws and scientific computing methods of artificial intelligence. By introducing physical dynamic constraints in the model training stage and constructing a large-step implicit sampling mechanism in the inference stage, it achieves spatiotemporal physical field reconstruction with both high computational efficiency and high physical fidelity.

[0020] like Figure 1 The physically constrained spatiotemporal sound field generation framework shown can be widely applied to the following specific industrial and scientific fields to solve the problem of high-fidelity physical field reconstruction under sparse sensor conditions: 1. Acoustic digital twin and non-destructive testing: The input data is a sparse scalar sound pressure time series collected by a small number of microphones / sensors distributed in space; the output data is a multimodal spatiotemporal physical field of sound pressure and particle velocity in full space with high resolution, which can be used for noise source localization or acoustic reconstruction of internal defects in materials.

[0021] 2. Fluid Mechanics and Aerodynamics: Input data consists of local wind speeds / vorticity collected by sparsely distributed Pitot tubes or hot-wire anemometers on the surface of the aircraft or in the wind tunnel; output data consists of the complete high Reynolds number turbulent flow field distribution around the aircraft, used for aerodynamic shape optimization design.

[0022] 3. Meteorological Forecasting and Earth Science: Input data consists of extremely sparse temperature and surface pressure observation data collected by meteorological stations or satellites; output data consists of highly continuous global-scale meteorological dynamic fields coupled with multiple physical quantities such as zonal / meridian wind speed and temperature (e.g., ERA5 climate reconstruction).

[0023] The technical solution adopted in this invention is as follows: I. Model Training with Regularization of Fractional Fock-Planck Equations To address the lack of dynamic constraints in traditional denoising score matching training objectives, this invention proposes a physically enhanced training strategy that aims to force the score function learned by the neural network to satisfy the underlying physical diffusion law.

[0024] 1. Constructing a Spatiotemporal Score Network: A score estimation model based on 3D U-Net is established. This model takes noisy spatiotemporal data and the corresponding noise scale as input and outputs the logarithmic density gradient (i.e., the score function) of the data distribution at the current time. The network architecture includes spatial convolutional modules and a spatiotemporal attention mechanism to capture the spatial structural features and temporal evolutionary dependencies of the physical field.

[0025] 2. Construct a composite loss function: Define the overall optimization objective of model training as a weighted sum of the denoising score matching (DSM) loss term and the score FPE regularization term: ,in This is the balance coefficient.

[0026] Denoising score matching term Used to guide the model to learn the statistical characteristics of the data distribution, ensuring the visual plausibility of the generated samples.

[0027] Regularization term of fractional Fock-Planck equation It is constructed based on the partial differential equations that a true fractional function must satisfy. Specifically, the true fractional function... Satisfying the following fractional Fock-Planck equations:

[0028] in, Indicates time The partial derivatives of . This represents the diffusion coefficient of the stochastic differential equation (SDE). In the variance explosion (VE-SDE) scenario applied in this application, The rate of change of the scale representing the injected noise over time, i.e. . This represents the gradient operator for differentiating spatial variables. Representing a fractional field (vector field) Divergence in spatial dimensions. Represents the computation of vectors Norm (i.e., 2-norm).

[0029] Force the model using a regularization term. It follows the fractional Fock-Planck equation. Regularization term. Defined as:

[0030] in, Indicates time Initial noiseless physical field data and the noisy physical field state at the corresponding time step Find the joint mathematical expectation. Indicates at time step The noisy physical field state at that time. Represents the spatial dimension of physical field data (used for scaling high-dimensional data). Indicates time step The relevant weighting function. and This is a hyperparameter for adjusting the weights of different regularization terms. This indicates the order of the norm (usually taken as 2, used in conjunction with the preceding 2-norm calculation).

[0031] in, This is for quantification Error terms introduced based on the degree of deviation from the above equation:

[0032] in, This represents the error term in the Fock-Planck equation that indicates the deviation of the model's estimated score from the true score.

[0033] in It is an operator that maps a vector field to a scalar function. The hyperparameter controls the strength of the regularization.

