Method of numerically simulating a system, model predictive control, operating a digital twin and virtual sensor

Abstract

The invention relates to a computer-implemented method for numerically simulating a system (SYS) under predefined system (SYS) boundary conditions (BCD), wherein said system (SYS) is comprising at least one physically moving element (MVE), in particular method for simulating a fluid-dynamic system (SYS), the method comprising: (a) generating of an element mesh (LTC) for the purpose of performing the simulation (SIM), wherein the element mesh (LTC) extends across the system (SYS), wherein said at least one physically moving element (MVE) moves relative to elements of the element mesh (LTC) during a period of the simulation (SIM); (b) running said simulation (SIM) to simulate the physical system (SYS) to determine at least one parameter value (PRV) of said system (SYS); (c) using a neural network (NNW), predicting at least one parameter correction value (PCV) representative of an error associated with a solution of the simulation (SIM) for the at least one parameter value (PRV) of said system (SYS); and (d) correcting the solution of the simulation (SIM) for the at least one parameter value (PRV) using the parameter correction value (PCV) to produce a corrected solution of said simulation (SIM) of the physical system (SYS). It is proposed, that said at least one parameter correction value (PCV) is assigned to the at least one physically moving element (MVE) and said at least one parameter correction value (PCV) being transported along said element mesh (LTC) elements together with said at least one physically moving element (MVE).

Description

[0001] Description

[0002] Method of numerically simulating a system, model predictive control, operating a digital twin and virtual sensor

[0003] The invention relates to a computer-implemented method for numerically simulating a system, wherein said system is comprising at least one physically moving element, wherein said method is for simulating a fluid-dynamic system with a flow medium, the method comprising:

[0004] (a) generating of an element mesh for the purpose of performing the simulation, wherein the element mesh extends across the system, wherein said at least one physically moving element moves relative to elements of the element mesh during a period of the simulation;

[0005] (b) running said simulation to simulate the physical system to determine at least one parameter value of said system;

[0006] (c) using a neural network, predicting at least one parameter correction value representative of an error associated with a solution of the simulation for the at least one parameter value of said system; and

[0007] (d) correcting the solution of the simulation for the at least one parameter value using the parameter correction value to produce a corrected solution of said simulation of the physical system.

[0008] Herein and in the full context of this document, the said at least one physical-moving element may be a physical unit or element, which may be ductile or rigid, and in particular it may be a fluid flow particle or volume element or voxel or element. Here, flow particle means a volume or voxel of the flow medium whose material identity does not change. Contrary to the usual use of this term, particle in this context does not refer to the fact that its material necessarily differs from the rest of the flow in the flow.

[0009] Speed, accuracy, robustness, and ease of use determine the value of simulation tools. A continuous endeavor of all computer-aided engineering [CAE] providers is better performance in terms of increasing speed, accuracy, robustness, and ease of use. Speed and accuracy are tightly coupled. Increasing one, typically leads to a decrease in the other. Making a significant advancement in one of these dimensions will directly lead to a better acceptance of the CAE-product, since it will allow to handle more complex engineering problems faster. Furthermore, increasing the efficiency of today’s simulation tools is crucial when it comes to more automated design optimization as recently pointed out by McKinsey & Company in https: / / www.mckinsey.com / capabilities / operations / our-insights / deep-learning-in- product-design.

[0010] Beyond the expressed need of current users, increasing the computational speed offers many opportunities to introduce simulation in new fields - ranging from interactive design tools for designers to operation parallel simulation with an enormous potential market as published under https: / / www.marketsandmarkets.com / Market- Reports / digital-twin-market-225269522.html.

[0011] Pushing the limitations in terms of accuracy and speed is at the heart of many innovation activities in the field of simulation technology. Specifically, Machine Learning [ML] based solutions are prominently positioned in these activities with an enormous potential. In this context 3D Computational Fluid Dynamics [CFD] is one of the most challenging problems. Due to their highly non-linear nature and need to resolve turbulent structures appropriately, CFD problems are among the most computationally resource intensive problems. While the preferred application field of the invention is 3D CFD the invention may as well be applied in other field of technology for example (non-linear) elasticity or electrodynamics.

