Predicting the feasible area for physical systems

Abstract

Predicting the feasible region of the design space for a physical system involves determining a set of design points for one or more iterations. This determination involves sampling each function for the output of the physical system, each function predicting the output value at the design point, and selecting design points based on an objective function that penalizes the distance from the boundary of the feasible region predicted by the value of each function. The selection of several design points is performed in parallel. In each iteration, data is obtained, including empirical values ​​of the output at the determined design points, and updated values ​​of the trainable parameters of the model are determined using the obtained data. After one or more iterations, the method includes predicting the feasible region of the design space using the data obtained during one or more iterations.

Description

[Technical Field]

[0001] This disclosure relates to the prediction of the feasible area of ​​design space for physical systems. More specifically, this disclosure relates to the prediction of the feasible area of ​​design space for vehicles. [Background technology]

[0002] The design of complex systems such as vehicles typically begins with defining top-level requirements and constraints, then progresses step-by-step to more detailed definitions of lower-level subsystems and components. Following this design phase, integration and testing begin at the lowest-level components, gradually reconstructing them into the top-level system. This is known as the "V-model" of the system development lifecycle.

[0003] During the design phase, feedback from downstream (lower-level) hierarchies is iteratively provided to upstream (higher-level) hierarchies until the design set of subsystems and components is determined. If a lower-level subsystem or component cannot meet the requirements imposed by a higher-level component, system, or subsystem, the higher-level design may need to be modified. Since other lower-level subsystems or components will compensate for this modification, this itself may change the requirements for them. As a result, costly rework and development delays may occur.

[0004] A set - based design is a design paradigm in which design decisions are delayed until a later stage in the design process when more information is available, rather than being fixed to a single design early in the design process. At a given stage of the process, for example using simulation tools, an executable set of designs is predicted and flexibility is provided to handle uncertainties or adjustments of lower - level components or subsystems that would occur. In this way, rework can be avoided and development time can be shortened. The problem with this approach is that for complex physical systems, the simulation tools used to predict the feasibility of the design at each stage are often very computationally expensive and time - consuming to execute. Moreover, especially when the dimensionality of the design space is large, a large number of simulation runs may be required to identify an executable set of designs. Due to these challenges, each company may be prevented from benefiting from the advantage of less delay and rework in the later stages of the system development life cycle. Summary of the Invention

[0005] One aspect of the present disclosure provides a computer implementation method for predicting a feasible region of a design space for a physical system. The method comprises, for one or more iterations: determining a set of design points in the design space; sampling a given design point in the set, for one or more outputs of the physical system, each of which has one set of trainable parameters, from each probabilistic model, each of which predicts the value of one or more outputs at the design point in the design space; and selecting a given design point based on an objective function that penalizes the distance from the boundary of the feasible region predicted by the value of each function. The selection of at least some of the design points in the set is performed in parallel. In each iteration, the method comprises obtaining data containing the value of one or more outputs of the physical system at each design point in the determined set of design points, and using the obtained data to determine updated values ​​of the trainable parameter set. After one or more iterations, the method comprises predicting a feasible region of the design space using the data obtained during one or more iterations.

[0006] One or more outputs of a physical system may each be subject to a constraint, and the feasible region may be a region of the design space in which all of the respective constraints for one or more outputs of the physical system are satisfied. By sampling each function from each probability model and independently selecting design points based on the predictions provided by each function, a set of design points can be determined that can be processed in parallel across potentially many nodes or processor cores. In each iteration, an arbitrarily large number of design points can be determined very efficiently with a processing cost that scales only linearly with the number of points. The determined design points are predicted to exist near the boundary of the feasible region by the probability models, and thus the method will automatically progress from exploring a large portion of the design space in early iterations to focusing on the most informative part of the design space in later iterations (i.e., near the ground truth boundary of the feasible region). As a result, the feasible region of the design space may be revealed using a relatively small number of rounds of empirical and / or simulated data collection.

[0007] Predicting the feasible region of the design space may use each probability model for one or more outputs of the physical system. Alternatively, without further use of the probability models, the feasible region of the design space may be predicted using data collected during one or more iterations, for example by providing the data to another model or algorithm. The data may be used, for example, to generate a representation or estimate of the boundary of the feasible region.

[0008] The boundary of the feasible region may depend on a variable threshold of one of the outputs of the physical system, and selecting a given design point may involve sampling the variable threshold from a threshold distribution to predict the distance from the boundary of the feasible region. In this way, the method can account for the uncertainty of the variable threshold when determining the set of design points and allow the threshold to be determined at a later stage. The distribution may be, for example, a uniform distribution over the entire given threshold range, or it may be a non-uniform distribution such as a Gaussian distribution. The resulting prediction of the feasible region is flexible and robust to changes in the variable threshold within the range corresponding to the distribution. In the case of a uniform distribution, the trained model is expected to be, on average, equally accurate for all thresholds within a given range.

[0009] Obtaining data may involve performing simulations of the physical system at a determined set of design points, or performing physical measurements of one or more mockups of the physical system or at least a part of the physical system. Data acquisition may be performed in parallel for at least some of the design points in the set. By obtaining data in parallel, the method can take advantage of the fact that many design points are determined in each iteration, thereby significantly reducing the time required for the data acquisition phase in each iteration.

[0010] The probabilistic models may include at least one of the following: sparse Gaussian processes, deep Gaussian processes, Bayesian neural networks, and deep ensembles. All of these models have the ability to predict the probability distribution of the output of a physical system, and all have the ability to operate within large data regimes, for example, with thousands to millions of design points. The inventors have found that deep Gaussian process models and deep ensembles are particularly well-suited to settings where the output of the physical system exhibits one or more discontinuities. Nevertheless, for small data regimes where accuracy is important, a “strict” Gaussian process regression (GPR) model may be preferable. In certain examples, the GPR model is used in early iterations while the volume of data obtained is still relatively small, and then switched to one or more other types of models when the GPR model becomes unfeasible in terms of processing resources, memory, or time due to the data volume.

[0011] Selecting a design point may involve sampling multiple candidate design points from within the design space, updating each of the candidate design points to reduce the corresponding value of the objective function, and then selecting one of the updated candidate design points based on the corresponding value of the objective function. In this way, the selected design point has a high probability of corresponding to the global minimum of the objective function predicted by the sampled function, and potentially less informative design points resulting from the local minimum of the sampled function are avoided. In other examples, design points can be selected using alternative optimization algorithms, such as evolutionary algorithms, or by comparing evaluations of the objective function on a regular grid.

[0012] For a given iteration, determining each design point in the set may involve optimizing each objective function within each confidence region of the design space, where each confidence region differs among at least several design points in the set. This can lead to a more efficient search of the feasible domain boundary, particularly for high-dimensional design spaces.

[0013] For a given iteration, determining each design point in the set is done using groups of one or more probabilistic models, each group corresponding to a different confidence region of the design space. In this way, each group of probabilistic models can model only a relatively small region of the design space, rather than modeling the entire design space as would be the case if a common group of probabilistic models were used for all design points in the set. As a result, the accuracy of the method can be further improved, and its scalability to higher-dimensional design spaces can be enhanced.

