PINN-based proxy modeling using local turbulence estimation

Abstract

Systems and methods optimize hydrodynamic modeling using a physical information neural network (FINN) and a deep neural network (DNN). The DNN is trained to create a data-driven mapping of invariants and Reynolds stress tensors associated with the fluid field based on the simulated data. The FINN is trained using losses based on the Navier-Stokes and continuity equations, using a Reynolds stress tensor correction term derived from the mapping, and invariants and tensors derived from the estimated flow field variables. The FINN input includes boundary conditions, as well as spatio-temporal coordinates of sampling points in the fluid flow field that describe temporal evolution of turbulence problems depending on geometric boundaries of the flow field.

Description

Technical Field

[0001] This application relates to physics-based modeling. More specifically, this application relates to applying deep learning frameworks to simulate fluid dynamics related to the engineering design and manufacturing of industrial components. Background Technology

[0002] The industrial design and manufacture of components that must withstand turbulent environments involve the simulation and analysis of fluid flows. Modeling examples include: performance modeling of wind turbines, particularly at blade boundaries; combustion modeling in gas turbines by modeling turbulence in air-fuel mixtures; modeling aircraft wing designs; and many other examples.

[0003] Turbulence is characterized by instability, chaotic-like flow states, and high nonlinearity. The structures involved exhibit a wide range of spatial and temporal scales, with the ratio of maximum to minimum structures scaling with the Reynolds number. To directly capture fluid motion at all scales, extremely fine computational grids and time steps are required. Therefore, in engineering-related problems (i.e., high Reynolds number situations), despite rapid advancements in computing power, the computational workload remains almost impossible to complete within a reasonable timeframe.

[0004] Recent advances in machine learning and deep learning methods have been largely driven by increased computational power and the availability of massive datasets, enabling solutions to this problem at a reasonable cost and improving the modeling of turbulence physics. However, data availability and the ability to generalize beyond available datasets remain challenging issues.

[0005] Modeling turbulence is challenging due to the wide range of active spatial and temporal scales and the potentially unpredictable nature of the phenomenon. Turbulent fluid flow problems have traditionally been solved using numerical partial differential equation (PDE) solvers and discretized domains. With the increasing availability of sufficient computing power, direct numerical simulations (DNS) of various turbulent processes involving numerical analysis at turbulent scales have seen significant growth. However, due to the high computational demands and performance issues involved in DNS, simplified approximation methods remain popular and widely used. Among these, Reynolds-averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES) methods are the most common. Summary of the Invention

[0006] The system and method utilize Physical Information Neural Networks (PINNs) and Deep Neural Networks (DNNs) to optimize fluid dynamics modeling. The DNN is trained to create data-driven mappings of invariants and Reynolds stress tensors associated with the fluid field based on simulated data. The PINN is trained using a loss based on the Navier-Stokes and continuity equations, and a Reynolds stress tensor correction term derived from the mappings and the invariants and tensors derived from the estimated flow field variables. PINN inputs include boundary conditions and the spatiotemporal coordinates of sampling points in the fluid flow field, which describe the temporal evolution of the turbulence problem depending on the geometric boundaries of the flow field. The trained PINN serves as a surrogate model capable of accelerating tasks related to computer-aided engineering (CAE) and computer-aided design (CAD) of turbomachinery. Attached Figure Description

[0007] Non-limiting and non-exhaustive embodiments of this disclosure are described with reference to the following accompanying drawings, wherein, unless otherwise specified, the same reference numerals refer to the same elements throughout the drawings.

[0008] Figure 1 An example framework of a two-stage training process performed on a proxy machine learning model according to an embodiment of this disclosure is shown.

[0009] Figure 2 An example framework of a trained agent-based machine learning model according to an embodiment of the present disclosure is shown.

