Wavefield simulation method based on progressive autoregressive fourier neural operator
By using a progressive autoregressive Fourier neural operator model, the problems of large computational cost in traditional methods and error accumulation in deep learning models are solved, achieving efficient and accurate wavefield simulation, which is suitable for full waveform inversion under complex geological conditions.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CHENGDU UNIVERSITY OF TECHNOLOGY
- Filing Date
- 2026-04-07
- Publication Date
- 2026-07-03
AI Technical Summary
In full waveform inversion, traditional numerical methods are computationally intensive and time-consuming, while deep learning models are difficult to follow the laws of physical evolution in wave field simulation and are prone to recursive cumulative errors, resulting in low computational efficiency and poor wave field quality.
A progressive autoregressive Fourier neural operator model is adopted. By constructing the neural operator model and introducing a progressive sampling strategy and a smoothing loss function, the model is controlled from supervision to autoregressive prediction, learning the single-step mapping relationship of the wave field and suppressing error accumulation.
It achieves efficient and stable wavefield simulation, significantly improves calculation speed, maintains high accuracy and physical realism, adapts to complex geological conditions, and has good generalization ability.
Smart Images
Figure CN122333992A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of exploration geophysics, and in particular relates to a wave field simulation method based on a progressive autoregressive Fourier neural operator. Background Technology
[0002] Full Waveform Inversion (FWI) technology has broad application prospects in exploration geophysics, and its iterative optimization process heavily relies on high-precision forward modeling of acoustic wave equations. The core efficiency bottleneck of traditional FWI lies in the fact that each iteration requires repeatedly solving the acoustic wave equations using numerical methods such as the finite difference method, finite element method, or spectral element method to generate a synthetic seismic wavefield to match actual observation data. To meet numerical stability and accuracy requirements, these traditional solvers are often limited by stringent spatial grids and temporal sampling rates, resulting in extremely high computational costs and significant time consumption when dealing with large-scale complex medium models. To overcome this bottleneck between high computational cost and industrial efficiency, the introduction of deep learning models enables "instantaneous forward modeling" after model training, thereby replacing traditional numerical solvers and significantly shortening the most time-consuming forward modeling calculation step in FWI. This aims to significantly improve FWI speed while maintaining physical accuracy.
[0003] In recent years, deep learning techniques, represented by the Fourier Neural Operator (FNO), have been introduced into solving partial differential equations and simulating wave fields. These data-driven methods can directly learn the mapping relationships between infinite-dimensional function spaces, offering orders of magnitude improvement in inference efficiency compared to traditional numerical algorithms. However, existing data-driven operators face two main dilemmas when dealing with the temporal evolution of wave fields: the first approach uses an end-to-end global mapping strategy, establishing a single mapping from the velocity model directly to the entire full-time wave field sequence. This mapping ignores the fact that wave field propagation is essentially a dynamic physical evolution process following strict time causality, resulting in generated wave fields lacking realistic dynamic characteristics and exhibiting poor generalization. The second approach employs a conventional autoregressive mechanism based on multi-step historical states, with input settings... arrive A continuous sequence of wavefield snapshots at each time point, directly output via a network. The drawback of predicting wavefield snapshots at a given moment is that the model is prone to severe recursive accumulation errors when faced with long time sequences of temporal evolution, which seriously damages the spatial structure and physical reality of the effective wavefield signal. Therefore, there is an urgent need for an efficient wavefield simulation method that can both follow the physical evolution of the wavefield and effectively suppress long-sequence recursive errors. Summary of the Invention
[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a wavefield simulation method based on a progressive autoregressive Fourier neural operator. This method balances the computational efficiency of seismic forward modeling with the numerical stability of long-term time-history evolution, effectively suppresses the cumulative error in the autoregressive recursion process, and ensures the prediction quality of full-time series wavefield data.
[0005] The objective of this invention is achieved through the following technical solution: a wavefield simulation method based on a progressive autoregressive Fourier neural operator, comprising the following steps:
[0006] S1. Construct a neural operator model;
[0007] S2. A progressive sampling strategy is used to train the model, controlling the transition from ground truth supervision to autoregressive prediction;
[0008] S3. By using the trained model and recursively feeding the current output back as subsequent inputs, a rapid inference of the full-time series wavefield snapshot under an unknown velocity model can be achieved.
