Physically enhanced fluid dynamics small sample solution method

Abstract

The application discloses a physical enhancement-based fluid dynamics small sample solving method, including: acquiring sparse fluid observation data in a complex geometric calculation domain, and mapping the observation data to a statistical average flow space described by a Reynolds-averaged Navier-Stokes equation; constructing a neural network model for predicting an average velocity field, an average pressure field and a spatially continuous and learnable eddy viscosity coefficient latent field; when calculating spatial derivatives in the Reynolds-averaged Navier-Stokes equation, a differentiable spectral operator module is used to replace a traditional automatic differentiation mechanism; based on geometric distance information of each spatial point in the calculation domain to a solid boundary, a geometry-aware mask is constructed; the mask is used to distinguish the calculation domain into a near-wall shear dominated region and a main flow region; a joint loss function containing a data term and a physical residual term is constructed, and the physical residual term is preconditioned; and a prediction result is output; the application realizes numerical stability and high prediction accuracy.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of scientific machine learning and computational fluid dynamics, and in particular to a physical enhancement-based method for solving fluid dynamics with small sample sizes. Background Technology

[0002] In engineering applications such as seawater desalination reverse osmosis systems, heat exchanger channels, aerospace aerodynamic layouts, and complex piping systems, fluid transport behavior and overall performance are often significantly influenced by factors such as inlet baffles, channel structures, membrane element arrangements, or complex geometries. Therefore, during the design and optimization of engineering systems, it is necessary to conduct high-precision evaluations of fluid flow characteristics, mass transfer processes, or momentum transfer. Traditionally, this is achieved by relying on high-fidelity computational fluid dynamics (CFD) simulations.

[0003] However, these engineering systems often possess complex three-dimensional geometries and involve numerous parametric design variables, such as mesh shape, channel dimensions, membrane element arrangement, and operating conditions. When performing systemic simulation analysis in such a high-dimensional parameter space, traditional mesh-based CFD methods require extensive repetitive calculations, resulting in extremely high computational costs. Especially under complex industrial geometries, high-fidelity simulations not only require long computation times but also demand high-performance computing resources, making the amount of high-quality flow field or mass transfer data extremely limited. Taking seawater desalination reverse osmosis membrane modules as an example, their channel and mesh structures are complex and diverse. Traditional CFD methods struggle to obtain sufficient flow information under multi-parameter design conditions within a short time, thus hindering the efficiency of engineering optimization design and rapid iteration. Therefore, how to achieve high-fidelity prediction and rapid evaluation of complex industrial flow systems under sparse observation or limited data conditions has become a critical technical problem that urgently needs to be solved in current engineering design.

[0004] In recent years, Physics-Informed Neural Networks (PINNs) have provided a new technical approach for constructing efficient physical surrogate models by directly embedding the partial differential equations controlling fluid motion into the loss function of the neural network. This allows the neural network to simultaneously satisfy data constraints and physical constraints during training. Therefore, this method is considered to have potential application value in the rapid prediction and design optimization of complex engineering systems.

[0005] However, in practical engineering flow problems, especially in convection-dominated flow states such as high Reynolds number turbulent flow, the training process of physical information neural networks still faces significant challenges. On the one hand, the data available in industrial scenarios is usually extremely sparse, making it difficult to provide sufficient supervision information for the model; on the other hand, the Navier-Stokes equations themselves have strong nonlinear and multi-scale characteristics, making it easy for physical surrogate models to encounter optimization difficulties when learning complex flow structures.

[0006] Existing physical information neural networks typically rely on automatic differentiation techniques to calculate the spatial derivatives of partial differential equations. However, in regions with strong gradient changes or multi-scale structures, such as near-wall boundary layers, the differential operators corresponding to automatic differentiation may amplify high-frequency approximation errors, introducing significant gradient noise during training. In sparse data scenarios, these errors are difficult to suppress effectively through data constraints, leading to model instability during optimization and significantly reducing the analytical capability for key physical structures (such as thin boundary layers).

[0007] Furthermore, under turbulent flow conditions, the transient Navier-Stokes equations describe a flow system with a high degree of freedom. Since sparse observational data cannot adequately constrain small-scale turbulent structures and their evolution, directly reconstructing the complete transient flow state mathematically using only a very small amount of observational data is often an ill-posed problem. Therefore, in practical engineering modeling, the Reynolds-Averaged Navier-Stokes (RANS) equations are typically used to statistically average the flow, reducing the system's degrees of freedom and obtaining a more stable flow description.

