Pinn-based surrogate modeling using local turbulence estimates

Abstract

System and method optimize fluid dynamics modeling using a physics informed neural network (FINN) and a deep neural network (DNN). The DNN is trained to create a data driven mapping for invariants and Reynolds stress tensors associated with the fluid field based on simulation data. The FINN is trained using Navier Stokes and continuity equation based losses, using Reynolds stress tensor correction terms derived from the mapping and from invariants and tensors derived from estimated flow field variables. The FINN inputs include boundary conditions, and time-space coordinates for sampling points in the fluid flow field which describe time evolution of the turbulence problem depending on geometric bounds for the flow field.

Description

PINN-BASED SURROGATE MODELING USING LOCAL TURBULENCE ESTIMATESTECHNICAL FIELD

[0001] This application relates to physics-based modeling. More particularly, this application relates to applying a deep learning framework for simulation of fluid dynamics related to engineering design and manufacture of industrial components.BACKGROUND

[0002] Industrial design and manufacturing of components that must endure a turbulent flow environment involves simulation and analysis of fluid flows. Modeling examples include performance modeling of wind turbines especially at the blade boundaries, combustion in gas turbines by modeling turbulence in air-fuel mix, modeling wing design of airplanes, and many others.

[0003] Turbulent flows are characterized by unsteadiness, chaotic-like flow states and high degree of non-linearity. The structures involved exhibit a wide range of spatial and temporal scales, with the ratio of largest to smallest structures scaling with the Reynolds number. In order to capture all scales of fluid motion directly, very fine computational meshes and time steps are required. Consequently, computational effort in the case of engineering-relevant problems (i.e., high Reynolds numbers) is nearly impossible to accomplish in reasonable time despite rapidly increasing capability of computers.

[0004] Recent advances in machine learning and deep learning methods, which are largely driven by increased computational power as well as the availability of exceptionally large data sets, make it possible to address this issue and improve the modeling of turbulence physics atreasonable costs. However, data availability and generalization beyond the available datasets remain challenging issues.

[0005] Modeling turbulence is challenging because of the broad range of active spatial and temporal scales and the underlying, unpredictable nature of the phenomenon. Turbulent fluid flow problems are solved traditionally using numerical partial differential equation (PDE) solvers and discretized domains. With increasing availability of adequate computing power, there has been a significant growth in direct numerical simulation (DNS) of a number of turbulent flows and processes involving the numerical resolution of turbulent scales. However, due to the high computing requirement and performance issues involved with DNS, simplified approximations continue to remain popular and widespread. Among these, Reynolds Averaged Navi er- Stokes (RANS) and Large Eddy Simulation (LES) approaches are the most common.SUMMARY

[0006] System and method optimize fluid dynamics modeling using a physics informed neural network (PINN) and a deep neural network (DNN). The DNN is trained to create a data driven mapping for invariants and Reynolds stress tensors associated with the fluid field based on simulation data. The PINN is trained using Navier Stokes and continuity equation based losses, using Reynolds stress tensor correction terms derived from the mapping and from invariants and tensors derived from estimated flow field variables. The PINN inputs include boundary conditions, and time-space coordinates for sampling points in the fluid flow field which describe time evolution of the turbulence problem depending on geometric bounds for the flow field. The trained PINN is a surrogate model capable of accelerated execution of tasks related to computer aided engineering (CAE) and computer aided design (CAD) of turbomachinery.BRIEF DESCRIPTION OF THE DRAWINGS

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

[0008] FIG. 1 illustrates a framework example for a two-stage training process performed for a surrogate machine learning model in accordance with embodiments of this disclosure.

[0009] FIG. 2 illustrates a framework example of a trained surrogate machine learning-based model in accordance with embodiments of this disclosure.

