Integrated algorithm: numerical simulation with two rounds of machine learning for monitoring and control devices

Abstract

A computer and machine-implemented method based on integrated numerical / machine learning for general-purpose simulation of real-world systems. The purpose is to allow machines to carry out simulations and analyze the surrounding environment for advanced monitoring and control tasks. The processors available in the devices are limited in their capabilities and they cannot perform complicated calculations. In addition, the boundary conditions of the real-world systems are in a continuous change, which makes the simulation task very difficult for machines. Moreover, validating the simulation is a human task. The method is set to overcome the challenges by compromising a numerical method with two- round machine learning. The first round helps to simplify the numerical model, whereas the second round allows the device to adjust the numerical algorithm in responding to any change in the boundary conditions. Additionally, the method can be used as computer software to reduce the computational and validation effort.

Description

[0001] Integrated algorithm: numerical simulation with two rounds of machine learning for monitoring and control devices Description Field of the invention

[0001] The present invention is related to a device-implemented and computer-implemented algorithm for simulating real-world systems. A two-round machine learning integrated into a numerical method for a fast and accurate simulation. The invention can be utilized on devices for control and monitoring real-world objects and engineering systems, besides computers for research and development purposes. Background of the invention

[0002] For control devices, the current state-of-the-art is based on devices equipped with sensors to gather data from real-world systems, the data then, is analyzed for further actions. The quality and quantity of collected data affect the overall quality of the control process.

[0003] Upgrading the potential of control devices to be capable of simulating real-world systems will open a new horizon for powerful monitoring and control. Computer-aided Engineering (CAE) and numerical methods help to provide comprehensive data about the system under study. Equipping monitoring and control devices with numerical methods will extend their resources beyond the sensors to obtain all the system's details. That, indeed, helps to build machines with more sophisticated functionality and higher efficiency.

[0004] Simulating physical phenomena becomes an important step in developing and improving engineering systems, it provides a lot of details about the phenomena under study. Prior art simulation of real-world objects is based on numerical methods such as finite difference, finite element, finite volume, etc. The vast implementation of simulation methods has been in the research and development and resulted in significant progress and improvement of now a days technologies (United States of America Patent No. US8126684B2, 2009), (United State Patent No. US10474773B2, 2018), (United States of America Patent No. US11403445B2, 2018) (United States of America Patent No. US11844570B2, 2021). Although the applications of the numerical methods include a wide spectrum of industries, their utilization is limited to computer-implemented applications like the design and development of products and predicting the behavior of natural or engineering systems. There are three obstacles to utilizing the simulation methods on devices:

[0005] The first one is the limited processing capabilities of the chipset on the devices and the relatively limited time to act. Numerical methods usually require good or even high- performance computing capabilities to simulate complicated systems, adding to that numerical solutions sometimes become a time-consuming process. Moreover, simulating complex systems with several phenomena that drive their behaviors requires special treatment and paying more attention to the model's accuracy.

[0006] The second obstacle is that the current numerical approaches are built to simulate cases where all the boundary conditions surrounding the system are well-defined. However, real- world systems are vulnerable to continuous variations in their boundary conditions, and that obligates frequent redefining and readjusting of the simulation inputs. Hence, the numerical methods in their current form cannot be equipped on devices to simulate real-world systems since the systems' boundaries are always subject to variations.

[0007] The last obstacle is about evaluating the results of numerical simulation, which usually requires more human effort. For many cases, the numerical solutions alone are insufficient for the reliability of the results, and they must be consolidated by empirical studies and practical measurements. The evaluation step is a task that is usually done by humans since it requires special technical knowledge to evaluate the solution's progress and the results.

[0008] Those issues are the major obstacles to expanding the utilization of numerical methods on other automated platforms than computers, like devices and machines since they still need a good definition of the system’s boundary conditions, moderate to high computing capabilities, and a man-based evaluation of the results. Solving the three major issues besides maintaining or even improving the accuracy of the simulation will help to implement the numerical methods on devices, machines, and instruments. That, indeed, will enhance the functionality of the devices.

[0009] The current invention in brief is an algorithm of integrated Numerical / Machine-Learning (NML) method that can be implemented by machines and devices, which are equipped with sensors. The sensors collect data from the real-world system, the numerical method works to simulate the system. The simulation results, then, will be compared with the sensors reading, and based on the difference between the numerical and actual data, the integrated machine learning (ML) techniques provide correction factors for the definitions of the numerical model and drive the numerical solution to be closer to the measured data and minimizing the error. This method is a fully automated approach that can evaluate the solution without the need for human intervention.