[0010] Figure 3 An example of a computing environment in which embodiments of the present disclosure can be implemented is shown. Detailed Implementation

[0011] Methods and systems are disclosed for solving classical problems in fluid dynamics related to turbulence (e.g., turbomachinery) and with real-world relevance to engineering problems such as the design of offshore structures installed in strong winds or needle-fin heat exchangers. The technical problems involve analyzing fluid flow around obstacles, which can affect multiple aspects related to the design and engineering of the target object. Unlike traditional computational fluid dynamics methods, the method disclosed in this paper is inherently meshless because it is based on a Physical Information Neural Network (PINN). Invariants derived from turbulent fluid field simulations are used as constraints to train a machine learning-based framework, which improves the accuracy of the learned surrogate model compared to traditional PINN methods. Convergence of the solution is accelerated and the accuracy of the surrogate model is improved by ensuring denser sampling of turbulent spots. Turbulence intensity algorithms are used to identify regions with turbulence, and these algorithms compute derivatives of the derived flow field values. Near real-time operational performance is achieved because the computational burden is primarily shifted to a two-stage training phase. Deep neural networks (DNNs) learn a data-driven mapping model of invariants and Reynolds stress tensors associated with the turbulent fluid field. This mapping was used to accelerate training and improve the accuracy of PINN with a loss function incorporating Reynolds stress correction. Utilizing this improved training method for both neural network-based and deep learning-based frameworks, a surrogate model was generated that can be used as a computer-implemented modeling tool for turbomachinery engineering. This surrogate model can perform tasks in a reasonable time while overcoming the complexity and high nonlinearity burden of chaotic flow states in such technical problems.

[0012] Figure 1 An example framework for a two-stage training process performed on a surrogate machine learning model according to an embodiment of this disclosure is shown. The training framework 100 includes a DNN 110 that learns a mapping 115 between invariants and Reynolds stress tensors (e.g., Reynolds stress anisotropy tensors) in a first training phase. In a second training phase, a PINN 120 is trained using the mapping 115 from the first training phase to learn a surrogate model for estimating flow field variables (e.g., velocity and pressure) for a given set of physical constraints (shown as boundary conditions 107 and geometric boundaries 109).

[0013] like Figure 1 As shown, the training framework 100 begins by acquiring a quantity of DNS data 101 representing at least two historical simulations of turbulence and processes. Multiple DNS simulation databases exist from which DNS data 101 can be obtained. The DNS data 101 is divided into Galilean invariants 102 and Reynolds stress tensor terms 103. For the Reynolds stress in the Navier-Stokes equations of turbulence, the training of the DNN 110 generates a mapping 115 between the invariants 102 and the tensor 103.

[0014] From the available historical DNS dataset 101, each data point represents a simulation case, and the preprocessing function derives the values ​​of the strain rate tensor s and the rotational stress tensor ω from the historical data. The invariants and basis tensors are functions of these two variables and their gradients. Additional basis tensors are used to compute the anisotropic tensor. An infinite tensor polynomial representation is created for the anisotropic tensor to establish the generalized viscosity assumption. The Reynolds stress, represented by the anisotropic tensor a, is a function of strain (an invariant represented by the strain rate tensor s and the rotational tensor ω). This representation includes coefficients and basis tensors, which are unknown functions of the invariants and nonlinear combinations of the invariants. For example, a three-dimensional model can have ten basis tensors and five invariants. The basis tensor T can be represented as follows:

[0015] Furthermore, the invariant can be represented by the following: {s 2},{ω 2}, {s 3},{ω 2 s},{ω 2 s 2}, Here, the strain rate tensor s is a dimensionless symmetric tensor with zero trace, the rotation tensor ω is dimensionless and antisymmetric, and I denotes Kronecker δ: I = δ ij .

[0016] An anisotropic tensor can be represented by the following expression: (1) For (1≤λ≤10), where a is the Reynolds stress anisotropic tensor polynomial with coefficients G and basis tensors T .coefficient G It is a function of a finite number of invariants. Because of the coefficients... G The relationship between the variables and the invariant is unknown. We use DNN 110 to map the relationship from the existing historical data 101.

[0017] DNN 110 training was used to estimate the function mapping 115 between the normalized stress tensor and the set of invariants / basis tensors. DNN 110 training converged when the value of the loss function 104 was minimized and reached a constant. This effectively reflects the DNN 110's performance on the coefficients. G The ability to model and predict Reynolds stress terms for any given invariant input. This model can then be used by the PINN 120 framework to compute the anisotropic tensor α of the flow field estimate 121 generated by PINN 120, as further described below.