[0010] FIG. 3 an example of a computing environment within which embodiments of the disclosure may be implemented.DETAILED DESCRIPTION

[0011] Methods and systems are disclosed to address classical problems in fluid dynamics related to turbulence (e.g., turbomachinery) and having real world relevance to engineering problems, such as design of offshore structure installation in high winds, or pin-fin heat exchangers to name examples. Technical problems include analysis of fluid flow around an obstacle which could affect a multitude of things related to design and engineering of a target object in interest. Unlike traditional computational fluid dynamics methods, the approach disclosed herein is an inherently mesh-free approach as it is a physics informed neural network (PINN) based approach. Invariants obtained from turbulent fluid field simulations are used as constraints in training the machine learning-based framework, which improves accuracy for the learned surrogate model compared to traditional PINN approaches. Convergence to a solution is accelerated and accuracy of the surrogate model is improved by ensuring turbulent spots are more densely sampled. Areaswith turbulence are identified using turbulence intensity algorithms which compute derivative of derived flow field values. Near real-time operational performance is realized as the computational burden is transferred primarily to a two stage training phase. A deep neural network (DNN) learns a data driven mapping model for invariants and Reynolds stress tensors associated with a turbulent fluid field. This mapping is used to accelerate training and improve accuracy of the PINN which incorporates a loss function for Reynolds stress correction. With this improved training approach for neural network-based and deep learning-based frameworks, a surrogate model is generated that is useful as a computer implemented modeling tool for engineering of turbomachinery, capable of executing tasks in a reasonable time while overcoming the burden of complexity of chaotic flow states and high degree of non-linearity for such technical problems.

[0012] FIG. 1 illustrates a framework example for a two-stage training process performed for a surrogate machine learning model in accordance with embodiments of this disclosure. Training framework 100 includes DNN 110 that learns a mapping 115 between invariants and Reynolds stress tensors (e.g., Reynolds stress anisotropy tensors) in a first training stage. In a second training stage, PINN 120 is trained using the mapping 115 from the fist training stage to learn a surrogate model for estimating flow field variables (e.g., velocity and pressure) for a given set of physics constraints, shown as boundary conditions 107 and geometric bounds 109.

[0013] As shown in FIG. 1, training framework 100 starts with obtaining a quantity of DNS data 101 representing at least two historical simulations of turbulent flows and processes. There exist a number of available DNS simulation databases from which DNS data 101 can be obtained. DNS data 101 is divided into Galilean invariants 102 and Reynolds stress tensor terms 103. the Reynolds stresses in the turbulent Navier Stokes equation the training of DNN 110 generates a mapping 115 between the invariants 102 and tensors 103.

[0014] From the available historical DNS dataset 101, each data point represents a simulation case, and a preprocessing function derives the values of strain rate tensor s and rotational stress tensor co from the historical data. The invariants and basis tensors are a function of these two variables and their gradients. Additional basis tensors are used to calculate the anisotropy tensors. An infinite tensor polynomial representation is created for the anisotropy tensors with the aim to make a generalized viscosity hypothesis. Reynolds stresses, represented by an anisotropy tensor a, are a function of strains (invariants represented by strain rate tensor s and rotational tensor co). This representation contains coefficients, which are an unknown function of the invariants, as well as basis tensors, which are non-linear combinations of the invariants. For example, a three dimensional model can have ten basis tensors with five invariants. Basis tensors T can be expressed as follows:and invariants can be expressed by the following: {<*>2},where strain rate tensor s is non-dimensional symmetric tensor with zero trace, rotational tensor co is non-dimensional and antisymmetric, and I represents Kronecker delta: I = 8^.

[0015] The anisotropy tensors can be represented by the following expression: a = S (G^) (1)for (1< A. <10) where a is the Reynolds Stress anisotropy tensor polynomial, with coefficients G and basis tensors T. The coefficients G are functions of a finite number of invariants. Since the relationship between the coefficients G and invariants are unknown, DNN 110 is used to map it from existing historic data 101.

[0016] DNN 110 training is used to estimate the functional mapping 115 between the normalized stress tensors and the sets of invariants / basis tensors. DNN 110 training is converged when the loss function 104 values are minimized and reach a constant. This effectively reflects the ability of the DNN 110 to model coefficients G and to predict Reynolds stress terms for any given invariant inputs. This model can then be used by the PINN 120 framework to calculate anisotropy tensors a for flow-field estimates 121 generated by the PINN 120, as described further below.

