Order reduction modeling and control of high-dimensional physical systems using neural network models

Abstract

A system and method are provided for training a neural network to control the operation of a system having nonlinear dynamics represented by a partial differential equation (PDE). The method includes collecting digital representations of time series data representing instances of the system's function space and measurements of the system's state of operation. Co-location points corresponding to solutions to the PDE are generated. The neural network is trained using training data including the collected time series data and the co-location points to train parameters of a nonlinear operator. The neural network has an autoencoder architecture including an encoder and a decoder. The encoder encodes each instance of the training data into a latent space, a linear operator propagates the encoded instance into the latent space using a transformation determined by the parameters of the nonlinear operator, and the decoder decodes the transformed encoded instances of the training data to minimize a hybrid loss function.

Description

[Technical Field] 【0001】 This disclosure relates in general to system modeling, prediction, and control, and more specifically to systems and methods for order reduction modeling and control of high-dimensional physical systems using neural network models. [Background technology] 【0002】 Control theory in control systems engineering is a branch of mathematics that deals with the control of continuously operating dynamic systems in engineering processes and machines. Its purpose is to develop control policies to control such systems using control actions in an optimal manner without delay or overshoot, and to ensure control stability. [Overview of the Initiative] [Problems that the invention aims to solve] 【0003】 Traditionally, some methods of controlling systems are based on techniques that enable model-based design frameworks in which system dynamics and constraints can be directly considered. Such methods can be used in many applications to control systems such as dynamic systems of varying complexity. Examples of such systems include production lines, automobile engines, robots, numerically controlled machining, motors, satellites, and generators. 【0004】 Furthermore, system dynamics models or system models describe the system's dynamics using differential equations. However, in many situations, system models can be nonlinear, difficult to design, difficult to use in real time, or inaccurate. Examples of such cases are frequently seen in specific applications such as robotics, building control including heating, ventilation, and air conditioning (HVAC) systems, gas leak detection, smart grids, factory automation, transportation, self-tuning machines, and traffic networks. In addition, even when nonlinear models are available, designing the optimal controller for system control can be an inherently difficult task. 【0005】 Furthermore, in the absence of an accurate model of the dynamic system, some control methods utilize behavioral data generated by the dynamic system to construct a feedback control policy that stabilizes the system dynamics or embeds quantifiable control-related performance. Typically, various types of methods can be used to control a system that utilizes behavioral data. In one embodiment, a control method may first construct a model of the system and then leverage that model to design a controller. However, such a control method results in a black-box design of the control policy that directly maps the system state to control commands. However, such a control policy is not designed with the physical properties of the system in mind. 【0006】 In another embodiment, the control method may construct a control policy directly from data without an intermediate model-building step for the system. A drawback of such a control method is that a large amount of data may be required in the model-building step. In addition, the controller is calculated from an estimated model, for example, according to the certainty equivalence principle, but in reality, the model estimated from the data may not capture the physical properties of the system's dynamics. Therefore, many of the control techniques for the system may not be usable in the constructed model of the system. 【0007】 Therefore, in order to address the aforementioned problems, there is a need for a method and system to control the system in an optimal manner. [Means for solving the problem] 【0008】 This disclosure provides a computer-based method and system for order reduction modeling and control of high-dimensional physical systems using neural network models. 【0009】 The objective of some embodiments is to train a neural network model so that the trained neural network model can be used to control the behavior of a system having nonlinear dynamics represented by partial differential equations (PDEs). The neural network model has an autoencoder architecture which may include an encoder, a linear predictor (such as a linear operator), and a decoder. In some embodiments, the linear predictor may be based on the Koopman operator. 【0010】 In other embodiments, the neural network model may have an autoencoder architecture that includes an encoder, a nonlinear predictor (such as a nonlinear operator), and a decoder. In one implementation example, the nonlinear predictor may be based on one of two intrusive or non-intrusive models of the system's higher-dimensional dynamics. For example, the linear or nonlinear operator may be based on a reduced-order model (ROM). 【0011】 To generate an intrusive ROM, a reduced-order solution of the system can be obtained by solving a reduced-order model, i.e., by projecting the original model onto a reduced space. In some embodiments, proper orthogonal decomposition (POD)-Gallerkin projection can be used to generate an intrusive ROM. In such cases, singular value decomposition (SVD) can be applied to a snapshot matrix of the system's dynamics data, and a POD basis can be extracted. Furthermore, the intrusive ROM can be constructed by applying Galerkin projection. 【0012】 Traditionally, POD-Galerkin projection-based model reduction is intrusive because numerical implementations of reduction models require access to the discretized PDE operator. This intrusiveness of ROM techniques limits the scope of conventional model reduction methods. The main drawback of such intrusive methods is the need for access to the full model. Furthermore, when proprietary software is used to solve PDEs where the details of the governing equations of the system's dynamics are unknown, conventional intrusive model reduction methods may not be applicable, as the solver may typically be unavailable during the process. 【0013】 To generate non-intrusive ROMs, reduction models can be learned from snapshots, i.e., from either numerical approximations or measurements of the dynamic system state, when operators for discretized systems are unavailable. In one embodiment, machine learning models may be used to analyze the underlying processes for generating non-intrusive ROMs. 【0014】 Another objective of some embodiments is to generate a model of the system's mechanics that captures the physical properties of the system's behavior. In this way, the embodiments simplify the system model design process while retaining the advantage of having a model of the system when designing the control application. 【0015】 Accordingly, one embodiment discloses a computer-implemented method for training a neural network model for controlling the behavior of a system having nonlinear dynamics represented by a partial differential equation (PDE). The neural network includes nonlinear operators of the system's dynamics, represented in latent space by a parameterized ordinary differential equation (ODE) with parameters determined by training. The computer-implemented method may include collecting digital representations of time-series data showing measurements of the system's behavior at different time instances. The computer-implemented method may further include generating collocation points corresponding to solutions of the PDE, which represent the nonlinear dynamics of a set of initial and boundary conditions in the system's operating state and constraints in the system's behavior, which are unfolded from the boundary conditions according to the PDE. The computer-implemented method may further include training the neural network with training data, including the collected time-series data and collocation points, to train the parameters of the nonlinear operators. The neural network includes an autoencoder architecture comprising an encoder and a decoder, wherein the encoder is configured to encode each instance of the training data into latent space, the nonlinear operator is configured to propagate the encoded instances of the training data into latent space using transformations determined by the parameters of the nonlinear operator, and the decoder is configured to decode the transformed encoded instances of the training data to minimize a hybrid loss function that includes a data-driven loss between the neural network's decoding and the collected time-series data, and a physical information loss between the neural network's decoding and the PDE solution at the colocation points. In one embodiment, the nonlinear operator may be a reduced-order model. Such a ROM may be useful for accurately representing a system with nonlinear dynamics. Such a low-order nonlinear operator or predictor may be designed to match desired properties such as linearity, reduced order, stability, and nonlinearity, while adhering to the physical laws of the system, etc. 