[0010] This invention provides a systematic automated method combining numerical approaches with machine learning techniques. The purpose of the integrated method is to allow the device (or the machine) to perform an accurate simulation even when some of the boundary conditions and / or the material properties are unavailable. The integrated approach complements the numerical methods with a two-round ML analysis. The first ML round helps to simplify the numerical model whereas the second ML round re-adjusts the definition of the numerical model and responds to any change in the boundary conditions and / or material properties. Summary of the invention

[0011] This invention is an algorithm that combines both numerical and machine-learning methods to provide a fast and reliable simulation of real-world objects and systems. The main implementation of the integrated method by machine, devices, and instruments for monitoring and control applications. Another utilization can be as computer software to reduce the computational cost and ease the validation of the simulation process.

[0012] The numerical methods are formulated by the laws of physics that describe the phenomenon under study i.e. energy conservation, mass conservation, hook’s law, etc. Therefore, the numerical methods are built on a robust basis that correlates all related parameters in a single frame. However, numerical solvers are iterative approaches that provide approximate solutions.

[0013] On the other hand, machine learning methods are used to train the machine by using data that is practically gathered, to find the correlations between different parameters (input parameters) that affect a certain phenomenon and provide predictions about it under different conditions. The size of training samples is proportional to the number of input parameters.

[0014] The invented method is formulated by combining the numerical and machine-learning methods in two rounds to get the advantage of the theoretical frame given by the numerical model and utilizes machine-learning techniques to predict the varied boundary conditions, improve the accuracy of the solution and reduce the size of the model. Also, the integrated method is capable of predicting the undefined or varied boundary conditions and accordingly updating the numerical model to maintain an accurate simulation.

[0015] The NML method (in most of the cases) does not require training samples to establish the ML models since that the correlations are set by the numerical model. The real-time data provided by sensors are adequate to establish a good ML model that gives full support to the numerical model. However, in case of very complicated problems and when the quantity of collected data is not adequate, the need to train the NML method for the given problem becomes important to maintain the accuracy of the results.

[0016] The invented method can be run on the computer to perform simulations. It can also be utilized in smart devices to give predictions of engineering systems without human interference. Brief description of the drawings

[0017] A detailed description of a preferred embodiment will follow, by way of example only, about the accompanying figures of the drawings, in which: FIGURE 1An illustration of the numerical method. For instance, obtaining the temperature variation within an object. FIGURE 2A schematic of the machine learning method. For instance, the temperature variation of an object. FIGURE 3A schematic of the invented integrated numerical / machine-learning method. For instance, the temperature variation of an object. FIGURE 4A flow chart of the integrated numerical / machine learning method to obtain the variation of the physical quantity ^^^^,^^through space and time. FIGURE 5The steps of two-round regression integrated with a numerical method FIGURE 6By using the integrated NML, the numerical model can be reduced by reducing the number of elements FIGURE 7By using the integrated NML, the numerical model can be reduced by simplifying the physics and considering only the parts / zones of interestFIGURE 8An application of the integrated NML method: Computer software FIGURE 9 An application of the integrated NML method: A code can be implemented by electronic chips for smart devices Detailed description of the embodiment or embodiments I. Numerical Methods:

[0018] Computer-aided engineering (CAE) has become a very important tool for simulating the behavior of engineering systems under different scenarios. CAE reduces the development time and investment cost besides providing comprehensive data about the system, which boosts the improvement process.

[0019] CAE is constructed based on the laws of principal physics to study natural phenomena and engineering systems. The laws of physics essentially describe the variation of the physical quantity (pressure, velocity, etc.) in space and time under the initial and boundary conditions of the system and for the material properties of the media within the system. Those laws of physics like conservation of energy, conservation of momentum, propagation of electromagnetic waves, Hook’s law, etc., are the best forms that human intellect developed to understand natural phenomena. CAE essentially utilizes numerical methods to solve the law of physics (considering the boundary conditions and material properties of the problem under study) (Raphael, 2003)

[0020] The constitutional laws in physics are formulated in the form of partial differential equations (PDE) to describe the spatial and temporal variation of quantities like temperature, fluid velocity, pressure, etc. Those partial differential equations have no exact solutions; however, the numerical methods bring approximate solutions.