[0014] The method may include determining that the distance between two design points in a determined set of design points is shorter than a threshold distance, and replacing one of the two design points with a replacement design sampled from a replacement point distribution. By replacing design points that are very close to each other in the design space, the method avoids unnecessarily aggregating redundant information. The replacement point distribution can be, for example, a normal distribution, or another probability distribution centered on the original design point (perhaps truncated so as not to extend beyond a suitable domain). Alternatively, the replacement point distribution may be, for example, a uniform distribution independent of the original point, in which case any redundant design points can be replaced with randomly sampled points predicted by a probabilistic model as being close to the feasible domain boundary. In another example, in a given iteration, a relatively large initial set of design points can be obtained, from which a subset can be algorithmically selected in such a way that any two design points within the subset are not too close to each other.

[0015] The method may include: obtaining a number of design points predicted or measured to exist within the feasible region of the design space based on data obtained during one or more iterations; and defining the boundaries of the number of design points using a set of geometric shapes with dimensionality equivalent to that of the design space, thereby determining a representation of the feasible region of the design space. For many physical systems, the design space can be high-dimensional, which means that it may be difficult to extract useful information about the predicted feasible region. By determining a representation of the feasible region using high-dimensional geometric shapes, it becomes possible to identify a set of feasible design points without requiring a human user to comprehensively inspect the high-dimensional space manually or visually. The determined representation may be used to determine various properties of the feasible region that may be relevant to downstream processes. The number of design points can be obtained, for example, by evaluating a probabilistic model for each of the one or more design points and / or based on data obtained during multiple iterations.

[0016] The physical system may include at least a portion of the vehicle, such as one or more of the following: a hybrid powertrain system, vehicle architecture, electric motor, battery system, thermal management system, regenerative braking system, tires, and aerodynamic components. More generally, the method can be used to assist in the design of the entire system (e.g., the vehicle) and / or subsystems or components at any level within a set-based design process.

[0017] In a further embodiment, a computer implementation method is provided for determining a representation of a feasible region of a design space for a physical system. The method includes: obtaining a plurality of design points that are predicted or measured to exist within the feasible region of the design space; and defining the plurality of design points using a set of geometric shapes (e.g., hyperellipsoids or hypercubes) of the same dimensionality as the design space to determine a representation of the predicted feasible region of the design space.

[0018] Using high-dimensional geometric shapes to determine the representation of the feasible domain allows human users to identify a set of feasible design points without having to comprehensively inspect high-dimensional space manually or visually.

[0019] Multiple design points are intermediate sets of design points, and defining multiple design points using geometric shapes may include defining the intermediate sets with intermediate geometric shapes of the same dimensionality as the design space, then applying a clustering algorithm according to the intermediate geometric shape to iteratively divide the intermediate sets into multiple intermediate sets, defining each of the multiple intermediate sets with its respective intermediate geometric shape, and, if the volume of design space occupied by each intermediate geometric shape after the division is not less than the volume of design space occupied by the intermediate geometric shape before the division, adding the intermediate geometric shape before the division to the set of geometric shapes. The set of shapes rapidly converges toward a set that closely corresponds to the feasible region of the design space.

[0020] Obtaining multiple design points that are predicted or measured to exist within the feasible domain of the design space may include obtaining a model for each of one or more outputs of the physical system, and evaluating each of the models for one or more outputs at the multiple design points in the design space in order to predict the multiple design points that exist within the feasible domain of the design space.

[0021] The method may include determining at least an uninterrupted portion of the feasible region of the design space by using geometric shapes to identify, for example, a connected set of geometric shapes. Such an uninterrupted region can be extremely advantageous because it defines a distinct set of feasible processes in which the design process can take place.

[0022] The method may further include determining at least one of the following for at least a portion of the feasible domain of the design space without interruption: a hypervolume, an external axis parallel boundary delimitation box, and a maximum internal axis parallel boundary delimitation box. These features may be critical to the design process. For example, the maximum internal axis parallel boundary delimitation box can determine a set of mutually independent constraints on individual design parameters, meaning that these design parameters can be independently varied within their corresponding ranges. Information indicating any of these features may be output, for example, via a user interface, or provided as input to an automated downstream design process.

[0023] In a further embodiment, a data processing system is proposed that includes means for carrying out any one of the above-described methods, and a computer program product (such as one or more non-transient storage media) is proposed that includes instructions causing the computer to carry out any one of the above-described methods when the program is executed by the computer.

[0024] In a further embodiment, a method is provided which includes predicting a feasible region of a design space for a physical system using one of the above-described methods for predicting a feasible region, and manufacturing a prototype of the physical system having design parameter values ​​that fall within the predicted feasible region of the design space.

[0025] In a further embodiment, a method is provided that includes determining a representation of a feasible region for a physical system using one of the above-described methods for determining a representation of a feasible region, and manufacturing a prototype of the physical system having design parameter values ​​that fall within the feasible region of the design space as indicated by the determined representation of the feasible region.

[0026] Further features and advantages of the present invention will become apparent from the following description of preferred embodiments of the invention, shown merely as examples, with reference to the accompanying drawings. [Brief explanation of the drawing]

[0027] [Figure 1] Figure 1 schematically shows a system for predicting the feasible area of ​​design space for a physical system. [Figure 2] Figure 2 shows an example of a feasible set of designs within the design space. [Figure 3] Figure 3 shows a method for predicting the feasible area of ​​design space for a physical system. [Figure 4] Figure 4 illustrates a function sampled from a probabilistic model of the output of a physical system. [Figure 5] Figure 5 illustrates the distribution of sampled design points around the output threshold using two different sampling techniques. [Figure 6] Figure 6 shows a two-dimensional plot illustrating the characteristics of the feasible domain in the six-dimensional design space. [Figure 7] Figure 7 is a flowchart illustrating a computer implementation method for determining the representation of the feasible domain of the design space for a physical system. [Figures 8A-8E] Figures 8A-8E illustrate the representation of the feasible domain of a design space based on a set of geometric shapes. [Figure 9] Figure 9 is a flowchart illustrating a computer implementation method for defining the boundaries of a design point set using a geometric shape set. [Figure 10A-10E] Figures 10A to 10E illustrate a method for defining the boundaries of the design point sets shown in Figures 8A to 8E using a set of geometric shapes. [Modes for carrying out the invention]

[0028] Details of the systems and methods relating to the embodiments will become apparent from the following description with reference to the figures. This description includes numerous specific details of several embodiments for illustrative purposes. References to “one embodiment” or similar terms in the specification mean that the features, structures, or properties described in relation to an embodiment are included in at least one embodiment, and not necessarily in other embodiments. Furthermore, it should be noted that some embodiments are described schematically, with some features omitted and / or necessarily simplified, in order to facilitate the explanation and understanding of the concepts inherent in those embodiments.

[0029] Embodiments of this disclosure relate to set-based system design. More specifically, embodiments described herein address challenges related to the time-consuming and resource-intensive nature of running simulations of complex physical systems, which can be a major obstacle in the development cycle of physical systems.

[0030] Figure 1 shows a predictive system 100 for predicting the feasible area of ​​design space for a target physical system. The target physical system could be a vehicle, e.g., a passenger car, motorcycle, truck, ship, aircraft, unmanned aerial vehicle (UAV) or drone, or a component thereof, e.g., an internal combustion engine, electric motor, or hybrid powertrain for a vehicle. Alternatively, the target physical system could be or include one or more of the following: a hydraulic system, an electrical system, or any other system under design. The target physical system may be formed in a hierarchy of subsystems and components, and the predictive system 100 can similarly be used for the purpose of predicting the feasible area of ​​design space for a subsystem or component at any level in the hierarchy. The design of the target physical system may not be determined at the time the method described herein is implemented, and the target physical system may not physically exist at the time the predictive system 100 is operated.

