A method and system for constructing regular structure features of stochastic partial differential equations

Abstract

The application belongs to the technical field of scientific computing and artificial intelligence, and in particular to a random partial differential equation regular structure feature construction method and system. The integral operation of the regular structure feature is transferred from the physical space to the Fourier space. The Fourier diagonalization property of the Laplace operator is used to convert the integral of the partial differential equation coupled at each space point into the accurate solution of the scalar ordinary differential equation independent of each wave number. On the applicable periodic domain or the region processed by the periodization, the spectral accuracy is realized. Further, the serial recursive format is expanded into the causal convolution matrix form in the Fourier space. The integral factor power is pre-calculated and the three-dimensional causal convolution tensor is assembled. Through the parallel pipeline of batch fast Fourier transform, de-aliasing, tensor contraction causal convolution and batch inverse fast Fourier transform, the memory occupation is significantly reduced and the calculation efficiency is improved. The feature calculation efficiency and approximation accuracy of the machine learning system based on the regular structure feature construction are improved.

Description

Technical Field

[0001] This invention belongs to the fields of scientific computing and artificial intelligence, as well as big data processing technology. Specifically, it is a method and system for constructing the regular structure features of stochastic partial differential equations, which can be used for climate simulation, turbulence simulation, stochastic reaction-diffusion systems, and other spatiotemporal field numerical prediction tasks involving stochastic forcing. Background Technology

[0002] Stochastic partial differential equations (SPDEs) are widely used to describe the evolution of uncertainties in complex physical phenomena such as climate simulation, turbulence, and stochastic reaction-diffusion systems. Although traditional numerical methods (such as finite difference and finite element methods) can solve SPDEs on a sample-by-sample basis, the computational cost increases dramatically when a large number of samples are needed for uncertainty quantification or ensemble forecasting, making it difficult to meet the needs of practical applications.

[0003] In recent years, deep learning-based neural operator methods (such as Fourier neural operators FNO and DeepOnet) have been used to directly learn the mapping from initial conditions and noise to solutions, avoiding discrete solution processing for each sample. However, these purely data-driven methods treat the input-output relationship as a black box and do not utilize the regularization structure inside SPDE (such as singular expansion and integral operators), resulting in the need for a large number of trainable parameters to achieve limited accuracy in low-dimensional problems, and long training time.

[0004] To overcome the aforementioned shortcomings, Gong et al. (2023) proposed a machine learning system based on regularized structural features (DLR-Net). This method integrates the regularized structural theory of mathematician Hairer into neural network feature engineering. By constructing noisy iterative integrals and tree-like nonlinear products as explicit features, and then inputting them into a lightweight decoding network, it significantly reduces the number of parameters and improves prediction accuracy. This provides a new technical path for machine learning solutions to SPDE.

[0005] However, DLR-Net still faces the following technical bottlenecks in practical numerical implementation: First, the spatial discretization accuracy is limited. DLR-Net uses a finite difference scheme to approximate the Laplace operator, and its spatial truncation error is only second-order, which is difficult to meet the requirements of high-precision predictions such as climate simulation.

[0006] Secondly, the computational memory overhead is enormous. This method requires storing a dense matrix in physical space, and the storage requirement increases quadratically with the number of discrete points in space and linearly with the number of time steps. When the spatial resolution or the number of time steps increases, the memory usage can easily exceed the typical GPU memory capacity, hindering its expansion to two-dimensional and three-dimensional high-resolution problems.

[0007] Third, it suffers from low computational efficiency. DLR-Net uses a serial recursive format to calculate integrals step by step, which cannot fully utilize the massive parallel capabilities of GPUs, making feature construction speed a bottleneck for overall training and inference.

[0008] The aforementioned shortcomings make existing methods for constructing regular structural features difficult to directly apply to practical tasks such as climate simulation, which require high resolution, long time-term forecasts, and large ensemble forecasts. In climate numerical simulations, the processing of stochastic forcing fields such as subgrid turbulence, cloud convection, and air-sea fluxes requires efficient integration over multiple random samples and time steps simultaneously, while demanding that the numerical format possess spectral accuracy and low memory consumption. Therefore, how to improve the numerical accuracy of integration operations, reduce pre-computation storage, and achieve high parallelization while maintaining the theoretical advantages of regular structural features is a pressing technical problem to be solved in this field. Summary of the Invention

[0009] To overcome the shortcomings of the prior art and solve at least one of the technical problems mentioned in the background art, the present invention provides a method and system for constructing the regular structure features of stochastic partial differential equations.

