Systems and methods for turbulent flow prediction using geometry-aware pinns

Abstract

Methods (300) for turbulent flow prediction and corresponding systems (100) and computer-readable mediums (126). A method (300) includes receiving (302) input features (240) for which turbulent flow prediction is to be produced. The method includes modeling (304) turbulent flow using a physics-informed neural network (PINN) (230) according to the input features (240) to generate an output (250) from the PINN (230). The method includes producing (310) the output (250), wherein the output (250) includes physical prediction values corresponding to the input features (240).

Description

202416541SYSTEMS AND METHODS FOR TURBULENT FLOW PREDICTION USING GEOMETRY- A WARE PINNSCROSS-REFERENCE TO OTHER APPLICATION

[0001] This application claims the benefit of the filing date of U.S. Provisional Patent Application 63 / 691,665, filed September 6, 2024, which is hereby incorporated by reference.TECHNICAL FIELD

[0002] The present disclosure is directed, in general, to computational fluid dynamics and, in particular, to systems and methods for predicting turbulent flow.BACKGROUND OF THE DISCLOSURE

[0003] Computational fluid dynamics, or CFD, is a field of engineering that uses computers to simulate how fluids like air or water move. One of the most difficult problems in CFD is predicting turbulence, which is a chaotic and unpredictable state of flow often seen in things like fast-moving water or air over an airplane wing. While computers have become powerful enough to simulate simple flows with great accuracy, turbulence remains a major challenge because it involves many swirling motions at different scales, all interacting with one another in complex ways.

[0004] To manage this complexity, scientists use models to approximate turbulence instead of simulating every detail directly. The most basic models, like those used in industry, are called Reynolds-averaged Navier-Stokes (RANS) models. These are fast and efficient but often not very accurate when the flow involves separation, sudden changes in direction, or swirling vortices. More advanced approaches, like large eddy simulation (LES), can capture some of the larger turbulent motions while modeling the smaller ones. LES is more accurate than RANS but also more computationally expensive, making it less practical for everyday engineering use.202416541

[0005] There is also growing interest in using machine learning to improve turbulence prediction. Over the last decade, deep learning has emerged as a vital tool for accelerating CFD simulations, particularly in applications where traditional solvers are computationally expensive or time-intensive. Several approaches have been developed, with some integrating neural networks to predict residuals or refine turbulence models within the framework of conventional solvers, effectively merging data-driven techniques with established numerical methods. Other strategies aim to replace the CFD solvers entirely by learning the flow fields directly through models that utilize convolutional neural networks (CNNs) or graph neural networks (GNNs). CNNs effectively capture the spatial features of the flow on regular grids, whereas GNNs excel at handling complex geometries and mesh representations, providing flexible and mesh-independent predictions.

[0006] Using CNNs and GNNs to predict turbulent flow shows promise but also comes with several significant challenges. One major problem is that turbulence involves a wide range of interacting scales in both space and time. CNNs are typically designed to capture local patterns in regular grids, such as images. However, turbulent flows often involve global interactions and non-local effects that are difficult for CNNs to model accurately, especially when the flow is highly three-dimensional or unsteady.

[0007] Another issue lies in the representation of fluid domains. CNNs require data to be laid out on structured grids, which limits their application to simple geometries. This becomes problematic for real-world fluid problems, which often involve irregular shapes like airplane wings or internal channels. GNNs can handle unstructured meshes and irregular domains better, but they introduce their own difficulties, such as defining meaningful message-passing rules that respect the physics of turbulence and maintaining numerical stability during training and inference.

[0008] Both CNNs and GNNs also struggle with generalization. A model trained on one type of turbulent flow may not perform well on a different type, especially if the Reynolds number, boundary conditions, or geometry changes. This lack of robustness makes it hard to rely on neural networks for practical engineering predictions without significant retraining or fine-tuning. Additionally, these networks can sometimes learn spurious202416541 correlations that do not reflect the underlying physics, leading to unphysical predictions. Improved systems are desirable.202416541SUMMARY OF THE DISCLOSURE

[0009] Various disclosed embodiments include a method for turbulent flow prediction that can be performed by one or more computer systems. The method includes receiving input features for which turbulent flow prediction is to be produced. The method includes modeling turbulent flow using a physics-informed neural network (PINN) according to the input features to generate an output from the PINN. The method includes producing the output, wherein the output includes physical prediction values corresponding to the input features.