[0008] On the other hand, while operator learning methods proposed in recent years (such as neural operator models) can learn partial differential equation solution operators to some extent, they usually rely on a large amount of high-quality training data to construct operator mappings. In complex geometric scenarios such as industrial membrane systems, the high cost of acquiring high-fidelity simulation data makes it difficult to construct sufficiently large datasets, thus limiting the applicability of such methods under extremely small sample conditions. Summary of the Invention

[0009] Purpose of the Invention: The purpose of this invention is to provide a physics-enhanced fluid dynamics method for solving small-sample problems, addressing the difficulty of high-fidelity flow field reconstruction in complex flow systems under conditions of extreme data scarcity. In complex industrial flow systems such as seawater desalination membrane modules, traditional computational fluid dynamics methods typically require a large amount of high-precision numerical simulation data. However, the observational data available in actual engineering environments is often extremely limited, leading to significant ill-posedness in fluid flow field reconstruction based on sparse data. Furthermore, existing physical information neural networks are prone to gradient instability, optimization stagnation, and high-frequency noise amplification when handling high Reynolds number convection-dominated flows, thus affecting the model's prediction accuracy and training stability.

[0010] Technical solution: The present invention provides a physical enhancement-based method for solving small-sample fluid dynamics problems, comprising the following steps:

[0011] (1) Obtain sparse fluid observation data in complex geometric computational domains and map the observation data to the statistical average flow space described by the Reynolds-averaged Navier-Stokes equations to reduce the solution degree of freedom of the fluid dynamics system;

[0012] (2) Construct a neural network model to simultaneously predict the mean velocity field, mean pressure field and a spatially continuous and learnable eddy viscosity coefficient latent field in each forward calculation; the eddy viscosity coefficient latent field is used to characterize the turbulent Reynolds stress that is difficult to analyze directly under sparse observation conditions, thereby achieving the closure of the Reynolds-averaged Navier-Stokes equations.

[0013] (3) When calculating the spatial derivative in the Reynolds-averaged Navier-Stokes equations, the differentiable spectrum operator module is used instead of the traditional automatic differentiation mechanism. The differentiable spectrum operator module transforms the physical field predicted by the neural network to the frequency space, solves the spatial derivative in the frequency space using algebraic relations, and then transforms the result back to the physical space to eliminate the gradient amplification problem caused by the recursive chain differentiation.

[0014] (4) Based on the geometric distance information from each spatial point in the computational domain to the solid boundary, construct a geometric sensing mask; use the mask to divide the computational domain into a near-wall shear-dominated region and a mainstream region, and actively apply spectral spatial damping in the mainstream region to suppress high-frequency numerical oscillations, while maintaining the original physical gradient structure in the near-wall region.

[0015] (5) Construct a joint loss function that includes data terms and physical residual terms, and precondition the physical residual terms to improve the condition number of the optimization problem; update the neural network parameters through iterative optimization, and finally output a high-fidelity flow field prediction result that satisfies both the physical equation constraints and the consistency of the observed data.

[0016] Furthermore, in step 3, the fluid physical field is mapped from the physical space to the frequency space through the fast Fourier transform, and the spatial derivative is calculated in the frequency space through the frequency multiplier. Then, the result is mapped back to the physical space through the inverse fast Fourier transform to realize the spectral calculation of the spatial derivative.

[0017] Furthermore, when solving for the spatial derivative, a boundary treatment method based on smooth Brinkman penalty is introduced. This method implicitly represents the existence of the solid region by defining a continuous penalty term within the computational domain, thereby reducing the spectral truncation error caused by the discontinuity of the physical interface and improving the computational stability of the spectral method at complex geometric boundaries.

[0018] Furthermore, in step 4, a geometrically aware mask is constructed using a signed distance function, where the value of the signed distance function represents the signed distance from any location within the computational domain to the nearest solid boundary, thereby achieving accurate and automated division between the near-wall region and the mainstream region.

[0019] Furthermore, in step 5, when calculating the nonlinear convection term residual in the joint loss function, an anti-aliasing mechanism based on the three-half rule is introduced. This involves performing an additional filling and truncation operation in the differentiable spectrum operator module to actively filter out high-frequency aliasing errors caused by nonlinear spectrum calculation, thereby improving the calculation accuracy of the physical residual.

[0020] Furthermore, in step 5, a dynamic spectral residual weighting mechanism is introduced: during the training process, the weights of the residuals of different frequency components in the joint loss function are dynamically adjusted according to their magnitudes, so as to balance the learning ability of the neural network to physical features at different scales and improve the convergence and stability of model training.