【0016】 In some embodiments, generating a parameterized ODE can be based on one or more model reduction techniques. These techniques may include at least one of the following: eigenorthogonal decomposition (POD)-Galerkin projection or dynamic mode decomposition (DMD). For control applications, ROM training methods need to find low-dimensional manifolds and dynamics that can yield both high-accuracy predictions and long-term stability. Generally, ROMs are projection-based, such as dynamic mode decomposition (DMD) and POD, which can transform the trajectories of high-dimensional dynamic systems into appropriate and optimal low-dimensional subspaces. One challenge with POD methods is their intrusion, i.e., the need for access to solver code. To overcome this drawback, operator inference approaches utilize singular value decomposition (SVD)-based model reduction, employing lifting to fit the system's latent space dynamics data into polynomials, for example, quadratic models. However, such polynomial models may have limited expressive power (e.g., up to quadratic in the case of lift-and-learn approaches), and custom-tuned SVD-based optimization techniques may be required. 【0017】 In some embodiments, to overcome these challenges, autoencoder-based reduced-order models can be used as nonlinear ROM techniques that can yield accurate and stable ROMs. However, autoencoder-based ROMs require a dataset that densely covers an infinite-dimensional phase portrait of the assumed dynamic system. Due to the large data demands, the use of such models is significantly limited in physical applications where data acquisition can be expensive. 【0018】 In some embodiments, the method may further include controlling the system by employing linear or nonlinear control laws. Thus, some embodiments are based on the recognition that a model of the system can be represented by a nonlinear reduced-order model. For example, if a complete physically based model of a system is typically captured by a PDE, then the ROM may be represented by an ordinary differential equation (ODE). While the ODE can express the system's dynamics as a function of time, it is less precise than the representation of dynamics using the PDE. In addition to or instead of this, some embodiments employ a model of the system determined by various model-based predictive controls, such as data-driven adaptation in Model Predictive Control (MPC). Such embodiments may allow leveraging the MPC's ability to consider constraints in system control. However, linearity may not adequately represent the complex dynamics of systems with multiple basins of attraction. In a dynamic system, a "basin of attraction" is a set of all starting points or initial conditions, usually close to each other, that, as the system unfolds over time, reach the same final state, which may be called equilibrium. On the other hand, nonlinear reduced-order methods can derive low-cost approximate models of nonlinear systems using projection techniques. 【0019】 In one embodiment, the residual factor of the PDE is based on the Lie operator. The method further includes performing an eigendecomposition on the Lie operator. Such a residual factor of the Lie operator can be used to learn the dynamics of either a linear or nonlinear model in the latent space. 【0020】 In some embodiments, the digital representation of time-series data can be obtained using computational fluid dynamics (CFD) simulations or experiments. CFD simulations and experiments are high-fidelity calculations for obtaining the digital representation of time-series data. CFD simulations or experiments enable improved accuracy and speed of complex simulation scenarios, such as transonic or turbulent fluid flow, for various applications of systems, such as heating, ventilation, and air conditioning (HVAC) applications, for describing airflow. 【0021】 In some embodiments, nonlinear operators are based on order reduction models. While the field of model reduction is mature for linear systems, reducing nonlinear models can be challenging. For example, the POD method can be used to reduce nonlinear models. To improve the computational efficiency of the resulting reduced model, the POD method can be combined with sparse sampling methods (also known as "hyper-reduction"). For example, sparse sampling methods may include missing point estimation (MPE), empirical interpolation (EIM), discrete empirical interpolation (DEIM), Gappy POD, or GNAT (Gauss-Newton with approximated tensors). Other methods for reducing nonlinear models employ data-driven approaches via dynamic mode decomposition (DMD) and operator inference. Recently, specific input-independent model reduction methods, such as balanced truncation and the iterative rational Krylov algorithm (IRKA), have been extended to quadratic bilinear systems. 【0022】 In one embodiment, the parameters of the non - linear operator are determined based on a probabilistic approach. The probabilistic approach can assume that the measurement data of the system is random and has a probability distribution that depends on the parameter of interest. 【0023】 In another embodiment, the non - linear operator is based on a continuous - time dynamic system. In some embodiments, the non - linear operator can be approximated in the latent space using data - driven approximation techniques. The data - driven approximation techniques can be generated using numerical or experimental snapshots. 【0024】 In some embodiments, the non - linear operator can be approximated using deep - learning techniques. By deep - learning techniques, the original dynamics of the system can be embedded in a significantly lower - order non - linear form. Deep - learning techniques for non - linear approximation can succeed in long - term dynamic prediction and control of the system. 【0025】 In some embodiments, the parameters of the non - linear operator can be fine - tuned in real - time based on a set of predicted measurements and the output of a neural network. When the ROM is trained, a neural network can be used for the reconstruction of the original dynamics of the system. Such ROM output can be projected onto appropriate measurement outputs based on a measurement model. Further, the difference between the actual measurements (such as a set of predicted measurements) and the output of the neural network can be used to fine - tune the parameters of the ROM. 【0026】 In some embodiments, the generation of colocation points may be based on a subset of initial and boundary conditions having a structure that reduces the complexity of solving the PDE, and the function space of the system that satisfies the subset of initial and boundary conditions. In some embodiments, the structure of the subset of initial and boundary conditions includes at least one of sine, harmonic, periodic, or exponential functions. For the PDE, the colocation points may be samples extracted from the domain of the system's function space such that the colocation points also satisfy boundary conditions or other constraints associated with the system. Advantageously, the generation of colocation points is computationally inexpensive compared to calculating snapshots of the CFD calculation. 【0027】 In some embodiments, estimation and control commands can be generated to control the operation of the system. In one or more embodiments, the generation of estimation and control commands to control the operation of the system is based on data-driven control and estimation techniques or optimization-based control and estimation techniques. Such techniques may be advantageous for controlling dynamic systems. For example, model-based control and estimation techniques enable model-based design frameworks in which system dynamics and constraints can be directly considered. 【0028】 In some embodiments, the generation of estimation and control commands to control the operation of the system is based on data-driven control and estimation techniques. The objective of data-driven control and estimation techniques is to design a control policy for the system from data and to control the system using that data-driven control policy. 