[0021] To solve the PDEs, the domain of study is discretized into a finite number of nodes (elements / volumes), then, the PDEs are converted to algebraic equations at each node (FIGURE 1). After that, algebraic linear equations (usually, the number of algebraic equations is proportional to the number of nodes) are solved using computers, whereas the accuracy of the solution is related to spatial and temporal discretization. Increasing the number of elements leads to reducing the error up to a certain limit (Australia Patent No. AU2015210607B2, 2015). Nevertheless, the problem will be computationally expensive in the case of a large number of linear equations. So, large-scale models and multi-physics problems require high-performance computational hardware and need more time to obtain the solution. II. Machine-Learning Methods:

[0022] Recently, machine learning (ML) techniques have been intensively utilized in solving engineering problems (Kelleher, 2020). Once enough data (empirically measured data) related to the system is provided, a machine-learning method can obtain correlations that link different parameters with the targeted output. The machine learning method can provide predictions on the system behavior under different scenarios. For instance, obtaining the temperature of a body of a system that is exposed to heat transfer and internal heat generation can be done by following ML techniques; as depicted in (FIGURE 2), all the parameters influencing the heat transfer process are listed in the left column. The initial temperature ^^0, heat generation rate, the material properties of the object like the density, thermal conductivity, specific heat, emissivity, etc., the dimensions of the body, and any other parameter that influences the temperature of the system.

[0023] The key factor of the machine learning techniques is providing adequate data (training samples), so, the ML algorithm can be used to infer the desired correlations.

[0024] Increasing the input parameters adds more difficulties to the ML technique since extra empirical data must be provided to ensure proper training for the ML algorithm. Training samples are the main challenge when building the ML model. It requires investment and technical skills to guarantee that each parameter is represented by an adequate number of values within the desired range of applications. III. Integrated Numerical / Machine-Learning (Integrated NML) Method:

[0025] In the current invention, it is suggested to implement the ML method starting from the law of physics to improve the performance of the numerical method. The integrated numerical / machine-learning (Integrated NML) method starts from the principal physical model that describes the physical systems in a robust way, where the law of physics theoretically can simulate any system. However, because of the complex nature of some problems, it is suggested here to include correction factors in the law of physics to ensure the results are aligned with the measured data. Those correction factors can be obtained by machine learning approaches. Implementing ML helps to reduce the problem, build a fast solver, and increase accuracy. This will be clarified in detail.

[0026] In FIGURE 3, the Integrated NML method is illustrated by giving the example of a heat transfer problem. Instead of applying the machine learning method on the input layer, it is suggested in the invented method to apply the ML on the principal physical model layer (the middle layer).

[0026] The first advantage of the intermediate layer is reducing the size of the required training samples and reducing the investment cost. It does not matter how many input parameters influence the output parameters, the energy balance concept represented by the intermediate layer is limited to only 4 terms (Heat generation rate ^^^^^^^^, conduction heat transfer ^^^^^^^^^^, convection heat transfer ^^^^^^^^^^, and radiation heat transfer ^^^^^^^^), and so, only four correction factors are required. The size of training samples is related to the number of correction factors. For instance, it is required to provide hundreds (or even thousands) of training samples for the ML described in FIGURE 2. However, only tens of the training samples are required to achieve good results for the improved ML algorithm shown in FIGURE 3.

[0027] In FIGURE 4, a flow chart clarifies how the integrated NML algorithm works to simulate a certain physical quantity ^^ (^^ is a temperature or a pressure or an electric field, etc..). In the beginning, a set of ^^ training samples are provided one by one, then, building the numericalmodel of case ^^ where ^^ ∈ {1,2,3, .. , ^^}, the initial conditions and the boundary conditions of theproblem will be provided based on the corresponding training sample ^^. After that, the correction factors will be defined with their initial values.

[0028] Two-round machine learning method consolidates the numerical methods by accelerating the solution and obtaining the missing boundary conditions / s.

[0029] To describe the temporal and spatial variations of a locally continuous field φ(^^, ^^, ^^, ^^)with a source term ^^, the below partial differential equation (PDE) is used:

[0030] Where ^^ is the diffusivity medium’s properties. In the NML model, two types of correction factors are introduced to equation 1. The first type (^^) is used to correct the solution and compensate for the large error caused by the simplifications applied to the model when using coarse temporal and spatial discretization: ^^2^^2^^2

[0031] Here, ^^1is a coarse ^^2, ^^3, ^^4 factors to correct the coarse discretization of the space. The first round of MLtechniques can be carried out when all boundary conditions and material properties are defined for a given case. The first round starts from assumed values of the factors ^^, then, solves the PDE by implementing numerical solvers like the finite element or finite difference or any other method. The numerical solution at certain points will be compared with the practically collected data at the same points and the factors ^^ will be optimized according to the ML technique (using linear regression, artificial neural network, etc.).