[0021] The present invention discloses a physical enhancement-based fluid dynamics small-sample solution system, comprising:

[0022] Physical dimensionality reduction module: used to acquire sparse observation data and map it to the statistically averaged flow space described by the Reynolds-averaged Navier-Stokes equations;

[0023] Learnable Closure Module: Contains a neural network model for simultaneously predicting the mean velocity field, mean pressure field, and spatially continuous learnable eddy viscosity coefficient latent field to achieve adaptive parameterized closure for turbulent stress.

[0024] Spectral Differentiation Module: Combined with neural network models, it is used to calculate the physical field spatial derivative in frequency space through algebraic operations, replacing automatic differentiation to provide stable physical residual calculation;

[0025] Geometric regularization module: used to construct geometrically aware masks based on the symbolic distance function and to implement differentiated spectral regularization strategies for the mainstream region and the near-wall region;

[0026] Training and optimization module: Used to construct and solve the pre-conditioned joint loss function, and optionally perform dynamic spectral residual weighting to drive parameter optimization of the entire system.

[0027] The present invention provides a storage medium storing a computer program, wherein the computer program is configured to execute the above-described method at runtime.

[0028] The apparatus of the present invention includes a memory and a processor, wherein the memory stores a computer program and the processor is configured to run the computer program to perform the above-described method.

[0029] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages:

[0030] (1) By mapping unsteady observation data to a steady-state average flow model, the degrees of freedom of the fluid dynamics system are effectively reduced, and the identifiability of the inverse physics solution problem under extremely sparse data conditions is improved.

[0031] (2) By introducing a learnable eddy viscosity latent field closure module, the turbulent Reynolds stress that is difficult to analyze under small sample conditions is parameterized, thereby achieving stable closure of the Reynolds-averaged equation;

[0032] (3) By constructing a differentiable spectrum operator calculation module to replace the traditional automatic differentiation mechanism, the gradient amplification problem caused by chain differentiation is effectively avoided, and the stability of partial differential operator calculation is improved.

[0033] (4) By introducing geometric perception mask and local spectrum regularization strategy, high-frequency noise in the mainstream region is suppressed while maintaining the physical gradient structure in the near-wall region, thereby improving the stability of the neural network training process;

[0034] (5) Even with very little observation data, the present invention can still achieve high-precision flow field prediction for complex industrial flow systems, and has good prospects for engineering applications. Attached Figure Description

[0035] Figure 1 This is a flowchart of the present invention;

[0036] Figure 2 This is a schematic diagram of the fluid computation domain and observation point distribution of the present invention;

[0037] Figure 3 This is a schematic diagram of the calculation mechanism for the derivative of the differentiable spectral operator of the present invention;

[0038] Figure 4 This is a schematic diagram of the learnable eddy viscosity latent field closure mechanism of the present invention;

[0039] Figure 5 This is a schematic diagram of the geometric sensing mask and local spectral regularization of the present invention;

[0040] Figure 6 This is a flowchart of the physical constraint training optimization process of the present invention. Detailed Implementation

[0041] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0042] like Figure 1 As shown, this embodiment of the invention provides a physical enhancement-based method for solving fluid dynamics with small sample sizes. In order to achieve stable reconstruction of complex flow fields under extremely sparse data conditions, the method specifically includes the following detailed steps: 1. Construct a computational domain containing complex geometric boundaries and obtain observation data in an extremely sparse state;

[0043] A mean flow physical model is established based on the Reynolds-averaged Navier-Stokes (RANS) equations to map transient flows to steady-state manifolds;

[0044] 2. Introduce a continuous latent field representation and construct a learnable eddy viscosity latent field closed module to meet the requirements of the physically permissible solution space;

[0045] 3. By combining Local Adaptive Spectral Projection (LASP), a bounded differentiable spectral operator derivative calculation module is constructed to replace the traditional automatic differentiation;

[0046] 4. Based on the symbolic distance field, construct a geometric sensing mask and local spectral regularization module to realize dual-domain spectral preconditioning;

[0047] 5. Construct a physical constraint loss function and use a dynamic spectral residual reweighting strategy for neural network training.

[0048] The specific process is as follows:

[0049] Step 1: Construct a computational domain with complex geometric boundaries and obtain extremely sparse observation data.

[0050] like Figure 2 As shown, this embodiment first establishes a three-dimensional fluid computation domain for the feed water mesh channel in the seawater desalination reverse osmosis membrane module. This computational domain not only includes the internal region where the fluid flows freely, but also explicitly includes the complex aperiodic solid boundary structure composed of cross-braced filaments and upper and lower membrane surfaces. Any point in space within the computational domain is precisely represented by a coordinate vector:

[0051]

[0052] Where: x represents the direction coordinate of the fluid's main convective transport; y represents the lateral coordinate within the flow channel; z represents the wall normal coordinate perpendicular to the membrane surface.

