Numerical simulation method using machine learning

Abstract

Numerical simulation method by machine learning. Numerical simulation method (500), for studying a physical system governed by at least one differential equation, comprising a simulation launch step (510), allowing the definition of a simulation domain, and a computation step (520) implementing a machine learning algorithm to predict a global solution of the equation in the simulation domain, the computation step (520) comprising n steps (521) of partitioning a piece in the simulation domain, n being a natural number strictly greater than 1, each followed by a step (522) of predicting a local solution in said piece from local boundary conditions, the prediction step being executed by a machine learning model (100) taking as input global boundary conditions on the simulation domain. Figure for the abstract: Figure 7

Description

Title of the invention: Digital simulation method using machine learning Technical field

[0001] The present invention belongs to the field of digital simulation, in particular the digital simulation of physical phenomena, and relates more particularly to a method of digital simulation, and more specifically of digital calculation, based on machine learning.

[0002] The invention also relates to the field of artificial neural networks implementing deep learning techniques. State of the art

[0003] Numerical simulation consists of running a computer program on a computer in order to simulate a real phenomenon from a theoretical model. This makes it an essential tool for studying complex physical phenomena such as fluid flow. More generally, numerical simulation makes it possible to study different systems in the natural sciences (physics, chemistry, biology, etc.), but also in the human sciences (economics, sociology, etc.).

[0004] As a general rule, numerical simulation is used to obtain an approximation of the solution of an equation when the analytical solution is complex, or even inextricable. For example, numerical simulation is almost essential in the case of nonlinear partial differential equations which govern certain physical principles and phenomenological behaviors, including a large family of transport and propagation phenomena.

[0005] Thus, digital simulation uses digital means implementing algorithms (calculation codes), sometimes sophisticated, for solving this type of equation. The evolution of digital simulation is therefore closely linked to the progress made in the field of computing, as evidenced by recent applications of artificial intelligence and big data in certain digital simulation methods (some patent documents).

[0006] Among the equations whose resolution requires numerical simulation, it is appropriate to cite the Navier-Stokes equations. The set of techniques for numerical resolution of these equations, in the context of fluid simulations, constitutes Computational Fluid Mechanics, commonly designated by its English acronym CFD (Computational Fluid Dynamics).

[0007] Conventional CFD methods are time-consuming and require significant computing resources. Typically, a routine industrial calculation can last between half a day and three months, knowing that a team of engineers in the aeronautics, automotive or related industries generally needs to carry out, on average, tens of thousands of simulations per year for the design, validation and certification phases accompanying the development of a product. As a result, current CFD methods prove to be extremely time-consuming, especially when they are accompanied by pre-processing phases, which are technically complex and therefore costly.

[0008] Despite the drawbacks mentioned above, CFD methods currently remain the most reliable and widely used numerical simulation methods in scientific and industrial circles. They allow very high accuracies to be achieved, but in return require more time and computing resources. In addition, the control of CFD simulations is often complex, which leads to frequent instabilities. In other words, small variations in initial conditions and / or parameters can lead to very different, or even completely different, solutions.

[0009] There is an alternative approach to CFD methods, consisting of developing surrogate models, which are, for example, linearized versions of the CFD equations adapted to specific application cases. This approach can be much faster than the CFD method due to its independent construction of the system studied and its modeling by analytical functions that are easy to evaluate, but it is just as less precise, or even unreliable, especially when the linearization assumptions are no longer respected.

[0010] All these known methods, CFD and substitution models, are based on physical modeling by mathematical formalism and therefore on a resolution of equations. Consequently, they use matrix inversions whose size and complexity depend on the volume and precision of calculation sought.

[0011] Nevertheless, statistical approaches can be deployed to model physical phenomena and rely on predictive models derived from machine learning techniques. These methods include methods based on statistical learning such as kernel methods, evolutionary optimization methods for building surrogate models, simulations by learning implemented by artificial neural networks, etc.