[0010] In various embodiments, the method also includes performing automatic differentiation functions to produce data loss and PDE loss. In various embodiments, the method also includes back-propagating data loss and PDE loss to the PINN to refine the output.

[0011] In various embodiments, the input features include physical parameters and geometry data. In various embodiments, the input features include geometry data including global geometric parameters, spatial and temporal coordinates, and an SDF field. In various embodiments, the input features include one or more of geometry shapes, sampling point clouds, and signed distance functions.

[0012] In various embodiments, the PINN models the input features using a continuity equation. In various embodiments, the PINN models the input features using a Navier- Stokes equation. In various embodiments, the PINN models the input features using a RANS model.

[0013] Disclosed embodiments also include a computer system having a processor and an accessible memory, configured to perform a method as discussed herein. Disclosed embodiments also include a non-transitory computer-readable medium encoded with executable instructions that, when executed, cause one or more computer systems to perform a method as discussed herein.

[0014] The foregoing has outlined rather broadly the features and technical advantages of the present disclosure so that those skilled in the art may better understand the detailed202416541 description that follows. Additional features and advantages of the disclosure will be described hereinafter that form the subject of the claims. Those skilled in the art will appreciate that they may readily use the conception and the specific embodiment disclosed as a basis for modifying or designing other structures for carrying out the same purposes of the present disclosure. Those skilled in the art will also realize that such equivalent constructions do not depart from the spirit and scope of the disclosure in its broadest form.

[0015] Before undertaking the DETAILED DESCRIPTION below, it may be advantageous to set forth definitions of certain words or phrases used throughout this patent document: the terms “include” and “comprise,” as well as derivatives thereof, mean inclusion without limitation; the term “or” is inclusive, meaning and / or; the phrases “associated with” and “associated therewith,” as well as derivatives thereof, may mean to include, be included within, interconnect with, contain, be contained within, connect to or with, couple to or with, be communicable with, cooperate with, interleave, juxtapose, be proximate to, be bound to or with, have, have a property of, or the like; and the term “controller” means any device, system or part thereof that controls at least one operation, whether such a device is implemented in hardware, firmware, software or some combination of at least two of the same. It should be noted that the functionality associated with any particular controller may be centralized or distributed, whether locally or remotely. Definitions for certain words and phrases are provided throughout this patent document, and those of ordinary skill in the art will understand that such definitions apply in many, if not most, instances to prior as well as future uses of such defined words and phrases. While some terms may include a wide variety of embodiments, the appended claims may expressly limit these terms to specific embodiments.202416541BRIEF DESCRIPTION OF THE DRAWINGS

[0016] For a more complete understanding of the present disclosure, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, wherein like numbers designate like objects, and in which:

[0017] FIG. 1 illustrates a block diagram of a computer system in which an embodiment can be implemented;

[0018] FIG. 2 illustrates a PINN architecture in accordance with disclosed embodiments; and

[0019] FIG. 3 illustrates a flowchart of a process in accordance with disclosed embodiments.202416541DETAILED DESCRIPTION

[0020] FIGS. 1 through 3, discussed below, and the various embodiments used to describe the principles of the present disclosure in this patent document are by way of illustration only and should not be construed in any way to limit the scope of the disclosure. Those skilled in the art will understand that the principles of the present disclosure may be implemented in any suitably arranged device. The numerous innovative teachings of the present application will be described with reference to exemplary non-limiting embodiments.

[0021] Design exploration or optimization using computational fluid dynamics (CFD) is commonly used in the industry. Geometric variation is a key component of such design problems, especially in turbulent flow scenarios, which involves running costly simulations at every design iteration.

[0022] Physics-Informed Neural Networks (PINNs) can be used to embed governing partial differential equations (PDEs) into the loss function, ensuring that the models comply with the fundamental physical laws, such as the Navier-Stokes equations. This physicsbased regularization allows PINNs to predict the spatiotemporal evolution of flow variables while maintaining consistency with the underlying conservation laws, making them especially useful when dealing with sparse or noisy data. By incorporating physical constraints, PINNs maintain a balance between data-driven model accuracy and the theoretical rigor of traditional approaches.