[0034] 3. Optimize the solution and approximate the calculation: Regarding the above... For high-dimensional divergence and time derivatives that are difficult to solve directly, this method employs the following approximate calculation strategy during training: Divergence term approximation: Using the Hutchinson trace estimator, a random vector following a standard Gaussian distribution is introduced. High-dimensional divergence Convert to a computable Jacobian-vector product: .

[0035] Time partial derivative approximation: The central finite difference method is used. The time grid is divided according to the noise scale discretization step size of the diffusion process, and a minimum time step size is selected. The difference between the outputs of the forward and backward networks is calculated:

[0036] In the formula, Indicates at time step The noisy physical field state at that time. This represents the estimated score of the deep neural network output (i.e., the log density gradient of the data distribution at the current time step). This represents a continuous time variable in the diffusion process. This represents the smallest time step used for central finite difference computation. The sign of the partial derivative. Indicates the estimated score versus time The partial derivatives of .

[0037] By minimizing this computable composite loss function, the theoretical stability of the generation process is ensured.

[0038] II. Implicit Fractional Sampling (ISS) for Variance Explosion Stochastic Differential Equations To address the reverse diffusion process in the form of variance explosion-SDE, this invention derives and proposes a non-Markovian implicit score sampler to solve the problem of traditional SDE solvers' dependence on extremely small time steps.

[0039] 1. Clean State Estimation: In each iteration of the reverse generation, based on the current noise state and the output of the trained score network, the potential initial clean physical field is analytically estimated using the Tweedie formula. This step transforms the denoising problem into an estimation problem of conditional expectation, providing accurate guidance for large-step updates.

[0040] 2. Constructing Implicit Iterative Rules: Unlike traditional SDE solvers, this method constructs a non-Markovian sampling recursive formula, which scales the noise sequence of VE-SDE. (in , The total number of discretized noise scale sequences (i.e., the total number of iterations in the reverse denoising process) is combined with the estimated clean state, and the complete single-step sampling recursive formula is as follows:

[0041] In the formula, For the first The noise state of the step, For the output of the fractional network, The injected standard Gaussian random noise. This is an adjustable random hyperparameter. When When the sampling process degenerates into a deterministic probabilistic flow ODE trajectory; when At this time, a controlled Gaussian perturbation is introduced to increase sample diversity and correct estimation bias caused by single-step discretization.

[0042] 3. Sequence generation strategy: For long-term physical fields, an overlapping block parallel generation strategy is adopted.

[0043] Blocking and overlapping: Dividing a long sequence into several blocks of length... The sub-blocks, with a length of [missing information] between adjacent sub-blocks. Overlapping area of ​​frames.

[0044] Consistency constraint mechanism: The consistency constraint is not a simple post-processing fusion, but rather a gradient alignment applied in each iteration of sampling. This occurs at each step of the reverse generation. Calculate the mean square error loss of two adjacent blocks in the overlapping region. Then, the gradient of the loss with respect to the current state is calculated. This is then added as a physical correction term to the aforementioned iteration rules to update the state. This forced alignment mechanism in the iteration fundamentally eliminates boundary artifacts between blocks.

[0045] Example 1 Taking "Turbulent Physical Field Reconstruction" as an example, the execution steps of the spatiotemporal physical field reconstruction method based on fractional Fokker-Planck regularization and implicit sampling are described in detail: Phase 1: Model pre-training incorporating physical regularization; Combined with appendix Figure 2 (System composition block diagram of the model training phase) This phase aims to train a deep neural network that embeds the laws of physical evolution.

[0046] Step 1: Acquire physical data and inject noise into the physical data.

[0047] High-fidelity physical simulation data (such as that obtained by solving the wave equation using the finite difference method FDTD) is used as initial noise-free physical field data. Define a discrete time step sequence. And define the noisy variance scheduling sequence of the variance explosion stochastic differential equation (VE-SDE). .Towards Injecting standard Gaussian noise Generate the noisy physical field state corresponding to the time step. .

[0048] Step 2: Construct the spatiotemporal feature extraction network structure.

[0049] Initialize a score network based on a 3D U-Net architecture. ,in These are the learnable weight parameters of the network. Internally, the network employs a U-Net encoder with shared weights to extract spatial features of the physical field frame by frame (spatial convolutional module), and embeds a spatiotemporal attention mechanism at the feature layer level to capture the dynamic evolution and interaction of fluids or sound waves between adjacent frames.