[0012] The advent of machine learning in the field of numerical simulations, particularly CFD, has made it possible to reduce computational time. For example, machine learning can be used to improve the rendering of computer-generated images and thus increase their realism, or to predict fluid motion. In the latter case, the objective of the learning model is to predict the future state of the fluid from its state at a previous time. This is made possible by means of training phases with a large number of CFD solutions. However, these methods are executed over the entire simulation domain in order to obtain a global solution and are therefore constrained by the size of the domain.

[0013] No digital simulation method by machine learning, to the applicant's knowledge, makes it possible to predict the solution locally and subsequently restore the global solution over the entire simulation domain. Presentation of the invention

[0014] The present invention aims to overcome the drawbacks of the prior art set out above, on the one hand the needs of conventional numerical simulation methods, such as CFD methods, in terms of time and computing resources, and on the other hand the limited accuracy of the substitution models. More specifically, an objective of the invention is to propose a numerical simulation method by automatic learning operating a piecewise prediction on the simulation domain to be faster than CFD methods and more accurate than substitution models.

[0015] To this end, the present invention relates to a numerical simulation method for studying a physical system governed by at least one differential equation, comprising a step of launching the simulation, making it possible to define a simulation domain, and a calculation step implementing an automatic learning algorithm for predicting a global solution of the equation in the simulation domain.This method is remarkable in that the calculation step comprises n steps of cutting a piece in the simulation domain, n being a natural integer strictly greater than 1, each followed by a step of predicting a local solution in said piece from local boundary conditions, the n cut pieces covering the entire simulation domain, in that the prediction step is executed by a machine learning model taking as input global boundary conditions on the simulation domain, and in that the global solution is reconstructed from the local solutions.

[0016] Advantageously, the machine learning model is a physically informed deep learning local network, trained using classical numerical simulations.

[0017] More particularly, the local boundary conditions are extracted by the machine learning model from classical numerical simulations divided into samples, each sample being associated with local boundary conditions so as to form training data.

[0018] Advantageously, each piece cut in the simulation domain overlaps with at least one other piece to allow an update of the local boundary conditions.

[0019] According to one embodiment of the invention, the steps of dividing the simulation domain are carried out from left to right and from top to bottom of said domain.

[0020] According to the invention, the calculation step is iterative, the iteration being conditioned by a convergence of the global solution of the numerical simulation.

[0021] Advantageously, the equation governing the simulated physical system is used to define a loss function.

[0022] For example, the differential equation of the physical system is a partial differential equation.

[0023] According to one embodiment of the invention, the physical system is a fluid in motion governed by the Navier-Stokes equations.

[0024] The numerical simulation method thus presented can be implemented by a calculator, such as a computer.

[0025] The invention also relates to a computer program product comprising a set of program code instructions which, when executed by a processor, configure said processor to implement a digital simulation method as presented.

[0026] The fundamental concepts of the invention having just been set out above in their most elementary form, other details and characteristics will emerge more clearly on reading the description which follows and with reference to the appended drawings, giving by way of non-limiting example an embodiment of a method of digital simulation by automatic learning in accordance with the principles of the invention. Presentation of the drawings

[0027] The figures are given purely for illustrative purposes for the understanding of the invention and do not limit the scope thereof. Throughout the figures, identical or equivalent elements bear the same reference sign.

[0028] It is thus illustrated in:

[0029] [Fig.l]: a numerical simulation obtained by a CFD method;

[0030] [Fig.2]: a block diagram of a machine learning model according to the invention for predicting a global solution from global boundary conditions;

[0031] [Fig.3]: a schematic diagram of the extraction of samples from a CFD solution and the association of local boundary conditions;

[0032] [Fig.4]: a simplified diagram of the piecewise simulation method according to the invention;

[0033] [Fig.5]: a schematic representation of a cutting step according to an embodiment of the invention;

[0034] [Fig.6]: a representation of a cutting step, on a simulation domain, according to another embodiment of the invention;

[0035] [Fig.7]: the main steps of a digital simulation method according to an embodiment of the invention;

[0036] [Fig.8]: simulation results of an air flow around an aircraft wing profile obtained by the method of the invention;

[0037] [Fig.9]: Simulation results of an air flow around a vehicle. Detailed description of embodiments

[0038] In the embodiment described below, reference is made to a method of numerical simulation by automatic learning intended mainly for the study of the movement of fluids, movement illustrated by the flow of air around an aerodynamic profile for example. This non-limiting example is given for a better understanding of the invention and does not exclude the use of the method to simulate other physical, economic, or other phenomena governed by differential equations.