[0010] The technical solution adopted by this invention to solve its technical problem is: a method for constructing the regular structure features of stochastic partial differential equations, comprising the following steps: Step 1: Obtain gridded climate state data as initial conditions, and random forced data representing subgrid turbulence, cloud convection, air-sea flux disturbances, or climate model errors as noise input; given the number of spatially discrete points and time step in the periodic domain, calculate the wavenumber array, parabolic linear sign, and single-step integration factor array, and construct a binary dealiasing mask. Step 2: Calculate the consecutive powers of the integral factors and store them as matrices, then assemble the 3D causal convolution tensor. Step 3: Differencing the random forced data along the time dimension to obtain the noise increment, performing a one-dimensional fast Fourier transform on the initial conditions, multiplying by the pre-calculated propagation factor step by step, and then performing an inverse fast Fourier transform to obtain the propagation initial condition characteristics. Step 4: Traverse each tree node in the topological order of the feature map, and perform element-wise exponentiation of the feature values ​​of its dependent nodes in the physical space to obtain the forcing term; perform batch one-dimensional fast Fourier transform along the spatial dimension on the active time step of the forcing term, multiply by the dealiasing mask, perform tensor shrinking causal convolution operation, and then perform batch one-dimensional inverse fast Fourier transform and take the real part to obtain the integral feature of the node. Step 5: Traverse the feature map and mark whether each node depends on the initial conditions. For nodes that only depend on noise, pre-calculate and cache the integral features before training, and directly read the cache during training. Step 6: Compress the multidimensional features output in Step 4 into single-channel latent variables using a fully connected neural network. Replace the original propagation initial conditions with these latent variables and recalculate the feature subset following the same process as in Step 4. Step 7: Concatenate all spectral integral features output in Step 4, the features recalculated in Step 6, the initial conditions, and the normalized spatiotemporal grid coordinates along the feature dimension. Then, pass them through a linear projection layer, a multi-layer Fourier neural operator layer, and a fully connected decoding network to output the future climate state field, subgrid parameterization correction terms, or ensemble forecast samples.

[0011] Preferably: the wavenumber array calculation in step 1 is applicable to one-dimensional, two-dimensional, or three-dimensional periodic domains. In the two-dimensional or three-dimensional case, the wavenumber field of the corresponding dimension is obtained by gridding the one-dimensional wavenumber array in each direction, and a one-dimensional fast Fourier transform is performed sequentially along each spatial direction; the gridded climate state data includes at least one of temperature field, wind speed field, humidity field, pressure field, precipitation field, or sea surface temperature field; the output includes at least one of the above physical fields or their corrections for one or more future time steps; wherein, the corrections include temperature trend correction, wind field trend correction, humidity field trend correction, pressure field trend correction, precipitation field trend correction, and sea surface temperature field trend correction; In step 1, the dealiasing mask adopts a 2 / 3 truncation rule, retaining only the low-frequency modes in each direction. The mask value is 1 to indicate retention and 0 to indicate truncation.

[0012] Preferably, the regular structural feature construction method is applied to the field of climate simulation.

[0013] Preferably: the elements of the causal convolution tensor C in step 2 are defined as follows: Otherwise, it is 0 and registered as a non-trainable buffer in complex format; is the integration factor, i represents the index of the output time step, j represents the index of the input time step, and k is the discrete index of the wavenumber array.

[0014] Preferably, when performing tensor shrinking operations in step 4, causal convolution is performed in parallel on the GPU using the Einstein summation convention, and the real part of the output is taken as the integral feature after inverse fast Fourier transform.