[0031] The design space for the target physical system may be parameterized by many design parameters (e.g., tens or hundreds of design parameters). A single design point within the design space can specify a value for each of the design parameters. The target physical system has one or more outputs that may depend on the values ​​of the design parameters. In a situation where the target physical system is a vehicle or a component of a vehicle, examples of outputs include carbon dioxide emissions, fuel consumption, maximum speed, and acceleration time from zero to a given speed, e.g., 100 km / h. Generally speaking, the output may be any combination or function of quantities whose values ​​may be measured empirically or determined from numerical simulation. A given output may be continuous or discrete, and for example, a given design point can directly indicate whether a numerical simulation of the target physical system generates a valid output value rather than a meaningless output value and / or error. One function of the prediction system 100 is to predict the value of one or more outputs for a given set of values ​​for the design variables for the target physical system. To do this, the prediction system 100 includes one or more probabilistic models having the ability to predict probability distributions for one or more outputs of a target physical system for a given design point. In some embodiments, a multi-output probabilistic model may be configured to predict probability distributions for some or all outputs of the target physical system. In other embodiments, separate probabilistic models may be provided for the distinct outputs of the target physical system. The probabilistic model may be used to directly model the outputs, or it may be used to model the residuals of a simple regression model, such as a linear regression model. This latter approach effectively removes the trend from the outputs of the target physical system, and the inventors have found that this can improve the accuracy of the probabilistic model in modeling the outputs.

[0032] The probabilistic model (or each probabilistic model) may be a sparse Gaussian process, a deep Gaussian process, a Bayesian neural network, an ensemble of neural networks (a so-called deep ensemble), and / or any combination of these models. These models remain computer-processable for a large number of data points (e.g., tens of thousands, hundreds of thousands, or millions of data points), in contrast to standard Gaussian process regression. Nevertheless, in other embodiments, standard Gaussian process regression or any other type of probabilistic model capable of predicting a probability distribution over the entire function corresponding to the output of a physical system may be used. For discrete outputs, the corresponding probabilistic model may be a classification model. For continuous outputs, the corresponding probabilistic model may be a regression model.

[0033] In detail, the inventors discovered that deep Gaussian process models and deep ensembles have the ability to faithfully capture the probability distribution of outputs even when the outputs exhibit discontinuities across a certain manifold within the design space. As described in the paper "Chained Gaussian Processes" by Saul et al., Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (2016), sparse Gaussian process or deep Gaussian process models can optionally utilize heterogeneous variance likelihood to assist the model in capturing fluctuating observational noise levels across different regions of the design space. In some embodiments, a multi-output probabilistic model may be configured to predict the probability distribution for some or all of the target physical systems. In some embodiments, separate probabilistic models may be provided for the distinct outputs of the target physical systems.

[0034] The output of the target physical system may be constrained. For clarity, only upper thresholds are considered in this disclosure, but the methods described herein can be generalized to address other types of constraints. For example, lower thresholds can be transformed into upper thresholds by inverting the corresponding output. Furthermore, a monotonic transformation can be applied to the output to transform constraints into upper thresholds. The transformation may be predetermined or learned along with the parameter values ​​of the corresponding probabilistic model. Nevertheless, in other embodiments, lower thresholds (or a defined range having both upper and lower thresholds) may be explicitly considered. A viable design for the target physical system may be defined as a set of values ​​for design parameters that satisfy all constraints. If the constraints are formulated as upper thresholds, the viable design may be a design that does not exceed any of the thresholds. The thresholds may be known a priori, for example, by rules or from modeling, simulation and / or experimentation performed before the design process occurs. Alternatively, a range of values ​​for a given threshold is initially provided, and the exact value is determined only at a later stage, for example, as a result of information gathered from experiments or simulations conducted during the design or testing process. Furthermore, the user can iteratively modify the threshold or threshold range as the learning process progresses. The overarching goal of the prediction system may be to assist in determining a viable set of designs for the target physical system.

[0035] Figure 2 shows an exemplary embodiment of a design space 200 for a target physical system with two design parameters and three outputs. In other embodiments, the number of design parameters and / or outputs may be significantly larger. The design space 200 includes a feasible region 202 defined by three boundary regions 204a, 204b, and 204c, each corresponding to a threshold for each output of the target physical system. Regions 204a and 204c have non-zero widths, meaning that the corresponding thresholds are not precisely known before the design process, and instead a threshold range is provided. For example, if the thresholds are fixed in a later testing phase, the three boundaries of the feasible region 202 will lie somewhere within regions 204a, 204b, and 204c. The lack of certainty about these boundaries presents a challenge to the design process, as it requires determining information about the feasibility of design points across the entire finite-width boundary region.

[0036] The prediction system 100 includes a processor 102 and memory 104. The processor 102 may include one or more of the following: a central processing unit (CPU), a graphics processing unit (GPU), a neural processing unit (NPU), a neural network accelerator (NNA), an application-specific integrated circuit (ASIC), an application-specific standard product (ASSP), a digital signal processor (DSP), a field-programmable gate array (FPGA), a system-on-a-chip (SoC), and any other form of integrated circuit. The prediction system 100 may be a single device or may include a number of devices or systems communicating with each other over a network. The processor 102 may include a number of processing nodes, which may include separate cores of a single processor and / or a number of processors distributed within a single device or across a number of devices. In this disclosure, the term memory is used to include both volatile and non-volatile working memory, as well as non-volatile storage. Memory 104 stores data obtained during the operation of the prediction system 100, as well as computer-readable instructions, in any format suitable for implementing the method described herein.

[0037] The prediction system 100 is communicatively connected to the data acquisition system 106. The data acquisition system 106 may include one or more computing devices. The data acquisition system 106 and the prediction system may be part of a single device or system, or they may be separate from each other. For example, the prediction system 100 and the data acquisition system 106 may be operated by different commercial entities, in which case the prediction system 100 may be configured to provide functionality to the operator of the data acquisition system 106 via a software-as-a-service or cloud-based model. The data acquisition system 106 is configured to obtain data representing the values ​​of one or more outputs of a target physical system. The data acquisition system 106 may be configured, for example, with software to perform numerical simulations of the target physical system. Advantageously, the data acquisition system 106 may have the ability to run many instances of numerical simulations in parallel with each other, with each instance using its respective set of values ​​for design points. Such simulations may be based on, for example, the finite element method (FEM), boundary element method (BEM), finite difference method (FDM), discrete element method (DEM), and / or discrete event-based simulation. The simulation may therefore be deterministic, such that a given set of inputs always produces the same set of outputs, or it may include randomness or pseudo-randomness, for example, to account for noise. In this disclosure, one or more output values ​​may include one or more simulated values ​​of outputs and / or may include one or more physical measurements of outputs obtained, for example, from a prototype of the target physical system. The data acquisition system 106 may be fully automated, or some manual input from a user, such as a test engineer, may be involved in its operation.