[0039] It should be noted at the outset that the invention is based on the fundamental property that any differential equation is verified locally, and can therefore be solved with boundary conditions, in which case it is a question of multi-scale modeling.

[0040] In the following description, we will consider the context of fluid mechanics, for the simulation of fluid flows around obstacles such as aircraft wing profiles. The acronym CFD is used to designate what relates to numerical fluid mechanics and the methods relating thereto. Fluids in motion obey the known Navier-Stokes equations and, depending on the physical variable considered, verify conservation and / or transport laws (advection, diffusion). These are non-linear partial differential equations (abbreviated as PDE).

[0041] [Fig.l] represents a CFD solution of the velocity field, and more precisely of the Vx component of the velocity vector, of the air flow flowing around a profile at zero incidence at a given instant. It is observed that this flow is accelerated on the extrados in a depression zone which, with an overpressure zone at the intrados, allows the range. However, this is not the object of the invention here. Such a CFD solution can be approximated with great precision thanks to machine learning technology.

[0042] The digital simulation method according to the invention is implemented by a machine learning model. This model is for example a deep learning artificial neural network.

[0043] With reference to [Fig.2], the method of the present invention relies on a neural network 100 implementing a machine learning technique, preferably deep learning, capable of locally learning the solution functions of the flow equation to construct a global solution 200 from local predictions as explained later. The network 100 is a physically informed deep learning local network, making it possible to learn and approximate piecewise the solutions of the Navier-Stokes equations without any matrix inversion, unlike conventional numerical simulation methods. In addition, convergence criteria are no longer required.

[0044] The network 100 is fed with CFD solutions, such as the solution of [Fig. 1], during the training phases as illustrated in [Fig. 2]. In the context of the development of the present invention, this network was trained by a massive data set representing tens of thousands of CFD solutions relating to more than 1500 different wing profiles, with a view to the application of said network in the field of aerodynamics to simulate airflow flows around wing profiles of different geometries.

[0045] Further, the physical equations are used to constrain the neural network 100 to improve its prediction as well as the efficiency of its learning as is customary in the art.

[0046] According to an advantageous aspect of the invention, the network 100 takes as input boundary conditions BC extracted from CFD solutions, and other input variables (initial conditions, geometric parameters, and other preprocessing variables), and makes it possible to obtain as output the global solution 200, restored from predicted local solutions, and other output variables including the other physical variables of the equation.

[0047] For example, boundary conditions include a Dirichlet boundary condition, a Neumann boundary condition, or a combination of the two.

[0048] Indeed, the network 100 operates a construction of the global solution 200 from “glued” local solutions 210, each of said local solutions being derived from a prediction of the behavior of the fluid within boundary conditions of a sample extracted from the CFD training solutions of the network.

[0049] [Fig.3] illustrates the extraction of local BC boundary conditions from a CFD solution. Indeed, the CFD solution is cut into pieces, each corresponding to a sample S, then local BC boundary conditions are extracted from each sample S, and will serve as training data in the network 100 when predicting the local solutions 210. In other words, the pairs (sample S - local BC boundary conditions of said sample) will serve as a basis for the classification carried out by the network 100.

[0050] Thus, from a single CFD solution, the network 100 can extract a determined number of samples S to create target local functions corresponding to the local solutions 210. This allows the network 100 to base its learning on elementary geometric patterns, such as portions of simple curvature profiles, to predict the behavior of the fluid around a complete geometry in other simulations.