[0009] The neural operator parameters included in the neural operator model constructed in step S1 are: The neural operator is The mapping relationship is as follows:
[0010]
[0011] in, Indicates the model prediction A snapshot of the wave field at a given moment; neural operators The inputs include: This represents the input formation velocity model, which is the seismic wave velocity at different spatial locations. The speed of spread, Indicates the current time ( A snapshot of the wave field at a given time. Represents the spatial coding state of the seismic source. Neural operator. The main structure consists of multiple consecutive Fourier neural operator blocks, and its core physics follows the following nonlinear iterative mapping formula:
[0012]
[0013] in, Indicates the first The output of the layer Fourier neural operator block It is a non-linear activation function. Linear transformation operators in the spatial domain are used for local residual learning of input features. and Let them represent the two-dimensional forward and inverse Fourier transforms in the spatial domain, respectively. It is a frequency domain parameterization operator used to learn the global wavefield mapping relationship in the frequency domain. These are the input features of the Fourier neural operator block in this layer.
[0014] Step S2 includes:
[0015] S201. Employ smoothing Loss function calculation predicts wavefield snapshot With truth labels The difference between them is defined by the loss function as:
[0016]
[0017] Among them, the smoothing threshold parameter Used to define the loss function in and The transition region between norms is used to balance the model's robustness to outliers. The closer the predicted value is to the true value, the smaller the loss function value. This is achieved by adjusting the model parameters through backpropagation. Perform iterative optimization;
[0018] S202. Introduce sampling probability values during model training. This value varies with the number of training rounds. The value decreases linearly with the increase of , and the calculation formula is:
[0019]
[0020] in, For the preset total number of training rounds, in each round of training, the model is probabilistically... Choose the true wave field from the previous time step as input, with probability By selecting the predicted wave field of the model at the previous time step as input, the training process can be controlled to gradually transition from supervised mode to autoregressive prediction mode, enabling the neural operator model to suppress the cumulative error of time evolution.
[0021] Step S3 includes:
[0022] S301. The formation velocity model and spatial information of earthquake source And the current Tron snapshot Input to the trained neural operator model In the process, obtain a snapshot of the predicted wave field at the next moment. The wave field at time zero is initialized to a zero matrix;
[0023] S302. Take a snapshot of the predicted wavefield generated in step S301. Recursion serves as the current time input for the next time step. Repeat the forward computation process of the neural operator until the iteration of the preset total time step is completed;
[0024] S303. The wavefield snapshots generated at each time point during the iteration process are stitched together in chronological order to synthesize a full-time sequence wavefield evolution data that reflects the propagation law of seismic waves under the unknown velocity model.
[0025] Compared with existing technologies, the beneficial effects of this invention are: the neural operator model learns the single-step evolution mapping relationship of the wave field at adjacent time points, which greatly reduces the model parameters and storage space occupation and is more flexible. It breaks through the limitation of spatiotemporal sampling rate of traditional numerical algorithms, which significantly improves the forward modeling simulation speed. At the same time, by introducing a progressive autoregressive strategy, it effectively alleviates the error accumulation of pure data-driven models in long sequence recursion, ensures high-fidelity simulation accuracy that is very close to that of traditional high-order numerical methods, and shows excellent generalization and adaptability to unknown complex velocity models. Attached Figure Description
[0026] Figure 1 This is a schematic diagram illustrating the principle of the progressive autoregressive training Fourier neural operator model of the present invention.
[0027] Figure 2 A schematic diagram illustrating the principle of autoregressive forward modeling inference for the Fourier neural operator model of this invention;
[0028] Figure 3 This is a comparison of the forward modeling results of the OpenFWI Style-A-28 velocity model at time 0.6s using the method of this invention;
[0029] Figure 4 This is a comparison of the forward modeling results of the OpenFWI Curve-A-02 velocity model at time 0.6s using the method of this invention;
[0030] Figure 5 The present invention compares the spectral domain results of the Curve-A-02 class velocity model in OpenFWI using the method of the present invention;
[0031] Figure 6 The present invention compares the phase domain results of the Curve-A-02 class velocity model in OpenFWI using the method of the present invention;
[0032] Figure 7 This invention provides a comparison of the simulation results of the Marmousi region velocity model A and its forward model at time 1.2s using the method of this invention.