[0053] To train the physical proxy model, a highly sparse set of observation points needs to be obtained within this computational domain, mathematically represented as:

[0054]

[0055] Where: N represents the total number of observation data points actually acquired. In this embodiment, which is designed for industrial scenarios with extremely scarce data, the value of N is very small, for example, only 80 or only covering about 8% of the degrees of freedom of the global computational domain; i is the observation point index, with a value range from 1 to N. This represents the three-dimensional spatial coordinate position of the i-th specific observation point. Representative at The three-dimensional average velocity vector observations obtained from the location. Representative at Average kinematic pressure observations obtained from the location.

[0056] In actual industrial deployments, it directly corresponds to the real-time physical measurement data collected by distributed micro-flow velocity sensors and differential pressure sensors inside the membrane module flow channel; in the engineering design and verification stage, it comes from the extremely sparse sampling snapshots extracted at specific spatial locations by a high-fidelity computational fluid dynamics simulation system.

[0057] These real or high-precision sparse observation data are directly input into the neural network as initial data constraints, which, together with the physical residuals of the subsequently constructed fluid dynamics partial differential equations, constrain the network's parameter space. Since the number of sensors that can be deployed or the number of high-precision snapshot points N that can be acquired in practical engineering applications is far less than the mesh degrees of freedom required to fully analyze complex multi-scale turbulent flows, directly using this extremely limited amount of discrete data to reconstruct continuous flow fields is mathematically a highly ill-posed inverse problem, highly susceptible to high-frequency noise interference that can induce non-physical oscillations. Therefore, it is necessary to introduce an average flow physical constraint model with dimensionality reduction and stability characteristics.

[0058] Step 2: Establish an average flow physical model and perform data mapping.

[0059] In traditional fluid dynamics, fluid motion can be described by the incompressible Navier-Stokes equations:

[0060]

[0061] in: Transient velocity field vector; containing average velocity components (u, v, w) in three directions; For transient kinematic pressure fields; t is the kinematic viscosity coefficient of the fluid; t is the time variable; For spatial gradient operators; For the Laplace operator.

[0062] Under extremely sparse observation conditions, directly reconstructing the transient flow field containing high-frequency turbulent fluctuations is mathematically a highly ill-posed problem. This is because the extremely limited number of observation points in engineering practice cannot capture complete small-scale transient fluid fluctuations, and forcing a physical surrogate model to fit these transient equations leads to severe instability and non-physical numerical oscillations. To address this issue, this embodiment employs the Reynolds-averaged method to perform time statistical averaging on the aforementioned Navier-Stokes equations, thereby obtaining the steady-state Reynolds-averaged Navier-Stokes equations:

[0063]

[0064] in: This is the time-averaged velocity field vector; The time-averaged pressure field; For the transient pulsating component of velocity; satisfying And its time average is zero; The term is the Reynolds stress tensor term caused by the fluctuating velocity component, representing the momentum transfer effect of turbulent fluctuations on the mean flow.

[0065] This time-based statistical averaging process transforms the original transient multi-scale flow problem with extremely high degrees of freedom into a statistical steady-state problem, restricting the flow field variables to a physically permissible steady-state average manifold. This not only effectively filters out high-frequency transient fluctuations that cannot be identified through sparse observations, significantly reducing the solution degrees of freedom of the fluid dynamics system, but also ensures that sparse observation data remains consistent with the steady-state average flow equations, thereby greatly improving the stability and identifiability of small-sample flow field reconstruction.

[0066] Step 3: Construct a learnable eddy viscosity latent field closed module.

[0067] While the transient Navier-Stokes equations, after Reynolds time statistical averaging, effectively filter out difficult-to-identify high-frequency transient fluctuations, the averaging of the nonlinear convection term introduces a nonlinear term caused by fluctuating velocity components, namely the Reynolds stress tensor term. This term represents the momentum transfer effect of transient turbulent fluctuations on the steady-state average flow. Because it includes the time average of the unknown fluctuating velocity product, the number of unknown variables in the established Reynolds-averaged equations exceeds the number of equations, making direct closure solution impossible. Therefore, a turbulence closure model must be introduced.

[0068] This embodiment uses the eddy viscosity assumption to parameterize the Reynolds stress, transforming the unknown Reynolds stress tensor into a product of the effective eddy viscosity coefficient and the average strain rate tensor:

[0069]

[0070] in: and These are transient pulsating velocity components in different directions; and These are time-averaged velocity components in different directions; The latent field for the turbulent eddy viscosity coefficient needs to be solved.