【0029】 Another embodiment discloses a training system for training a neural network to control the behavior of a system having nonlinear dynamics represented by a partial differential equation (PDE). The neural network may include nonlinear operators of the system's dynamics, represented in latent space by a parameterized ordinary differential equation (ODE) with parameters determined by training. The training system may include at least one processor and a memory storing instructions, which, when executed by at least one processor, cause the training system to collect digital representations of time-series data showing instances of the system's function space and corresponding measurements of the system's behavior at different time instances. The training system may further generate collocation points corresponding to solutions of the PDE representing the nonlinear dynamics of a set of initial and boundary conditions in the system's behavior and constraints on the system's behavior, which are unfolded from the boundary conditions according to the PDE. Furthermore, the training system may train the neural network using training data, including the collected time-series data and collocation points, to train the parameters of the nonlinear operators. The neural network has an autoencoder architecture including an encoder and a decoder, the encoder configured to encode each instance of the training data into a latent space, a nonlinear operator configured to propagate the encoded instances of the training data into the latent space using transformations determined by the parameters of the nonlinear operator, and the decoder configured to decode the transformed encoded instances of the training data to minimize a hybrid loss function that includes a data-driven loss between the neural network's decoding and the collected time-series data, and a physical information loss between the neural network's decoding and the PDE solution at the colocation points. 【0030】 Another embodiment discloses a non-temporary computer-readable storage medium containing a processor-executable program for performing a method for controlling the behavior of a system having nonlinear dynamics represented by a partial differential equation (PDE). The neural network may include nonlinear operators of the system's dynamics, represented in latent space by a parameterized ordinary differential equation (ODE) with parameters determined by training. The method may include collecting digital representations of time-series data showing measurements of the system's behavior at different time instances. The method may further include generating collocation points corresponding to solutions of the PDE representing the nonlinear dynamics of a set of initial and boundary conditions in the state of the system's behavior and constraints in the system's behavior, which are unfolded from the boundary conditions according to the PDE. The method may further include training the neural network with training data, including the collected time-series data and collocation points, to train the parameters of the nonlinear operators. The neural network includes an autoencoder architecture, which comprises an encoder and a decoder, the encoder configured to encode each instance of the training data into a latent space, a nonlinear operator configured to propagate the encoded instances of the training data into the latent space using transformations determined by the parameters of the nonlinear operator, and the decoder configured to decode the transformed encoded instances of the training data to minimize a hybrid loss function, which comprises a data-driven loss between the neural network's decoding and the collected time-series data, and a physical information loss between the neural network's decoding and the PDE solution at the colocation points. 【0031】 Nonlinearity (or nonlinear operators) in latent space plays a crucial role in the development of non-intrusive ROMs. For example, if the original system may contain several retraction regions, finite-dimensional approximations of linear operators in latent space may fail to find various equilibria of the system. Furthermore, linear operators that can approximate the Koopman operator may not be compressible in latent space. The advantages of nonlinear operators in latent space are, on the one hand, that an integrated model with several retraction regions can be constructed, and on the other hand, that due to nonlinearity, a ROM that may be more accurate for a given reduction order (or a lower-order model for a given accuracy) may be obtained. 【0032】 In some embodiments, nonlinear ROMs can be generated. In other embodiments, a lift-and-learn approach may be used for nonlinear model reduction, enabled by data-driven learning of the reduced model through a projection structure-preserving lens. The lift-and-learn approach typically has two key components: data-driven learning of the reduced model via operator inference, and variable transformations and lifting to reveal the system polynomial structure. However, such nonlinear model reduction may employ an inductive bias, for example, assuming that the ROM model must be quadratic or polynomial, which may not be the most efficient scenario. 【0033】 This disclosure proposes the generation of a non-intrusive ROM that does not require access to the original finite element or finite volume solver to determine the system's dynamics. 【0034】 The Disclosure will be further described below with reference to several drawings shown as non-limiting examples of exemplary embodiments of the Disclosure. In the drawings, similar reference numerals represent similar parts throughout several drawings. The drawings shown are not necessarily to a constant scale, but rather are generally exaggerated to illustrate the principles of the embodiments disclosed herein. [Brief explanation of the drawing] 【0035】 [Figure 1A] This is a two-stage block diagram for training a neural network in an offline phase to be used in an online phase to control the operation of a system, according to one embodiment of the present disclosure. [Figure 1B] This is a schematic diagram of a nonlinear operator architecture for a neural network according to some embodiments of the present disclosure. [Figure 1C] This is a block diagram for fine-tuning the parameters of a nonlinear operator in real time to train a neural network, according to some embodiments of the present disclosure. [Figure 1D] This figure shows a flowchart for generating a colocation point according to one embodiment of the present disclosure. [Figure 2A] This is a schematic diagram of a principle used to control the operation of a system, according to some embodiments of this disclosure. [Figure 2B] This figure shows a flowchart illustrating an exemplary method for approximating a nonlinear operator according to some embodiments of the present disclosure. [Figure 2C] This is a schematic diagram of a neural network autoencoder architecture according to some embodiments of the present disclosure. [Figure 3] This is a block diagram of a device for controlling the operation of a system, according to some embodiments of the present disclosure. [Figure 4] This figure shows a flowchart of a principle for controlling the operation of a system, according to some embodiments of this disclosure. [Figure 5] This is a block diagram illustrating the generation of an order reduction model according to some embodiments of the present disclosure. [Figure 6] This is a schematic diagram of a neural network according to some embodiments of the present disclosure. [Figure 7A] This figure illustrates the input of a digital representation in an encoder of a neural network model according to some embodiments of the present disclosure. [Figure 7B]This figure illustrates the propagation of an encoded digital representation of a neural network model into a latent space by a nonlinear operator, according to some embodiments of the present disclosure. [Figure 7C] This figure illustrates the decoding of a transformed encoded digital representation by a decoder of a neural network model according to some embodiments of the present disclosure. [Figure 8] This is an illustrative diagram of a real-time implementation of a device for controlling the operation of an air conditioning system, according to some embodiments of the present disclosure. [Figure 9] This is an illustrative diagram of a real-time implementation of a device for reconstructing the gas distribution from a camera, according to some embodiments of the present disclosure. [Figure 10] This figure shows a flowchart illustrating a method for training a neural network according to some embodiments of this disclosure. Description of Embodiments 【0036】 The following description includes numerous specific details for illustrative purposes to ensure that the disclosure is fully understood. However, it will be apparent to those skilled in the art that the disclosure can be implemented without these specific details. Where else, the apparatus and methods are shown in block diagram form, solely to avoid obscuring the disclosure. Various modifications to the function and configuration of the elements are intended without departing from the spirit and scope of the disclosed subject matter as set forth in the appended claims. 【0037】 As used herein and in the claims, the terms “for example,” “as an example,” and “like,” as well as the verbs “equip,” “have,” and “include,” and each of their other verbal forms, should be interpreted as open-ended when used with an enumeration of one or more components or other items, meaning that the enumeration should not be considered as excluding other further components or items. The term “based on” means based on at least partially. Furthermore, it should be understood that the style and terminology used herein are for illustrative purposes only and should not be considered restrictive. Any headings used herein are for convenience only and have no legal or restrictive effect. 