[0017] Training of PINN 120 involves the following steps. In an embodiment, a multi-layer perceptron (or feed forward neural network) is used for the PINN architecture. PINN 120 of the framework 100 is used to model an estimate of the flow field profile 121, which may be expressed in terms of velocity variables u, v, and pressure variables p. Inputs to PINN 120 include boundary conditions (e.g., constant velocity inlet, pressure outlet, no slip walls, etc.) 107 and time and spatial coordinates 108 depending on the geometric bounds 109 (i.e., solution domain which defines flow geometry, obstacle shapes, etc.). Time and spatial coordinates 108 for sampling points in the turbulence field describe time evolution of the turbulence problem The neural network of PINN 120 is defined by a function of parameters (weights and biases) which are optimized during training to obtain the field variable estimates 121 that are useful to solve PDE related to the turbulence problem at hand.

[0018] Based on a sampling strategy, temporal and spatial sampling of time-space coordinates108 are carried out to determine points at which the PDE loss function 124 used to train the PINN120 are computed. During training of PINN 120, time averaging engine 130 calculates time averaged quantities for each field variable (u, v, p ). Invariant and tensor engine 140 is configured to compute the invariants and basis tensors using the output of time averaging engine 130.

[0019] During training of PINN 120, one or more loss functions are implemented to find the best set of parameters (i.e., weights and biases of the neural network) using gradient descent. In an embodiment, PDE loss 116 is implemented to represent how well at each of the sampled points of the estimated fluid flow field 121 satisfies the physics based constraints (e.g., conservation of mass and momentum balance as expressed by Navier-Stokes equations).

[0020] In an embodiment, Reynolds stress correction loss 122 is implemented as a loss function configured to determine how well the mean flow of flow field estimates 121 satisfies Reynolds averaged Navier Stokes equations. Correction terms module 141 is configured to execute equation (1) to solve for the anisotropy tensors a. This operation is based on input received from mapping 115 and invariant and tensor engine 140. Mapping 115 provides values of coefficient G and the tensors T are obtained from the invariant and tensor engine 140.

[0021] In an embodiment, boundary conditions loss function 123 is implemented as a loss function that represents how well at each of the sampled points of the estimated fluid flow field variables 121 satisfy boundary conditions imposed by the problem definition for the flow field related to the target design.

[0022] In an embodiment, training of PINN 120 is performed using feedback from one or more loss values of loss functions 122, 123, 124, either alone or in combination, until convergence is reached.

[0023] In addition to the above steps, the sampling schemes for time-spatial inputs 108 are also improved by sampling more points in areas of high turbulence intensity within the flow field. Turbulence intensity is calculated by selecting a limited number of random points that cover the entire field. At a given point, the turbulence kinetic energy is divided by the mean flow bulk velocity squared. In an aspect, sampling points having higher turbulence are selected for spatial inputs 108 which improves the accuracy and convergence of the approach. These turbulence calculations can be performed at chosen times during the training phase of framework 100 to check for or introduce improvements in the various loss functions. Any sampled spatial point must be within the geometric bounds 109.

[0024] FIG. 2 illustrates a framework example of a trained surrogate machine learning-based model in accordance with embodiments of this disclosure. Trained PINN 220 can be operated as s surrogate model to provide flow field variable estimates 221 (e.g., velocity and pressure profile of the fluid flow) for a turbulence problem given new boundary conditions 207 and samples of time space coordinates 208 from given geometric bounds 209. In an embodiment, trained PINN 220 is a surrogate model add-on tool for CAE / CAD platforms used for design and engineering of turbomachinery and dynamic fluid handling systems involving simulation of turbulent flow fields.

[0025] FIG. 3 shows an example of a computer environment within which embodiments of the disclosure may be implemented. A computing device 310 includes a processor 315 and memory 311 (e.g., a non-transitory computer readable media) on which is stored various computer applications, modules or executable programs. In an embodiment, memory 311 includes one or more of the following modules: a DNN module 110, a PINN module 120 / 220, time averaging engine module 130 and invariant and tensor engine module 140.

