Asynchronous GMRES precondition method for AI acceleration of pressure Poisson equation in incompressible flow numerical calculation

Abstract

The invention discloses an asynchronous GMRES precondition method for AI acceleration of a pressure Poisson equation in incompressible flow numerical calculation, and belongs to the field of computational fluid mechanics. Aiming at the calculation bottleneck of a traditional GMRES method when a pressure Poisson equation is solved, the method adopts an asynchronous parallel AI precondition framework, neural network training and flow field solving are decoupled, and concurrent execution is achieved. The solver thread performs real-time inference by using the current model, and provides high-quality initial guess; the weight is optimized in the background of the trainer thread, and the lagging is tolerated by using the flow field time coherence. Experimental results show that the method shortens the solving time by 67%-82% on the premise of ensuring convergence precision, and is suitable for large-scale fluid simulation.

Description

Technical Field

[0001] This invention belongs to the field of computational fluid dynamics (CFD), specifically involving an artificial intelligence-based method for accelerating numerical calculations of incompressible flows, particularly the optimization of solving the pressure Poisson equation. Background Technology

[0002] Numerical simulations of incompressible fluid dynamics are widely used in turbulence research, aerodynamic optimization, and other fields. In solving the Navier-Stokes equations, the pressure Poisson equation (PPE) accounts for the majority of the computation time, typically reaching 80%-90%. Traditional GMRES methods rely on initial guesses, and their convergence efficiency is limited by the spectral distribution and condition number. Existing online deep learning methods employ a serial blocking mode, leading to a conflict between training and solution efficiency and failing to fully utilize hardware resources. Summary of the Invention

[0003] The purpose of this invention is to provide an asynchronous parallel AI preconditioning method that eliminates training waiting delays, enables concurrency between computation and learning, and improves the efficiency of PPE solution.

[0004] To achieve the above objectives, the present invention adopts the following technical solution: An AI-accelerated method for the pressure Poisson equation in numerical calculations of incompressible flows includes: (1) Construct a neural network to learn the inverse PPE mapping and provide an initial guess; (2) Asynchronous architecture decouples training and solving; (3) Solve the non-blocking inference of threads; (4) Continuous optimization of training threads; (5) Utilize time coherence to tolerate lag.

[0005] The beneficial effects of this invention are: significantly reducing the time of simulated wall clocks and improving the practicality of AI in CFD. Attached Figure Description

[0006] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0007] Figure 1 This is a schematic diagram of the asynchronous architecture of the present invention; Figure 2 This is a diagram of the neural network structure. Figure 3To compare the convergence history and iteration time of standard GMRES, serial MLGMRES and the proposed parallel MLGMRES on the 2D pressure Poisson equation flow test problem (40×40 grid); Figure 4 As the time consumed per iteration varies with the number of iterations, the parallel MLGMRES (green) maintains the lowest and most stable single-step time, significantly outperforming the serial MLGMRES (red) and standard GMRES (blue). Figure 5 To accelerate the comparison chart through iterations. Detailed Implementation

[0008] The technical solution of the present invention is further illustrated below through specific embodiments. These embodiments are only for illustrating the present invention and are not intended to limit its scope of protection. Those skilled in the art can make various modifications or improvements to the following embodiments based on the basic idea of ​​the present invention, and all such modifications or improvements should fall within the scope of protection of the present invention.

[0009] Example 1: Accelerated Solving of Pure Elliptic Problems under the 2D Poisson Equation This embodiment focuses on the classic 2D Poisson equation as an independent benchmark problem. This problem is a purely elliptic partial differential equation, often used to test the performance of linear solvers without involving the time evolution of fluid dynamics. The 2D Poisson equation has the form ∇²p = f(x,y), where p is the unknown field and f is the source term function. The boundary conditions are of the Dirichlet type (boundary values ​​are fixed at 0).

[0010] (1) Computational domain: unit square [0,1]×[0,1].

[0011] (2) Mesh size: 50×50 structured mesh (2500 degrees of freedom), using central difference to generate a five-point template matrix A (sparse, symmetric positive definite).

[0012] (3) Source term function: f(x,y) = -2π² sin(πx) sin(πy), exact solution p(x,y) = sin(πx) sin(πy) / (2π²), used to verify accuracy.

[0013] (4) GMRES solution parameters: restart dimension m=20, convergence threshold ε=10^{-8}, maximum number of iterations 500 (no time step, only single solution repeated test).