[0015] A system for constructing the regular structure features of stochastic partial differential equations includes: The data acquisition module is used to acquire gridded climate state data as initial conditions, and random forced data representing subgrid turbulence, cloud convection, air-sea flux disturbances or climate model errors as noise input. The spectral integration factor module is used to perform exact integration of linear operators in Fourier space. Its configuration is as follows: calculate the wavenumber array, linear sign, single-step integration factor, and construct the dealiasing mask. The vectorized causal convolution module is used to compute the integral results at all time steps in parallel. Its configuration is as follows: pre-compute the successive powers of the integral factors, assemble the three-dimensional causal convolution tensor C, register the tensor as a non-trainable buffer of the neural network, and perform batch fast Fourier transform, dealiasing, tensor shrinking causal convolution and batch inverse fast Fourier transform in the forward propagation. The pseudospectral nonlinear product module is used to calculate nonlinear products in physical space. Its configuration is as follows: traverse the tree nodes in the topological order of the feature map, perform element-wise exponential multiplication on the feature values ​​of the dependent nodes in physical space to obtain the forced term, and input the result into the spectral integral factor module. The feature compression and quadratic integration module is used to compress multidimensional features into single-channel latent variables through a fully connected neural network, and then use these latent variables to replace the original propagation initial conditions to recalculate the feature subset. The decoder module is used to output future climate state fields, subgrid parameterization corrections, or ensemble forecast samples.

[0016] Preferably: In the spectral integration factor module, the integration factor is... ,in, It is applicable to parabolic stochastic partial differential equations.

[0017] Preferably: In the vectorized causal convolution module, the dimension of the three-dimensional causal convolution tensor C is... ,in Let N be the number of time steps, and N be the number of spatial discrete points in each spatial direction.

[0018] Preferably, the decoder module includes four Fourier neural operator layers, each layer including a spectral convolution branch and a pointwise convolution residual branch. The spectral convolution branch is configured as a one-dimensional, two-dimensional, or three-dimensional spectral convolution branch according to the input space dimension; the non-final layer uses the GeLU activation function.

[0019] Preferably, the fully connected neural network in the feature compression and quadratic integration module is a three-layer fully connected structure, which compresses multidimensional features into single-channel latent variables for quadratic spectral integration.

[0020] The beneficial effects of this invention are as follows: This invention transfers the integral operation of the regular structural features from physical space to Fourier space. Utilizing the diagonalization property of the Laplacian operator under Fourier bases, it transforms the linear diffusion integral problem coupled at each spatial point in the original DLR-Net into a propagation problem of scalar ordinary differential equations with independent wavenumbers. For the case of periodic boundaries and sufficient smoothness, it fundamentally eliminates the problems caused by finite-difference Laplacian matrices. The spatial truncation error was reduced, achieving spectral accuracy.

[0021] Based on this, the original serial recursive format is expanded into a causal convolution matrix form in Fourier space. Utilizing the independent structural characteristics of each mode, the pre-computation storage is reduced from a dense matrix in physical space. Reduced to the causal convolution tensor corresponding to Fourier space (Actual measurement reduced from 173 MB to 2.6 MB), and replaced the original single large matrix multiplication with a fully parallel pipeline of batch FFT-tensor shrinking-batch IFFT, achieving comparable training speed on GPUs (less than 5% overhead). Furthermore, since C has a lower triangular Toeplitz structure, another implementation can store only the power matrix of the integral factor. During computation, it is accessed by diagonal index, thereby further reducing the storage overhead of explicit tensors.

[0022] Experiments show that, under matched data conditions, this invention achieves prediction accuracy comparable to or better than the pure data-driven FNO method with 13.1M parameters on the task of predicting solutions to parabolic stochastic partial differential equations with only 133K parameters. At the same time, the training time on this task is reduced by about 40 times and the feature caching speed is increased by about 3 times, verifying the comprehensive advantages of Fourier space regular structure features in terms of accuracy, memory efficiency and computational efficiency.

[0023] Furthermore, in climate simulation applications, this invention maps the climate state field and subgrid random perturbations into parallel-computable regularized structural features, which can serve as modules for random subgrid parameterization, bias correction, regional downsampling, or ensemble forecast generation in climate numerical models. Since the feature construction performs batch FFT, dealiasing, tensor shrinking, and batch IFFT in Fourier space, it can process multiple climate samples and multiple time steps in parallel on GPU computing devices, reducing the computational overhead of solving stochastic partial differential equations sample-by-sample and improving the computational efficiency and stability of spatiotemporal field predictions such as temperature, wind field, humidity, precipitation, or sea surface temperature. Attached Figure Description

[0024] The invention will now be further described with reference to the accompanying drawings.