[0026] As shown in FIG. 3, as an alternative computer implementation of DNN module 110, and PINN module 120 / 220, one or more of a DNN module 110 and PINN module 120 / 220 may be deployed as cloud-based or web-based operations, or as a divided operation shared by local modules 110, 120 / 220, and web-based modules 341, 342.

[0027] A network 360, such as a local area network (LAN), wide area network (WAN), or an internet- based network, connects training data 351 to modules 110, 120 / 220 of computing device 310 and to cloud based modules 341, 342.

[0028] User interface module 314 provides an interface between modules 110, 120 / 220 and user interface 330 devices, such as display device 331 and user input device 332. GUI engine 313 drives the display of an interactive user interface on display device 331, allowing a user to receive visualizations of analysis results and assisting user entry of learning objectives and domain constraints for DNN module 110, 341, and PINN module 120 / 220, 342.

[0029] Computer readable medium instructions for carrying out operations of the present disclosure may be assembler instructions, instruction-set-architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state-setting data, or either source code or object code written in any combination of one or more programming languages, including an object oriented programming language such as Smalltalk, C++ or the like, and conventional procedural programming languages, such as the "C" programming language or similar programming languages. The computer readable program instructions may execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or theconnection may be made to an external computer (for example, through the Internet using an Internet Service Provider). In some embodiments, electronic circuitry including, for example, programmable logic circuitry, field-programmable gate arrays (FPGA), or programmable logic arrays (PLA) may execute the computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry, in order to perform aspects of the present disclosure.

[0030] Aspects of the present disclosure are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the disclosure. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, may be implemented by computer readable medium instructions.

[0031] The program modules, applications, computer-executable instructions, code, or the like depicted in FIG. 3 as being stored in the system memory 311 are merely illustrative and not exhaustive and that processing described as being supported by any particular module may alternatively be distributed across multiple modules or performed by a different module. In addition, various program module(s), script(s), plug-in(s), Application Programming Interface(s) (API(s)), or any other suitable computer-executable code hosted locally on the computer system 310, remote network devices storing modules 341, 342 and / or hosted on other computing device(s) accessible via one or more of the network(s) 360, may be provided to support functionality provided by the program modules, applications, or computer-executable code and / or additional or alternate functionality. Further, functionality may be modularized differently such that processing described as being supported collectively by the collection of program modules depicted in FIG.3 may be performed by a fewer or greater number of modules, or functionality described as being supported by any particular module may be supported, at least in part, by another module. In addition, program modules that support the functionality described herein may form part of one or more applications executable across any number of systems or devices in accordance with any suitable computing model such as, for example, a client-server model, a peer-to-peer model, and so forth. In addition, any of the functionality described as being supported by any of the program modules depicted in FIG. 3 may be implemented, at least partially, in hardware and / or firmware across any number of devices.

[0032] It should further be appreciated that the computer system 310 may include alternate and / or additional hardware, software, or firmware components beyond those described or depicted without departing from the scope of the disclosure. More particularly, it should be appreciated that software, firmware, or hardware components depicted as forming part of the computer system 310 are merely illustrative and that some components may not be present 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 appreciated that functionality described as being supported by the program modules may be enabled by any combination of hardware, software, and / or firmware. It should further be appreciated that each of the above-mentioned modules may, in various embodiments, represent a logical partitioning of supported functionality. This logical partitioning is depicted for ease of explanation of the functionality and may not be representative of the structure of software, hardware, and / or firmware for implementing the functionality. Accordingly, it should be appreciated that functionality described as being provided by a particular module may, in various embodiments, be provided at least in part by one or more other modules. Further, one or more depicted modules may not bepresent in certain embodiments, 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. Moreover, while certain modules may be depicted and described as sub-modules of another module, in certain embodiments, such modules may be provided as independent modules or as sub-modules of other modules.