【0038】 Specific details are provided in the following description to ensure a full understanding of the embodiments. However, those skilled in the art will understand that the embodiments can be carried out even without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in block diagrams to avoid obscuring the embodiments with unnecessary details. In other examples, well-known processes, structures, and techniques may be shown without unnecessary details to avoid obscuring the embodiments. Furthermore, similar reference numbers and names in different drawings refer to similar elements. 【0039】 In describing embodiments of the disclosure, the following definitions apply throughout this disclosure: “Control system” or “controller” may mean a device or set of devices for managing, directing, supervising, or controlling the behavior of other devices or systems. A control system may be implemented in either software or hardware and may include one or more modules. A control system including a feedback loop may be implemented using a microprocessor. A control system may be an embedded system. 【0040】 A “heating, ventilation, and air conditioning (HVAC) system” may refer to a system that uses a vapor compression cycle to move a refrigerant through the system’s components based on the principles of thermodynamics, fluid dynamics, and / or heat transfer. HVAC systems encompass a wide range of systems, from those that supply only outside air to building occupants to those that control only the building’s temperature, and those that control both temperature and humidity. 【0041】 The term "Central Processing Unit (CPU)" or "processor" may refer to a computer or a component of a computer that reads and executes software instructions. Furthermore, a processor can be defined as "at least one processor" or "one or more processors." 【0042】 Figure 1A shows a two-stage block diagram 100A for training a neural network model in an offline stage for use in an online stage to control the operation of a system, according to one embodiment of the present disclosure. Block diagram 100A may include two stages, such as an offline stage 102 and an online stage 104. Block diagram 100A depicts the control and estimation of large-scale systems, such as systems with nonlinear dynamics represented by partial differential equations (PDEs), using a two-stage apparatus, namely an offline stage 102 and an online stage 104. 【0043】 The offline stage 102 (or stage I) may include a neural network 106. The neural network 106 has an autoencoder architecture. The neural network 106 comprises an autoencoder 108 including an encoder and a decoder. The neural network 106 further includes nonlinear operators 110 of the system's dynamics, which are represented in latent space by parameterized ordinary differential equations (ODEs) having parameters determined by training the neural network 106. The offline stage 102 may further include a computational fluid dynamics (CFD) simulation or experiment module 112, differential equations 114 for representing the system's nonlinear dynamics, a digital representation 116 of time-series data, and a collocation point 118. The time-series data may represent the temporal development of a vector field, such as a wind field, which unfolds over time due to initial and boundary conditions or external forcing. Alternatively, the time-series data may represent the temporal change in the spatial distribution of the density of an entity, such as a liquid, a crowd concentration, a flock of birds, or a school of fish. Online stage 104 (or stage II) may include a data assimilation module 120 and a control unit 122 for controlling the system. 【0044】 In the offline stage 102, offline tasks for system control and estimation may be performed to derive the nonlinear operator 110 (or nonlinear predictor). In some embodiments, the nonlinear operator 110 may be based on a reduced-order model (ROM). For example, the ROM may be represented by a neural ordinary differential equation (NODE) ​​operator. Such a ROM is sometimes called a latent space model. Generally, it is desirable that the dimensions of the latent space are significantly smaller than the input. Further details of the architecture of the nonlinear operator 110 for representing the ROM in latent space are provided, for example, in Figure 1B. 【0045】 Typically, data for developing latent space models (represented by nonlinear operators 110) can be generated by performing high-fidelity CFD simulations and experiments using CFD simulation or experimental module 112. 【0046】 Generally, CFD refers to a branch of fluid dynamics that uses numerical analysis and data structures to analyze and solve problems that may involve fluid flow. For example, computers can be used to perform the calculations necessary to simulate the free flow of fluids, as well as the interaction between fluids (such as liquids and gases) and surfaces defined by boundary conditions. Furthermore, various software has been designed to improve the accuracy and speed of complex simulation scenarios associated with transonic or turbulent flow that may occur in system applications such as HVAC applications to describe airflow within a system. In addition, initial validation of such software can typically be performed using equipment such as wind tunnels. Moreover, analytical or empirical analyses previously performed on specific problems related to airflow associated with a system can be used for comparison in CFD simulations. 【0047】 In some embodiments, a digital representation 116 of time-series data is obtained using a CFD simulation or experimental module 112. The CFD simulation or experimental module 112 may output a dataset such as the digital representation 116 of time-series data, which can be used to develop a latent space model (or nonlinear operator 110). The nonlinear operator 110 may be constructed for several trajectories generated by the CFD simulation. In an exemplary scenario, an HVAC system may be installed in a room. This room may have various scenarios, such as the windows being open or the doors being closed. The CFD simulation may be performed for rooms with closed windows, rooms with open windows, rooms with one, two, or more occupants, etc. In such cases, the autoencoder 108 may be valid for all such conditions associated with the room. Tasks such as the CFD simulation may be performed in an offline stage 102. 【0048】 In some embodiments, the collocation points 118 associated with the function space of the system may be generated based on the PDE, the digital representation 116 of the time-series data, and the nonlinearly transformed encoded digital representation (such as the output of the nonlinear operator 110). The neural network 106 may be trained based on the generated collocation points 118. Specifically, the neural network 106 may be trained based on the difference between the predicted values ​​of the nonlinear operator 110 and a dataset including the physical information portion in addition to the digital representation 116 of the time-series data, i.e., based on the differential equation 114 that represents the nonlinear dynamics of the system that generates the collocation points 118. 【0049】 Furthermore, the output of the neural network 106 can be utilized by the data assimilation module 120 in the online stage 104. The data assimilation module 120 may output, for example, a reconstruction model of temperature and velocity in a region such as a room associated with a system such as an HVAC system. The reconstruction model of temperature and velocity can be utilized by the control unit 122. The control unit 122 may generate control commands to control the operation (such as airflow) of the system such as an HVAC system. 【0050】 The data assimilation module 120 utilizes a data assimilation process, which involves assimilating accurate information from sensors with potentially inaccurate model information. For example, sensors may be installed in a room to monitor specific sensory data. Examples of sensory data installed in a room for HVAC applications include, but are not limited to, thermocouple readings, thermal camera measurements, velocity sensor data, and humidity sensor data. Information from sensors can be assimilated by the data assimilation module 120. 【0051】 Typically, data assimilation refers to the mathematical field that attempts to optimally combine predicted values ​​(usually in the form of numerical models) with observed values ​​associated with a system. Data assimilation can be used for a variety of purposes, such as finding optimal state estimates for a system, determining initial conditions for numerical predictive models of a system, interpolating sparse observed data using observed system knowledge, and identifying numerical parameters of a model from observed experimental data. Different solution methods may be employed depending on the purpose. 