[0032] After optimizing the first type of correction factors, another round of ML will be applied when some boundary conditions start to vary from the first case, and when the medium properties are not the same as the one used in the first ML round. However, the geometry must be maintained as the one used to solve the first ML round. The second type of correction factor (^^) is used to predict the undefined boundary conditions for the system, they are introduced to the PDE as: ^^2^^2^^2 Where a factor to compensate for the undefined boundary conditions. The second of ML starts from assumed values for ^^ factors, then, solving the PDE, comparing the numerical solution with the collected data, and implementing the ML technique to optimize ^^ factors.

[0034] The first round of the ML is run for only one time. Once the correction factors of the spatial and temporal discretization are optimized, there will be no need to re-optimize them again unless the discretization model is changed. In contrast, the second ML round is designated to be continuously running since the boundary conditions are in a continuous change for the real-world systems. For simple systems, the real-time collected data are adequate to run the ML techniques without the need for pre-training of the model. However, for complicated systems, and / or when the number of sensors does not provide enough data to effectively operate the ML techniques, the need for pre-training of the model becomes mandatory for maintaining the accuracy of the solution within reasonable limits.

[0035] To demonstrate the NML method with more details, an example of a thermal analysis will be presented. For a system shown in FIGURE 5, the thermal behavior of the system can be described by the law of physics (Lienhard, 2005): ^^^^)

[0036] The main goal of this model is to obtain the temperature ^^ field in space and time within the system, where ^^, ^^^^, ^^: are the density, heat capacity and thermal conductivity of thematerial. ^^ represents the heat gain and loss through the boundary of the system (including the volumetric heat generation / loss). ∇ represents the spatial variation of the physical quantity ∇= ^^ ^^ ^^ 4, then, can be written for the system with its boundary conditions as: ^^^^)

[0037] Up to here, the model is a typical system that can be simulated by implementing the finite element method (or any other numerical method) to calculate the temperature distribution in space and time.

[0038] With an assumption that all boundary conditions and material’s properties are defined except two boundaries (^^^^and ^^^^), also, the temperature is defined for 3 points inside the system ^^1,̂ ^^2̂ and ^^3̂, which represent the dependent variables (3 points are given for example).In this case, it is no longer possible to follow the numerical methods to predict the temperature distribution in the system. A two-round machine learning will be introduced to simulate the temperature distribution in the system, the correction factors will be introduced to equation 5. • The first ML round:

[0039] The first ML round aims to compensate for the coarse mesh and to correct all the errors resulting from the simplification of the model. The first ML round is mandatory to minimize the error of the numerical solution and it is carried out only once for a given model. Equation 5 is modified as: (the underlines are only for clarification) ^^^^)

[0040] ^^1is the correction factor of the coarse time step ∆^^ (it also can be used to correct the material’s properties ^^ and / or ^^^^wherein they are undefined for the material).

[0041] ^^2is the correction factor of the coarse spatial discretization ∆^^, ∆^^ and ∆^^ (it also can be used to correct the material’s properties ^^ where it is undefined for the material).

[042] ^^3, ^^4, ^^5, ^^6 and ^^7 are the correction factors for the coarse spatial steps and theboundary conditions ^^^^ , ^^^^ , ^^^^ , ^^^^ and ^^^^ (required to compensate for the simplification of themodel and assumptions that are necessary to reduce the model complexity).

[0043] The first ML round starts by assuming values for the missing boundary conditions ^^^^and ^^^^, also, all the correction factors start with an initial assumption of 1.

[0044] With these assumptions, equation 6 can be solved by applying a numerical method (for example FEM) and accordingly, the temperature distribution is obtained for the whole system.

[0045] After solving the numerical model, ^^1, ^^2 and ^^3 are calculated, so, a comparison withempirical data (^^1,̂ ^^2̂ and ^^3̂) can be done.

[0046] It should be noted that empirical data are a set of data for (^^) number of empirical measurements for ^̂^ that corresponded to ^^ number of different sets of boundary conditions,the temperature data for the three points inside the system ^^^̂^, ^^^̂^ and ^^^̂^:^^^̂^ = [^^11 , ^^21 , ^^31 , … , ^^1^^] (7A) ^^^̂^ = [^^1 2 32, ^^2, ^^2 , … , ^^2^^] (7B) ^^^̂^ = [^^13, ^^23, ^^33 , … , ^^3^^] (7C)

[0047] Those measured temperatures correspond to other sets of data about the boundary conditions influencing the system: ^^^̂^ = [^^1^^, ^^2^^, ^^3^^, … , ^^^^^^ ](8A) ^^^̂^ = [^^1^^ , ^^2^^ , ^^3^^ , … , ^^^^^^] (8A) . . .