[0012] From US 10,751 ,879 B2 a method is known combining a physical process simulation with a real-world-data-trained machine learning model correcting the simulation output to control the physical process performance.

[0013] The below references relate to additional background information as outlined further below. [1] X Guo, W Li, F Iorio (2016): Convolutional Neural Networks for Steady Flow Approximation. Autodesk (https: / / damassets.autodesk.net / content / dam / auto- desk / research / publications-assets / pdf / convolutional-neural-networks-for.pdf)

[0014] [2] D Kochkov, JA Smith, A Alieva, Q Wang, MP Brenner, S Hoyer (2021): Machine learning-accelerated computational fluid dynamics.

[0015] PNAS (Google) (https: / / doi.org / 10.1073 / pnas.2101784118)

[0016] [3] D Kochkov et al. (2023): Neural General Circulation Models. arXiv (https: / / ar- xiv.org / abs / 2311.07222)

[0017] [4] A Sanchez-Gonzalez, J Godwin, T Pfaff, R Ying, J Leskovec, PW Battaglia (2020): Learning to Simulate Complex Physics with Graph Networks. DeepMind (https: / / arxiv.org / abs / 2002.09405)

[0018] [5] Bjorn List, LW Chen, N Thuerey (2022): Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimization horizons. Journal of Fluid Mechanics (https: / / doi.org / 10.1017 / jfm.2022.738)

[0019] [6] N Margenberg, D Hartmann, C Lessig, T Richter (2022): A neural network multigrid solver for the Navier-Stokes equations. Journal of Computational Physics (https: / / doi.Org / 10.1016 / j.jcp.2O22.110983)

[0020] Next to many other approaches, in the last decade many ML concepts have been proposed to improve computational speed and accuracy of 3D simulation tools, e.g., [1] [2] [3] [4] [5] [6], Out of the many approaches, solver in the loop approaches, combining classical solvers with ML enrichments, seem to be the most promising ones [2][5][6],

[0021] The publication of CORNELL UNIVERSITY LIBRARY, 2023, HONGWEI TANG ET AL, "Discovering explicit Reynolds-averaged turbulence closures for turbulent sep- arated flows through deep learning-based symbolic regression with non-linear corrections" deals with applying the Reynolds-averaged Navier-Stokes equations [RANS] to solve turbulent flow problems.

[0022] A major challenge in all state-of-the-art approaches addressing CFD-applications is the need to resolve the Lagrangian nature within Eulerian formulations. That is, while the physics, and as such any intrinsic properties as turbulent structures in the case of CFD, are transported or advected with the fluid or material, any information is stored usually in a grid fixed in space and not moving with the fluid.

[0023] Since many fluid properties, e.g., due to turbulence, usually have a non-linear behavior and depend on the history of the flow, e.g., what has happened further down a streamline or pathline, Machine Learning approaches need to take these non-linear history-dependent relationships into account. Therefore, existing approaches, e.g. [1][2][5], often consider Convolutional Neural Networks [CNN] for determining a local value based in a larger neighborhood. Herein, the convolution typically covers a neighborhood of 10+ discretization points in any spatial direction. This allows to access corresponding information further down pathlines or streamlines.

[0024] Alternatively, an explicit history dependence can be introduced via corresponding Neural Networks [NN], as outlined in [6] such as recurrent Neural Networks [RNN], Gated Recurrent Units Neural Networks [GRU], or Long Short-Term Memory Neural Networks [LSTM],

[0025] While these approaches work well in reference implementations of CFD solvers, they fail in industrial high end CFD solvers. Given the computational requirements of industrial CFD problems, which are typically facing memory limitations as well as the necessity of parallel computing on distributed processors or clusters as in High Performance Computing, most solvers take a matrix free approach and provide only efficient access to information for nearest neighbors and only very few steps back in time. Therefore, state of the art approaches based on convolutional NN as well as history dependent neural networks cannot be employed. The invention is based on the problem of improving the known method such that accurate determination results are obtained faster. The invention aims to avoid unreasonable long time for technical calculations and increasingly high computational power demand for technical determination of system parameters in particular in the field of CFD as outlined above.