[0023] Disclosed embodiments include a novel geometry-aware parametric PINN surrogate model that can predict turbulent flows. Disclosed embodiments overcome limitations of relying on the Reynolds-stress formulation, which demands high-fidelity data from costly simulations such as Direct Numerical Simulations, Large Eddy Simulations, or high-resolution flow measurements, and therefore has limited scalability to practical applications.

[0024] Disclosed embodiments improve upon other approaches by using Signed Distance Functions (SDFs) to incorporate geometry into PINN-based RANS modeling. By embedding SDFs into the PINN framework, disclosed systems and methods enhance202416541 prediction accuracy over complex geometries and generalize flow reconstruction to novel geometries of the solution domain.

[0025] Specific embodiments can predict flow fields for NACA 4-digit airfoils in turbulent conditions, for unseen shapes as well as inlet flow conditions. NACA airfoils are a series of airfoil shapes developed by the National Advisory Committee for Aeronautics (NACA) in the early to mid-20th century to provide standardized, mathematically defined profiles for aircraft wing design. Each NACA airfoil is identified by a code that describes its geometric properties, such as maximum camber, location of maximum camber, and thickness. For example, the NACA 2412 airfoil has a maximum camber of 2 percent located 40 percent back from the leading edge, and a maximum thickness of 12 percent of the chord. These airfoils were created to simplify aerodynamic analysis and testing, and many are still widely used today in both experimental and production aircraft.

[0026] Disclosed embodiments use a combination local and global approach for embedding, where known global design parameters for an airfoil as well as local SDF values can be used as inputs to the model along with velocity inlet / Reynolds number (Re) to predict the flow fields. A RANS formulation of the Navier-Stokes equations with a two- equation k-epsilon turbulence model can be used for the PDE losses, in addition to limited CFD data from multiple, such as six, different NACA airfoils for training the PINN model.

[0027] FIG. 1 illustrates a block diagram of a computer system in which an embodiment can be implemented, for example as a computer system particularly configured by software or otherwise to perform the processes as described herein, and in particular as each one of a plurality of interconnected and communicating systems as described herein. The computer system depicted includes a processor 102 connected to a level two cache / bridge 104, which is connected in turn to a local system bus 106. Local system bus 106 may be, for example, a peripheral component interconnect (PCI) architecture bus. Also connected to local system bus in the depicted example are a main memory 108 and a graphics adapter 110. The graphics adapter 110 may be connected to display 111.

[0028] Other peripherals, such as local area network (LAN) / Wide Area Network / Wireless (e.g. WiFi) adapter 112, may also be connected to local system bus 106.202416541Expansion bus interface 114 connects local system bus 106 to input / output (I / O) bus 116. I / O bus 116 is connected to keyboard / mouse adapter 118, disk controller 120, and I / O adapter 122. Disk controller 120 can be connected to a storage 126, which can be any suitable machine usable or machine readable storage medium, including but not limited to nonvolatile, hard-coded type mediums such as read only memories (ROMs) or erasable, electrically programmable read only memories (EEPROMs), magnetic tape storage, and user-recordable type mediums such as floppy disks, hard disk drives and compact disk read only memories (CD-ROMs) or digital versatile disks (DVDs), and other known optical, electrical, or magnetic storage devices.

[0029] Storage 126 can store any data necessary or useful for carrying out processes as described herein, such as executable code 152, PINN 154 (which can be, for example, a PINN architecture 200), geometry data 156, model(s) 158, and other data 160.

[0030] Also connected to I / O bus 116 in the example shown is audio adapter 124, to which speakers (not shown) may be connected for playing sounds. Keyboard / mouse adapter 118 provides a connection for a pointing device (not shown), such as a mouse, trackball, trackpointer, touchscreen, etc.

[0031] Those of ordinary skill in the art will appreciate that the hardware depicted in FIG. 1 may vary for particular implementations. For example, other peripheral devices, such as an optical disk drive and the like, also may be used in addition or in place of the hardware depicted. The depicted example is provided for the purpose of explanation only and is not meant to imply architectural limitations with respect to the present disclosure.