[0033] Figure 8 This paper compares the simulation results of the Marmousi region velocity model B and its forward model at time 0.8s using the method of this invention. Detailed Implementation
[0034] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings, but the scope of protection of the present invention is not limited to the following description.
[0035] This invention innovatively introduces a progressive autoregressive training strategy and a smoothing loss function based on Fourier neural operators, thereby achieving high-efficiency and high-precision long-time-history evolution simulation of seismic wavefields and effectively avoiding the defects existing in existing pure data-driven operator techniques. In other words, the wavefield simulation method provided by this invention not only breaks through the computational bottleneck of traditional numerical algorithms to achieve ultra-fast forward modeling inference by hundreds of times, but also fundamentally suppresses the recursive accumulation error that is easily generated during autoregressive prediction, ensuring the quality and physical authenticity of the wavefield data after long-sequence inference. Specifically:
[0036] like Figure 1 and Figure 2 As shown, a wavefield simulation method based on a progressive autoregressive Fourier neural operator includes the following steps:
[0037] S1. Construct a neural operator model;
[0038] S2. A progressive sampling strategy is used to train the model, controlling the transition from ground truth supervision to autoregressive prediction;
[0039] S3. By using the trained model and recursively feeding the current output back as subsequent input, a rapid inference of the full-time series wavefield snapshot under an unknown velocity model can be achieved.
[0040] The neural operator parameters included in the neural operator model constructed in step S1 are: The neural operator is The mapping relationship is as follows:
[0041]
[0042] in, Indicates the model prediction A snapshot of the wave field at a given moment; neural operators The inputs include: This represents the input formation velocity model, which is the seismic wave velocity at different spatial locations. The speed of spread, Indicates the current time ( A snapshot of the wave field at a given time. Represents the spatial coding state of the seismic source. Neural operator. The main structure consists of multiple consecutive Fourier neural operator blocks, and its core physics follows the following nonlinear iterative mapping formula:
[0043]
[0044] in, Indicates the first The output of the layer Fourier neural operator block It is a non-linear activation function. Linear transformation operators in the spatial domain are used for local residual learning of input features. and Let them represent the two-dimensional forward and inverse Fourier transforms in the spatial domain, respectively. It is a frequency domain parameterization operator used to learn the global wavefield mapping relationship in the frequency domain. These are the input features of the Fourier neural operator block in this layer.
[0045] Step S2 includes:
[0046] S201. Employ smoothing Loss function calculation predicts wavefield snapshot With truth labels The difference between them is defined by the loss function as:
[0047]
[0048] Among them, the smoothing threshold parameter Used to define the loss function in and The transition region between norms is used to balance the model's robustness to outliers. The closer the predicted value is to the true value, the smaller the loss function value. This is achieved by adjusting the model parameters through backpropagation. Perform iterative optimization;
[0049] S202. Introduce sampling probability values during model training. This value varies with the number of training rounds. The value decreases linearly with the increase of , and the calculation formula is:
[0050]
[0051] in, For the preset total number of training rounds, in each round of training, the model is probabilistically... Choose the true wave field from the previous time step as input, with probability By selecting the predicted wave field of the model at the previous time step as input, the training process can be controlled to gradually transition from supervised mode to autoregressive prediction mode, enabling the neural operator model to suppress the cumulative error of time evolution.
[0052] Step S3 includes:
[0053] S301. The formation velocity model and spatial information of earthquake source And the current Tron snapshot Input to the trained neural operator model In the process, obtain a snapshot of the predicted wave field at the next moment. The wave field at time zero is initialized to a zero matrix;
[0054] S302. Take a snapshot of the predicted wavefield generated in step S301. Recursion serves as the current time input for the next time step. Repeat the forward computation process of the neural operator until the iteration of the preset total time step is completed;
[0055] S303. The wavefield snapshots generated at each time point during the iteration process are stitched together in chronological order to synthesize a full-time sequence wavefield evolution data that reflects the propagation law of seismic waves under the unknown velocity model.