[0026] Another object of the invention is to enable an iterative design optimization.

[0027] The term “mesh” as used herein refers to the discretization or subdivision of the system into smaller elements for numerical analysis. The mesh may be structured into mesh elements or an element mesh, wherein the single mesh elements are not necessarily shaped identically but can have various shapes and configurations, such as triangles, quadrilaterals, tetrahedra, hexahedra, or a combination of these.

[0028] The at least one parameter value may be the value of any parameter or physics variable or any other entity which belongs to the results of the simulation.

[0029] The neural network preferably generates the parameter correction value on the basis of at least one of:

[0030] - the boundary conditions of the simulation, wherein said boundary conditions include the full system description like the geometry, the initial or starting conditions of the system, the factors entering the system during the simulation period, like material / fluid inflow, energy entering the system and the like.

[0031] - the current parameter value of the simulation,

[0032] - any other part of the current simulation solution.

[0033] To solve the objective the invention proposes a method defined at the beginning is proposed, characterized in that said at least one parameter correction value is as- signed to the at least one physically moving element and said at least one parameter correction value being transported along said mesh elements together with said at least one physically moving element.

[0034] The at least one parameter correction value is a numerical value that is used to correct the one or more parameter values determined by the simulation. Here, the parameter correction value can serve as a difference that corrects the parameter value by summing, or as a correction parameter that is used as a multiplication factor, or as a correction parameter that corrects the parameter directly or indirectly by any other mathematical operation.

[0035] One preferred embodiment of the invention provides that the neural network is a variant of a Long Short-Term Memory neural network [LSTM], This embodiment proposes using a method applying a NN-architecture which takes into account in addition to the short-term physical states u(t) also long-term states c(t). By assigning the long-term state c(t) as a parameter correction value to a physically moving element the long-term state c(t) is transported with the flow along said mesh elements. This feature makes the new method well suited for hybrid ML-approaches for improvement of speed and accuracy of 3d multi-physics applications modelled, which may be formulated in Eulerian coordinate systems preferably for CFD. The LSTM is a type of recurrent neural network [RNN] architecture that is particularly effective in handling and processing sequential data. The LSTM is designed to address the limitations of traditional RNNs, which struggle with capturing long-term dependencies in sequences. Conventionally LSTMs are used in natural language processing, speech recognition, time series analysis or machine translation. The key feature of LSTMs is their ability to retain information over long sequences by using memory cells and gates. These memory cells allow LSTMs to selectively remember or forget information from previous time steps, enabling them to capture dependencies over longer time horizons.

[0036] One preferred embodiment of the invention provides that said LSTM architecture comprises an update gate, and an output gate. This update gate enables to adjust the long-term memory to long term changes of the physical system. Preferably the LSTM architecture comprises three main components: the input gate, the forget gate, and the output gate. These gates control the flow of information into, out of, and within the memory cell.

[0037] At each time step, the input gate determines which parts of the input should be stored in the memory cell. The forget gate decides which information from the previous time step should be discarded or forgotten. The output gate determines which parts of the memory cell should be outputted as the final prediction.

[0038] In LSTM architectures the long-term state undergoes some updating or evolution handled by so-called forget and input-gates. The role of the forget gate is to decide how much information from the previous time step is kept and the input gate inserts new information to the long-term memory.

[0039] Moreover, the consideration of a peephole architecture as outlined in FA Gers, J Schmidhuber (2000), IEEE: “Recurrent nets that time and count” may be beneficial. In such peephole architecture the update of the long-term states also takes the long term-state itself into account. Furthermore, an optimal implementation will rather use a single update gate combining the forget and input gate into a single update gate, like in the approach of GRUs es explained in more detail in arXiv: K Cho, B van Merrienboer, C Gulcehre, D Bahdanau, F Bougares, H Schwenk, Y Bengio (2014): “Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation”.