[0050] Step 3: Calculate and optimize the composite loss function.

[0051] Noisy data and noise scale Input the network, output the predicted score field (i.e., the log density gradient of the data distribution). Calculate the composite loss function. .in: The denoising fractional matching loss is used to fit the statistical distribution characteristics of the physical signal; To achieve the ScoreFPE regularization loss, a high-dimensional divergence approximation is calculated using a Hutchinson trace estimator combined with Gaussian random vectors, and the time partial derivatives are approximated using the central finite difference of the forward / backward network. This term forces the network output to satisfy the Fock-Planck partial differential equation. To balance the hyperparameters of the two loss terms, the Adam optimizer is used for backpropagation updates. Continue until the model converges.

[0052] Phase Two: Implicit Sampling Inference for Multimodal Physics Combined with appendix Figure 3 (Iterative update logic diagram of the implicit fractional sampler ISS), which performs efficient generation with large steps during the inference phase.

[0053] Step 4: Input physical observation boundary conditions.

[0054] Load the trained score network Collect sparse sensor observation data in real industrial environments. (e.g., sound pressure at measurement points), and construct the corresponding physical space observation mask matrix. The observed data is input into the fractional network, which outputs a predicted fractional field from pure Gaussian noise. Start reverse iteration.

[0055] Step 5: Cleanliness estimation based on the Tweedie formula.

[0056] At the current reverse time step The network outputs an estimated score. Using the Tweedie formula, the current noise state is... Mapped to the estimated noiseless physical field .

[0057] Step 6: Observe the gradient calculation.

[0058] against Calculate the data consistency loss on sparse observation points. Differentiating the noisy physical field state yields the comprehensive gradient guiding term used for correction.

[0059] Step 7: Implicit state update.

[0060] Figure 3 As shown, the deterministic drift term is calculated according to the deterministic and stochastic combination update rule. and controlled random Gaussian perturbation term Update the noise state for the next time step. .

[0061] The deterministic and stochastic update rule is as follows: the predicted noiseless state is linearly combined with the current noisy state to form a deterministic drift component, and then controlled Gaussian noise is superimposed on this as a stochastic perturbation component. Specifically, the deterministic drift term is calculated. (Guiding samples to move towards high probability density regions) and controlled random Gaussian perturbation term (Correcting the large step size discretization bias in a single step), update the noise state for the next time step. , is represented as: in, Injected standard Gaussian random noise, It is the identity matrix; It is an adjustable random hyperparameter; = ; = ; = ; and They represent the first Step and the first The noise scale of the step.

[0062] Decrease the number of steps in a loop until To obtain a noise-free physical field .

[0063] Phase 3: Application Implementation and Effect Verification Referring to Figure 4 (Evaluation of Turbulent Flow Field Reconstruction Effect), taking the Kolmogorov turbulent system with highly nonlinear and chaotic characteristics as an example, the engineering application process and technical effects of the present invention in complex physical systems such as fluid mechanics are verified.

[0064] Step 8: Input of physical observation data and reconstruction of full-field spatiotemporal dynamics.

[0065] In the application implementation, the input data consists of extremely sparse flow field observation points (local data retained after spatial downsampling of the target area by 6 times). System parameters (such as high Reynolds number) are set. And external force wavenumber After that, the implicit fractional sampler (ISS) in stage two is invoked for inverse denoising inference. The final output data of the system is a complete, high-fidelity two-dimensional vortex field sequence containing multi-scale vortex structures.

[0066] Step 9: Verify the reconstruction effect and assess physical consistency.

[0067] The physical field output by this invention exhibits significant technical advantages in the following three dimensions: Sparse condition structure completion capability: When inputting extremely sparse test samples, compared with LNO (prediction failure), FNO (structural distortion) and standard SDE (accompanied by high frequency noise and artifacts), the vorticity field reconstructed by this invention (ISS) is closest to the reference true value in terms of morphology and detail, accurately capturing the complex vortex evolution characteristics under high Reynolds number.