[0018] Training the PINN 120 involves the following steps: In one embodiment, a multilayer perceptron (or feedforward neural network) is used for the PINN architecture. The PINN 120 of framework 100 is used to model an estimate of a flow field profile 121, which can be represented by velocity variables u, v, and pressure variable p. The inputs to the PINN 120 include boundary conditions (e.g., constant velocity inlet, pressure outlet, no-slip wall, etc.) 107 and temporal and spatial coordinates 108 depending on geometric boundaries 109 (i.e., the solution domain defining the flow geometry, obstacle shape, etc.). The temporal and spatial coordinates 108 of the sampling points in the turbulent field describe the temporal evolution of the turbulence problem. The neural network of the PINN 120 is defined as a function of parameters (weights and biases), which are optimized during training to obtain field variable estimates 121 useful for solving the PDE associated with the current turbulence problem.

[0019] Based on a sampling strategy, temporal and spatial sampling is performed on spatiotemporal coordinates 108 to determine the points for calculating the PDE loss function 124 used to train PINN 120. During the training of PINN 120, a temporal averaging engine 130 is used for each field variable. Calculate time-averaged quantities. The invariant and tensor engine 140 is configured to use the output of the time-averaged engine 130 to calculate invariants and basis tensors.

[0020] During the training of PINN 120, gradient descent is used to implement one or more loss functions to find the optimal set of parameters (i.e., the weights and biases of the neural network). In one implementation, the PDE loss 116 is implemented to represent the degree to which physics-based constraints (e.g., mass conservation and momentum balance as expressed by the Navier-Stokes equations) are satisfied at each sampling point of the estimated fluid flow field 121.

[0021] In one embodiment, the Reynolds stress correction loss 122 is implemented as a loss function configured to determine the extent to which the average flow of the flow field estimate 121 satisfies the Reynolds-averaged Navier-Stokes equations. The correction term module 141 is configured to execute equation (1) to solve for the anisotropic tensor a. This operation is based on inputs received from the map 115 and the invariants and tensor engine 140. The map 115 provides the values ​​of the coefficients G, and the tensor T is obtained from the invariants and tensor engine 140.

[0022] In one embodiment, the boundary condition loss function 123 is implemented as a loss function representing the degree to which the boundary conditions imposed by the problem definition of the flow field associated with the target design are satisfied at each of the sampling points of the estimated fluid flow field variable 121.

[0023] In one implementation, PINN 120 is trained using feedback from one or more loss values ​​from loss functions 122, 123, 124, individually or in combination, until convergence is achieved.

[0024] In addition to the steps described above, the sampling scheme for the spatiotemporal input 108 is further improved by sampling more points in regions of high turbulence intensity within the flow field. Turbulence intensity is calculated by selecting a finite number of random points covering the entire field. At a given point, the turbulent kinetic energy is divided by the square of the average overall flow velocity. In one aspect, selecting sampling points with higher turbulence for the spatial input 108 improves the accuracy and convergence of the method. These turbulence calculations can be performed at selected times during the training phase of frame 100 to check for or introduce improvements in various loss functions. Any sampling spatial points must be within the geometric boundary 109.

[0025] Figure 2 An example framework of a trained surrogate machine learning model according to an embodiment of this disclosure is shown. The trained PINN 220 can operate as a surrogate model to provide estimates 221 of flow field variables (e.g., velocity and pressure profiles of fluid flow) for turbulence problems, given new boundary conditions 207 and samples 208 of spatiotemporal coordinates based on given geometric boundaries 209. In one embodiment, the trained PINN 220 is a surrogate model add-on tool for a CAE / CAD platform used for the design and engineering of turbomachinery and dynamic fluid handling systems involving the simulation of turbulent fields.

[0026] Figure 3 An example of a computer environment in which embodiments of the present disclosure may be implemented is shown. The computing device 310 includes a processor 315 and a memory 311 (e.g., a non-transitory computer-readable medium) on which various computer applications, modules, or executable programs are stored. In one embodiment, the memory 311 includes one or more of the following modules: a DNN module 110, a PINN module 120 / 220, a time averaging engine module 130, and an invariant and tensor engine module 140.

[0027] like Figure 3 As shown, as an alternative computer implementation of DNN module 110 and PINN module 120 / 220, one or more of DNN module 110 and PINN module 120 / 220 can be deployed as a cloud-based or network-based operation, or as a partitioned operation shared by local modules 110, 120 / 220 and network-based modules 341, 342.