[0040] By allowing for the retention of relevant information over a longer time period or long sequences, LSTMs are very suitable for being the memory of the moving element or the fluid particle since they are effective in modeling and predicting sequential data.

[0041] According to a preferred embodiment the LSTM may be trained using backpropa- gation through time, which may be an extension of the backpropagation algorithm for training recurrent networks. During training, the network learns to adjust the weights and biases of the LSTM cells to optimize the prediction performance on the given task.

[0042] Thereby the short-term physical states as well as the long-term states are defined cell-wise or discretization-point-wise. Given the Lagrangian nature of CFD- problems, short-term and long-term physical states are transported with the flow. The invention may be categorized as a fusion of hybrid ML architectures enhancing simulation algorithms for 3d multi-physics problems formulated in a Eulerian coordinate system with LSTM-type architectures.

[0043] One essential feature in this context is the transport of the flow particle memory. According to the essential properties of the LSTM herein proposed to be used in simulation, in particular in dynamic flow simulation, the specific NN architecture according to the invention may hereinafter be referred to as Transported Memory Neural Networks [TMNN],

[0044] A preferred embodiment of the invention provides a memory-behavior of the respective flow element or flow particle. Basically, the history of the flow particle is memorized or better be transported with the flow or flow particle along the flow or flow lines. The herein explained flow-particle memory or transport gate may be understood as a complementary function to the forget gate of a general LSTM.

[0045] Generally, said numerical simulation may be based on at least one governing equation. This may be a single equation - in most cases a partial differential equation [PDE] - like the Navier-Stokes-equation or it may be a set of equations in particular PDEs.

[0046] According to a preferred embodiment transporting said at least one parameter correction value moving along said element mesh elements together with said at least one physically moving element can be realized such that as an advection equation may be inserted into the at least one governing equation. Said advection equation may be expressed like: at c(t)=u(t)-vc(t))

[0047] Possible advection equations may be, wherein C is the quantity being transported and t represents time:

[0048] - Linear advection equation, where the velocity field is assumed to be constant: ac / at + c ac / ax = o wherein c represents the constant velocity of the fluid flow in the x-direction.

[0049] - Non-linear advection equation: ac / at + v(C) ac / ax = o wherein V(C) is the velocity field that depends on a quantity C.

[0050] - multidimensional advection equation wherein the advection equation may be extended to multiple spatial dimensions: ac / at + v • vc = o wherein, V is the velocity vector and VC represents the gradient of C with respect to spatial coordinates.

[0051] Another preferred embodiment of the invention provides a diffusive term being inserted into the at least one governing equation, wherein said diffusive term is a derivative of the parameter correction value with respect to space: au / at = a (a2u / ax2 + a2u / ay2 + a2u / az2). Beneficially adding this at least one diffusive term into the at least one governing equation spreads the information from the corrective parameter term, which stabilizes the intended transport operation. The diffusive term could be also spatially dependent, i.e.,:

[0052] V • (K • Vu) with a matrix K depending on other quantities, e.g. on spatial position, fluid velocity of the quantity c itself.

[0053] Based on the underlying idea of the invention of including neural network-elements or learnable components into the governing equation giving the moving element a transported memory e.g. turbulent flows may be simulated with less computer power requirement accurately.

[0054] According to one embodiment it is proposed that the method further comprises a training step wherein the training comprising:

[0055] - providing reference data as training data by comprising predefined system boundary conditions and a system simulation solution including said at least one parameter value;

[0056] - training said neural network predicting said at least one parameter correction value while said neural network is integrated in said system simulation, which training is performed using said reference data.

[0057] Another teaching of the invention relates to a method of controlling a system by model predictive control [the control method]. This model predictive control of a system, wherein said system is comprising at least one physically moving element, in particular, wherein said at least one moving element is a fluid flow, comprises:

[0058] (i) defining a control objective;

[0059] (ii) defining a system parameter to be a control parameter;

[0060] (iii) defining a control action.