[0025] Figure 1 This is a block diagram of a system for constructing regular structural features of stochastic partial differential equations according to the present invention. Detailed Implementation

[0026] To make the technical means, creative features, objectives and effects of this invention easier to understand, the invention will be further described below in conjunction with specific embodiments.

[0027] Example 1 like Figure 1As shown in the embodiment of the present invention, a system for constructing regular structural features of stochastic partial differential equations includes the following modules: The data acquisition module is used to acquire gridded climate state data as initial conditions, and random forced data representing subgrid turbulence, cloud convection, air-sea flux disturbances or climate model errors as noise input. Spectral integral factor module: for those defined in the periodic domain The parabolic SPDE on the t+10 ... , Where L is the side length; d represents the spatial dimension; The partial derivative operator with respect to time t; The diffusion coefficient controls the intensity of the diffusion term; Δ is the Laplace operator, representing the sum of the second derivatives in space; I[f] is the integral operator acting on the function f, where f is the source term of the integral operator; Represents spatial variables.

[0028] set up In Fourier space, the Laplace operator Diagonalization is performed under the Fourier basis. The evolution equations for each Fourier mode k (scalar in 1D, vector in 2D and 3D) are independent of each other: Where v(x,t) is the representation of the integration result in physical space (spatial-temporal coordinates); x is a spatial coordinate vector; t: represents the time coordinate; v(k,t) is the coefficient of v(x,t) at the Fourier mode k after performing a Fourier transform along the spatial variable x; k is the wavenumber vector in Fourier space (a scalar in one dimension, and a vector in two / three dimensions). |k| is the magnitude of the wavenumber vector k; f(k,t) is the coefficient of the source term f(x,t) at the Fourier mode k.

[0029] The above equations can then be recursively simplified to discrete time steps using the integral factor method combined with Euler discretization: Where n is the index of the discrete time step; Here are the Fourier coefficients of wavenumber k at time step n+1; It is an integral factor, and its core is an exponential decay / growth function; Here are the Fourier coefficients of wavenumber k at time step n; Let f be the Fourier coefficients of the source term f at wavenumber k at time step n; Δt is the discrete time step; It is the integrating factor; The specific calculation steps are as follows: Step 1: Wavenumber Calculation Given N discrete points in each spatial direction and a periodic region of length L, calculate the wavenumber array using the Fast Fourier Transform: Where j is the discrete index of the wavenumber array, ; N is a discrete point in each spatial direction; FFT stands for Fast Fourier Transform; This step generates a real number array of length N. For 2D and 3D cases, a one-dimensional wavenumber array is generated in each spatial direction, and then a 2D or 3D wavenumber field is obtained through meshing.

[0030] Step 2: Calculate the integrating factor: For parabolic equations, calculate the linear sign. This leads to the single-step integration factor. The length is N.

[0031] Step 3: Constructing the dealiasing mask: A dealiasing mask D(k) is constructed using a 2 / 3 truncation rule, retaining only the low-frequency modes in each direction: D(k) is set to 1 when the wavenumber of the mode is within the allowable range in each coordinate direction; otherwise, D(k) is set to 0. In the 1D case, D is a binary array of length N; in the 2D and 3D cases, D is respectively... and A binary tensor. Step 4: Pre-calculation of initial condition propagation factor: For initial conditions Time propagation, pre-calculation of propagation factors at each time point. ,in Store as (1D case) (For 2D and 3D storage as...) An array of N is used for subsequent modal multiplication calculations. t t represents the total number of discrete time steps. n This is the physical time corresponding to the nth time step.

[0032] The following steps are presented in the 1-dimensional case. The 2-dimensional and 3-dimensional cases can be naturally generalized in the tensor form.