[0048] Simulating with the consideration of the sets of boundary conditions ^^^̂^, ^^^̂^, ^^^̂^, ^^^̂^ and^^^̂^ results in a set of simulation results for the three points ^^^^, ^^^^ and ^^^^, the differencebetween the simulation results and empirical measurements is: ^^

[0049] If the difference is greater than an accepted limit, i.e. ^^ ≥ ^^ (^^ = 1, or ^^ = 0.1 or anyaccepted limit based on the application), then, the accuracy of the numerical solution is not accepted, and the numerical model requires a correction. To correct the model, a linear regression method (one of the ML techniques) can be used to adjust the correction factors^^1, ^^2, .. , ^^7 to calculate how much the correction factors must be varied to minimize the error(the underlining is only for clarification): ^^ . ^^

[0050] Where 3 in the equations is the number of dependent variables (^^^^, ^^^^ and ^^^^) and thenew values for the correction factors will be: ^^1 = ^^1 + ^^^^ × ^^^^1 (11A)^^2 = ^^2 + ^^^^ × ^^^^2 (11B). . ^^7 = ^^7 + ^^^^ × ^^^^7 (11G)

[0051] ^^^^ is the learning rate, which is a defined value to control how fast or slow the regression process will be. After getting the modified correction factors, the simulation will be repeatedto get new values for the three points ^^^^, ^^^^ and ^^^^. The difference ^^ between the simulationand empirical values ^^^̂^, ^^^̂^ and ^^^̂^ must be calculated from equation 8 and based on thedifference, the correction factors will be modified again by equations 9 and 10. A series of simulations and modifications of the correction factors will be conducted until an acceptederror is achieved (i.e. ^^ < ^^).

[0052] Up to here, the correction factors ^^1, ^^2, .. , ^^7 are optimized and they will be fixed forthe given system. This is the first ML round which is adequate to give accurate results for the simulation and accurate temperature distribution inside the system and only the temperature distribution has been well-simulated. • The second ML round:

[0043] Even though the first ML round is adequate to simulate the system, it is not enough for the numerical model to respond to any change in the boundary conditions.

[0054] When some of the boundary conditions are undefined (For instance ^^^^and ^^^^) and / or one of the material properties is / are not available, there must be another set of correction factorsthat can fulfill the condition ^^ < ^^.

[0055] Therefore, another round of regression is necessary to accurately obtain the boundary conditions (if that is required), and / or, to respond to any variation in any of the boundary conditions. The law of physics will be modified by adding another type of correction factor, for the given example, ^^^^and ^^^^will be introduced to equation 5 (the underlines are only for clarification): ^^^^

[0056] ^^^^and ^^^^will be given initial values, all the correction factors from the first ML round^^1, ^^2, optimized and now have fixed values, also, the undefined boundary conditions^^^^and ^^^^have initial assumed values from the first ML round. Hence, a simulation can be performed to obtain the temperature distribution inside the system.

[0057] Once the simulation is done, the difference between the simulation results (^^1, ^^2 and ^^3)and another set of empirical measurements (^^1,̂ ^^2̂ and ^^3̂) will be calculated as depicted inequation 8. If the difference is greater than an accepted limit, i.e. ^^ ≥ ^^ (^^ < ^^, i.e. the secondML round has higher accuracy than the first round), then, the accuracy of the undefined boundary conditions is not accepted, and the numerical model requires a correction. To correct the model, a linear regression method (for example) can be used to adjust the correction factors ^^^^and ^^^^by using the linear regression method to calculate how much the correction factors must be varied to minimize the error.

[0058] Equation 9 can be implemented to calculate ^^^^^^and ^^^^^^, then equation 10 shall be applied to obtain the modified ^^^^and ^^^^.

[0059] The simulation then shall be carried out again with the modified correction factors (^^^^and ^^^^), the difference with the empirical measurements will be again calculated and the required modification will be applied for ^^^^and ^^^^. A series of simulations and ML steps willbe carried out until reaching an accepted error, i.e. ^^ < ^^. Once that is achieved, the correctionfactors ^^^^ and ^^^^ are now optimized and (^^^^ . ^^4. ^^^^) and (^^^^ . ^^5. ^^^^) represent the undefinedboundary conditions.