[0071] To avoid the excessive empirical assumptions and model errors introduced by traditional empirical turbulence models when dealing with complex industrial flow channel geometries, this invention constructs a physical information deep feedforward neural network to achieve adaptive latent field closure. The specific model prediction process is as follows:

[0072] The spatial coordinates within the computational domain are used as input to this deep feedforward neural network. During each forward propagation calculation, the network not only predicts the conventional flow field but also simultaneously outputs multiple consecutive spatial physics fields:

[0073]

[0074] in: Representative parameterization The neural network surrogate model; x, y, z are the spatial coordinates of the input; , , The average velocity components of the fluid in three spatial directions (flow direction, lateral direction, and wall normal direction) are predicted by the neural network. The average pressure field predicted by the neural network; The potential field for spatially continuous and learnable eddy viscosity coefficients predicted by a neural network.

[0075] Furthermore, to ensure the mathematical enforceability of the viscous diffusion operator and the well-posedness of the solution to the governing equations, the eddy viscosity coefficient latent field must satisfy the physical property of non-negativity. Therefore, in this embodiment, the neural network corresponds to... At the output channel, a strictly positive activation function mapping (e.g., Softplus mapping) is applied to ensure the effective kinematic viscosity of the fluid. ( (where is the inherent kinematic viscosity coefficient of the fluid itself). This characteristic fundamentally avoids numerical divergence caused by non-physical "negative viscosity," ensuring that the flow field solution predicted by the network always lies within the physically permissible steady-state RANS manifold space.

[0076] Step 4: Construct the derivative calculation mechanism of the local adaptive spectral projection operator.

[0077] In traditional physical information neural networks, partial differential operators are typically computed using automatic differentiation mechanisms. However, in flow problems dominated by high Reynolds number convection, automatic differentiation is essentially equivalent to performing unbounded recursive chain differentiation in the learned function space. This mechanism is highly susceptible to amplifying high-frequency approximation errors uncontrollably, leading to severe gradient noise and training instability.

[0078] To fundamentally eliminate this problem, this embodiment abandons the automatic differentiation mechanism and the global Fourier transform that cannot adapt to complex non-periodic boundaries, and innovatively adopts the local adaptive spectral projection operator to calculate the spatial derivative.

[0079] First, the entire 3D computational domain containing complex obstacles is divided into multiple non-overlapping local geometric sub-regions:

[0080]

[0081] in: M represents the global computational domain of the fluid; D represents the total number of local geometric sub-regions. m Let m be the m-th local geometric sub-region.

[0082] Within each local sub-region, the local spectral substrate is adaptively selected based on the ratio of the shortest distance from that region to the nearest solid boundary to the local boundary layer thickness. Specifically, in the algorithm implementation, the following is defined: For subregion D m The shortest distance to the nearest solid boundary. The thickness is the characteristic boundary layer. The substrate selection strategy strictly follows the following criterion:

[0083] when When (C is a constant, usually on the order of O(1)), it is determined to be a viscous sub-region near the solid wall and affected by strong shear. Chebyshev polynomial basis is used to stabilize the non-periodic physical boundary.

[0084] when If the region is determined to be a smooth mainstream region far from the wall, then Fourier or wavelet basis is used.

[0085] Next, the continuous average velocity field predicted by the neural network is projected onto the truncated local spectral space, and the local modal coefficients are calculated:

[0086]

[0087] in: The j-th local modal coefficient is located within the m-th sub-region. The continuous average velocity field vector containing three directional components is predicted by the neural network. Let j be the local spectral basis function selected in the m-th subregion; Indicates in sub-region The inner product operation is performed by taking the product of numerical points.

[0088] To filter out unreasonable high-frequency components and prevent aliasing, the local substrate truncation order is adjusted. Local modal energy decay tolerance Adaptive determination. Define the energy of the j-th mode. truncation order Take the minimum value that satisfies the following formula:

[0089]

[0090] This truncation process decouples the network's representational capability from the frequency of derivative calculation, ensuring the boundedness of the derivative.

[0091] Within this local finite-dimensional spectral space, the calculation of the spatial derivative is transformed into an algebraic multiplication operation (or spectral differential matrix operation) with respect to the modal coefficients:

[0092]

[0093] in: For in sub-region The spatial analytic derivative of the velocity field obtained from internal calculations; The spatial derivatives of the basis functions are obtained through direct mapping using a pre-calculated algebraic differential matrix.

[0094] Through this local algebraic mapping mechanism, partial differential operators are transformed into bounded linear operations in a finite-dimensional space. Their computational accuracy is completely decoupled from the depth of the neural network, fundamentally avoiding the gradient amplification phenomenon caused by deep chain differentiation.