【0052】 It should be noted that the offline stages 102 and online stages 104 are examples of the development of a simplified, robust neural network 106, which can then be used for estimation and control of a system with nonlinear dynamics by a control unit 122. Typically, estimation and control of a system involves estimating the values ​​of parameters of a nonlinear operator 110 based on measured empirical data that may have random components. The parameters describe the underlying physical setting such that the parameter values ​​may influence the distribution of the measured data. Furthermore, an estimator such as the control unit 122 attempts to approximate unknown parameters using the measured values. Generally, there are two possible approaches to approximation. According to one embodiment, the parameters of the nonlinear operator 110 for approximation may be determined based on a probabilistic approach. The probabilistic approach may assume that the measured data of the system are random, having a probability distribution that depends on the parameter in question. In some embodiments, the parameters of the nonlinear operator 110 for approximation may be determined based on a set membership approach, which may assume that the measured data vectors belong to a set that depends on the parameter vectors. 【0053】 In some embodiments, particularly when it is difficult for the user to obtain data covering the entire distribution of possible data inputs, poor out-of-distribution performance can pose significant challenges when utilizing the ROM of the nonlinear operator 110. In an exemplary scenario, in an HVAC application, data may be collected from a room with two windows, but not from one room for every possible number of windows. In such a thermal fluid application, experiments may be performed for specific parameters, but it may be difficult to perform experiments for all parameters. In such situations, it becomes necessary to embed knowledge of physical properties into the system model to improve extrapolation performance. 【0054】 For example, some embodiments use symbolic regression to determine the underlying structure of a nonlinear dynamic system from data. In other embodiments, symbolic regression can be used in conjunction with a graph neural network (GNN) to extract the explicit physical relationships of a system, while facilitating sparse latent representations. Generally, symbolic representations extracted from GNNs generalize better to out-of-distribution data than GNNs do. However, symbolic regression also suffers from the drawback of excessive computational cost and a tendency to overfit. Typically, symbolic regression is a type of regression analysis that explores a space of mathematical formulas to find a model that best fits a given dataset in terms of both accuracy and simplicity. No specific model is provided as a starting point for symbolic regression. Instead, the initial formula is formed by randomly combining mathematical components such as mathematical operators, analytic functions, constants, and state variables. 【0055】 Furthermore, GNNs are a class of artificial neural networks for processing data that can be represented as a graph. Generally, certain existing neural network architectures can be interpreted as GNNs operating on a well-defined graph. In the context of computer vision, a convolutional neural network can be considered a GNN applied to a graph structured as a pixel grid. In the context of natural language processing, a transformer can be considered a GNN applied to a complete graph where words in a sentence are nodes. 【0056】 It should be noted that by incorporating knowledge of the physical information portion or differential equations associated with the system, the need for large training datasets, such as digital representations 116 of time-series data for identifying nonlinear operators 110, can be reduced. Furthermore, since the neural network 106 performs operator learning, it becomes capable of predicting beyond the training horizon, which can then be used for compressed sensing, estimation, and control of the system. 【0057】 Another example of incorporating physical properties into ROM is the use of parametric models in latent space by employing sparse identification of nonlinear dynamics (SINDy), which relies on the fact that most dynamic systems under consideration have relatively few nonlinear terms in their family of possible terms (i.e., polynomial nonlinearity). Such methods can utilize sparsity enhancement techniques to find models that automatically balance sparsity in some terms with model accuracy. For example, some embodiments may use chain law-based losses that link latent space derivatives to observable space derivatives for simultaneous training of the latent dynamics of the autoencoder 108 and the nonlinear operator 110. However, such chain law-based losses can be sensitive to noise in the data, especially when it is necessary to evaluate time derivatives with finite differences. Such numerical difficulties can sometimes be addressed by coordinating physical properties based on coordinating, i.e., projecting candidate functions onto the governing equations to coordinate chain laws instead of finite differences. Furthermore, neural ODEs (NODEs) can be used to fit arbitrary nonlinear models (e.g., networks) as latent spatial dynamics models (or nonlinear operators 110), significantly expanding the set of models for latent dynamics that can be efficiently trained. 【0058】 The computer-implemented method of this disclosure uses an autoencoder to perform nonlinear model reduction with NODE in latent space to model complex nonlinear dynamics. Such a method is used with the aim of reducing data demand and improving the overall predictive stability of the system under strict training conditions. To achieve this objective, a numerical analysis collocation method is used to embed knowledge from known governing equations into the latent space dynamics of ROM, as shown in Figure 1B. 【0059】 As shown in Figure 1B, the nonlinear operator 110 of the neural network 106 can be represented by NODE. 【0060】 Figure 1B shows schematic diagram 100B of the architecture of a nonlinear operator 110 represented by NODE according to some embodiments of the present disclosure. Schematic diagram 100B shows a nonlinear ROM for latent space. Schematic diagram 100B may include a data-driven loss model 124, a physical information loss model 126, and a hybrid model 128. The data-driven loss model 124 is a data-driven loss (L data ) shows 130. The physical information loss model 126 shows the physical information loss (L physics )132 is shown. Hybrid model 128 shows hybrid loss (L hybrid ) indicates 134. 【0061】 A method for training the neural network 106 may include collecting digital representations 116 of time-series data. These digital representations 116 represent instances of the system's function space and corresponding measurements of the system's behavior at different time instances. The collection of digital representations 116 of time-series data by CFD simulation or experimental module 112 is further described, for example, in Figure 1A. 【0062】 The method may further include generating collocation points 118 corresponding to a solution of the PDE representing the nonlinear dynamics of a set of initial and boundary conditions in the operating state of the system and constraints in the operating state of the system, which are unfolded from the boundary conditions according to the PDE. In some embodiments, the generation of collocation points 118 may be based on a subset of the set of initial and boundary conditions having a structure that reduces the complexity of solving the PDE. The generation of collocation points 118 may further be based on the function space of the system that satisfies the subset of initial and boundary conditions. Details of the generation of collocation points 118 are further provided, for example, in Figure 1D. 【0063】 【number】 【0064】 【Number】 【0065】 【Number】 【0066】 【Number】 【0067】 In the actual application of the system, it may be necessary to use a ROM instead of directly integrating Equation 1. For example, integrating Equation 1 may be computationally difficult on platforms with limited computing power such as embedded devices and autonomous devices. For example, in an HVAC system, solving Equation 1 means solving the Navier-Stokes equations on a fine grid in real time, which may exceed the computing power of the HVAC system's processor. On the other hand, when m << n, Equation 3 can be integrated at a low cost. Finally, even if it is possible to solve Equation 1 in real time (for example, by using a remote cluster), it may still be difficult to execute the control for the resulting model, which is the ultimate goal of the HVAC system. In fact, to execute the control, it is necessary to repeatedly evaluate Equation 1 many times for each iteration of the control. 【0068】 【Number】 【0069】 【Number】 【0070】 【Number】 【0071】 【number】 【0072】 【number】 【0073】 【number】 【0074】 【number】 【0075】 In particular, NODE is a nonlinear operator (h θ ) is used to represent 110. NODE is a neural network model that generalizes standard interlayer propagation to continuous depth models. Starting from the observation that forward propagation in a neural network is equivalent to one step of discretization of ODE, a model can be built and effectively trained via ODE. In addition to providing a novel family of architectures, particularly for reversible density models and continuous time series, NODE can further provide improved memory efficiency in supervised learning tasks. 