[0068] Physical consistency guarantee in the frequency domain: By comparing the kinetic energy spectrum of the flow field, it can be found that the energy spectrum curve of the reconstructed flow field of the model after fine-tuning by FPE (fractional Fock-Planck equation) regularization in stage one can maintain a high degree of overlap with the true value in the high wavenumber range (especially the inertial subregion and dissipation region), successfully correcting the energy distribution deviation. This proves that FPE regularization effectively ensures that the model accurately follows the high-frequency statistical characteristics of the physical field while reproducing large-scale motion.

[0069] Inference Efficiency and Sparse Robustness: In evaluating the error evolution of sequence generation, the ISS method of this invention exhibits extremely high computational efficiency. The prediction accuracy achieved with only 100 denoising steps surpasses that of the standard SDE method requiring 500 denoising steps, and the error level is close to that of the computationally expensive 1000-step SDE method. Simultaneously, as the spatial downsampling factor (observation sparsity) increases, this invention consistently maintains extremely low normalized root mean square error (nRMSE), overcoming the drawback of traditional operator networks where errors rise sharply under sparse data, demonstrating excellent robustness for industrial applications.

[0070] Example 2 This application also provides a spatiotemporal physics field reconstruction system based on fractional Fokker-Planck regularization and implicit sampling. Implemented using the above method, the system includes: The module for obtaining the predicted fractional field is used to input sparse sensor observation data into a trained deep neural network embedded with physical evolution laws and output the predicted fractional field. The module for obtaining noiseless physics is used to iterate backward from pure Gaussian noise to obtain noiseless physics.

[0071] This application may also provide a computer device, including: at least one processor, memory, at least one network interface, and a user interface. The various components in this device are coupled together via a bus system. It is understood that the bus system is used to implement communication between these components. In addition to a data bus, the bus system also includes a power bus, a control bus, and a status signal bus.

[0072] The user interface can include a display, keyboard, or clicking device. Examples include a mouse, trackball, touchpad, or touchscreen.

[0073] It is understood that the memory in the embodiments disclosed in this application may be volatile memory or non-volatile memory, or may include both volatile and non-volatile memory. The non-volatile memory may be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), or flash memory. The volatile memory may be random access memory (RAM), which is used as an external cache. By way of example, but not limitation, many forms of RAM are available, such as Static Random Access Memory (SRAM), Dynamic Random Access Memory (DRAM), Synchronous DRAM (SDRAM), Double Data Rate SDRAM (DDRSDRAM), Enhanced Synchronous DRAM (ESDRAM), Synchlink DRAM (SLDRAM), and Direct Rambus RAM (DRRAM). The memories described herein are intended to include, but are not limited to, these and any other suitable types of memory.

[0074] In some implementations, the memory stores elements such as executable modules or data structures, or subsets thereof, or extended sets thereof: operating systems and applications.

[0075] The operating system includes various system programs, such as the framework layer, core library layer, and driver layer, used to implement various basic business functions and handle hardware-based tasks. The application programs include various applications, such as media players and browsers, used to implement various application functions. Programs implementing the methods of the embodiments of this disclosure can be included in the application programs.

[0076] In the above embodiments, the processor can also invoke programs or instructions stored in memory, specifically programs or instructions stored in an application program, for the following purposes: Follow the steps described above.

[0077] The above methods can be applied to or implemented by a processor. The processor may be an integrated circuit chip with signal processing capabilities. During implementation, each step of the above methods can be completed by integrated logic circuits in the processor's hardware or by software instructions. The processor can be a general-purpose processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components. It can implement or execute the methods, steps, and logic diagrams disclosed above. The general-purpose processor can be a microprocessor or any conventional processor. The steps of the disclosed methods can be directly implemented by a hardware decoding processor, or by a combination of hardware and software modules in the decoding processor. The software modules can reside in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory; the processor reads information from the memory and, in conjunction with its hardware, completes the steps of the above methods.

[0078] It is understood that the embodiments described in this application can be implemented using hardware, software, firmware, middleware, microcode, or a combination thereof. For hardware implementation, the processing unit can be implemented in one or more application-specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), general-purpose processors, controllers, microcontrollers, microprocessors, other electronic units for performing the functions described in this application, or combinations thereof.

[0079] For software implementation, the technology of this application can be implemented by executing the functional modules (e.g., procedures, functions, etc.) of this application. The software code can be stored in memory and executed by a processor. The memory can be implemented in the processor or outside the processor.