[0014] (5) Neural network model: Simple multilayer perceptron (MLP), the structure includes an input layer (the grid is flattened into a vector, dimension 2500), 3 hidden layers (neurons 512-256-128), and an output layer (dimensional 2500). The activation function is Leaky ReLU (α=0.01), the total number of parameters is about 1.5M, and it is suitable for purely static problems.

[0015] The asynchronous architecture implementation steps are as follows (adapted to pure Poisson problems, without time stepping, but simulating multiple independent solutions to accumulate data): Initialization phase: The solver thread generates the initial source term f and uses the traditional method (zero initial guess) to complete the first solution Ap=f, obtaining p. The trainer thread initializes the network weights and sets up a buffer to store (f,p) pairs.

[0016] Solver thread execution: To simulate the online scenario, repeatedly generate mutation source terms (e.g., add random noise Gaussian perturbation σ=0.1): Calculate the source term f(baseline + disturbance).

[0017] Obtain the current network parameter θ in a non-blocking manner.

[0018] Inferring the initial guess p0 = f θ (f).

[0019] Call GMRES to solve Ap=f, using p0 as the initial value.

[0020] Send the (f,p) pair to the trainer buffer.

[0021] Trainer thread execution: background optimization.

[0022] Batch size: 16.

[0023] Loss function: L = MSE(f θ (f), p) + λ||θ|| 2 (λ=0.0005).

[0024] Optimizer: SGD with momentum (learning rate η=0.005, momentum=0.9).

[0025] Update frequency: Update once every 5 solutions, epochs=8 per update, lag tolerance of 10-15 solutions.

[0026] Early Stop: Monitoring based on the validation set (20% of the data). Inverter: SGD with momentum (learning rate η=0.005, momentum=0.9).

[0027] Experimental Results: To highlight the actual performance of parallel MLGMRES, the table below provides a detailed breakdown of computation time for various parameters on highly localized peak problems with mesh sizes ranging from 20×20 to 40×40. On highly localized spike-type right-hand term test problems, the total solution time (in seconds) of standard GMRES, multilevel MLGMRES (parallel version), neural network training time, and the proposed AI-accelerated GMRES method are compared at different grid resolutions (20×20 to 40×40). Although the AI-accelerated method includes neural network training overhead, its overall runtime is significantly lower than that of standard GMRES (saving up to 50% on fine grids), demonstrating the efficiency improvement gained from learning-based preconditioning or acceleration.

[0028] Example 2: Implementation of an asynchronous AI preconditioning method based on lid-driven cavity flow This embodiment uses the classic cap-driven square cavity flow problem as the test benchmark. This problem is a typical two-dimensional incompressible fluid dynamics simulation scenario, often used to verify the accuracy and efficiency of numerical solvers. Cap-driven square cavity flow describes a square cavity with a cap plate moving at a constant velocity, driving the internal fluid to generate vortex flow. The simulation parameters are set as follows: Mesh size: 40×40 structured mesh (approximately 1600 degrees of freedom) to ensure a moderate computational scale and facilitate demonstration.

[0029] Reynolds number (Re): 1000, representing moderate turbulence intensity, with the flow field containing obvious vortex structures and boundary layers.

[0030] The time step (Δt) is 0.01 s, which satisfies the CFL condition (CFL<1) and ensures numerical stability.

[0031] Projection framework: The Navier-Stokes equations are solved using the Chorin projection method. The intermediate velocity field is calculated using the explicit Euler scheme. The pressure Poisson equation (PPE) is discretized using a five-point difference to form a sparse linear system Ax=b, where A is the Poisson operator matrix and b is the source term (the divergence of the intermediate velocity field).

[0032] GMRES solution parameters: Restart dimension m=30, convergence threshold ε=10 -6 (Relative residual norm ||r|| / ||r0||<ε), maximum number of iterations 1000.

[0033] Neural Network Model: A lightweight convolutional neural network (CNN) is used, consisting of three convolutional layers (3×3 kernel size, 16-32-64 channels), followed by two fully connected layers (128-64 hidden units). The output is an initial pressure field guess. The activation function is ReLU. The input is the source term field b (40×40 grid), and the output is an approximation of the pressure field p (of the same size). The total number of network parameters is approximately 50,000 to avoid overfitting.