[0038] Obtaining output values ​​for a target physical system can be an extremely time-consuming and resource-intensive process. Specifically, numerical simulations of complex systems can require long processor times, even days, to run. Similarly, building and adjusting physical prototypes can be very expensive in terms of materials and labor costs. Therefore, it is desirable to determine the feasible area of ​​design space for the target physical system using only a relatively small volume of empirical and / or simulated data. To facilitate this ultimate goal, the prediction system 100 and the data acquisition system 106 participate in an iterative process to reveal the feasible area of ​​design space. In each iteration, the prediction system 100 determines a set of design points 108 for the target physical system and provides this set of design points 108 to the data acquisition system 106. Each design point (or query point) corresponds to the respective design or configuration of the target physical system. The set of design points 108 may be provided in any preferred format, for example, in the form of a comma-separated values ​​(CSV) file where the input corresponds to the design parameter values ​​for each design point. The design point set 108 may contain tens, hundreds, or thousands of design points. The design points may have coordinates restricted to a predetermined domain within the range [0,1] for each dimension of the design space, for example. The design points may include continuous and / or discrete input dimensions. In some embodiments, the design parameters may include one or more categorical variables, in which case the mapping of the categorical variables to the (latent) input dimensions of the design space may be learned concurrently with the parameter values ​​of the probabilistic model.

[0039] For each design point in the design point set 108, the data acquisition system 106 obtains a set of data points 110, each data point representing one or more output values ​​of the target physical system for each of the design points 108. The data acquisition system 106 provides the data points 110 to the prediction system 100, for example, as a further CSV file. The output values ​​may be normalized to be within or near a given range, such as the range [0,1]. The data transmitted between the prediction system 100 and the data acquisition system 106 may be authenticated at each iteration to ensure, for example, that the data has the expected format, contains the expected number of data points, and the values ​​are within the expected domain and / or range. Other processing such as normalization and outlier removal may be performed.

[0040] In each iteration, the design point 108 and data point 110 may be transferred between the prediction system 100 and the data acquisition system 106 by electronically transmitting the data over a network, for example, via an application programming interface (API) or by email or any other preferred form of electronic communication. The transmission may be automated or performed manually by a human user. In other embodiments, the data indicating the design point 108 and / or data point 110 may be transferred via a wired connection between the prediction system 100 and the data acquisition system 106, or by manual transfer of data using a removable data storage device. The time it takes for the prediction system 100 and the data acquisition system 106 to perform their respective tasks may be approximately several days for each iteration.

[0041] As will be explained in more detail below, the prediction system 100 determines a design point that provides a high degree of information about the boundary of the feasible region of the design space. In the early iterations, when the prediction system 100 has received only a small number of data points from the data acquisition system 106 and therefore has relatively little information about the dependence of the output of the target physical system on the design variables, the design point determined by the prediction system exhibits a high degree of random variance. In later iterations, when the prediction system 100 has received more data points from the data acquisition system 106 and therefore has more information about the dependence of the output of the target physical system on the design variables, the prediction system 100 will determine a design point closer to the boundary of the feasible region, as predicted by one or more probabilistic models. In this way, the prediction system 100 considers the search-exploit dilemma problem in a reasonable and efficient manner.

[0042] After a certain number of iterations of an iterative process involving the prediction system 100 and the data acquisition system 106, the prediction system 100 generates data 112 indicating a feasible region of the design space for the target physical system, as predicted by one or more probabilistic models. This may be done, for example, when certain stopping criteria are met, or in response to user input. The data 112 may indicate, for example, the predicted boundaries of the feasible region and / or another preferred representation of the feasible region. The method for determining the representation of the feasible region of the design space is described in further detail below. The data 112 may be provided to an engineering system 114 that can perform many functions related to the predicted feasible region, such as displaying information about the feasible region via a graphical user interface (GUI) or generating control data for an automated manufacturing process. In other embodiments, the data 112 indicating the predicted feasible region of the design space is provided for use in a downstream design process, for example, to design lower-level components or subsystems of the target physical system. In that case, the prediction system 100 may be deployed again to predict the feasible area of ​​design space for subsequent lower-level components or subsystems, etc.

[0043] Figure 3 shows an example of a method for predicting a feasible area of ​​design space for a physical system, such as the target physical system in Figure 1. This method may be carried out by, for example, a prediction system 100. The method begins by initializing one or more probabilistic models in 302. Initializing the probabilistic models may include determining initial values ​​for trainable parameters of the probabilistic models, such as connection weights and / or biases in the case of a Bayesian neural network or a deep ensemble, or inducing variables in the case of a sparse Gaussian process or a deep Gaussian process. Initializing the probabilistic models may further include setting values ​​for trainable or fixed hyperparameters, such as kernel hyperparameters in the case of a sparse Gaussian process or a deep Gaussian process. The trainable parameters may be initialized with random values ​​sampled from, for example, a uniform distribution or any other suitable distribution over a suitable range of values, or with values ​​corresponding to a preceding design process.

[0044] An optional additional initialization step may include sampling an initial set of design points within a preferred domain of the design space (e.g., a hypercube). The initial set of design points may be randomly sampled, for example, using uniform sampling or any other arbitrary random sampling method, or deterministically sampled according to a space-filling method such as Sobol sampling. Next, the output values ​​of the physical system in the initial set of design points can be obtained, for example, using numerical simulation. Then, to best fit the probabilistic model to the obtained output values, the values ​​of the trainable parameters of the probabilistic model can be determined. This may include, for example, maximizing the variational lower bound (ELBO) of the obtained values ​​using reverse-mode differentiation and stochastic gradient descent or variations thereof.

[0045] The method continues in 304 by sampling a function from a probabilistic model. Each probabilistic model may be viewed as a probability distribution over multiple functions, meaning that a single function can be sampled from the probabilistic model in a manner similar to how a single point can be sampled from a finite-dimensional probability distribution. Each sampled function predicts one or more output values ​​of a physical system for a design point within the design space. Sampling a function does not necessarily require determining the function's value for the design point across the entire design space, but it may involve determining the values ​​of one or more variables that determine the function's value for the design point across the entire design space. For certain types of probabilistic models, accurate sampling of the function may not be possible, in which case an approximate sample function can be obtained. In the case of Gaussian process models, the approximate sample function can be obtained, for example, using the decoupling sampling approach described in Wilson et al.'s paper "Efficiently sampling functions from Gaussian process posteriors," International Conference on Machine Learning (2020). In embodiments where different probabilistic models are used for different outputs or different groups of outputs of a physical system, the sampled functions can be grouped such that a given group predicts the value of each output of the physical system for a design point in design space. If a batch of N design points must be determined and J outputs of the physical system are considered, then the set of functions f j (n) However, sampling may be performed for j=1,...,J,n=1,...N, where the function for a given n value is either from a single multi-output model or 1 <n f n about ≤ J f Samples may be taken from individual, independent single or multi-output models.

[0046] FIG. 4 shows an exemplary plot 400 of a probability model of the output of a physical system with a single design variable (i.e., a single input dimension), assuming deterministic data. The shaded region of the plot represents two standard deviations of the probability model. In this example, it is observed that the standard deviation drops to zero at each data point of the data point set, indicating that the value of the output is known at these points and thus there is no uncertainty in the output at these points. The curve passing through the data points is a function sampled from the probability model, and the horizontal line 402 represents the output threshold.