[0028] Network 360 (such as a local area network (LAN), wide area network (WAN), or Internet-based network) connects training data 351 to modules 110, 120 / 220 of computing device 310 and cloud-based modules 341, 342.

[0029] User interface module 314 provides an interface between modules 110, 120 / 220 and user interface devices 330 (such as display device 331 and user input device 332). GUI engine 313 drives the interactive user interface to be displayed on display device 331, thereby allowing users to receive visualizations of analysis results and assisting users in inputting learning objectives and domain constraints for DNN modules 110, 341 and PINN modules 120 / 220, 342.

[0030] Computer-readable medium instructions used to perform the operations of this disclosure may be assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, status setting data, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk, C++, and conventional procedural programming languages ​​such as "C" or similar programming languages. The computer-readable program instructions may be executed entirely on the user's computer, partially on the user's computer as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer via any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be to an external computer (e.g., via the Internet through an Internet service provider). In some embodiments, electronic circuit systems including, for example, programmable logic circuit systems, field-programmable gate arrays (FPGAs), or programmable logic arrays (PLAs) may execute computer-readable program instructions by utilizing status information from the computer-readable program instructions to personalize the electronic circuit system for performing aspects of this disclosure.

[0031] This document describes aspects of the disclosure with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the disclosure. It should be understood that each block in the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented using computer-readable medium instructions.

[0032] Figure 3The program modules, applications, computer-executable instructions, code, etc., described as stored in system memory 311 are merely illustrative and not exclusive, and are described as being supported by any particular module, which may alternatively be distributed across multiple modules or executed by different modules. Additionally, various program modules, scripts, plug-ins, application programming interfaces (APIs), or any other suitable computer-executable code may be provided locally hosted on computer system 310, remote network devices of storage modules 341, 342, and / or hosted on one or more other accessible computing devices via network 360 to support the functionality and / or additional or alternative functionality provided by the program modules, applications, or computer-executable code. Furthermore, functionality may be modularized differently, such that what is described as being supported by... Figure 3 The processing collectively supported by the set of program modules described herein can be performed by fewer or more modules, or functionality described as being supported by any particular module can be at least partially supported by another module. Furthermore, the program modules supporting the functionality described herein can form part of one or more applications executable across any number of systems or devices according to any suitable computational model (e.g., client-server model, peer-to-peer model, etc.). Additionally, the processing described as being supported by... Figure 3 Any functionality supported by any program module described herein can be implemented, at least in part, in the hardware and / or firmware across any number of devices.

[0033] It should also be understood that, without departing from the scope of this disclosure, computer system 310 may include alternative and / or additional hardware, software, or firmware components other than those described or depicted. More specifically, it should be understood that the software, firmware, or hardware components depicted as forming part of computer system 310 are merely illustrative, and some components may be absent or additional components may be provided in various embodiments. While various illustrative program modules have been depicted and described as software modules stored in system memory 311, it should be understood that functionality described as supported by program modules can be implemented by any combination of hardware, software, and / or firmware. It should also be understood that, in various embodiments, each of the modules mentioned above may represent a logical division of supported functionality. Such logical division is depicted for ease of interpretation of functionality and may not represent the structure of the software, hardware, and / or firmware used to implement functionality. Accordingly, it should be understood that, in various embodiments, functionality described as provided by a particular module may be provided at least partially by one or more other modules. Furthermore, in some embodiments, one or more depicted modules may be absent, while in other embodiments, additional modules not depicted may be present and may support at least a portion of the described functionality and / or additional functionality. Furthermore, while some modules may be described and depicted as submodules of another module, in some implementations, such modules may be provided as independent modules or submodules of other modules.

[0034] While specific embodiments of this disclosure have been described, those skilled in the art will recognize that numerous other modifications and alternative implementations are within the scope of this disclosure. For example, any functionality and / or processing capability described with respect to a particular device or component can be performed by any other device or component. Furthermore, while various illustrative implementations and architectures have been described according to embodiments of this disclosure, those skilled in the art should understand that numerous other modifications to the illustrative implementations and architectures described herein are also within the scope of this disclosure. Additionally, it should be understood that any operation, element, component, data, etc., described herein as being based on another operation, element, component, data, etc., may be additionally based on one or more other operations, elements, components, data, etc. Therefore, the phrase "based on" or variations thereof should be interpreted as "at least partially based on".