[0032] A computer system in accordance with an embodiment of the present disclosure includes an operating system employing a graphical user interface. The operating system permits multiple display windows to be presented in the graphical user interface simultaneously, with each display window providing an interface to a different application or to a different instance of the same application. A cursor in the graphical user interface may be manipulated by a user through the pointing device. The position of the cursor may be changed and / or an event, such as clicking a mouse button, generated to actuate a desired response.202416541

[0033] One of various commercial operating systems, such as a version of Microsoft Windows™, a product of Microsoft Corporation located in Redmond, Wash, may be employed if suitably modified. The operating system is modified or created in accordance with the present disclosure as described.

[0034] LAN / WAN / Wireless adapter 112 can be connected to a network 130 (not a part of computer system 100), which can be any public or private computer system network or combination of networks, as known to those of skill in the art, including the Internet. Computer system 100 can communicate over network 130 with server system 140, which is also not part of computer system 100, but can be implemented, for example, as a separate computer system 100.

[0035] A PINN machine learning model as disclosed herein incorporates physical laws, typically expressed as partial differential equations PDEs, directly into the training process. Unlike traditional neural networks that rely purely on data, PINNs are trained to satisfy both observational data and the governing equations of the physical system. This is achieved by constructing a loss function that penalizes deviations from known data points and also from the residuals of the PDEs, boundary conditions, and initial conditions. As a result, PINNs can generalize well from sparse data while respecting known physics, making them especially useful for problems in fluid dynamics, heat transfer, and other areas governed by well-established equations.

[0036] The structure of a PINN in accordance with disclosed embodiment is a feedforward neural network that takes spatial and temporal coordinates (such as x, y coordinates, global geometric parameters, physical parameters such as inlet velocity, and SDF field as inputs and outputs the predicted physical quantities, including velocity and pressure.

[0037] During training, automatic differentiation can be used to compute the derivatives of the network output with respect to its inputs. These derivatives are then plugged into the governing equations, such as the Navier-Stokes equations, and used to calculate a physicsbased loss term. The total loss combines this term with the traditional data-driven loss, and optimization is carried out using gradient descent or similar techniques.202416541

[0038] In particular, various embodiments use an SDF -based geometric embedding along with a set of physically interpretable shape parameters to predict flow fields over geometry, such as any NACA 4-digit airfoil geometry, and for any value of the inlet velocity.

[0039] FIG. 2 illustrates a PINN architecture 200 in accordance with disclosed embodiments. As illustrated in FIG. 2, PINN architecture 200 includes geometry data 202, which can include geometry shape(s) 204, sampling point cloud(s) 206, and SDFs 208.

[0040] PINN architecture 200 also includes PINN training function 220, which itself includes the PINN 230. PINN 230 receives input features 240, which can include geometry data 202 and its derived data such as global geometric parameters 242, spatial and temporal coordinates 244, and SDF field 246, along with physical parameters 248 such as inlet velocity.

[0041] PINN 220 performs automatic differentiation functions 232 and produces data loss 234 and PDE loss 236, which are fed back to PINN 200 as back-propagation.

[0042] PINN 220 also produces outputs 250, which in various embodiments can include velocity data 252 and pressure data 254 for the flow over the geometry represented by geometry data 202.

[0043] PINN architecture 200 can generate physical predictions including velocity and pressure predictions for various surfaces, such as NACA airfoils, at different Reynolds numbers.

[0044] According to disclosed embodiments, the constant density fluid flow is governed by the continuity equation (to conserve mass) and Navier-Stokes equation (to conserve momentum). A 2-equation eddy viscosity model describes the closureterms associated with the RANS formulation that is needed to capture the effect of turbulence. In example embodiments, a standard k-e turbulence mode can be used. Symbols U and p denote the flow velocity (x component u and y component v) and pressure, respectively. Continuity and Navier-Stokes equations, at a density of 1 kg / m3 can be expressed as:Continuity: V(t / ) = 0202416541Navier-Stokes: (U • V)t / + Vp — \ief^2U = 0 where V denotes the vector differential operator, and p := p + pt = p + 0.09k2 / e represents the effective viscosity, i.e., the sum of molecular viscosity («) and turbulent viscosity (jut). In addition, the k-e turbulence model can be expressed as: k - equation:e - equation:where, Cl = 1.44, C2 = 1.92, GI< = 1, and= 1.3 are empirical model constants. Pk represents the production term.