[0095] In particular, when calculating the nonlinear convection terms in the fluid momentum equation, the direct multiplication of physical field variables can cause frequency doubling, leading to severe high-frequency spectral aliasing. To address this issue, this embodiment introduces an aliasing mechanism based on the three-half rule in the differentiable spectrum operator calculation module, combined with local Galerkin projection:

[0096]

[0097] in: The nonlinear convection term residuals after removing aliasing errors; This represents the physical space multiplication and spectral space truncation operators performed on the expanded local configuration point grid. Before computation, this dealiasing mechanism first expands the discrete spectrum of each sub-region to 1.5 times the original resolution by frequency zero-padding, then performs an inverse transformation to the local physical space for point-by-point multiplication, and finally transforms back to the spectral space and truncates to the original highest frequency. This physical-level spectral filtering operation strictly maintains the orthogonality of the basis modes and completely eliminates spurious high-frequency aliasing errors caused by nonlinear interactions.

[0098] Finally, based on the aforementioned locally bounded derivatives and dealiasing convection terms, in each subregion D... m The internal assembly yields discretized and physically consistent local residuals of the steady-state RANS equations. :

[0099]

[0100] The local residual This was then used to construct a global physical constraint loss function.

[0101] Step 5: Construct a geometry-aware mask and a dual-domain local spectral regularization module.

[0102] Even with the use of local spectral projection operators, under extremely sparse observation conditions, due to the inherent spectral bias of neural networks, non-physical high-frequency numerical oscillations can still be easily induced in the mainstream region where data constraints are weak. However, near the solid boundary (i.e., the viscous sublayer), high wavenumber components correspond to the real physical shear stresses that transfer fluid momentum. If a globally uniform regularized damping is used, the physical gradients necessary near the wall will be incorrectly smoothed out.

[0103] To resolve this spatial scale contradiction, this embodiment extracts the geometric features of the computational domain and constructs a geometrically perceptual mask based on the signed distance function for each local sub-region:

[0104]

[0105] Where: M m (x) is the geometry-aware mask function in the m-th local sub-region, whose value approaches 1 near the wall and approaches 0 in the mainstream region far from the wall; x is the spatial coordinate point in the computation domain; Let be the signed distance function from a point in space to the nearest solid boundary. This is an adaptive control parameter whose value is positively correlated with the local boundary layer thickness scale, used to control the attenuation range of the mask. To achieve precise adaptation of geometric and physical features, The specific value depends on the local boundary layer thickness. The physical scale mapping is performed, and the specific calculation logic is as follows: First, based on the characteristic length L and Reynolds number Re of the fluid system, the characteristic thickness of the local boundary layer is estimated. and set the initial mapping relationship. (in (This is a constant scaling factor). Secondly, to ensure the spatial smoothness of the mask function is compatible with the truncated local spectral basis in step 4, and to prevent additional spatial discretization errors caused by excessive mask attenuation, this embodiment mandates that the adaptive parameter satisfy the lower bound constraint:

[0106]

[0107] in, This represents the maximum truncation order of the local spectral base within this subregion.

[0108] Using the mask function with strict physical and numerical constraints, a physical soft partitioning of the near-wall and mainstream regions of the computational domain was achieved, and a dual-domain local spectral regularization strategy was implemented accordingly: for the mainstream region, 1-M... m (x) Extract the physical residuals of this region and apply local Sobolev spectral spatial damping to forcefully penalize and attenuate spurious high-frequency numerical oscillations caused by data sparsity. For the near-wall region, use M... m (x) Extract the residuals of this region, preserving its complete physical high-frequency modes without imposing additional frequency attenuation, thus perfectly maintaining the true physical shear gradient necessary for momentum transfer. This geometry-aware dual-domain regularization mechanism, as a spatial projection strategy, can accurately filter out non-physical numerical noise while maintaining high-fidelity analysis of the boundary layer, providing excellent numerical stability for subsequent loss function optimization.

[0109] Step 6: Construct a joint loss function that incorporates spectral preconditioning and drive neural network parameter optimization.

[0110] This embodiment constructs the following joint loss function:

[0111]

[0112] in: The total joint loss function is derived from the data loss. and physical loss constitute; and These are adaptive weighting coefficients for the data loss term and the physical residual loss term, respectively, used to balance data fitting and physical constraints. In the specific engineering deployment of this embodiment, the initial... and The weights are determined during the pre-training phase using a Bayesian optimization algorithm. During training, a dynamic weight balancing strategy based on neural tangent kernel eigenvalue tracking or gradient statistics is adopted. The weights are scaled and updated in real time according to the relative magnitudes of the two gradients during backpropagation to avoid the gradient of the physical loss term dominating the optimization direction in the early stages of training.