[0034] The steps for implementing an asynchronous architecture are as follows: Initialization Phase: Two independent threads are started. The solver thread loads the initial flow field (velocity u=0, v=0, except for the top cover u=1) and uses the traditional method (pressure from the previous time step as a guess) to solve for the PPE at the first time step. At the same time, the trainer thread initializes the neural network weights (random Gaussian distribution) and prepares a data buffer to store historical source-pressure pairs (initially empty).

[0035] Solver thread execution: at each time step t n : Calculate the intermediate velocity fields u and v (ignoring the pressure gradient).

[0036] Calculate the source term b = ∇·(u*,v*).

[0037] Query the latest neural network parameters in a non-blocking manner (if the update is lagging, use the previous available version).

[0038] Using a neural network to infer the initial guess p0 = f θ (b), where θ is the current parameter.

[0039] Call GMRES to solve Ap = b, using p0 as the initial value.

[0040] Update flow field: Correct velocity u n+1 = u* - Δt ∇p, v n+1 = v* - Δt ∇p.

[0041] The (b, p) pair is sent asynchronously to the buffer of the trainer thread (queue size 100, to avoid blocking).

[0042] Experimental Results: In a simulation of 1000 time steps, this method was compared with serial online learning (Luna method) and the standard GMRES: In the 2D Taylor-Green vortex test problem, the total computation time (in seconds) of the serial MLGMRES method (original author's results) and the proposed parallel MLGMRES method are compared at different mesh resolutions (20×20 to 40×40). The parallel implementation continuously reduces the runtime, with a speedup from 1.25x to 1.61x (average approximately 1.41x), and the performance gap widens further as the mesh size increases.

[0043] These embodiments fully demonstrate the feasibility and advantages of the present invention, eliminating blocking through an asynchronous architecture and improving GMRES efficiency by 67%-82%. Those skilled in the art can adjust the parameters according to specific flow scenarios.

[0044] It should be noted that in this application, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0045] The above description is merely a specific embodiment of this application, enabling those skilled in the art to understand or implement this application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of this application. Therefore, this application is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features claimed herein.

Claims

1. An AI-accelerated method for the pressure Poisson equation in numerical calculations of incompressible flows, characterized in that, Includes the following steps: Step 1: Construct a neural network model to capture flow field features in real time and generate an initial guessed solution to the pressure Poisson equation; Step 2: Adopt an asynchronous parallel architecture to decouple the training process of the neural network from the time-step solution process of the flow field, so as to realize the concurrent execution of computation and learning; Step 3: In the solver thread, non-blocking inference is performed using the current neural network parameters to provide an initial guess for the GMRES solution; Step 4: In the trainer thread, continuously optimize the neural network weights using historical flow field data; Step 5: Based on the temporal coherence of the flow field, tolerate the time lag in model parameter updates to ensure improved GMRES convergence efficiency.

2. The AI ​​acceleration method for the pressure Poisson equation in numerical calculation of incompressible flow according to claim 1, characterized in that, In step one, the neural network model is a lightweight convolutional neural network, with the input being the source term of the pressure Poisson equation and the output being an initial pressure field guess generated by an approximate inverse mapping.

3. The AI ​​acceleration method for the pressure Poisson equation in numerical calculation of incompressible flow according to claim 1, characterized in that, In step two, the asynchronous parallel architecture includes two independent threads: the solver thread is responsible for solving the Navier-Stokes equations using the projection method and the GMRES iteration; the trainer thread performs backpropagation optimization in the background to avoid solver waiting delays.

4. The AI ​​acceleration method for the pressure Poisson equation in numerical calculation of incompressible flow according to claim 1, characterized in that, In step three, the non-blocking inference utilizes the latest available model parameters to reduce the number of GMRES iterations; the initial residuals are reduced by 2-3 orders of magnitude through AI preconditioning.

5. The AI ​​acceleration method for the pressure Poisson equation in numerical calculation of incompressible flow according to claim 1, characterized in that, In step four, the historical flow field data includes source-pressure pairs, and the training process utilizes an online learning framework, increasing the cumulative training amount by 15-40 times compared to the serial method.

6. The AI ​​acceleration method for the pressure Poisson equation in numerical calculation of incompressible flow according to claim 1, characterized in that, In step five, the time lag is several to tens of steps, which is applicable to flow scenarios such as high Reynolds number turbulence, cap-driven square cavity flow, and Taylor-Green vortex, ensuring numerical stability.

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