[0033] Although specific embodiments of the disclosure have been described, one of ordinary skill in the art will recognize that numerous other modifications and alternative embodiments are within the scope of the disclosure. For example, any of the functionality and / or processing capabilities described with respect to a particular device or component may be performed by any other device or component. Further, while various illustrative implementations and architectures have been described in accordance with embodiments of the disclosure, one of ordinary skill in the art will appreciate that numerous other modifications to the illustrative implementations and architectures described herein are also within the scope of this disclosure. In addition, it should be appreciated that any operation, element, component, data, or the like described herein as being based on another operation, element, component, data, or the like can be additionally based on one or more other operations, elements, components, data, or the like. Accordingly, the phrase “based on,” or variants thereof, should be interpreted as “based at least in part on.”

[0034] The flowchart and block diagrams in the Figures 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 the flowchart or block diagrams may represent a module, segment, or portion of instructions, which comprises one or more executable instructions for implementing the specified logical function(s). In some alternative implementations, the functions noted in the block mayoccur out of the order noted in the Figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and / or flowchart illustration, and combinations of blocks in the block diagrams and / or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts or carry out combinations of special purpose hardware and computer instructions.

Claims

CLAIMSWhat is claimed is:

1. A computer based system of neural networks for creating a surrogate model used for estimating turbulent fluid flow field variables related to engineering and design of turbomachinery, comprising: a processor; and a non-transitory memory having stored thereon modules executed by the processor, the modules comprising: a deep neural network (DNN) trained to create a mapping for invariants and Reynolds stress tensors associated with the fluid flow field based on simulation data; and a physics informed neural network (PINN) trained as a surrogate model for estimating turbulent fluid flow field variables, wherein the training is based on at least one loss function using gradient descent, the at least one loss function including a Reynolds stress tensor loss function configured to determine how well a mean flow of the estimated flow field variable satisfies Reynolds averaged Navier Stokes equations determined in part by comparison with the mapping provided by the DNN, wherein the PINN inputs comprise: time and space coordinates for sampling points in the turbulence field which describe time evolution of the turbulence problem, geometric bounds for defining flow geometry of the turbulent flow field, and boundary conditions of the turbulent flow field.

2. The system of claim 1, wherein the PINN is further trained according to a loss function that represents how well at each of the sampled points of the estimated fluid flow field variables satisfy physics based constraints.

3. The system of claim 1, wherein the PINN is further trained according to a loss function that represents how well at each of the sampled points of the estimated fluid flow field variables satisfy boundary conditions imposed by a problem definition for the flow field.

4. The system of claim 1, wherein training inputs to the DNN include 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, Reynolds stress terms are predicted accurately for any given invariant.

5. The system of claim 1, wherein sampling points are selected based areas in the turbulent fluid flow field having higher turbulence intensity.

6. A computer-implemented method for deriving a surrogate model that estimates a turbulent fluid flow field, comprising: using deep neural network (DNN) trained to create a mapping for invariants and Reynolds stress tensors associated with the fluid flow field based on simulation data; and using a physics informed neural network (PINN) trained as a surrogate model for estimating turbulent fluid flow field variables, wherein the training is based on at least one loss function using gradient descent, the at least one loss function including a Reynolds stress tensor loss function configured to determine how well a mean flow of the estimated flow field variable satisfies Reynolds averaged Navier Stokes equations determined in part by comparison with the mapping provided by the DNN, wherein the PINN inputs comprise: time and space coordinates for sampling points in the turbulence field which describe time evolution of the turbulence problem, geometric bounds for defining flow geometry of the turbulent flow field, and boundary conditions of the turbulent flow field.

7. The method of claim 6, wherein the PINN is trained according to at least a loss function that represents how well at each of the sampled points of the estimated fluid flow field variables satisfy physics based constraints.

8. The method of claim 6, wherein the PINN is trained according to at least a loss function that represents how well at each of the sampled points of the estimated fluid flow field variables satisfy boundary conditions imposed by a problem definition for the flow field.

9. The method of claim 6, wherein training inputs to the DNN include 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, Reynolds stress terms are predicted accurately for any given invariant.

10. The method of claim 6, wherein sampling points are selected based areas in the turbulent fluid flow field having higher turbulence intensity.

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