[0033] Vectorized causal convolution module: Expanding the above recursive format, the solution at the nth time step can be expressed as: Where m is the summation index (accumulation variable) of the discrete time step, and its value ranges from 1 to n; To achieve vectorized computation for time recursion, a causal convolution matrix is ​​defined. In the one-dimensional case, matrix C can be viewed as a three-dimensional tensor, where the three dimensions have the following meanings: the first dimension... The second dimension is used to output the time step index. The input time step index is given, and the third dimension k is the Fourier mode index. The matrix elements are defined as follows: When j > i, due to the causality requirement that future moments cannot affect the current moment, the corresponding element is set to 0. The specific calculation steps are as follows: Step 1: Pre-calculation of the power of the integral factor: calculate Successive powers: Store as matrix .

[0034] Step 2: Assembling the causal convolution matrix: Initial size is A complex tensor C consisting of all zeros. For each diagonal offset ,Will Assign the value to the d-th lower diagonal of C, i.e. .

[0035] Step 3: Register the causal convolution matrix as a GPU buffer: Register C in complex64 format as a non-trainable buffer for the neural network module, so that it can be automatically transferred to the GPU with the model and referenced in the forward propagation.

[0036] Step 4: Parallelize integration execution: During the forward propagation, the following pipeline is executed on the input forcing term f: (a) Batch FFT: Performs a one-dimensional FFT on all spatial data at the active time step. This operation is performed in parallel on the last dimension (spatial dimension), and all batches and time steps are processed simultaneously; (b) Dealiasing: Multiplying each element by the dealiasing mask D truncates the high-frequency components, resulting in... ; (c) einsum causal convolution: performs tensor union operation This operation is highly parallelized on the GPU; (d) Batch Inverse FFT: for Perform an inverse FFT and extract the real part to obtain the integral result in physical space. , The 0th time step is filled with 0.

[0037] The pseudospectral nonlinear product module: The feature map of the regular structural features is indexed according to a tree structure, where the depth of the tree represents the feature order, and each tree node records a feature value. During the construction of the regular structural features, the nonlinear product in the tree structure requires calculating the pointwise power product of multiple existing features. This invention employs a pseudospectral method: the nonlinear product is performed pointwise in physical space, and then the result is transformed to Fourier space for subsequent spectral integration. For each tree node to be calculated in the feature map G... Its dependency relationship is represented as in, It is the j-th tree node in the feature map G; For nodes The set consisting of the child nodes it depends on and their powers; ( ) represents a dependency pair, where It is a dependent child node. It is the power of the corresponding child node's feature value; The meaning of this dependency is: The corresponding forced terms are ;in, It is a lookup table with tree nodes as keys and the corresponding spacetime function value tensor as values, which is gradually filled during the traversal.

[0038] The specific calculation steps are as follows: Step 1: Include the mandatory items Initialize as a tensor of all 1s.

[0039] Step 2: For each In physical space, perform element-wise operations: if p=1, then If p>1, then All operations are element-wise and are fully parallelized on the GPU.

[0040] Step 3: Multiply the physical space results Input the spectrum integration module to obtain the integration result. .

[0041] Step 4: For noise-dependent applications The features are cached in advance before training, and the table is directly looked up during training to avoid repeated calculations.

[0042] Feature compression and quadratic integration module: The feature map topology traversal and complete construction process is as follows: Given a batch of input data: noise and initial conditions Feature construction is performed according to the following steps: Step 1: Noise Differentiation. Calculate the noise increment. As the root node The value of .

[0043] Step 2: Propagation of initial conditions. Get by performing FFT Subsequently, at each time step... The initial propagation conditions are obtained by multiplying the initial propagation conditions by the pre-calculated propagation factor modally and then returning to the physical space via the inverse fast Fourier transform. .

[0044] Step 3: Traverse the feature graph in topological order. For each node in the graph except the root node... If the node depends only on noise and is pre-cached, read directly from the cache; otherwise, calculate the nonlinear product as described in Section 4. Then perform spectral integration to obtain .

[0045] Step 4: Feature Stacking. Stack the values ​​of all nodes along the last dimension to obtain the output feature tensor. .

[0046] Decoder Module: The complete spectral depth implicit regularization network consists of the following six functional layers, connected sequentially from input to output: First layer—Spectral integral feature construction layer: Input noise and initial conditions Construct a regular structural feature tensor according to the above steps, and denote its regular feature number (excluding noise itself) as F.