[0061] Furthermore, the method of controlling a system by model predictive control comprises: (iv) simulating the system answer to said control action according to a method according to the herein explained method of numerically simulating a system;

[0062] (v) evaluating the simulated system answer as to whether the control objective was reached by the defined control action according to said simulation;

[0063] (vi) repeating steps (iii) - (v) with variations of said control action until a predefined termination criterion is fulfilled in such a way that the control objective is sufficiently achieved by the control action;

[0064] (vii) controlling said system on basis of the last control action of step (vi).

[0065] According to one embodiment the control method further comprises:

[0066] - measuring at least one physical parameter value of said system;

[0067] - using the measured physical parameter value as a boundary condition for the simulation or for calibrating and / or training the neural network.

[0068] Regarding the control method, all explanations given with regard to the computer- implemented method for simulation are fully applicable.

[0069] Another teaching according to the invention relates to a method for operating a digital twin of a physical system, wherein said digital twin is based on a computer- implemented method for numerically simulating a system as explained herein.

[0070] All explanations given with regard to the computer-implemented method and said control method are fully applicable for the method for operating a digital twin.

[0071] Another teaching according to the invention relates to a virtual sensor being applied to a physical system, the virtual sensor comprising a processor being configured to operate a digital twin of said physical system according to said method for operating a digital twin of a physical system as explained herein. The virtual sensor is configured such that an output of said sensor is a parameter being determined as an output of the simulation of the physical system. Herein a virtual sensor may be understood as a software-based system that utilizes a model, in particular said simulation model to predict a parameter, meaning a physical quantity which would be a result of a real measurement. The virtual sensor can simulate the behavior of a physical sensor without the need for an actual physical sensor device at the location of the parameter determination. By analyzing the available data from other sources, such as existing real sensors, virtual sensors can provide real-time or historical information on various parameters, such as temperature, pressure, humidity, or even complex variables like energy consumption or equipment performance. This enables ‘measurements’, where sensors may be difficult to install, expensive, or impractical.

[0072] All explanations given with regard to the simulation method, the control method or the digital twin are fully applicable with regard to the virtual sensor.

[0073] The properties, features and advantages of this invention described above, as well as the manner they are achieved, become clearer and more understandable in the light of the following description and embodiments, which will be described in more detail in the context of the drawings. This following description does not limit the invention on the contained embodiments. Same components or parts can be labeled with the same reference signs in different figures. In general, the figures are not for scale. It shall be understood that a preferred embodiment of the present invention can also be any combination of the dependent claims or above embodiments with the respective independent claim.

[0074] Figure 1 : shows a schematic illustration of a method of numerically simulating a system, model predictive control, operating a digital twin and virtual sensor;

[0075] Figures 2, 3, 4 respectively show examples of how the parameter correction value is assigned to the fluid flow;

[0076] Figure 5 shows an initialization of the parameter correction value.

[0077] Figures 6, 7 show potential architectures for individual gates.

[0078] Figures 8, 9, 10 show potential architectures for individual gates. Detailed Description

[0079] Figure 1 shows a schematic illustration of a computer-implemented method for numerically simulating a system SYS under predefined system SYS boundary conditions BCD.

[0080] The numerical simulation SIM is based on at least one governing equation GVE.

[0081] The system SYS comprises a physically moving element MVE, which is herein a fluid-dynamic system SYS, respectively a fluid flow along an obstacle OBC.

[0082] A first step (a) of the method provides generating of an element mesh LTC for the purpose of performing the simulation SIM. The element mesh LTC extends across the system SYS. The physically moving element MVE moves relative to elements of the element mesh LTC during a time period of the simulation SIM.

[0083] During a second step (b) the simulation SIM to simulate the physical system SYS is done to determine one or several parameter values PRV or parameter fields of said system SYS. This may be for example a velocity field, the overall mass flow across the system, the viscosity anywhere or everywhere or the Reynolds-number or the type of the flow, e.g. laminar or turbulent.

[0084] During a third step (c) using a neural network NNW, which is a Long-Short-Term- Memory LSM neural network NNW, a parameter correction value PCV is predicted corresponding to said parameter value to be determined by said simulation. The parameter correction value PCV is representative of an error associated with a solution of the simulation SIM for the parameter value PRV.