[0051] [Fig. 4] schematizes the operation of the network 100, according to a simplified example, to generate a global solution from the initial conditions BC as input (see [Fig. 2]). The method implemented by the network 100 corresponds to a piecewise approach in which a first piece T1 is delimited from the boundary of the simulation domain, said piece therefore including a part of the input boundary conditions BC, and associated with a learning pair BC1 - SI representing local boundary conditions BC1 and an associated local solution SI. The local solution SI is therefore injected by inference into the piece T1 of the simulation domain. A second piece T2 overlapping the first piece T1 is then delimited in the simulation domain, the overlap making it possible to update the boundary conditions, and associated with a pair BC2 - S2. The operation is continued until the simulation domain is completely covered.Then, this construction of the piecewise global solution is iterated until convergence.

[0052] Thus, the physically informed deep learning local network 100 carries out a piecewise digital simulation according to a precise division of the simulation domain.

[0053] [Fig. 5] schematizes an example of piecewise simulation on a simulation domain Q, allowing, via a precise division of the domain Q into elementary pieces T and a prediction of the local solution on each of said pieces as the division progresses, to obtain the prediction of the global solution on the entire domain Q. Indeed, according to the simplified example of the figure, a first piece at the top left of the domain Q is initially delimited, the local solution is then estimated there by means of the network 100 by deep learning, the filling points indicating the local solution on each piece concerned. The second piece is delimited in the domain Q and must overlap with the previous piece on a zone Z for an update of the boundary conditions as previously highlighted. In the example of [Fig.5], the division of the domain into pieces is carried out from left to right and from top to bottom, the lines being scanned first, like the relative movement of the air flow around the profile. The accuracy and speed of the calculation depend on the number of pieces, and therefore on the size of the pieces for a given domain. The circular arrow at the bottom right of [Fig.5] indicates that the piecewise simulation operation is iterative.

[0054] [Fig.6] gives another example of the piecewise simulation method according to a different division of the domain, carried out from bottom to top and from left to right, with the columns scanned first.

[0055] The division of the domain in the context of the implementation of piecewise simulation can be carried out in different ways provided that the overlap between the pieces is respected, and more precisely between each piece and a piece cut before, necessary for updating the boundary conditions as the division progresses. For example, the division can be linear from left to right, from top to bottom or vice versa, diagonal or in a spiral converging towards the center of the simulation domain, or even arbitrary. Other ways of segmenting the domain can be used provided that the overlap is respected.

[0056] [Fig.7] represents the main steps of a method 500 of digital simulation by automatic learning, according to an embodiment of the invention, said method comprising: - an initial step 510 for launching the simulation; - an iterative step of piecewise calculation 520, comprising sub-steps of cutting 521 and prediction 522; - an iterative step of reconstitution 530 of the global solution; - a 540 convergence test; and - a post-processing step 550.

[0057] The step 510 of launching the simulation comprises for example the preprocessing, the adjustment of the boundary conditions and the initial conditions (boundary conditions and initial field for example), the adjustment of the physical properties, and the temporal control (adjustment of the time step), depending on the nature of the phenomenon studied. The boundary conditions will make it possible to define the input of the neural network 100 which executes the calculation step 520.

[0058] The piecewise local calculation step 520, shown diagrammatically in Figures 4 to 6, comprises consecutive cutting-prediction operations consisting of predicting a local solution on each piece cut in the simulation domain. In a generic manner, the piecewise local calculation step 520 comprises steps 521 of cutting the piece i, with i being a natural integer between 1 and n inclusive, and n the number of pieces chosen. Each cutting step 521 is followed by a step 522 of predicting the local solution in the cut piece i.

[0059] The global solution (over the entire simulation domain) is then reconstructed during the reconstruction step 530, which may be implicit.

[0060] The piecewise calculation steps 520 and reconstruction 530 are repeated until the global solution converges.