[0047] The second layer—the feature compression layer—is a three-layer fully connected network that compresses the F-dimensional features into single-channel latent variables U. The function of this layer is to extract core information related to the initial conditions from the regularized features.

[0048] The third layer—the quadratic spectral integral layer—uses the same spectral integral module as the first layer, replacing the original propagation initial conditions with the compressed latent variable U, and recalculates the dependencies. A subset of features.

[0049] Fourth layer—Feature concatenation layer: Combines all features from the first layer (F-dimensional) and the third layer... By concatenating the dependent features (F(U) dimension) and the two-dimensional spatiotemporal grid coordinates (2 dimension), we obtain (F+F(U)+2) dimension features.

[0050] Fifth layer—FNO decoding layer: contains L( () A series of Fourier neural operator layers.

[0051] The sixth layer—output decoding layer: a two-layer fully connected network (with GeLU activation function) that projects the multi-channel output of the FNO layer into a single-channel predicted solution.

[0052] An example of a specific application in climate simulation: It receives gridded spatiotemporal data formed from climate models, reanalysis data, or observational data, including one or more of the following: temperature field, east-west wind field, north-south wind field, humidity field, pressure field, precipitation field, sea surface temperature field, topographic height, and land-sea mask; and simultaneously receives a stochastic forcing field representing subgrid turbulence, cloud convection, air-sea flux disturbances, radiative forcing disturbances, or climate model errors.

[0053] In this embodiment, the aforementioned gridded climate state variables are used as initial conditions or conditional inputs for stochastic partial differential equations, and the stochastic forcing field is used as noise input. For two-dimensional or three-dimensional spatial grids, wavenumber arrays are constructed according to each spatial direction, and corresponding parabolic linear symbols and integration factors are formed in Fourier space. For non-periodic regional data, the study area can be periodically extended, windowed, or divided into local blocks to meet the input format of the spectral integration module.

[0054] In climate simulation applications, the aforementioned functional modules are invoked to construct a regularized structural feature that includes noise integral, nonlinear interaction, propagation initial conditions, and quadratic spectral integral results according to the topological order of the feature map. The regularized structural feature is then concatenated with normalized coordinates or static variables such as latitude and longitude, vertical layer, altitude, time, topography, and land-sea mask, and input into the Fourier neural operator decoding layer to output the climate state field for the next time step or multiple future time steps.

[0055] In one implementation, the output is a subgrid parameterization correction term of a low-resolution climate model, including temperature trend correction, wind field trend correction, humidity field trend correction, pressure field trend correction, precipitation field trend correction, and sea surface temperature field trend correction; the correction term is fed back to the time-progression process of the climate numerical model to compensate for the influence of unresolved turbulence, cloud convection, or ocean eddies on the large-scale climate field.

[0056] In another implementation, the output is a bias correction field or a high-resolution regional climate field; the low-resolution climate model output is added to the bias correction field to obtain the corrected temperature, wind, humidity, precipitation or sea surface temperature prediction results; or the low-resolution input is mapped to a high-resolution output for regional climate downsampling.

[0057] In another implementation, multiple future climate state field samples are generated by repeatedly sampling the random forced field and repeatedly executing the regular structure feature construction and decoding process, forming an ensemble forecast result; the ensemble forecast result is used to output the average prediction field, variance field, confidence interval, or the probability of occurrence of events such as extreme precipitation, heat waves, and droughts.

[0058] In this way, the present invention does not simply provide mathematical rules or abstract algorithms, but combines the construction of regular structural features of stochastic partial differential equations, GPU parallel spectral integration calculation, and numerical prediction of climate state fields to form a specific technical processing flow for climate simulation equipment or climate prediction computing platforms. This can reduce the overhead of repeated solutions in climate ensemble simulation and improve the ability to represent multi-scale stochastic disturbances.

[0059] Example 2 The method for constructing regular structural features of stochastic partial differential equations according to an embodiment of the present invention is characterized by comprising the following steps: Step 1: Wavenumber and Integral Factor Construction: Obtain gridded climate state data as initial conditions, and random forced data representing subgrid turbulence, cloud convection, air-sea flux disturbances, or climate model errors as noise input; given N spatially discrete points on the periodic domain [0,L] and a time step. Calculate the wavenumber array ; Calculate the sign of the parabolic linear equation ; Calculate the single-step integration factor array; Construct a binary dealiasing mask.