[0085] During a next step (d) the solution of the simulation SIM for the at least one parameter value PRV is corrected using the parameter correction value PCV. This way a corrected solution of said simulation SIM of the physical system SYS is obtained. This correction isn’t done in a conventional - kind of static - way but the transportation of this information by the fluid flow is considered as well. In this example the parameter correction value PCV is transported together with the flow (=> said at least one physically moving element MVE). The transport is done by insertion of the parameter correction value PCV via an advection equation term AET in at least one governing equation GVE of the simulation SIM. Furthermore, a diffusive term DFT containing the parameter correction value PCV is inserted into the at least one governing equation GVE of the simulation SIM.

[0086] One essential point is that the parameter correction value PCV is assigned to the movement of the flow such that said at least one parameter correction value PCV is transported along said element mesh LTC elements together with the fluid flow (=>said at least one physically moving element MVE).

[0087] To prepare the simulation with the Long-Short-Term-Memory LSM neural network NNW for the determination of system parameters the method further comprises a training TRN step. The training TRN comprises - as shown in figure 1 - the following steps:

[0088] - providing reference data RFD as training data by comprising predefined system SYS boundary conditions BCD and a system SYS simulation SIM solution including said at least one parameter value PRV;

[0089] - training said neural network NNW predicting said at least one parameter correction value PCV while said neural network NNW is integrated in said system SYS simulation SIM, which training is performed using said reference data RFD.

[0090] Figure 1 further illustrates a method of controlling a system by model predictive control MPC. Here, a fluid flow (=>MVE) is controlled via an actuator CTA, which is a valve. In detail the method comprises:

[0091] (i) defining a control objective CTO;

[0092] (ii) defining a system SYS parameter to be a control parameter CTP;

[0093] (iii) defining a control action CTA; characterized in (iv) simulating the system SYS answer ANW to said control action CTA according to a method according to at least one preceding claim 1-6;

[0094] (v) evaluating the simulated system SYS answer ANW as to whether the control objective CTO was reached by the defined control action CTA according to said simulation SIM;

[0095] (vi) repeating steps iii - (v) with variations of said control action CTA until a predefined termination criterion TRC is fulfilled in such a way that the control objective CTO is sufficiently achieved by the control action CTA;

[0096] (vii) controlling said system SYS on basis of the last control action CTA of step (vi).

[0097] The control objective CTO may be for example a certain mass flow through the system SYS or the avoidance of turbulence.

[0098] To improve accuracy the method of controlling a system according to comprises further:

[0099] - measuring MES at least one physical parameter value PRV of said system SYS (real);

[0100] - using the measured physical parameter value PRV as a boundary condition BCD for the simulation SIM or for calibrating and / or training the neural network NNW.

[0101] Figure 1 further shows a method for operating a digital twin DTW of a physical system SYS, wherein said digital twin DTW is based on a computer-implemented method for numerically simulating a system SYS according as explained above.

[0102] A virtual sensor VRS is provided to the system SYS measuring the system answer to the control action CTA. The virtual sensor VRS comprises a processor PRC configured to operate a digital twin DTW of said physical system SYS as illustrated in figure 1 . An output OTP of said sensor is a parameter being determined as an output OTP of the simulation SIM of the physical system SYS.

[0103] The Long-Short-Term-Memory (LSM) neural network (NNW) requires permanent updating the long-term-memory. This is shown in Figure 2. The LSTM architecture comprises an update gate, and an output gate. The update gate changes the longterm memory according to long term developments of the physical system. In detail the LSTM architecture comprises three main components: the input gate and the forget gate as part of the update gate, and the output gate, wherein the input gate selects the inputs to be stored in the memory cell. The forget gate discards information from the previous time step from the memory. The output gate selects the parts of the memory cell to be outputted as the prediction.

[0104] Figures 2, 3, 4 respectively show examples of how the parameter correction value PCV is assigned to the at least one physically moving element MVE, respectively to the fluid FLD flow such that said correction value PCV being transported along said element mesh LTC elements together with said at least one physically moving element MVE (=> fluid FLD flow).