【0076】 In the numerical analysis of ODEs, the Runge-Kutta method is a family of implicit and explicit iterative methods, including the Euler method, used in time discretization for approximate solutions of systems of nonlinear equations such as ODEs. In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical method for solving ODEs with given initial values. The Euler method is the basic explicit method for numerical integration of ODEs and is the simplest Runge-Kutta method. 【0077】 Furthermore, residual networks (ResNet) can be used to train deep networks. ResNet can be used to approximate ODEs and provide a training approach for neural network 106. Given network parameters, any residual network can be considered an explicit Euler discretization of a particular ODE, and any numerical ODE solver can be used to evaluate the output layer of neural network 106. To efficiently propagate (and thus train) neural network 106 in reverse, the adjoint method can be further used. Such methods can be further used to train time-continuous normalization flows. In such cases, using a continuous formulation avoids the calculation of the Jacobian determinant, which is one of the main bottlenecks in the normalization flow of the system. Also, irregularly sampled data can be easily handled by using NODE to model latent dynamics in time-series modeling. 【0078】 The training task for learning an autoencoder 108 and a neural network 106, which both have ROM, a digital representation 116 of time-series data, and a colocation point 118, is a data-driven loss (L data )130 and physical information loss (L physics )132 can be used. Nonlinear operator 110 is trained and h θ It is represented as a NODE indicated by . 【0079】 【number】 【0080】 A collocation point 118 is proposed that differs from conventional collocation points in terms of the sample space. For example, instead of sampling from a spatiotemporal domain as in the conventional method, the collocation point 118 of this disclosure samples from an appropriate function space. The selection of the collocation point 118 is further explained in Figure 1D. 【0081】 Figure 1C shows the fine-tuning of the neural network 106 parameters to minimize the hybrid loss 134. 【0082】 Figure 1C shows a block diagram 100C for real-time fine-tuning of the parameters of a nonlinear operator 110 to train a neural network 106. Block diagram 100C may include a digital representation 116 of time-series data, a colocation point 118, the neural network 106, and a data assimilation module 120. Block diagram 100C may further include measurements 136 based on the output of the neural network 106 and measurements 138 based on the output of the system. 【0083】 【number】 【0084】 It should be noted that the measurement model describes the relationship between sensor data output and the system's state variables. The simplest model is linear, and such a relationship can be represented by a measurement matrix. The state trajectory can be measured during the system's online functioning. For example, an apparatus (or training system) may include an input interface configured to acquire measurement data from sensors placed within an HVAC system, such as system velocity and temperature data. In such a case, the measurement matrix has a size corresponding to the "number of sensors" multiplied by the "number of original dynamics n", and entries in the measurement matrix are "0" where no sensors are present in the system's physical domain and "1" where sensors are present. 【0085】 The generation of collocation points 118 used to train the neural network 106 is further described in Figure 1D. 【0086】 Figure 1D shows a flowchart 100D for generating a colocation point 118 according to one embodiment of the present disclosure. Flowchart 100D may include steps 140, 142, 144, 146, 148, 150, and 152. Fewer or more steps may be provided. In addition, one or more steps may be combined or separated without departing from the scope of the disclosure. 【0087】 In general, a naive selection of colocation points 118 (or samples) can lead to inaccurate latent space dynamics, even when the dimensions of the latent space are large, i.e., r > n. In some embodiments, greedy selection of colocation points 118 or samples can yield significantly fewer degrees of freedom than a full-order simulation, while achieving higher accuracy than naive uniform sampling. Furthermore, in some embodiments related to hyperreduction of the model, stochastic sampling can eliminate errors caused by uniform sampling. To better control hyperreduction errors, a greedy algorithm for selecting colocation points 118 is proposed, which can augment the colocation points 118 set to satisfy the target residuals of the system. 【0088】 In step 140, the method may include obtaining differential equation 114 for the governing model of the system. Details of obtaining differential equation 114 are further provided, for example, in Figure 1A. 【0089】 In step 142, the method may include randomly selecting a collocation point 118. The selected collocation point 118 is used for the physical information loss 132. To robustly select the collocation points 118 used in the physical information loss 132, it may be necessary to balance computation speed and accuracy. In one example, "N" samples may be used, and the computation speed scales almost linearly with the number of samples used. To maximize computation speed, the fewest possible samples may be selected. Given a target accuracy, a greedy algorithm may select the fewest possible collocation points 118 from a suitable function space of full-order PDE solutions to achieve the target accuracy. 【0090】 In step 144, the neural network 106 may be executed, and the residual of the physical information loss 132 may be calculated for each of the selected colocation points 118. Based on the randomly selected colocation points 118, the neural network 106 may be executed. Based on the output of the neural network 106, the residual of the physical information loss 132 may be calculated for each of the selected colocation points 118. In all iterations, the greedy algorithm may add one colocation point to the sample set of N″ samples, reducing the error of the latent spatial dynamics associated with the physical information loss 132. 【0091】 In step 146, the method may include checking the convergence criterion. For example, it may be necessary to check the convergence of errors in latent spatial dynamics related to the physical information loss 132. The convergence criterion may be defined as the average of the individual residuals of the physical information loss 132. 【0092】 In step 148, the method may include terminating the sampling of collocation points 118. For example, the selection of random collocation points 118 may be terminated when the desired error convergence is achieved. In one example, the sampling of collocation points 118 may be terminated when the error convergence falls below a predetermined threshold error. 【0093】 In step 150, the method may include sampling of the colocation points 118. Sampling of the colocation points 118 may continue until the desired convergence of the error is achieved, for example, until the convergence of the error exceeds a predetermined threshold error. In such a case, the greedy algorithm loops over the "Q" colocation points 118 where the individual residuals of the physical information loss 132 are the largest. 【0094】 In step 152, the method may include adding each sample of the "Q" collocation points 118 to the sample set. Based on the determination that the desired convergence has not been achieved, the "Q" collocation points 118 are successively added to the sample set. The greedy algorithm is repeated until the target precision is met. For example, the loop continues from step 144 to step 152 until the target precision is met. 【0095】 It should be noted that the proposed approach using a greedy algorithm yields significantly more accurate results compared to conventional naive uniform sampling approaches. Such sampling can be performed either before or after the calculation of trajectories obtained from CFD simulations or experimental module 112, which are then required for the calculation of the data-driven loss 130. 【0096】 The control of the system based on the output of the neural network 106 is further described in Figure 2A. Figure 2A shows a schematic diagram 200A of a principle used to control the operation of the system in some embodiments of the present disclosure. Schematic diagram 200A depicts a control device 202 and a system 204. System 204 may be a system having nonlinear dynamics. The control device 202 may include a nonlinear operator 110 and a control unit 206 that communicates with the nonlinear operator 110. The control unit 206 is similar to the control unit 122 in Figure 1A. 