[0045] The PINN can create a CFD model with wake refinement and prism layers, and all wall y+ can be created for each NACA airfoil. The PINN can include a velocity inlet boundary condition along with pressure outlets, no-slip walls, and free stream side walls.

[0046] The PINN can be trained, for example, using six randomly-generated Reynolds numbers ranging from 200k to 700k for eight such airfoils.

[0047] For a first model (L), an SDF can be used to embed the distance of each domain point from the geometry, thereby creating an invariant for each shape. In addition to this, the 4 digits from each NACA airfoil can be used as a design input for a second model (L+G). By using the inlet velocity as an additional input to the PINN, the surrogate has been parameterized to Re variation.

[0048] However, in case of a lack of knowledge of such design parameters for an unknown geometry, disclosed embodiments can produce flow prediction with only the SDF fields. These results are comparable to an ablation model (G) which has only the design parameters as an input in addition to the inlet velocity.

[0049] The disclosed PINN architecture can use a warm start approach, where only the data losses are used to pre-train the PINN before introducing the PDE losses, in order to avoid convergence issues with a set of complex governing PDEs.202416541

[0050] The disclosed PINN architecture can model and analyze carious turbulent flow features in a flow over an airfoil case. Due to acceleration of flow over the top surface, a pressure differential is formed, which then results in lift. There is a stagnation zone formed at the nose, and a separation wake at the tail region.

[0051] These features can be modeled captured by the PINN architecture in a local SDF model for all geometries, which can model and produce pressure and velocity predictions as well as corresponding plots and graphics as output. Pressure plots, for example, can show a peak at the nose, representing stagnation, a large low magnitude zone on the upper surface representing flow acceleration, and pressure differential between the two surfaces.

[0052] Disclosed embodiment can also produce output such as error plots that represent where predicted or modeled flows differ from observed flows.

[0053] For example, an output of the PINN architecture can include plots that represent good match between the predicted and true cases, showing areas such as narrow regions of high error at the walls, where the thin turbulent boundary layer is formed.

[0054] Disclosed embodiments produce much more accurate results by embedding local information, such as the geometric boundary information, into the model.

[0055] FIG. 3 depicts a flowchart of a process 300 in accordance with disclosed embodiments that may be performed, for example, by a computer system as disclosed herein implementing a PINN architecture as disclosed herein (generically referred to as the “system” below).

[0056] At 302, the system receives input features for which turbulent flow prediction is to be produced. The input features can include, for example, physical parameters such as an inlet velocity and can include other geometry data. The geometry data can include such elements as geometric shapes, sampling point clouds, signed distance functions, global geometric parameters, spatial and temporal coordinates, and an SDF field. “Receiving,” as used herein, can include loading from storage, receiving from another device or process, receiving via an interaction with a user, or otherwise.202416541

[0057] At 304, the system models turbulent flow using a PINN according to the input features to generate an output from the PINN. The output can include physical prediction values such as velocity and pressure that correspond to the input features.

[0058] At 306, the system can also perform automatic differentiation functions to produce data loss and PDE loss.

[0059] At 308, the system can back-propagate the data loss and PDE loss to the PINN to refine the output.

[0060] At 310, the system produces the output, such as storing the output in a memory or other storage, displaying the output to a user, and / or transmitting the output to another device, system, or process.

[0061] Disclosed embodiments use parametric PINNs to predict flow fields at unseen Reynolds numbers and unseen airfoil geometries for incompressible turbulent flows. Where a conventional RANS formulation is expensive for repeated simulations in case of design exploration, disclosed embodiments achieve a significant technological improvement by leveraging an SDF embedding of airfoil (and other) geometries, in addition to physics losses from RANS formulation with 2-equation turbulent models. Parameterized with respect to varying geometries and inlet velocities, disclosed embodiments can predict flows over unknown velocities and geometries.

[0062] Disclosed embodiments thereby provide distinct technical improvements over other approaches. For example, disclosed embodiments can produce accurate prediction of velocity and pressure fields for any Reynolds number (within the range of prediction) over any given shape of a NACA 4-digit airfoil. Disclosed embodiments can incorporate a k-e turbulence model and RANS based conservation equations to ensure data efficiency and improved generalization.