[0035] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present disclosure. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of instructions, including one or more executable instructions for implementing a specified logical function. In some alternative implementations, the functions indicated in the blocks may not occur in the order shown in the figures. For example, two blocks shown consecutively may actually be executed substantially simultaneously, or these blocks may sometimes be executed in reverse order, depending on the functionality involved. It will also be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, may be implemented by a dedicated hardware-based system that performs the specified function or action or executes a combination of dedicated hardware and computer instructions.

Claims

1. A computer-based system for creating a neural network for estimating turbulent fluid flow field variables relevant to the engineering and design of turbomachinery, including: processor; as well as A non-transitory memory storing modules executed by the processor, the modules comprising: Deep neural networks (DNNs) are trained to create mappings of invariants and Reynolds stress tensors associated with fluid flow fields based on simulated data; and A Physical Information Neural Network (PINN) is trained as a surrogate model for estimating turbulent fluid flow field variables. The training is based on at least one loss function using gradient descent, including a Reynolds stress tensor loss function configured to determine the extent to which the average flow of the estimated flow field variables satisfies the Reynolds-averaged Navier-Stokes equations determined by comparison with the mapping provided by the DNN. The inputs to the PINN include: The time and space coordinates of points in the turbulent field are used to sample the turbulent flow field, and the time and space coordinates describe the time evolution of the turbulence problem; Geometric boundaries used to define the flow geometry of a turbulent flow field; and The boundary conditions of the turbulent flow field.

2. The system according to claim 1, wherein, The PINN is also trained based on a loss function that represents the degree to which physical constraints are satisfied at each of the sampling points of the estimated fluid flow field variables.

3. The system according to claim 1, wherein, The PINN is also trained based on a loss function that represents the degree to which the boundary conditions imposed by the problem definition of the flow field are satisfied at each of the sample points of the estimated fluid flow field variables.

4. The system according to claim 1, wherein, The training input of the DNN includes invariants and Reynolds stress tensor terms obtained from the simulation data, wherein the training of the DNN is based on a loss function using gradient descent, such that, upon convergence, the Reynolds stress term is accurately predicted for any given invariant.

5. The system according to claim 1, wherein, The sampling points are selected based on regions with high turbulence intensity in the turbulent fluid flow field.

6. A computer-based method for deriving a surrogate model for estimating the flow field of a turbulent fluid, comprising: Using deep neural networks (DNNs), the DNNs are trained to create mappings of invariants and Reynolds stress tensors associated with the fluid flow field based on simulation data; as well as Using a Physical Information Neural Network (PINN), PINN is trained as a surrogate model for estimating turbulent fluid flow field variables. This training is based on at least one loss function using gradient descent, including a Reynolds stress tensor loss function configured to determine the extent to which the average flow of the estimated flow field variables satisfies the Reynolds-averaged Navier-Stokes equations, determined in part by comparison with the mapping provided by the DNN. The inputs to PINN include: The temporal and spatial coordinates of sampling points in the turbulent field, wherein the temporal and spatial coordinates describe the temporal evolution of the turbulence problem; Geometric boundaries used to define the flow geometry of a turbulent flow field; and The boundary conditions of the turbulent flow field.

7. The method according to claim 6, wherein, The PINN is trained at least according to a loss function that represents the degree to which physical constraints are satisfied at each of the sampling points of the estimated fluid flow field variables.

8. The method according to claim 6, wherein, The PINN is trained at least according to a loss function that represents the degree to which the boundary conditions imposed by the problem definition of the flow field are satisfied at each of the sample points of the estimated fluid flow field variables.

9. The method according to claim 6, wherein, The training input of the DNN includes invariants and Reynolds stress tensor terms obtained from the simulation data, wherein the training of the DNN is based on a loss function using gradient descent, such that, upon convergence, the Reynolds stress term is accurately predicted for any given invariant.

10. The method according to claim 6, wherein, The sampling points are selected based on regions with high turbulence intensity in the turbulent fluid flow field.

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