【0097】 The control device 202 may be configured to control continuously operating dynamic systems such as system 204 in engineering processes and machines. Hereinafter, "control device" and "device" may be used interchangeably and have the same meaning. Hereinafter, "continuously operating dynamic system" and "system" may be used interchangeably and have the same meaning. Examples of system 204 include, but are not limited to, HVAC systems, LIDAR (light detection and ranging) systems, condensing units, production lines, self-tuning machines, smart grids, automobile engines, robots, numerically controlled machining, motors, satellites, generators, and transportation networks. In some embodiments, the control device 202 may be configured to generate estimation and control commands to control the operation of the system. For example, the control device 202 may develop control policies, such as estimations and control commands, to control system 204 using control actions in an optimal manner without delay or overshoot, and to ensure control stability. 【0098】 In some embodiments, the generation of estimation and control commands for controlling the operation of the system may be based on model-based control and estimation techniques. For example, the control unit 206 may be configured to generate control commands for controlling the system 204 based on at least one of model-based control and estimation techniques or optimization-based control and estimation techniques. For example, the optimization-based control and estimation technique may be model predictive control (MPC) technique. 【0099】 Model-based control and estimation techniques may be advantageous for controlling dynamic systems such as system 204. For example, MPC techniques can enable a model-based design framework in which the dynamics and constraints of system 204 can be directly considered. MPC techniques can develop control commands for controlling system 204 based on a latent space model or a model of the nonlinear operator 110. The nonlinear operator 110 of system 204 refers to the dynamics of system 204 described using nonlinear differential equations. 【0100】 In some embodiments, the control unit 206 may be configured to generate estimation and control commands for controlling the system 204 based on data-driven control and estimation techniques. The data-driven control and estimation techniques may utilize the operational data generated by the system 204 to construct a feedback control policy that stabilizes the system 204. For example, each state of the system 204 measured during its operation may be given as feedback for controlling the system 204. 【0101】 【number】 【0102】 In mathematics and computer algebra, automatic differentiation (AD) (also known as algorithmic differentiation, computational differentiation, autodiff, or simply autodiff) is a set of techniques for a computer program to evaluate the derivative of a given function. AD takes advantage of the fact that even the most complex computer programs perform sequences of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exponential, logarithmic, sine, cosine, etc.). By repeatedly applying chain laws to such operations, derivatives of any order can be automatically calculated with working precision, using at most a constant multiple of the arithmetic operations of the original computer program. 【0103】 Typically, the use of behavioral data to design control policies or control commands is called data-driven control and estimation techniques. Data-driven control and estimation techniques can be used to design control policies from data, and data-driven control policies can be further used to control system 204. Furthermore, in contrast to such data-driven control and estimation techniques, some embodiments may use behavioral data to design models such as nonlinear operators 110. Data-driven models such as nonlinear operators 110 can be used to control system 204 using various model-based control methods. Furthermore, data-driven control and estimation techniques can be used to determine a real model of system 204 from data, i.e., a model that can be used to estimate the behavior of system 204 having nonlinear dynamics. In one example, a model of system 204 may be determined from data that can capture the dynamics of system 204 using differential equations. Furthermore, a model with physics-based PDE model accuracy can be learned from behavioral data. 【0104】 Furthermore, to simplify the model generation calculations, an ordinary differential equation (ODE) of the nonlinear operator 110 may be formulated to describe the dynamics of system 204. In some embodiments, the parameterized ODE may be generated based on one or more model reduction techniques. For example, one or more model reduction techniques may include at least one of the following: proper orthogonal decomposition (POD)-Galerkin projection or dynamic mode decomposition (DMD) methods. In addition, the ODE may be part of a PDE that describes, for example, boundary conditions. However, in some embodiments, the ODE may not be able to reproduce the actual dynamics of system 204 (i.e., the dynamics described by the PDE) in the case of uncertainty conditions. Examples of uncertainty conditions may be that the boundary conditions of the PDE may change over time, or that one of the coefficients included in the PDE may change. 【0105】 An exemplary method for approximating the nonlinear operator 110 is further described in Figure 2B. 【0106】 Figure 2B shows flowchart 200B illustrating an exemplary method for approximating a nonlinear operator 110 according to some embodiments of the present disclosure. Flowchart 200B may include steps 208, 210, 212, and 214. Fewer or more steps may be provided. In addition, one or more steps may be combined or separated without deviating from the scope of the disclosure. Flowchart 200B illustrates an example of a data-driven ROM using POD and Galerkin projection. In such a case, the POD ROM derivation requires a snapshot that can be state parameter values ​​over a period of time. Furthermore, a finite volume method or finite element method may be used to solve the original PDE or large-scale system to find the snapshot. The basis functions may be described as a singular value decomposition (SVD) of the snapshot matrix. 【0107】 In step 208, a finite volume model or a finite element model may be obtained. "Finite volume" may refer to the small volume surrounding each node point on the mesh. The finite volume method (FVM) is a method for expressing and evaluating the PDE in the form of algebraic equations. Furthermore, the finite element method (FEM) can be used to numerically solve differential equations that arise in engineering and mathematical modeling. 【0108】 In step 210, snapshots can be determined based on a finite volume model or a finite element model. In the finite volume method, the divergence theorem is used to convert volume integrals in partial differential equations that may contain divergence terms into surface integrals. These terms are then evaluated as fluxes at the surface of each finite volume. Such a method can be conservative because the flux entering a given volume is identical to the flux leaving adjacent volumes. Another advantage of FVM is that it can be easily formulated to allow for unstructured meshes. Such methods are used in many computational fluid dynamics packages. 【0109】 Furthermore, typical problem domains include traditional fields such as structural analysis, heat transfer, fluid flow, mass transfer, and electromagnetic potential. FEM is a general numerical method for solving partial differential equations (i.e., some boundary value problems) in two or three spatial variables. To solve a problem, FEM subdivides a large system into smaller, simpler parts called finite elements. Such finite elements can be achieved by a specific spatial discretization in a spatial dimension, which is realized by constructing a mesh of objects with a finite number of points, i.e., a numerical domain of the solution. The finite element method formulation of boundary value problems ultimately yields a system of algebraic equations. This method approximates an unknown function on the domain. The simple equations that model these finite elements are then assembled into a larger system of equations that model the entire problem. FEM then approximates the solution by minimizing the associated error function using variational calculus. 【0110】 In step 212, a base function can be applied to the snapshot. In linear algebra, SVD is the factorization of a real or complex matrix. This generalizes the eigenvalue decomposition of a square orthogonal matrix with orthonormal eigenbases to any matrix. Specifically, the SVD of an m×n complex matrix M is M = UΣV * is a factorization of the form, where U is an m×m complex unitary matrix, Σ is an m×n rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an n×n complex unitary matrix, and V * is the conjugate transpose of V. Such a decomposition always exists for any complex matrix. If M is real, U and V can be guaranteed to be real orthogonal matrices, and in such a context, the SVD is often shown. 