[0060] The main advantage of the integrated NML algorithm is its capability to simulate the whole system when empirical data are provided for points inside the system, even when some or all the boundary conditions are unidentified. The conventional numerical simulation methods always require obtaining the boundary conditions that affect the system. However, for the integrated NML algorithm, the training step is mandatory to obtain the correction factors for the numerical model. The unidentified boundary conditions can be predicted during the training process, once they are obtained, boundary conditions can be used later in the simulation.

[0061] Another advantage of the integrated NML method is its capability to reduce the size of the problem since there are correction factors that ensure the solution is close to the empirical data. Reducing the numerical model can strongly reduce the solver time and the required hardware resources to solve the problem. One reduction method is reducing the number of discretization elements (FIGURE 6). Another reduction method is to eliminate the unrequired part / s of the system and keep only the one / s needed for the study and simplify the physics (FIGURE 7). Both reduction methods can be applied without affecting the outputs in many cases. Both reduction methods were used by the inventors to solve the heat transfer problem for a complicated system (Mallah, 2022). Embodiment of the two-round NML method • As a computer software:

[0062] The integrated NML method can be implemented as a simulation software that runs on normal computational units (computers, workstations, and servers) FIGURE 8. Thanks to the first round of ML which allows for simplifying complicated systems, provides a fast solution and ensures the solution is aligned with the empirical data.

[0063] The integrated method also can predict the undefined boundary conditions and simulate the system. Thanks to the second round of ML which is elaborated to consolidate the simulation in case data inside the system are available instead of the boundary conditions.

[0064] It also can reduce the human efforts spent on setting up the simulation model since analysts usually spend much time on several iterations of the simulation to fine-tune the numerical model and ensure the numerical results are close to reality. Thanks to the two-round NML model, the numerical model can be automatically corrected once it is fed with the empirical data for a few points inside the system. • As a code operated by devices:

[0065] The main goal of developing the two-round NML method is the utilization of simulation methods by machines (hardware / controllers / smart devices) FIGURE 9. Indeed, this expands the capabilities of the hardware and allows devices to do more complicated tasks based on the rich data they get when they are capable of simulating the surrounding environment.

[0066] devices are usually equipped with sensors to get data from the surrounding environment, they also have a processor to analyze the data and to give instructions to the actuators and make a required action.

[0062] The limited capability of the hardware and controllers’ processors (they are usually way lower than the capability of personal computers) and the continuous changes occurring in the surrounding environment are the main challenges that do not allow hardware to use conventional simulation methods. However, the two-round NML is elaborated to be operated by devices. It is worth to mention how the trained algorithm works in the hardware: - The algorithm must be set by defining the system’s dimensions, materials’ properties and all known boundary conditions, - Most cases need to make the simulation model as simple as possible and consider a relatively coarse mesh, - Then, empirical data must be fed to train the algorithm by the first ML round integrated NML, - After the first round of training, another set of training data must be fed to train the algorithm to respond to the variation of boundary conditions, - Once the algorithm is trained for the desired system, it can be set in the device and run, - The device receives data from a limited number of sensors connected to it, and then, the processors run the trained algorithm to analyze the whole system, - The simulation results then will be compared with the standard data provided by sensors, - If the difference between the simulation and the training data is significant, the integrated NML with the second round of ML will be implemented until minimizing the difference, - Once the integrated method refines the simulation, the results can be used for monitoring and / or control. - The boundary conditions influencing the system are in continuous change. Hence, the second round of ML integrated with the numerical method will be continuously implemented to predict the new boundary condition and to refine the correction parameters of the numerical algorithm.

[0067] The coupling of the numerical model with the second ML round and for special cases can be performed without previous training. The ML analysis will be performed on the basis of the real-time data collected by sensors to force the simulation to match with measured data. The special cases are: - Only one boundary condition is undefined and / or only one variable boundary condition. In this case, the second ML pre-training can be neglected, and the real-time second ML round (3) is enough for correcting the simulation and predicting the unknown boundary condition, - More than one boundary condition is unknown, but the interest is only in simulating the system accurately. In this case, the second ML pre-training can be neglected, and the real- time second ML round (3) is enough for correcting the simulation, but not to obtain the unknown boundary conditions. References Chen, H. (2018). United State Patent No. US10474773B2. Hipsley, A. (2021). United States of America Patent No. US11844570B2. Joseph F. Hair, W. C. (2019). Multivariate Data Analysis [8th Edition]. Mason, United States: Cengage. Joshua L. CAMP, A. L. (2015). Australia Patent No. AU2015210607B2. Kelleher, J. D. (2020). Fundamentals of machine learning for predictive data analytics: algorithms, worked examples, and case studies. MIT press. Lienhard, J. H. (2005). A heat transfer textbook. Phlogiston Press. Mallah, A. R. (2022). A hybrid numerical / machine learning model development to improve the bimetal performance in the electric circuit breakers. Scientific Reports, 12(1), 18087. Raphael, B. &. (2003). Fundamentals of computer-aided engineering. John wiley & sons. Roux, T. G. (2009). United States of America Patent No. US8126684B2. Zheng, X. Z. (2018). United States of America Patent No. US11403445B2.