[0080] This application may also provide a non-volatile storage medium for storing a computer program. When the computer program is executed by a processor, it can implement the steps in the above method embodiments.

[0081] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this application and are not intended to limit it. Although this application 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 this application do not depart from the spirit and scope of the technical solutions of this application, and should all be covered within the scope of the claims of this application.

Claims

1. A spatiotemporal physics field reconstruction method based on fractional Fock-Planck regularization and implicit sampling, comprising: The sparse sensor observation data is input into a trained deep neural network with embedded physical evolution laws, and the output is a predicted fractional field. Starting from pure Gaussian noise, it is iterated in reverse to obtain a noise-free physical field. The reverse iteration includes: Using the Tweedie formula, the noise state of the current step is mapped to the estimated noise-free physical field, and multi-physics coupled gradient calculation is performed. Calculate the deterministic drift term and the controlled random Gaussian perturbation term, superimpose the physical correction gradient, update the noise state for the next time step, and iterate until the step count is 0 to obtain the noise-free physical field.

2. The spatiotemporal physics field reconstruction method based on fractional Fock-Planck regularization and implicit sampling according to claim 1, characterized in that, The deep neural network that embeds the laws of physical evolution is a fractional network based on the 3D U-Net architecture.

3. The spatiotemporal physics field reconstruction method based on fractional Fock-Planck regularization and implicit sampling according to claim 1, characterized in that, The training process of the deep neural network embedding physical evolution laws includes: Obtain physical simulation data as initial noise-free physical field data. Set a discrete time step sequence , The total number of discretized noise scale sequences; the noise variance scheduling sequence of the variance explosion stochastic differential equation is defined. ;Towards Injecting standard Gaussian noise Generate the noisy physical field state corresponding to the time step. ; Noisy physical field state and noise scale Input a deep neural network that embeds physical evolution laws, and output a predicted score field.

4. The spatiotemporal physics field reconstruction method based on fractional Fock-Planck regularization and implicit sampling according to claim 3, characterized in that, The loss function of the deep neural network embedding the laws of physical evolution is: ； in, The denoising fractional matching loss is used to fit the statistical distribution characteristics of the physical signal; The regularization loss for the fractional Fock-Planck equation is constructed based on the partial differential equations that the true fractional function must satisfy.

5. The spatiotemporal physics field reconstruction method based on fractional Fock-Planck regularization and implicit sampling according to claim 1, characterized in that, The Tweedie formula is used to map the noise state of the current step to the estimated noise-free physical field, expressed as: ； in, Represents the estimated noiseless physical field; Indicates the first i The noisy physical field state of the step; Indicates the first i The noise scale of the step; The estimated score represents the output of a deep neural network that incorporates the laws of physical evolution.

6. The spatiotemporal physics field reconstruction method based on fractional Fock-Planck regularization and implicit sampling according to claim 1, characterized in that, The multiphysics coupled gradient calculation includes: Calculation of physical residual loss based on the partial differential equations of mass and momentum conservation discretized by the finite difference method. ; Calculate the data consistency loss on sparse observation points : ； in, This represents sparse sensor observation data; Represents the physical space observation mask matrix; Represents the 2-norm of a matrix; Differentiating the noisy physical field state yields the comprehensive gradient guiding term used for correction.

7. The spatiotemporal physics field reconstruction method based on fractional Fock-Planck regularization and implicit sampling according to claim 1, characterized in that, The updated noise state for the next time step is expressed as: ； in, Injected standard Gaussian random noise, It is the identity matrix; It is an adjustable random hyperparameter; Represents the estimated noiseless physical field; Indicates the first i The noisy physical field state of the step; = ; = ; = , and They represent the first Step and the first The noise scale of the step.

8. A spatiotemporal physics field reconstruction system based on fractional Fock-Planck regularization and implicit sampling, implemented according to the method described in any one of claims 1-7, characterized in that, The system includes: The module for obtaining the predicted fractional field is used to input sparse sensor observation data into a pre-trained deep neural network embedded with physical evolution laws, and output the predicted fractional field; and The module for obtaining noiseless physics is used to iterate backward from pure Gaussian noise to obtain noiseless physics.

Images