[0060] Step 2: Pre-computation of the causal convolution matrix: Calculate the consecutive powers of the integral factors. Store as The matrix; assembles a three-dimensional causal convolution tensor C; this tensor is registered as a non-trainable buffer for the neural network in complex64 format.

[0061] Step 3: Noise Differentiation and Initial Condition Propagation: Differentiate the input noise along the time dimension to obtain the noise increment. Perform a one-dimensional FFT on the initial conditions, multiply by the pre-calculated propagation factor step by step, and then perform an IFFT to obtain the propagation initial condition features. ; Step 4: Constructing spectral integral features: Traverse each tree node in topological order of the feature map. The forced term is obtained by exponentiation of the eigenvalues ​​of the dependent nodes in physical space element by element. ;right Perform a batch one-dimensional FFT along the spatial dimension at the active time step; multiply the FFT result element-wise by the dealiasing mask D; and perform tensor shrinking operation. Perform a batch one-dimensional IFFT on the results to extract the real part, and obtain the integral feature of the node. Step 5: Feature caching and reuse: Traverse the feature map and mark whether each node depends on the initial conditions. For nodes that only depend on noise, pre-calculate and store the tensor according to step 4 before training. During training, directly read the cache and skip the real-time spectral integral calculation. Step 6: Feature compression and quadratic integration: The multidimensional features output in Step 4 are compressed into single-channel latent variables through a fully connected neural network; these latent variables replace the original propagation initial conditions, and the feature subset is recalculated following the same process as in Step 4. Step 7: FNO Decoding and Output: All spectral integral features output in Step 4, features recalculated in Step 6, initial conditions, and normalized spatiotemporal grid coordinates are concatenated along the feature dimension. After passing through a linear projection layer, multiple Fourier neural operator layers, and a fully connected decoding network, the future climate state field, subgrid parameterization correction terms, or ensemble forecast samples are output.

[0062] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for constructing the canonical structure features of stochastic partial differential equations, characterized in that: Includes the following steps: Step 1: Obtain gridded climate state data as initial conditions, and random forced data representing subgrid turbulence, cloud convection, air-sea flux disturbances, or climate model errors as noise input; given the number of spatial discrete points and time step in the periodic domain, calculate the wavenumber array, parabolic linear sign, and single-step integral factor array, and construct a binary dealiasing mask. Step 2: Calculate the consecutive powers of the integral factors and store them as matrices, then assemble the three-dimensional causal convolution tensor; Step 3: Difference the random forced data along the time dimension to obtain the noise increment, perform a one-dimensional fast Fourier transform on the initial conditions, multiply by the pre-calculated propagation factor step by step, and then perform an inverse fast Fourier transform to obtain the propagation initial condition characteristics. Step 4: Traverse each tree node in the topological order of the feature map, and perform element-wise exponentiation of the feature values ​​of its dependent nodes in the physical space to obtain the forcing term; perform batch one-dimensional fast Fourier transform along the spatial dimension on the active time step of the forcing term, multiply by the dealiasing mask, perform tensor shrinking causal convolution operation, and then perform batch one-dimensional inverse fast Fourier transform and take the real part to obtain the integral feature of the node. Step 5: Traverse the feature map and mark whether each node depends on the initial conditions. For nodes that only depend on noise, pre-calculate and cache the integral features before training, and directly read the cache during training. Step 6: Compress the multidimensional features output in Step 4 into single-channel latent variables using a fully connected neural network. Replace the original propagation initial conditions with these latent variables and recalculate the feature subset following the same process as in Step 4. Step 7: Concatenate all spectral integral features output in Step 4, the features recalculated in Step 6, the initial conditions, and the normalized spatiotemporal grid coordinates along the feature dimension. Then, pass them through a linear projection layer, a multi-layer Fourier neural operator layer, and a fully connected decoding network to output the future climate state field, subgrid parameterization correction terms, or ensemble forecast samples.