Claims

AMENDED CLAIMS received by the International Bureau on 08 October 2024 (08.10.2024)1. A numerical method (1) for real-time simulation of real-world systems, wherein the improvement is by the integration with a functional two-round (2.1) and (2.2) machine learning technique (NML), characterized by operational capabilities in:An apparatus (3) for monitoring and control,A computer (4) or apparatus (3) for simulating systems with undefined boundary conditions,A computer (4) or apparatus (3) for simulating systems with reduced human dependency for results validation,A computer (4) for simulating large or complex systems, offering reduced solution time compared to conventional methods.

2. The system according to claim 1, comprises any object or assembly subject to specific circumstances, boundary conditions, and considering the medium’s physical properties.

3. The numerical method (1) according to claim 1, is configured for solving partial differential equations that describe the variation of a physical field within a system over space and time, executed by:Selecting partial differential equations representing the physical field (Poisson’s equation, conservation of mass, momentum and energy, Hooke’s law, etc.), Establishing equations for the system under given boundary conditions, Discretizing the system into a finite number of nodes and elements,Transforming integral and differential equations into a series of linear equations over the nodes,Iteratively solving these equations to fully describe the field distribution.

4. The physical field according to claim 3, comprises physical quantities like temperature, force, pressure, stress, velocity, electric and magnetic fields, acoustic waves, light beams, etc., where variation depends on the system's geometry, time, medium properties, and boundary conditions. The variation of the field cp within the system is described by the equation: dtp—K is the dissipation coefficient and Sgcontains all the surface and volumetric source terms.

5. The machine learning (ML) technique according to claim 1, is constituted by regression analysis and / or statistical techniques to establish relationships between dependent and independent variables, utilizing training samples to determine the correlation coefficients.

6. The two-round ML (2.1) and (2.2) according to claim 1, is performed in sequential steps, contingent on the availability of training samples, with each analysis round involving multiple iterations to refine the coefficients.

7. The two-round ML (2.1) and (2.2) according to claims 5 and 6 wherein said the set of training samples is organized data for the dependent and independent variables of the system. The dependent variables are usually measured quantities for a limited number of selected points inside the system, while the independent variables are the medium’s properties, and the boundary conditions applied to the system.

8. The integration of the first ML round (2.1) with the numerical method (1) according to claim 1 helps to simplify the system by reducing the required number of nodes / elementsbesides reducing the number of the sub-parts and internal units, thereby accelerating the solution process.The integration of the first ML round (2.1) is conducted only one time to add correction factors for the numerical model due to the coarse discretization by the steps:The numerical method formulates the correlations of the dependent variable / s in terms of the independent variables,Building the numerical method on the simplified system, initially inadequate for precise results,Employing ML (2.1) to adjust the numerical method (1) by adding correction factors, the correction factors have initial assumed values, and they are denoted here by (a). The equation of the field variation within the system after adding the correction factors (a) is:The numerical solution will be carried out,The results of the numerical method then, will be compared with the training samples, if a large difference is found, then, the ML technique will be carried out,The ML technique (2.1) modifies the correction factors in a way that reduces the difference between the solution and the training samples,Another iteration of the numerical solution will be carried out with the updated correction factors,Iteratively update and optimize the correction factors by solving the numerical model, then update the correction factors based on comparison with training samples until minimized discrepancies are achieved, and then fix these optimized factors in the system model.