[0047] The method of FIG. 3 continues with the step of selecting design points 306 using the sampled function f j (n) In this example, design points n = 1,..., N are selected, and the nth design point is selected based on an objective function that evaluates the sampled function f j (n) for j = 1,..., J. The objective function can impose a penalty on the difference or distance between the threshold for a given output and the prediction of the given output provided by the sampled function. The step of selecting design points based on the objective function may include performing an optimization routine to find design points that are close to an extreme value (e.g., corresponding to the minimum distance between the threshold and the output of the sampled function). The nth selected design point x n ∈argmin x d (n) (x), where d (n) (x) is the objective function for the nth design point. For example, x exists along the predicted boundary of the feasible region nThere may be an infinite number of solutions for this. In that case, the choice of optimization routine will determine which design points are selected. Those skilled in the art will know of suitable optimization algorithms for this purpose, such as L-BFGS-B, CMA-ES, differential evolution, or any other suitable optimization algorithm. In some embodiments, the optimizer can be run several times (e.g., tens, hundreds or thousands of times) starting from different random locations in the design space, resulting in a number of candidate design points. Then, a design point can be selected from the candidate design points, for example, randomly and / or based on the value of the objective function and / or according to the variance of the probabilistic model and / or in a way that ensures diversity among the selected design points.

[0048] In some embodiments, further steps can be taken to avoid design point duplication, both within a single batch and optionally in relation to points in preceding batches. In this regard, design points that are too close to each other can result in redundant and resource-intensive identical observations, especially when collected using a deterministic simulator with limited accuracy. To avoid this, if a duplicate design point is detected (i.e., determined to be too close to another design point), the duplicate design point may be replaced by a replacement design point sampled from a replacement point distribution. The replacement point distribution may depend on the location of the duplicate design point and could be, for example, a normal distribution centered on the duplicate design point and truncated so as not to extend beyond the design space. Alternatively, the replacement point distribution could be independent of the duplicate design points. In this case, a set of candidate replacement design points could be randomly (e.g., uniformly) sampled within the design space and then filtered to remove candidates that the model predicts are sufficiently close to the boundary. If a duplicate design point exists, it can then be replaced with a point from the filtered set of candidate replacement design points.

[0049] Objective function d (n)(x) may be chosen to reflect the predicted distance from the boundary of the feasible region. Boundary B is,

number

number

[0050] For a single output, the objective function may be provided by or depended upon by any suitable distance metric that measures the distance between the function output and the predicted boundary. For example, the objective function d for the nth design point. (n) (x) is d (n) (x) = ||f (n) This can be obtained by (x)-T||, where x represents a point in the design space and f (n) (x) is the value of the function at point X, T is the threshold of the output of the physical system, and ||-|| represents the distance metric, e.g., L1 distance, L2 distance, or smoothed L1 distance.

[0051] When a threshold is applied to multiple outputs of a physical system, a suitable example of an objective function is d for q=1, 2, ... (n)(x) = |max i (f i (n) (x)-T i )| q It is given by. To avoid possible numerical problems, the objective function is arbitrarily d (n) (x) = |max i ([f i (n) (x)-T i ] / scale i )| q Each positive scaling factor i This can be corrected by rescaling one or more operands of the max operator. Other corrections are also possible, such as substituting the max operator with a smoothing approximation of the maximum operator, such as one of the Boltzmann, LogSumEpx, mellowmax, p-Norm, or Smooth Maximum Unit (SMU) operators. These objective functions have a threshold T depending on their relative value. i and function f i We selectively penalize the distance between two points. By selectively penalizing distance in this way, the objective function can be concentrated at the intersections between boundary segments corresponding to different thresholds (as it might be if we were to penalize the sum of all such distances), or we can guide the design points to exist along the boundary of the feasible region rather than extending to manifolds far from the boundary of the feasible region (as might be if we were to penalize the smallest such distance).

[0052] Another example of an objective function with suitable characteristics is d (n) (x) = min 1≦j≦J (|f j (n) (x)-T j | q +Σ i≠j max(0,f i (n) (x)-T i ) q ) is given by, where T jis the threshold for the j-th output, where q > 0. If q = 1 or q = 2, then this is d (n) (x) = Σ i max(0,f i (n) (x)-T i ) q +min 1≦j≦J (|f j (n) (x)-T j | q -max(0,f j (n) (x)-T j ) q ) can be written as equivalently.

[0053] Optimizing any of the objective functions described above from a given point in the design space will trace a path toward the predicted boundary of the feasible region. In Figure 2, thick white arrows illustrate the possible directions of the negative gradient of the objective function at various points in the design space.

[0054] The optimization of the objective function may be applied comprehensively across the entire design space, or alternatively, locally to individual design points within a batch, for example, within separate "confidence regions" of the design space. For example, the nth design point is constrained such that x lies within its respective confidence region, and the corresponding objective function d (n) (x) may be determined by optimizing it. Each confidence region can be used to determine one or more design points, and a different function is sampled from the probabilistic model for each design point within each confidence region.

[0055] The confidence regions may be sub-regions of the design space, each centered at a different location and having a defined shape, such as a hypersphere or hypercube. The location and dimensions of the confidence regions may evolve between iterations according to the method described, for example, in the paper "Scalable Global Optimization via Local Bayesian Optimization" by Eriksson et al., Conference on Neural Information Processing, 2019. According to this method, the size of a given confidence region may be increased after a predetermined number of "successful" iterations in which the value of the objective function improves, and decreased after a predetermined number of "failed" iterations in which the value of the objective function does not improve. Whenever a confidence region satisfies a predetermined reinitialization condition, that confidence region can be terminated and a new confidence region can be initialized.

[0056] Currently, reinitialization conditions may include, for example, the size of the confidence region falling below the threshold size, the objective function reaching zero (indicating that the design point lies on the predicted boundary), the confidence region moving too close to another confidence region, and / or the optimizer converging to an already determined design point. These last two conditions may further help promote diversity within the determined set of design points. Currently, it is advantageous to initialize new confidence regions at locations predicted by the probabilistic model to be near the boundary of the feasible region, or otherwise expected to be close to the boundary of the feasible region, for example, locations determined by randomly perturbing from randomly selected design points determined in previous iterations. Locally, the objective function d is determined within each confidence region. (n) Optimizing (x) can, as a result, identify local optima along the estimated boundaries of the feasible domain, potentially leading to more efficient boundary searches, especially for high-dimensional design spaces.

[0057] In some implementations, the same probabilistic model (or set of probabilistic models) can be used to determine all design points in a given batch. In other implementations, different probabilistic models can be used to determine different design points in a batch. For example, N independent probabilistic models (or sets of N independent probabilistic models) can each be responsible for a different confidence region. In this way, in a given iteration, each probabilistic model can be responsible only for predicting the output of the physical model for a relatively small region of the design space, resulting in a simpler optimization surface for fitting the probabilistic models, and potentially further improving the accuracy of the method and its scalability to higher-dimensional design spaces.

[0058] As mentioned above, in some cases, the threshold values ​​for one or more outputs of a physical system may not be precisely known during a stage of the design process. In this case, the objective function described above is applied to the threshold T independently for each design point n in the batch. i The threshold can be modified so that it is sampled. The threshold may be sampled from a probability distribution, such as a uniform distribution over the entire range of a given output value. By sampling the threshold along with the function, the result of this method is that design points are selected across the entire region where the (currently uncertain) boundary of the feasible region may exist. In the embodiment of Figure 2, it can be seen that the design points are selected across the entire regions 204a, 204b, and 204c, which represent the uncertain boundary of the feasible region 202.