[0056] To verify the effectiveness and high fidelity of the method proposed in this invention, a rigorous comparative test was conducted between the proposed progressive autoregressive Fourier neural operator and two existing techniques (the traditional global FNO method that directly maps the velocity model to the full sequence wavefield, and the pure autoregressive FNO method without progressive sampling training). Figure 3 and Figure 4 As shown, the present invention and comparison method are demonstrated on OpenFWI datasets such as... Figure 3 and Figure 4 The image (h) shows a comparison of wavefield snapshots at time 0.6 s in forward modeling under the Style-A and Curve-A velocity models. This contrasts with the traditional global mapping FNO method. Figure 3 and Figure 4 (a) and (e), lacking the constraint guidance of wavefield propagation time causality, suffer from severe wavefront amplitude spread problems, resulting in low wavefield resolution and physical fidelity, and poor generalization; while the autoregressive FNO method without progressive sampling training... Figure 3 and Figure 4 Although methods (b) and (f) employ a time-recursive architecture consistent with the physical evolution of the wave field, the model encounters rapid and severe accumulation of recursive errors during long-sequence autoregressive inference without effective feedback control. At 0.6 s, strong noise and high-frequency artifacts erupt in the prediction results, causing the spatial structure of the effective wave field signal to completely diverge. The progressive autoregressive wave field simulation method proposed in this invention... Figure 3 and Figure 4 (c) and (g) enable the model to adaptively learn and predict the errors generated by the model, thereby maintaining the consistency of wave field iteration during autoregressive inference and achieving high-fidelity reconstruction that is highly consistent with the true value.
[0057] like Figure 5 and Figure 6 As shown, the traditional global FNO method, which directly maps the velocity model to the full-sequence wavefield, and the forward modeling results of the proposed progressive autoregressive Fourier neural operator for the Curve-A-02 velocity model are compared in the frequency and phase domains. The solid red line represents the true value, the dashed green line represents the global FNO method, and the dashed blue line represents the proposed method. During the 0.96s wavefield simulation, it can be observed that after 0.40s, with the increase in wavefield propagation depth and the increasing complexity of reflection characteristics, the difference between the traditional global FNO method and the true value fit widens significantly. However, the proposed method, throughout the entire long-sequence inference period, consistently shows a high degree of agreement between the blue dashed line and the red true value curve. This indicates that the progressive autoregressive mechanism and sampling strategy introduced in this invention successfully overcome the inherent spectral bias of traditional deep learning models in long-term wavefield prediction, and more accurately and stably preserves the dynamic and kinematic characteristics of seismic wave propagation in both the frequency and phase domains.
[0058] To further verify the effectiveness and generalization ability of the method of this invention under complex geological structures (such as the Marmousi velocity model), this embodiment extracts 230 local region samples of 200×200 pixels from a 767×270 two-dimensional Marmousi velocity model with a step size of 10 (210 for training and 20 for testing). Random left-right flipping is applied during training to augment the data. The grid spacing is set to... =10m, The earthquake was conducted at a depth of 10m, using a Ricker wavelet with a dominant frequency of 10Hz. The source coordinates were set at lateral distances of 500m, 1000m, and 1500m, and a depth of 10m. The total simulation time was set to 1.3s, with a time step of [missing information]. =0.001s. Under this parameter configuration, the forward simulation using the 8th-order finite difference method takes approximately 1.0s per run. For example... Figure 7 and Figure 8As shown, the forward modeling results of the proposed method on the velocity models (a) of two tests in the Marmousi data are presented respectively. The proposed progressive autoregressive Fourier neural operator has a simulation time of approximately 0.1 s, which is nearly 10 times more efficient. By comparing the network simulation results (c) with the ground truth generated by the traditional finite difference method (b), it can be clearly seen that the wavefront morphology of the wavefield snapshot generated by the proposed method is sharp and the energy distribution is continuous, accurately capturing the subtle reflection and scattering characteristics generated at the complex strata boundary. However, the error diagram (d) shows that there is still a certain gap in the absolute fitting of the amplitude. Nevertheless, this fully demonstrates that the proposed progressive autoregressive neural operator still has excellent cross-model generalization ability and high physical fidelity when facing unknown and structurally complex geological conditions, and achieves a significant improvement in forward modeling efficiency while well preserving the seismic wave dynamic characteristics.