[0113] The data loss term is used to quantify the error between the neural network prediction result and the sparse observation data, and the calculation formula is as follows:

[0114]

[0115] Where: N is the total number of sparse observation points; x i Let i be the three-dimensional spatial coordinates of the i-th observation point; For neural networks in x i Predicted fluid physical quantities; For x i The actual fluid physical quantity observations obtained at the location.

[0116] The physical residual loss term combines a local spectral projection mechanism with a two-domain geometrically perceptual mask, and is computed in a truncated local spectral space:

[0117]

[0118] in: R is the physical residual loss at the t-th training iteration; m For the m-th local sub-region, the Reynolds-averaged Navier-Stokes equations combined with the eddy viscosity latent field are the physical residuals.

[0119] Furthermore, the acquisition and update mechanism of the two key spectral weight parameters in formula (13) is as follows:

[0120] This is the local Sobolev penalty coefficient applied to a specific high-frequency mode in the mainstream region. This coefficient is not an empirical constant, but rather rigorously determined based on the modal eigenvalues, and its specific calculation formula is as follows: ,in Let j be the analytical eigenvalue corresponding to the j-th basis mode within the local sub-region. To adjust the scalar, this formula ensures that the penalty received by the high-frequency mode increases non-linearly.

[0121] Let be the spectral residual weighted tensor dynamically updated for local modes at the t-th iteration. Its update mechanism is as follows: it is used as a learnable adaptive penalty parameter, based on the local mode residual energy from the previous iteration. The moving average is updated in real time (the frequency component with the larger error is assigned a larger weighted tensor), thereby adaptively amplifying the loss contribution of the underanalytical mode in the frequency domain and guiding the network to converge quickly.

[0122] By minimizing this joint loss function, the parameters of the neural network are continuously updated using a gradient descent optimization algorithm (such as a combination of Adam and L-BFGS optimizers).

[0123]

[0124] The process for determining the above three optimization parameters is as follows:

[0125] Let be the set of weights and bias parameters of the neural network surrogate model at the t-th iteration. Before the iteration begins (t=0). Random initialization can be performed using the Xavier or Kaiming initialization strategy;

[0126] The learning rate is set by the optimizer. In actual training, the base learning rate is set a priori and equipped with an exponentially decaying or cosine annealing learning rate scheduler, which dynamically decreases as the number of iterations t increases. The value of is determined to ensure fine convergence in the later stages of training;

[0127] This represents the gradient of the partial derivatives of the joint loss function with respect to the network parameters. Since the spatial differentiation mentioned above has already been algebraically calculated by the local spectral projection operator, only the gradient propagation in the parameter space is retained here. This gradient is precisely obtained by the automatic differentiation backend of the deep learning framework using the standard backpropagation algorithm.

[0128] By iterating continuously until the loss function converges, the neural network can finally output a high-fidelity flow field prediction that simultaneously satisfies the strict constraints of the fluid dynamics equations and is highly consistent with the extremely sparse observation data. Throughout the optimization and iteration process of the loss function, this invention introduces a crucial dynamic spectral residual preconditioning mechanism. By monitoring the residual energy distribution of different frequency components in real time, the spectral loss weight tensor is adaptively and dynamically adjusted. In the frequency domain, a geometrically weighted similarity transformation is performed on the induced neural tangent kernel. This achieves a powerful bidirectional compression of the eigenvalue spectrum of the neural tangent kernel at the algorithm's underlying level: on the one hand, dynamic weights suppress the largest eigenvalues ​​that trigger gradient explosion; on the other hand, the aforementioned geometrically perceptual mask applies a penalty to the mainstream region to clamp the smallest eigenvalues ​​that cause optimization stagnation. This synergistic effect reduces the condition number of the optimization problem from the traditionally extremely ill-conditioned order O(10^2 / 20^2). 9 The O(10) level was significantly reduced to a healthy state. 2This approach fundamentally and thoroughly suppresses the gradient ill-conditioning and optimization stagnation phenomena of existing physical information neural networks when dealing with high Reynolds number convection-dominated problems.

[0129] Through the coordinated execution of the above six steps, this invention can overcome the numerical computation bottleneck of existing technologies under extremely sparse observation conditions, achieving high-precision and high-stability reconstruction of complex industrial multi-scale flow systems, while strictly ensuring the physical consistency and mass conservation of the predicted flow field. This method can be widely applied in cutting-edge fields such as flow channel optimization for seawater desalination membrane modules, aerospace heat exchanger design, engineering turbulence proxy modeling, and data-driven digital twins of industrial flow fields, possessing extremely high engineering practical value.