[0105] In all figures 2, 3, 4 a time step (t => t+Dt) of the method according to the invention and the integrated simulation SIM is illustrated. Herein c(t) represents a long-term- state and u(t) a short-term-state. The Long-Short-Term-Memory LSM neural network NNW receives boundary condition BCD - like geometry information and the current simulation SIM solution to generate an updated parameter correction value PCV to correct the at last one parameter value PRV.

[0106] The transportation via the fluid FLD or the moving element MVE is input into the neural network NNW. In figure 2 a learned correction is performed while in figure 3 a learned interpolation is done.

[0107] Figures 2, 3 respectively show an extension of the architectures introduced in above referenced [2],

[0108] In an LSTM architecture, the short-term state is updated by means of the output gate I network. But in the context of hybrid ML architectures enhancing simulation (SIM) solvers, the output step corresponds to the enhancement step. E.g., it could correspond to the so-called Learned Correction or Learned Interpolation as illustrated in figures 2, 3. In the Learned Correction (fig. 2) for example an additional NN term may be added to the momentum part of the Navier-Stokes equation. In the Learned Interpolation (fig. 3), the underlying advection is modified by means of NNs. The two variants are well suited for the correction of 3d multi-physics simulations SIM. The new architecture requires an additional update gate or update network as described above. Other variants of this architecture are possible as well, e.g., the update network coming further down in the flow using already updated physics u(t+At), while the correction works with advected c(t) only.

[0109] Figure 4 shows a more general architecture as a most preferred implementation according to the invention, wherein the simulation incorporates an integrated parameter correction value PCV-process.

[0110] Figure 5 shows an initialization process. The long-term state c(t) may initially c(t=O) be not defined, such that an initialization process as illustrated can be performed. The initial long-term-state c(t=O) may be determined from boundary conditions BCD (such as geometry g) and from the current short-term-state u(t=O).

[0111] Figures 6 -10 relate to possible LSTM architectures. The dependence of the various gates I network on the different states can be multifold, including the shortterm physical state, the long-term state, a state representing the geometry, as well as other potential states such as parameters.

[0112] As for other LSTM variants (e.g., Convolutional LSTMs) the networks can depend on the local as well as neighboring cells I discretization points (ideal case). The exact nature of this dependence can vary from situation to situation including an on- the-fly calculation of appropriate (learned) features (e.g., discretely evaluated gradients - ideal case)

[0113] The individual gates or networks can be of different natures as shown in Figures 6, 7. Figures 6, 7 both illustrate simplified the contribution of said Long-Short-Term- Memory LSM neural network NNWwhen assigning parameter correction values PCV to single mesh LTC elements. The Long-Short-Term-Memory LSM neural network NNW may - next to its own current state OCS - take neighboring states into account and / or a graph neural network may take neighboring states NST into account and / or a multi-layer perceptron may take neighboring states into account via appropriate on-the fly calcu- lations CLC, e.g., discretely evaluated gradients GRD.

[0114] Figures 8, 9, 10 show potential architectures of neural network NNW for individual gates as possible implementations of the invention. Input into the gates are preferably the long-term-state c(), the short-term-state u() and the geometry GMT. If tak- ing multiple states into account, the states can be simply concatenated, but an ideal realization would consider an individual pre-processing of the different states which are then combined by means of a Hadamard product (as symbolized in the figures 8, 9, 10 by “X”). Most favorable neural network NNW layers are added as shown in figure 9 or 10.

[0115] Independent of the grammatical term usage, individuals with male, female or other gender identities are included within the term.