[0059] In early iterations, where the probabilistic model may have relatively high variance, significant diversity is expected in the sampled function, and therefore significant diversity exists within the selected design points, thereby exploring a wide area of ​​the design space. In later iterations, where more empirical and / or simulated data are collected, the probabilistic model may have significantly lower variance, in which case the diversity within the sampled function also decreases, and therefore the selected design points are concentrated around the boundary of the feasible region, making it possible to accurately predict the precise location of the feasible region boundary. Figure 5 shows the results of 20 iterations of an exemplary experiment in which 1000 design points are selected in each iteration for a single physical system with a single output subject to one threshold. The left frame 502 shows the distribution of output values ​​for the selected points in each iteration using Sobol sampling. The distribution remains qualitatively similar across iterations and is observed to be independent of the output threshold (indicated by the vertical line 506). In contrast, the right frame 504 shows the distribution of output values ​​for the selected points in each iteration using the method described herein. In this case, even in the zeroth iteration, the distribution of output values ​​is close to the threshold (indicated by vertical line 508), indicating that the probabilistic model, as initially initialized, can reasonably accurately predict the output values. The distribution will have sharper peaks around the threshold in later iterations as the probabilistic model becomes more confident in its predictions.

[0060] As explained above, the functions sampled in 304 are independent of each other, meaning that the corresponding objective functions can be optimized in parallel. As a result, large batches of design points (e.g., thousands or tens of thousands of design points) can be determined in a relatively short time. This is in contrast to other batch Bayesian devices that typically rely on multipoint approaches, where the point batch is determined by optimizing an acquisition function that assigns a single score to the entire batch, or greedy approaches, where the acquisition function is optimized to determine the first point, then modified to penalize locations close to the first point before being optimized again to determine the second point, and this continues until the desired number of points are obtained. At the time of writing this specification, the current state of greedy approaches is computationally unfeasible for batches larger than around 20 points, and the sequential nature of the algorithm hinders parallelization. Multipoint algorithms tend to be extremely computationally intensive. This sampling-based approach, in contrast, allows for the determination of large point batches in parallel.

[0061] The method in Figure 3 continues in step 308, where the output values ​​of the physical system are obtained at the design points determined in 306. As discussed above, this may include a step of performing physical measurements of one or more prototypes of the physical system, and / or a step of performing numerical simulations of the physical system. In some embodiments, the raw data obtained in 308 can be processed, for example, by normalization and / or outlier removal before proceeding to the next step.

[0062] The method continues in step 310 with the step of updating the probabilistic model using the data obtained in step 308. The step of updating the probabilistic model may include determining updated values ​​for the trainable parameters of the probabilistic model using gradient-based optimization with respect to the maximum posterior probability or maximum likelihood objective function (which may be equivalent to maximizing the ELBO). The step of determining the updated parameter values ​​in a given iteration involves performing an optimization procedure starting from the parameter values ​​determined in a preceding iteration (or during initialization). In this way, optimization does not need to start from scratch each time, and training can be made faster. However, for at least some iterations, the step of updating the model may include a step of reinitializing the model and a step of retraining from scratch using all the accumulated data.

[0063] The method continues in step 312 with a step of determining whether a stopping condition has been met. The stopping condition may include, for example, that one or more convergence criteria have been met and / or that a predetermined number of iterations have been performed, or that user input has been provided. The stopping condition may depend on the evaluation of the probabilistic model. For example, the stopping condition may depend on a metric that evaluates the classification accuracy of the model, such as a Briar score or an F1 score. Alternatively or additionally, the stopping condition may depend on the predicted variance associated with one, some or all of the probabilistic models in a given set of design points below a threshold. In this way, it is possible to self-assess the convergence of the model in each iteration using uncertainty estimates built into the probabilistic model.

[0064] If the termination condition is not met, the method returns to step 304. Steps 304–310 are repeated iteratively until the termination condition is met, with new data being collected and the probabilistic model being updated in each iteration. When the termination condition is met, the method in Figure 3 terminates in step 314, in which the probabilistic model is used to predict the feasible region of the design space. The step of predicting the feasible region may include identifying a region of the design space in which the probabilistic model predicts design points to satisfy feasibility criteria, such as the predicted probability of feasibility being greater than 0.5 or another selected value. To achieve this, the probabilistic model may be used to evaluate the feasibility criteria at design points within the design space (e.g., some or all of the design points selected during the preceding steps, or a new set of design points selected according to a space-filling plan, for example). From clusters of design points that satisfy the feasibility criteria, the feasible region can be inferred. Such regions may be presented via a user interface, for example, using 2D or 3D plots in which certain dimensions are suppressed by marginalization or values ​​within certain dimensions are fixed.

[0065] In some embodiments, the user may want to examine the view or representation of the feasible region in more detail after one or more initial iterations, and then one or more further iterations may be performed (as indicated by the dashed arrows in Figure 3). More specifically, the user may decide to fine-tune the thresholds of one or more design variables depending on the characteristics or appearance of the feasible region after the initial iterations.

[0066] Figure 6 shows an array of two-dimensional plots, each displaying two dimensions of a six-dimensional design space, where the values ​​of the other four dimensions are defined as the coordinates of the slicing points (represented as diamond-shaped points in the plot). The plot in the upper right shows the predicted feasibility probabilities for a given threshold of output, with darker shading corresponding to higher feasibility probabilities. The curves represent the contours where the estimated feasibility probabilities are given above by 90%, 50%, and 10%. The curve in the lower left plot represents the contour where the estimated feasibility is given above by 50% for thresholds that vary by ±∈, similar to the same threshold of output mentioned above.

[0067] For users, manually exploring the design space using 2D or 3D plots to detect and characterize the feasible region of a high-dimensional design space can be a challenging task. Therefore, one objective of this disclosure is to provide a method for automatically determining a representation of the feasible region of a design space, such as a feasible region predicted using the method described above. Such a method is described with reference to Figure 7. The method begins in 702 with the step of obtaining a set of design points predicted or measured as feasible. This set of design points may include, for example, design points predicted or measured as feasible based on experiments or simulations performed in step 308 in one or more iterations of the method in Figure 3. Alternatively or additionally, the set of design points may include points predicted as feasible based on one or more predictive models, such as a probabilistic model trained according to the method in Figure 3. These probabilistic models have the ability to predict the probability distribution of the output of a physical system at a given design point and may therefore be used to predict whether a given design point lies within the feasible region (i.e., whether the feasibility criterion is met). Other types of probabilistic models may simply predict the output value of the physical system at a given design point, or they may directly classify the point as feasible or unfeasible. In any of these cases, the predictive model may optionally be trained using data collected during the method shown in Figure 3.

[0068] To obtain a predicted set of feasible design points using a predictive model, the predictive model can be evaluated on a set of candidate design points that can be selected randomly or deterministically according to a space-filling method, for example. However, for high-dimensional design spaces, a vast number of candidate design points may be required to obtain a sufficient set of predicted feasible design points. Therefore, a preferred method is to use a predictive model to guide the selection of design points using an optimization procedure similar to that described in relation to step 306 in Figure 3, for example.