Claims

1. A physical enhancement-based method for solving small-sample fluid dynamics problems, characterized in that, Includes the following steps: (1) Obtain sparse fluid observation data in complex geometric computational domains and map the observation data to the statistical average flow space described by the Reynolds-averaged Navier-Stokes equations to reduce the solution degree of freedom of the fluid dynamics system; (2) Construct a neural network model to simultaneously predict the mean velocity field, mean pressure field and a spatially continuous and learnable eddy viscosity coefficient latent field in each forward calculation; the eddy viscosity coefficient latent field is used to characterize the turbulent Reynolds stress that is difficult to analyze directly under sparse observation conditions, and to achieve the closure of the Reynolds-averaged Navier-Stokes equations. (3) When calculating the spatial derivative in the Reynolds-averaged Navier-Stokes equations, the differentiable spectrum operator module is used instead of the traditional automatic differentiation mechanism. The differentiable spectrum operator module transforms the physical field predicted by the neural network to the frequency space, solves the spatial derivative in the frequency space using algebraic relations, and then transforms the result back to the physical space to eliminate the gradient amplification problem caused by the recursive chain differentiation. (4) Construct a geometrically perceptual mask based on the geometric distance information from each spatial point in the computational domain to the solid boundary; The computational domain is divided into near-wall shear-dominated region and mainstream region by using a mask. Spectral spatial damping is actively applied in the mainstream region to suppress high-frequency numerical oscillations, while the original physical gradient structure is maintained in the near-wall region. (5) Construct a joint loss function that includes data terms and physical residual terms, and precondition the physical residual terms to improve the condition number of the optimization problem; update the neural network parameters through iterative optimization, and finally output a high-fidelity flow field prediction result that satisfies both the physical equation constraints and the consistency of the observed data.

2. The method for solving small-sample fluid dynamics problems based on physical enhancement according to claim 1, characterized in that, In step 3, the fluid physical field is mapped from the physical space to the frequency space through the fast Fourier transform, and the spatial derivative is calculated in the frequency space through the frequency multiplier. Then, the result is mapped back to the physical space through the inverse fast Fourier transform to realize the spectral calculation of the spatial derivative.

3. The method for solving small-sample fluid dynamics problems based on physical enhancement according to claim 2, characterized in that, In step 3, when solving for the spatial derivative, a boundary treatment method based on smooth Brinkman penalty is introduced, that is, the existence of the solid region is implicitly represented by defining a continuous penalty term within the computational domain.

4. The method for solving small-sample fluid dynamics problems based on physical enhancement according to claim 1, characterized in that, In step 4, a geometrically aware mask is constructed using a signed distance function, where the value of the signed distance function represents the signed distance from any location in the computational domain to the nearest solid boundary, thereby achieving accurate and automated division of the near-wall region and the mainstream region.

5. The method for solving small-sample fluid dynamics problems based on physical enhancement according to claim 1, characterized in that, In step 5, when calculating the residual of the nonlinear convection term in the joint loss function, an anti-aliasing mechanism based on the three-half rule is introduced. This involves performing an additional filling and truncation operation in the differentiable spectrum operator module to actively filter out high-frequency aliasing errors caused by nonlinear spectrum calculation.

6. The physical enhancement-based fluid dynamics small-sample solution method according to claim 5, characterized in that, In step 5, a dynamic spectral residual weighting mechanism is introduced: during the training process, the weights of different frequency components in the joint loss function are dynamically adjusted according to the magnitude of their residuals, so as to balance the learning ability of the neural network to physical features at different scales.

7. A physical enhancement-based fluid dynamics small-sample solution system, characterized in that, include: Physical dimensionality reduction module: used to acquire sparse observation data and map it to the statistically averaged flow space described by the Reynolds-averaged Navier-Stokes equations; Learnable Closure Module: Contains a neural network model for simultaneously predicting the mean velocity field, mean pressure field, and spatially continuous learnable eddy viscosity coefficient latent field to achieve adaptive parameterized closure for turbulent stress. Spectral Differentiation Module: Combined with neural network models, it is used to calculate the physical field spatial derivative in frequency space through algebraic operations, replacing automatic differentiation to provide stable physical residual calculation; Geometric regularization module: used to construct geometrically aware masks based on the symbolic distance function and to implement differentiated spectral regularization strategies for the mainstream region and the near-wall region; Training and optimization module: Used to construct and solve the pre-conditioned joint loss function, and optionally perform dynamic spectral residual weighting to drive parameter optimization of the entire system.

8. A storage medium storing a computer program, wherein, The computer program is configured to execute the method as described in any one of claims 1-6 at runtime.

9. An apparatus comprising a memory and a processor, wherein, The memory stores a computer program, and the processor is configured to run the computer program to perform the method as described in any one of claims 1-6.

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