[0061] Calculation is the step requiring the most time and computing resources. Different techniques allow for optimizing the use of computing resources, such as parallel computing.

[0062] The post-processing step 550 finally makes it possible to exploit the results of the digital simulation via physical and / or statistical analyses and corresponds for example to the visualization of different fields of variables (speed, pressure, etc.).

[0063] [Fig.8] represents simulation results of the velocity field of the relative movement of an air flow around a profile, obtained by the machine learning simulation method according to the invention, at different iterations: 1, 25 and 50. In front of each iteration, the boundary conditions of the input velocity, the prediction obtained by the method and the target CFD solution are represented, the latter being the same at all iterations. It can be noted that the difference between iteration 25 and iteration 50 is relatively small and that the convergence of the solution is established quickly thanks to the method of the invention.

[0064] The method has also been extrapolated to other obstacle geometries for satisfactory results. An example of simulation of the air flow around a motor vehicle is given in [Fig.9].

[0065] To train the network 100, any CFD solution can be used and cut into pieces of different sizes to obtain the training samples, said pieces being used as supervised classifiers.

[0066] The physical equations of the theoretical model are used in loss functions in both supervised and unsupervised learning, using the residuals of said equations in the latter case.

[0067] Indeed, if we consider V the unknown vector of the system equation, we can construct a cost function as follows: C “ ^math ' pred-' targ> ^sup ) + dphys ( H ( Vpred ) ~^supH ( V targ ) ) In which, the indices targ and pred correspond respectively to the target solution and the solution predicted by the network, a Sup is a coefficient between 0 and 1 to activate supervised learning, a math is a coefficient between 0 and 1 to add a usual loss in machine learning (L2 or L3 norm for example), a phys is a coefficient between 0 and 1 to activate the physical loss (i.e. the residual of the system equation), and M is a linear operator allowing to couple the different machine learning losses.

[0068] The numerical simulation method according to the present invention has proven to be thirty times faster than a conventional CFD method, with 98% accuracy compared to said CFD method on 128*128 pixel images.

[0069] In view of the description, the machine learning simulation method can be slightly modified and / or adapted without departing from the scope of the invention. This method finds direct, non-limiting applications in technological industries such as aeronautics, space, automobiles, energy, naval, multimedia (video games, special effects, etc.).

Claims

Claims

1. Method for designing a product such as an aircraft wing profile, comprising a computer-implemented numerical simulation method (500) for predicting the movement of a fluid around said product, governed by at least one differential equation, said simulation method comprising a step (510) of launching the simulation, making it possible to define a simulation domain, and a calculation step (520) implementing a machine learning algorithm for predicting a global solution of the equation in the simulation domain, characterized in that the calculation step (520) comprises n consecutive sequences each comprising a step (521) of cutting out a piece in the simulation domain followed by a step (522) of predicting a local solution in said piece from local boundary conditions, n being a natural integer strictly greater than 1, the n cut pieces covering the entire simulation domain,in that the prediction step is executed by a machine learning model (100) taking as input global boundary conditions on the simulation domain, and in that the global solution is reconstructed from the local solutions.,

2. The method of claim 1, wherein the machine learning model (100) is a physically informed deep learning local area network trained using classical numerical simulations.

3. Method according to one of claims 1 or 2, in which the local boundary conditions are extracted by the machine learning model (100) from classical numerical simulations cut into samples, each sample being associated with local boundary conditions so as to form training data.

4. A method according to any preceding claim, wherein each sliced ​​piece in the simulation domain overlaps with at least one other piece to allow updating of local boundary conditions.

5. Method according to any one of the preceding claims, in which the steps (521) of cutting the simulation domain are carried out from left to right and from top to bottom of said domain.

6. Method according to any one of the preceding claims, in which the calculation step (520) is iterative, the iteration being conditioned by a convergence of the global solution.

7. A method according to any preceding claim, wherein the equation is used to define a loss function.

8. A method according to any preceding claim, wherein the differential equation is a partial differential equation.

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