[0069] The method continues in step 704 with the step of defining design points predicted or measured as feasible using a set of geometric shapes having the same dimensionality as the design space. The shapes may include, for example, hypercubes or other hyperpolygons, hyperellipses, or any combination thereof or other shapes. The number, location, orientation, and dimensions of the geometric shapes may be algorithmically determined to most densely define the feasible region. The properties of the feasible region, such as its boundary within any given dimension, its (hyper)volume, and its shape, can be readily derived from the parameter values ​​of the set of geometric shapes. The shape of the feasible region can provide information about the dependencies between design variables.

[0070] A method for defining the boundaries of a predicted set of feasible design points using a set of geometric shapes is provided below, with reference to Figure 9.

[0071] Figure 8A shows an example of a two-dimensional design space in which a probabilistic model has been evaluated at design points across the entire design space. Design points predicted to be feasible are shown in three clusters 802, 804, and 806, corresponding to the feasible regions of the design space. Figure 8B shows clusters 802, 804, and 806 demarcated by a set of six ellipses.

[0072] The method continues in step 706 with the step of determining connected or uninterrupted portions of the executable region. This can be done by determining which of the determined sets of geometric shapes overlap or intersect, and then appropriately grouping them. This grouping may be represented using a graph with nodes corresponding to the geometric shapes. If two shapes are determined to intersect, an edge is added between them. If an edge has been added for all pairs of intersecting shapes, each set of connected nodes in the graph represents separate disprime executable portions of the design space. Algorithms for determining whether geometric shapes intersect are trivial for certain shapes (e.g., hypercubes) and are known for other shapes as well. For hyperellipses, the method outlined in the paper "A robust computational test for overlap of two arbitrary-dimensional ellipsoids in fault-detection of Kalman filters" by Gilitschenski and Hanebeck, 15th International Conference on Information Fusion, 2012, can be used. As an example, Figure 8B shows that cluster 802 is demarcated by four intersecting ellipse groups.

[0073] The method ends in 708 with the step of determining one or more properties of the feasible region of the design space. These properties may include, for example, the volume, shape, and / or centroid of individual portions of the feasible region or corresponding external boundary delimiting boxes. Additionally or alternatively, the properties may include the centroid and dimensions of the largest internal axis parallel boundary delimiting box. For a given set of intersecting geometric shapes, the largest internal axis parallel boundary delimiting box can define a set of mutually independent constraints on individual design parameters, meaning that these design parameters can vary independently within their corresponding ranges without necessarily having to vary the values ​​of other design parameters. Information indicating properties such as centroids, external boundary delimiting boxes, and / or the largest internal axis parallel boundary delimiting box can be output, for example, via a user interface, or provided as input to an automated downstream design process. Given a description of the geometric shape of a given feasible portion of the design space, it is possible to formulate the detection of the largest internal axis-parallel boundary delimiting box as a single optimization problem by maximizing the volume of a d-dimensional axis-parallel hypersphere while ensuring that no region of the design space outside the connected geometric shape is stored. As with many optimization problems, the optimizer may enter a local minimum. This can be mitigated by optimizing from a number of different starting points and selecting the largest resulting hypersphere. Optionally, different optimization runs may be performed in parallel.

[0074] In Figure 8C, the regular dashed lines indicate the (external) axis-parallel boundary delimitation boxes corresponding to three separate clusters 802, 804, and 806 (reproduced in Figure 8D for clarity). The irregular dashed lines in Figure 8C indicate the largest internal axis-parallel boundary delimitation box corresponding to clusters 802, 804, and 806 (reproduced in Figure 8E for clarity, along with the centroid of the largest internal axis-parallel boundary delimitation box).

[0075] Figure 9 shows an iterative method in which a subset of design points predicted to be feasible can be bounded by a set of geometric shapes. The method begins with the step of bounding a subset of design points having an intermediate geometric shape in 902. In the first iteration of the method, the intermediate geometric shape can cover all or most of the design space and may be selected to enclose all subsets of points predicted to be feasible. The intermediate geometric shape may be, for example, a hyperellipse with the same dimensionality as the design space, but other shapes may also be used. A hyperellipse is given by equation (xv) T It can be defined by A(xv)<1, where v is the centroid (mean) and A is the covariance matrix of the hyperellipse. Figure 10A shows an example of a two-dimensional ellipse that covers most of the design space in Figure 9. It is observed that the ellipse encloses all the design points in clusters 802, 804, and 806.

[0076] The method in Figure 9 continues in step 904, dividing the point subset into multiple point subsets according to the intermediate geometric shape. The split or decomposition step may be performed by clustering the points at two or more cluster centers (e.g., using K-means clustering) at locations according to the intermediate geometric shape. In embodiments where the intermediate geometric shape is a hyperellipse, the two K-means cluster centers may be initialized at the endpoints of the major axis of the hyperellipse. In one embodiment where the intermediate geometric shape is a hypercube, the two K-means cluster centers may be initialized at the centers of each end of the hypercube. In other embodiments, alternative clustering methods such as mean-shift clustering or density-referenced spatial clustering for noisy applications (DBSCAN) may be used.

[0077] The method continues in step 906, defining each of the subsets of points obtained as a result of the division using each intermediate geometric shape, and in step 908, determining whether the intermediate geometric shape determined in 906 occupies a smaller overall volume than the intermediate geometric shape determined in 902, or in other words, whether the division resulted in a reduction in the volume of the intermediate geometric shape. If the volume is reduced, the method returns to 904 for each subset of points obtained as a result of the division in 904. If the volume is not reduced, the intermediate geometric shapes that defined the subset of points before the division are added to the geometric shape set. If no further reduction in volume is obtained as a result of the division of any of the remaining subsets, the method terminates and the geometric shape set is completed.

[0078] In the embodiment shown in Figure 10B, cluster 802 is divided from clusters 804 and 806, and the resulting point subset is demarcated by two ellipses. In the subsequent Figures 10C-10E, the resulting subset is further divided until no further volume reduction is considered to result from further division, and is demarcated by further ellipses. The set of ellipses remaining in Figure 10E represents the viable region occupied by clusters 802, 804, and 806.

[0079] Modifications of the method in Figure 9 are possible, for example, in which non-executable design points are explicitly excluded from the representation of the executable region, as opposed to simply defining the boundaries of executable points. The method in Figure 9 may be implemented using parts of the open-source Dynesty software package as described in the paper "dynesty: A Dynamic Nested Sampling Package for Estimating Bayesian Posteriors and Evidences" by J. Speagle, MNRAS (2019), which provides an algorithm for iteratively decomposing hyperellipses for use in different contexts, namely for nested sampling for the purpose of estimating posterior distributions and evidence for Bayesian inference.

[0080] At least some aspects of the embodiments described herein with reference to Figures 1-10 include computer processes or methods performed within one or more processing systems and / or processors. However, in some embodiments, the disclosure also extends to computer programs, more specifically computer programs on or within devices, adapted to put the disclosure into practice. Such programs may be in the form of non-temporary source code, object code, code intermediate source, and object code in any other non-temporary form, such as a partially compiled form or any other form suitable for use in implementing the processes relating to the disclosure. A device may be any entity or device capable of carrying a program. For example, a device may include storage media, such as solid-state drives (SSDs) or other semiconductor-based RAM; ROMs, such as CD-ROMs or semiconductor ROMs; magnetic recording media, such as hard disks; and optical memory devices in general.