9. The integration of the second ML round (2.2) with the numerical method (1) according to claim 1 is to predict the undefined or variable boundary conditions.The integration of the second ML round (2.2) is conducted by steps:Adding another type of correction factor to the numerical model where undefined or variable boundary conditions are involved, the correction factors will have assumed initial values. Also, the undefined or variable boundary conditions will have assumed initial values. The correction factors here are denoted by (P). The equation of the variation of field (cp) within the system after adding the correction factors (P) is: dcp d2cp d2cp d2cp a1.— = p1. K. (a2.^ + a3.— + a4. —) + ^2. aJr. SgThe numerical solution will be performed,The results of the numerical method, then, will be compared with a set of training samples, if a large difference is found, then, the ML technique (2.2) will be carried out, The ML technique (2.2) modifies the correction factors in a way that reduces the difference between the solution and the training samples,Another iteration of the numerical solution will be carried out with the updated correction factors,Iteratively update and optimize the correction factors by solving the numerical model, then update the correction factors based on comparison with training samples untilminimized discrepancies are achieved. The optimized correction factors help to determine the undefined or variable boundary condition.

10. The apparatus (3) according to claim 1 is a device or instrument comprises:Processor (3.1),Memory (3.2),NML algorithm (2.1) and (2.2),General-Purpose Input / Output (GPIO) interface (3.3) to read data from sensors and give instructions to the actuators that are connected to the device,Sensors (3.4), andActuator (3.5) (optional item, its utilization is for control).

11. The apparatus (3) according to claim 1 functions in the following steps:Step 1 : Initially, define the system under study by setting up the 3D model, determining the relevant physics (such as heat transfer, fluid flow, stress analysis, etc.), material properties, and boundary conditions. It is also essential to identify which boundaries may change during the real-time simulation.Step 2: Configure the model discretization (mesh), ensuring that the number of discretization elements is minimized for faster simulation.Step 3: Establish the equations of the numerical model (refer to Equation 1).Step 4: Incorporate the correction factors for the first round of the integrated NML by adding the first type of correction factors corresponding to the first ML round (2.1).Step 5: Provide the training samples, including a set of well-defined boundary conditions for the system, as well as solutions at specific points within the system (e.g., values of temperature, stress, fluid velocity, pressure, or any other required quantities for the simulation). The solutions at specific points within the system can be measured data or simulation results of a fine mesh model (i.e. large number of elements).Step 6: Conduct the first round of ML calculations, as per Claim 8. By the end of this step, the numerical model will include equations with optimized correction factors (a). Step 7: Load the trained model (the numerical model with the optimized correction factors a) into the simulation device. Add the second type of correction factors (P) to the numerical model (second ML round (2.2)), focusing only on the terms related to boundaries that are susceptible to changes.Step 8: Position the sensors of the simulation device at designated locations within the system under study.Step 9: The sensors will collect data from the system and supply this data to the NML model as training samples for the second round of ML as per Claim 9.Step 10: Execute the second round of ML calculations, as per Claim 9. By the end of this step, the numerical model will incorporate equations with optimized correction factors (P).Step 11 : The optimized correction factors (P) enable accurate simulation with minimal deviation from the data measured by the sensors, thereby achieving validation of the simulation results without human intervention.Step 12: The sensors continuously monitor the real-world system. If there is a significant discrepancy between the real-time data and the last simulation, this indicates a change in the boundary conditions. Steps 9, 10, and 11 will be repeated, and the correction factors (P) will be re-optimized to determine the new boundary conditions and simulate the system accordingly.

12. The integrated NML method according to claim 1 can be operated as executing software instruction wherein can be applied in several ways:Software to facilitate the simulation of very large systems wherein hardware resources cannot do the simulation within a reasonable time.Software to facilitate the simulation wherein one boundary condition or more is not defined for the system.A simulation software with a self-validation feature. By the integration of two-round ML, the numerical method can automatically refine the discretization, the material properties and the boundary conditions to drive the simulation with the desired results.

13. The integration of the numerical method (1) with the second ML round (2.2) according to claims 10 and 11, in many cases can be performed without previous training, i.e. without the first ML round (2.1). That can be possible only when the system under study is simple, resulting in fast solutions.

14. The computer (4) according to claim 1 is a computing unit that has a processor (4.1) and a memory (4.2).

15. The monitoring according to claim 1 is a process of presenting data related to the real-world system. The data is related to the distribution of a certain physical field like temperature, pressure, electric field, velocity, etc. The data is obtained by the integrated method and includes all the points inside the system and can be for certain points. The monitoring also includes presenting data about the variable or unknown boundary conditions.

16. The control according to claim 1 is a process of modifying the physical status of the real- world system. The physical field can be temperature, pressure, electric field, velocity, etc. The control process is based on the monitoring process as per claim 19, when the physical status at a certain point or zone within the system exceeds a certain limit, the control process uses the actuator to adjust some of the applied boundary conditions in a way that maintains the monitored physical quantity within a desired range.

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