2. A method for constructing the canonical structure features of stochastic partial differential equations according to claim 1, characterized in that: The wavenumber array calculation in step 1 is applicable to one-dimensional, two-dimensional, or three-dimensional periodic domains. In the two-dimensional or three-dimensional case, the wavenumber field of the corresponding dimension is obtained by gridding the one-dimensional wavenumber array in each direction, and a one-dimensional fast Fourier transform is performed sequentially along each spatial direction. The gridded climate state data includes at least one of the following: temperature field, wind speed field, humidity field, pressure field, precipitation field, or sea surface temperature field. The output includes at least one of the above physical fields or their corrections for one or more future time steps. Among them, the corrections include temperature trend correction, wind field trend correction, humidity field trend correction, pressure field trend correction, precipitation field trend correction, and sea surface temperature field trend correction. In step 1, the dealiasing mask adopts a 2 / 3 truncation rule, retaining only the low-frequency modes in each direction. The mask value is 1 to indicate retention and 0 to indicate truncation.

3. A method for constructing the canonical structure features of stochastic partial differential equations according to claim 2, characterized in that: The method for constructing regular structural features is applied in the field of climate simulation.

4. A method for constructing the canonical structure features of stochastic partial differential equations according to claim 3, characterized in that: In step 2, the elements of the causal convolution tensor C are defined as follows: Otherwise, it is 0 and registered as a non-trainable buffer in complex format; is the integration factor, i represents the index of the output time step, j represents the index of the input time step, and k is the discrete index of the wavenumber array.

5. A method for constructing the canonical structure features of stochastic partial differential equations according to claim 4, characterized in that: In step 4, when performing tensor shrinking operations, causal convolution is performed in parallel on the GPU using the Einstein summation convention. The real part of the output is then taken as the integral feature after inverse fast Fourier transform.

6. A system for constructing the canonical structure features of stochastic partial differential equations, characterized in that: include: The data acquisition module is used to acquire gridded climate state data as initial conditions, and random forced data representing subgrid turbulence, cloud convection, air-sea flux disturbances or climate model errors as noise input. The spectral integration factor module is used to perform exact integration of linear operators in Fourier space. Its configuration is as follows: calculate the wavenumber array, linear sign, single-step integration factor, and construct the dealiasing mask. The vectorized causal convolution module is used to compute the integral results at all time steps in parallel. Its configuration is as follows: pre-compute the successive powers of the integral factors, assemble the three-dimensional causal convolution tensor C, register the tensor as a non-trainable buffer of the neural network, and perform batch fast Fourier transform, dealiasing, tensor shrinking causal convolution and batch inverse fast Fourier transform in the forward propagation. The pseudospectral nonlinear product module is used to calculate nonlinear products in physical space. Its configuration is as follows: traverse the tree nodes in the topological order of the feature map, perform element-wise exponential multiplication on the feature values ​​of the dependent nodes in physical space to obtain the forced term, and input the result into the spectral integral factor module. The feature compression and quadratic integration module is used to compress multidimensional features into single-channel latent variables through a fully connected neural network, and then use these latent variables to replace the original propagation initial conditions to recalculate the feature subset. The decoder module is used to output future climate state fields, subgrid parameterization corrections, or ensemble forecast samples.

7. A system for constructing the canonical structure of stochastic partial differential equations according to claim 6, characterized in that: In the spectral integration factor module, the integration factor is... ,in, It is applicable to parabolic stochastic partial differential equations.

8. A system for constructing the canonical structure of stochastic partial differential equations according to claim 7, characterized in that: In the vectorized causal convolution module, the dimension of the 3D causal convolution tensor C is... ,in Let N be the number of time steps, and N be the number of spatial discrete points in each spatial direction.

9. A system for constructing the regular structure features of stochastic partial differential equations according to claim 8, characterized in that: The decoder module contains four Fourier neural operator layers, each including a spectral convolution branch and a pointwise convolution residual branch. The spectral convolution branch is configured as a one-dimensional, two-dimensional, or three-dimensional spectral convolution branch according to the input space dimension; the non-final layers use the GeLU activation function.

10. A system for constructing the canonical structure of stochastic partial differential equations according to claim 9, characterized in that: The fully connected neural network in the feature compression and quadratic integration module is a three-layer fully connected structure that compresses multidimensional features into single-channel latent variables for quadratic spectral integration.

Images