[0081] The embodiments described above should be understood as exemplary embodiments of the present invention. Further embodiments of the present invention are contemplated. For example, the above-described method for determining the representation of the feasible region of the design space may be carried out independently of the iterative sample-based method for training a probabilistic model. The probabilistic model may alternatively be obtained by any other preferred method, for example, by fitting the probabilistic model to a single dataset of empirical or simulated values, and / or constructed on physical principles. Furthermore, the method for determining the representation of the feasible region may be carried out using any type of model, including a deterministic model having the ability to predict whether a design point is feasible or not.

[0082] Any feature described in relation to any one embodiment may be used alone or in combination with other described features, or in combination with one or more features of any other embodiment or any combination of any other embodiment. Furthermore, equivalents and modifications not described herein may also be used without departing from the scope of the invention as defined in the appended claims.

Claims

1. In a computer implementation method for predicting the feasible area of ​​design space for a physical system: For one or more repetitions: Determining a set of design points within the aforementioned design space, and determining a given design point within the said set: Sampling a function from a probabilistic model, each having a set of trainable parameters, for one or more outputs of a physical system, wherein each function predicts the value of the one or more outputs at a design point in the design space; and, Selecting a given design point based on an objective function that penalizes the distance from the boundary of the feasible region predicted by the values ​​of each of the aforementioned functions, This includes, and in parallel, the selection of at least some of the design points within the set is carried out. To decide; Obtaining data including the values ​​of one or more outputs of the physical system at each design point in the determined set of design points; Using the obtained data, the updated values ​​of the trainable parameter set are determined; After the one or more iterations, the feasible region of the design space is predicted using the data obtained during the one or more iterations; Computer implementation methods including

2. The computer implementation method according to claim 1, wherein predicting the executable region of the design space is done by using the respective probabilistic models for one or more outputs of the physical system.

3. The boundary of the executable region depends on a variable threshold of one of the outputs of the physical system; Selecting the given design point includes sampling the variable threshold from a threshold distribution to predict the distance from the boundary of the feasible region. The computer implementation method according to claim 1 or 2.

4. The computer implementation method according to any one of claims 1 to 3, wherein obtaining the aforementioned data includes performing a simulation of the physical system in the determined set of design points.

5. The computer implementation method according to any one of claims 1 to 4, wherein obtaining the aforementioned data is carried out in parallel for at least some of the design points in the set.

6. A computer implementation method according to any one of claims 1 to 5, wherein determining each design point in the set for a given iteration includes optimizing each objective function within each confidence region of the design space, wherein each confidence region differs among at least several design points in the set.

7. The computer implementation method according to claim 6, wherein determining each design point in the set for the given iteration is done using each group of one or more probabilistic models, each of which corresponds to the respective confidence region of the design space.

8. The computer implementation method according to any one of claims 1 to 7, wherein the probabilistic model includes at least one of Gaussian process regression, sparse Gaussian process, deep Gaussian process, Bayesian neural network, and deep ensemble.

9. Using the respective probabilistic models for one or more outputs of the physical system, a plurality of design points predicted to exist within the feasible region of the design space are obtained; The plurality of design points are defined using a set of geometric shapes with the same dimensionality as the design space, thereby determining the representation of the feasible region of the design space; A computer implementation method according to any one of claims 1 to 8, further comprising:

10. Based on the data obtained during the one or more iterations, to obtain a plurality of design points that are predicted or measured to exist within the feasible region of the design space; and The plurality of design points are defined using a set of geometric shapes with dimensions equivalent to the design space, thereby determining the representation of the feasible region of the design space; A computer implementation method according to any one of claims 1 to 8, further comprising:

11. One or more outputs of the physical system are multiple outputs; and The objective function selectively penalizes the distance between the respective thresholds of the multiple outputs and the respective functions; The computer implementation method according to any one of claims 1 to 10.

12. The computer implementation method according to claim 11, wherein the selective penalty application is selective depending on the distance between the respective thresholds and respective functions of the plurality of outputs.

13. The computer implementation method according to any one of claims 1 to 12, wherein the physical system includes at least a part of the vehicle.

14. The computer implementation method according to claim 13, wherein at least a portion of the vehicle includes one or more of a hybrid powertrain system, a vehicle architecture, an electric motor, a battery system, a thermal management system, a regenerative braking system, tires, and aerodynamic components.

15. The computer implementation method according to any one of claims 1 to 14, wherein each of the functions sampled for any given design point in the set is independent of one or more functions used to determine any other given design point in the set.

16. The computer implementation method according to any one of claims 1 to 15, wherein the selection of at least some of the design points within the set is performed in parallel across multiple processor cores or processor nodes.

17. In a computer implementation method for determining the representation of the executable domain of the design space for a physical system, Obtaining a plurality of design points that are predicted or measured to exist within the feasible region of the design space; The plurality of design points are defined using a set of geometric shapes with the same dimensionality as the design space, thereby determining the representation of the feasible region of the design space; Computer implementation methods, including those mentioned above.

18. The plurality of design points are an intermediate set of design points, and the plurality of design points can be defined using the geometric shape: The intermediate set is defined by an intermediate geometric shape having the same dimensionality as the aforementioned design space; For the aforementioned intermediate set or each intermediate set, iteratively: In order to divide the aforementioned intermediate set into multiple intermediate sets, a clustering algorithm is applied according to the intermediate geometric shape; To define each of the aforementioned sets of intermediates by its respective intermediate geometric shape; If the volume of the design space occupied by each of the intermediate geometric shapes after the division is not less than the volume of the design space occupied by the intermediate geometric shapes before the division, then the intermediate geometric shapes from before the division are added to the set of geometric shapes; A computer implementation method according to claim 9, 10, or 17, including the method described in claim 9, 10, or 17.

19. Obtaining the plurality of design points predicted or measured to be located within the feasible region of the design space: To obtain a model for each of one or more outputs of the aforementioned physical system; To predict the plurality of design points located within the feasible region of the design space, the model for one or more outputs at the plurality of design points in the design space is evaluated; A computer implementation method according to claim 17 or 18, including the method described in claim 17 or 18.

20. A computer implementation method according to any one of claims 9, 10, or 17 to 19, further comprising determining at least an uninterrupted portion of the executable region of the design space by identifying at least a concatenated subset of the set of geometric shapes.

21. The computer implementation method according to claim 20, further comprising determining at least one of the supervolume, the external axis parallel boundary delimitation box, and the maximum internal axis parallel boundary delimitation box for at least the uninterrupted portion of the executable area of ​​the design space.

22. The computer implementation method according to claim 20, comprising outputting information indicating at least one of the supervolume, the outer axis parallel boundary delimitation box, and the maximum inner axis parallel boundary delimitation box via a user interface.

23. The computer implementation method according to any one of claims 9, 10, or 17 to 22, wherein the set of geometric shapes includes at least one of a set of hyperellipses and a set of hypercubes.

24. Predicting a feasible area of ​​the design space for a physical system using the computer implementation method described in any one of claims 1 to 16; To manufacture a prototype of a physical system having design parameter values ​​that fall within the predicted feasible region of the design space; A method that includes this.

25. Determining a representation of a feasible region of a design space for a physical system using the computer implementation method described in any one of claims 9, 10, and 17 to 23; To manufacture a prototype of the physical system having design parameter values ​​that fall within the executable region of the design space as indicated by the determined representation of the executable region; A method that includes this.

26. A data processing system comprising means for carrying out the method described in any one of claims 1 to 23.

27. A computer program product that, when the program is executed by the computer, includes an instruction causing the computer to perform the method according to any one of claims 1 to 23.

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