[0059] The underlying mechanism of neural learning, which establishes mappings in an infinite-dimensional function space, allows the model trained in this invention to inherently possess resolution invariance, meaning it can perform inference directly at different grid resolutions without retraining. This flexibility, which completely eliminates the stringent spatiotemporal grid constraints of traditional numerical algorithms, coupled with high computational speed, provides a potentially powerful intelligent acceleration solution for future large-scale industrial-grade 3D seismic forward modeling and full-waveform inversion.
[0060] The foregoing description illustrates and describes a preferred embodiment of the present invention. However, as previously stated, it should be understood that the present invention is not limited to the forms disclosed herein and should not be construed as excluding other embodiments. It can be used in various other combinations, modifications, and environments, and can be altered within the scope of the inventive concept described herein through the foregoing teachings or techniques or knowledge in related fields. Any modifications and variations made by those skilled in the art that do not depart from the spirit and scope of the present invention should be within the protection scope of the appended claims.
Claims
1. A wavefield simulation method based on progressive autoregressive Fourier neural operators, characterized by: Includes the following steps: S1. Construct a neural operator model; S2. A progressive sampling strategy is used to train the model, controlling the transition from ground truth supervision to autoregressive prediction; S3. By using the trained model and recursively feeding the current output back as subsequent input, a rapid inference of the full-time series wavefield snapshot under an unknown velocity model can be achieved.
2. The wavefield simulation method based on a progressive autoregressive Fourier neural operator according to claim 1, characterized in that: The neural operator parameters contained in the neural operator model constructed in step S1 are , the neural operator is , and the mapping relationship is: ； where, represents the model-predicted wavefield snapshot at time The input of the neural operator represents the input velocity model of the subsurface, which is the propagation velocity of seismic waves at different spatial locations, represents the wavefield snapshot at the current time, i.e. represents the source spatial encoding state; the neural operator The main structure is composed of multiple continuous Fourier neural operator blocks, and the core physics follows the following nonlinear iterative mapping formula: ； in, Indicates the first The output of the layer Fourier neural operator block It is a non-linear activation function. Linear transformation operators in the spatial domain are used for local residual learning of input features. and Let them represent the two-dimensional forward and inverse Fourier transforms in the spatial domain, respectively. It is a frequency domain parameterization operator used to learn global wavefield mapping relationships in the frequency domain. These are the input features of the Fourier neural operator block in this layer.
3. The wavefield simulation method based on a progressive autoregressive Fourier neural operator according to claim 2, characterized in that: Step S2 includes: S201. Employ smoothing Loss function calculation predicts wavefield snapshot With truth labels The difference between them is defined by the loss function as: ； Among them, the smoothing threshold parameter Used to define the loss function in and The transition region between norms is used to balance the model's robustness to outliers; the closer the predicted value is to the true value, the smaller the loss function value, and the model parameters are adjusted through backpropagation. Perform iterative optimization; S202. Introduce sampling probability values during model training. This value varies with the number of training rounds. The value decreases linearly with the increase of , and the calculation formula is: ； in, For the preset total number of training rounds, in each round of training, the model is probabilistically... Choose the true wave field from the previous time step as input, with probability Choose the predicted wave field from the model at the previous time step as input.
4. The wavefield simulation method based on a progressive autoregressive Fourier neural operator according to claim 3, characterized in that: Step S3 includes: S301. The formation velocity model and spatial information of earthquake source And the current Tron snapshot Input to the trained neural operator model In the process, obtain a snapshot of the predicted wave field at the next moment. The wave field at time zero is initialized to a zero matrix; S302. Take a snapshot of the predicted wavefield generated in step S301. Recursion serves as the current time step input for the next time step. Repeat the forward computation process of the neural operator until the iteration of the preset total time step is completed; S303. The wavefield snapshots generated at each time point during the iteration process are stitched together in chronological order to synthesize a full-time sequence wavefield evolution data that reflects the propagation law of seismic waves under the unknown velocity model.