[0063] Disclosed embodiments can use warm-start based training phase to ensure convergence when dealing with multiple PDE -based loss terms along with limited and shape varying CFD data. Disclosed embodiments improve model performance by taking202416541 into account local and global geometry information in form of SDF fields, and NACA 4- digit shape parameters.

[0064] Of course, those of skill in the art will recognize that, unless specifically indicated or required by the sequence of operations, certain steps in the processes described above may be omitted, performed concurrently or sequentially, or performed in a different order.

[0065] Those skilled in the art will recognize that, for simplicity and clarity, the full structure and operation of all computer systems suitable for use with the present disclosure is not being depicted or described herein. Instead, only so much of a computer system as is unique to the present disclosure or necessary for an understanding of the present disclosure is depicted and described. The remainder of the construction and operation of computer system 100 may conform to any of the various current implementations and practices known in the art.

[0066] It is important to note that while the disclosure includes a description in the context of a fully functional system, those skilled in the art will appreciate that at least portions of the mechanism of the present disclosure are capable of being distributed in the form of instructions contained within a machine-usable, computer-usable, or computer-readable medium in any of a variety of forms, and that the present disclosure applies equally regardless of the particular type of instruction or signal bearing medium or storage medium utilized to actually carry out the distribution. Examples of machine usable / readable or computer usable / readable mediums include: nonvolatile, hard-coded type mediums such as read only memories (ROMs) or erasable, electrically programmable read only memories (EEPROMs), and user-recordable type mediums such as floppy disks, hard disk drives and compact disk read only memories (CD-ROMs) or digital versatile disks (DVDs).

[0067] Although an exemplary embodiment of the present disclosure has been described in detail, those skilled in the art will understand that various changes, substitutions, variations, and improvements disclosed herein may be made without departing from the spirit and scope of the disclosure in its broadest form.202416541

[0068] None of the description in the present application should be read as implying that any particular element, step, or function is an essential element which must be included in the claim scope: the scope of patented subject matter is defined only by the allowed claims. Moreover, none of these claims are intended to invoke 35 USC §112(f) unless the exact words "means for" are followed by a participle. The use of terms such as (but not limited to) “mechanism,” “module,” “device,” “unit,” “component,” “element,” “member,” “apparatus,” “machine,” “system,” “processor,” or “controller,” within a claim is understood and intended to refer to structures known to those skilled in the relevant art, as further modified or enhanced by the features of the claims themselves, and is not intended to invoke 35 U.S.C. §112(f).

Claims

202416541WHAT IS CLAIMED IS:

1. A method (300) for turbulent flow prediction, the method performed by a computer system (100) and comprising: receiving (302) input features (240) for which turbulent flow prediction is to be produced; modeling (304) turbulent flow using a physics-informed neural network (PINN) (230) according to the input features (240) to generate an output (250) from the PINN (230); and producing (310) the output (250), wherein the output (250) includes physical prediction values corresponding to the input features (240).

2. The method of claim 1, further comprising performing (306) automatic differentiation functions (232) to produce data loss (234) and PDE loss (236).

3. The method of claim 1, further comprising back-propagating (308) data loss (234) and PDE loss (236) to the PINN (230) to refine the output (250).

4. The method of any of claims 1-3, wherein the input features (240) include physical parameters (248) and geometry data (202).

5. The method of any of claims 1 -3, wherein the input features (240) include geometry data (202) including global geometric parameters (242), spatial and temporal coordinates (244), and a signed distance function field (246).

6. The method of any of claims 1-3, wherein the input features (240) include one or more of geometry shapes (204), sampling point clouds (206), and signed distance functions (208).2024165417. The method of any of claims 1-3, wherein the PINN (230) models the input features (240) using a continuity equation.

8. The method of any of claims 1-3, wherein the PINN (230) models the input features (240) using a Navier-Stokes equation.

9. The method of any of claims 1-3, wherein the PINN (230) models the input features (240) using a Reynolds-averaged Navier-Stokes (RANS) model.

10. A computer system (100) comprising: a processor (102); and an accessible memory (108), the computer system (100) particularly configured to perform a method (300) according to any of claims 1-3.

11. A non-transitory computer-readable medium (126) encoded with executable instructions (152) that, when executed, cause one or more computer systems (100) to perform a method (300) according to any of claims 1-3.

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