Claims

Patent claims1. A computer-implemented method for numerically simulating a system (SYS) under predefined system (SYS) boundary conditions (BCD), wherein said system (SYS) is comprising at least one physically moving element (MVE), wherein said method is for simulating a fluid-dynamic system (SYS) with a flow medium, the method comprising:(a) generating of an element mesh (LTC) for the purpose of performing the simulation (SIM), wherein the element mesh (LTC) extends across the system (SYS), wherein said at least one physically moving element (MVE) moves relative to elements of the element mesh (LTC) during a period of the simulation (SIM);(b) running said simulation (SIM) to simulate the physical system (SYS) to determine at least one parameter value (PRV) of said system (SYS);(c) using a neural network (NNW), predicting at least one parameter correction value (PCV) representative of an error associated with a solution of the simulation (SIM) for the at least one parameter value (PRV) of said system (SYS); and(d) correcting the solution of the simulation (SIM) for the at least one parameter value (PRV) using the parameter correction value (PCV) to produce a corrected solution of said simulation (SIM) of the physical system (SYS); characterized in: that said at least one parameter correction value (PCV) is assigned to the at least one physically moving element (MVE) and said at least one parameter correction value (PCV) being transported along said element mesh (LTC) elements together with said at least one physically moving element (MVE), wherein said moving element (MVE) is a volume of the flow medium, whose material identity does not change.

2. A computer-implemented method according to claim 1 , wherein the neural network (NNW) is a Long-Short-Term-Memory (LSM) neural network (NNW).

3. A computer-implemented method according to claim 1 or 2, wherein said numerical simulation (SIM) is based on at least one governing equation (GVE) and wherein the transportation of said at least one parameter correction value (PCV) together with said at least one physically moving element (MVE) is done by inserting the parameter correction value (PCV) via an advection equation term (AET) containing the parameter correction value (PCV) in said at least one governing equation (GVE) of the simulation (SIM).

4. A computer-implemented method according to at least one of the preceding claims, wherein said numerical simulation (SIM) is based on at least one governing equation (GVE) and said parameter correction value (PCV) is inserted into said at least one governing equation (GVE) via a diffusive term (DFT) containing the parameter correction value (PCV) is inserted into the at least one governing equation (GVE) of the simulation (SIM).

5. A computer-implemented method according to at least one of the preceding claims, wherein the update gate has a peephole architecture such that the update of the long-term states also takes the current long term-state into account.

6. A computer-implemented method according to at least one of the preceding claims, wherein the method further comprising a training step, comprising:- providing reference data (RFD) as training data by comprising predefined system (SYS) boundary conditions (BCD) and a system (SYS) simulation (SIM) solution including said at least one parameter value (PRV);- training said neural network (NNW) predicting said at least one parameter correction value (PCV) while said neural network (NNW) is integrated in said system (SYS) simulation (SIM), which training is performed using said reference data (RFD).

7. Method of controlling a system by model predictive control (MPC), wherein said system (SYS) is comprising at least one physically moving element (MVE), in particular, wherein said at least one moving element (MVE) is a fluid flow, the method comprising:(i) defining a control objective (CTO);(ii) defining a system (SYS) parameter to be a control parameter (CTP);(iii) defining a control action (CTA); characterized in(iv) simulating the system (SYS) answer (ANW) to said control action (CTA) according to a method according to at least one preceding claim 1-6,(v) evaluating the simulated system (SYS) answer (ANW) as to whether the control objective (CTO) was reached by the defined control action (CTA) according to said simulation (SIM);(vi) repeating steps (c) - (e) with variations of said control action (CTA) until a predefined termination criterion (TRC) is fulfilled in such a way that the control objective (CTO) is sufficiently achieved by the control action (CTA);(vii) controlling said system (SYS) on basis of the last control action (CTA) of step (f).

8. Method of controlling a system according to claim 7, the method comprising:- measuring at least one physical parameter value (PRV) of said system (SYS);- using the measured physical parameter value (PRV) as a boundary condition (BCD) for the simulation (SIM) or for calibrating and / or training the neural network (NNW).

9. A method for operating a digital twin (DTW) of a physical system (SYS), wherein said digital twin (DTW) is based on a computer-implemented method for numerically simulating a system (SYS) according to at least one of the preceding claim 1-6.

10. Virtual sensor being applied to a system (SYS) comprising a processor (PRC) being configured to operate a digital twin (DTW) of said physical system (SYS) according to a method according to claim 9,wherein an output (OTP) of said sensor is a parameter being determined as an output (OTP) of the simulation (SIM) of the physical system (SYS).

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