【0111】 In step 214, a reduced-order model can be obtained. For example, the non-linear operator 110 can be approximated to obtain a reduced-order model. If the governing equation of the field is known, the Galerkin method can be used to derive a system of ordinary differential equations for expanding the time-dependent amplitude. The tools of proper orthogonal decomposition (POD) and Galerkin projection provide a systematic method for generating a reduced-order model from data. The central idea of POD is to determine a nested family of increasing (finite) dimensional subspaces that optimally span the data in the sense that the error of the projection onto each subspace is minimized. Next, Galerkin projection determines the dynamics on each subspace by the orthogonal projection of the governing equation. 【0112】 【Number】 【0113】 However, ROM solutions can lead to unstable solutions that do not always reproduce the physical properties of the original PDE model, which have a viscous term that makes the solution stable, i.e., bounded over bounded-time support (for example, the solution may diverge over finite-time support). Furthermore, POD-based model reduction methods restrict the states to unfold in a linear subspace (linear trial subspace), which imposes fundamental limitations on the efficiency and accuracy of the resulting ROM. Such linear trial subspaces also exist in other model reduction methods, such as equilibration censoring, rational interpolation, and reduction-based methods. To address these limitations, some embodiments of this disclosure propose a data-driven, non-intrusive model reduction framework that tackles linear trial subspace problems using autoencoder network methods. Such deep learning-based models project the original high-dimensional dynamic system into a nonlinear subspace and predict the nonlinear dynamics. 【0114】 An example of using deep learning techniques (or neural network model 106) to approximate the nonlinear operator 110 for model reduction is further provided in Figure 2C. 【0115】 Figure 2C shows a schematic diagram 200C of an autoencoder architecture for a neural network according to some embodiments of the present disclosure. For example, a deep neural network model may be used to train nonlinear basis and nonlinear operators 110 using snapshot data. Schematic diagram 200C includes an autoencoder 108. The autoencoder 108 includes an encoder 216, a decoder 218, and a nonlinear operator 220. The nonlinear operator 220 may be identical to the nonlinear operator 110 in Figure 1A. Schematic diagram 200C further includes the nonlinear operator 220 and the nonlinear operator 222. 【0116】 【number】 【0117】 Furthermore, within the latent space of the autoencoder 108, such as the nonlinear operator 220, the dynamics of the system 204 are constrained to be represented by a NODE. 【0118】 Typically, the autoencoder 108 can be trained in several ways. Usually, the training dataset X is arranged as a three-dimensional (3D) tensor, with its dimensions being the number of sequences (with different initial states), the number of snapshots, and the dimensionality of the measurements, respectively. 【0119】 A block diagram of the device for controlling the operation of system 204 is further shown in Figure 3. 【0120】 Figure 3 shows a block diagram 300 of a device 302 for controlling the operation of system 204 according to some embodiments of the present disclosure. Block diagram 300 may include device 302. Device 302 may include an input interface 304, a processor 306, memory 308, and storage 310. Storage 310 may further include model 310a, a controller 310b, an update module 310c, and a control command module 310d. Device 302 may further include a network interface controller 312 and an output interface 314. Block diagram 300 includes a network 316, a state track 318, and an actuator 32 associated with system 204. 2 It may further include the following. 【0121】 The device 302 includes an input interface 304 and an output interface 314 for connecting the device 302 to other systems and devices. In some embodiments, the device 302 may include a plurality of input interfaces and a plurality of output interfaces. The input interface 304 is configured to receive the status trajectory 318 of system 204. The input interface 304 includes a network interface controller (NIC) 312 adapted to connect the device 302 to the network 316 via a bus. Furthermore, the device 302 receives the status trajectory 318 of system 204 via the network 316, either wirelessly or wired. 【0122】 The state trajectory 318 can be a set of states of system 204 that define the actual behavior of the system's dynamics. For example, the state trajectory 318 can function as a reference continuous state space for controlling system 204. In some embodiments, the state trajectory 318 can be received from real-time measurements of some of the states of system 204. In some other embodiments, the state trajectory 318 can be simulated using a PDE that describes the dynamics of system 204. In some embodiments, the shape of the received state trajectory 318 can be determined as a function of time. The shape of the state trajectory 318 can represent an actual pattern of the system's behavior. 【0123】 The device 302 further includes memory 308 for storing instructions executable by the processor 306. The processor 306 may be a single-core processor, a multi-core processor, a computing cluster, or any number of other configurations. The memory 308 may include random-access memory (RAM), read-only memory (ROM), flash memory, or any other suitable memory system. The processor 306 is connected to one or more input / output devices via a bus. Furthermore, the stored instructions implement a method for controlling the operation of the system 204. The memory 308 may be further extended to include storage 310. The storage 310 may be configured to store (310) a model 310a, a controller 310b, an update module 310c, and a control command module 310d. 【0124】 The controller 310b may be configured to store instructions in the storage 310 when executed by the processor 306, which runs one or more modules. Furthermore, the controller 310b manages each module in the storage 310 to control the system 204. 【0125】 Furthermore, in some embodiments, the update module 310c may be configured to update the gain associated with the model of system 204. The gain may be determined by reducing the error between the state of system 204 estimated using model 310a and the actual state of system 204. In some embodiments, the actual state of system 204 may be a measured state. In some other embodiments, the actual state of system 204 may be a state estimated using a PDE that describes the dynamics of system 204. In some embodiments, the update module 310c may update the gain using extreme value search. In some other embodiments, the update module 310c may update the gain using a Gaussian process-based optimization technique. 【0126】 The control command module 310d may be configured to determine control commands based on model 310a. The control command module 310d can control the operation of system 204. In some embodiments, the operation of system 204 may be constrained. Furthermore, the control command module 310d determines control commands using predictive model-based control techniques while enforcing constraints. The constraints include state constraints in the continuous state space of system 204 and control input constraints in the continuous control input space of system 204. 【0127】 The output interface 314 is configured to send control commands to the actuator 322 of system 204 in order to control the operation of system 204. Some examples of the output interface 314 include a control interface that sends control commands to control system 204. 【0128】 The control of system 204 will be further explained in Figure 4. 【0129】 Figure 4 shows a flowchart 400 of a principle for controlling the operation of system 204 according to some embodiments of the present disclosure. Flowchart 400 may include steps 402, 404 and 406. 【0130】 In some embodiments, system 204 can be modeled from physical laws. For example, the mechanics of system 204 can be expressed by mathematical equations using physical laws. 【0131】 【number】 【0132】 【number】 【0133】 【number】 【0134】 In some embodiments, such abstract dynamics can be obtained from the numerical discretization of nonlinear partial differential equations (PDEs), which typically require a large number of n state dimensions. 【0135】 In some embodiments, a physical-based high-dimensional model of system 204 needs to be solved in order to control the operation of system 204 in real time. For example, in the case of an HVAC system, the Boussinescu equations need to be solved to control the airflow dynamics and temperature of the room. In some embodiments, the physical-based high-dimensional model of system 204 includes a large number of equations and variables that may be complex to solve. For example, greater computational power is required to solve the physical-based high-dimensional model in real time. Therefore, the physical-based high-dimensional model of system 204 may be simplified. 【0136】 In step 404, the apparatus 302 is provided to generate a reduced-order model to reproduce the dynamics of system 204 so that the apparatus 302 can efficiently control system 204. In some embodiments, the apparatus 302 may simplify a physical-based high-dimensional model using model reduction techniques to generate a reduced-order model. In some embodiments, model reduction techniques reduce the dimensionality of the physical-based high-dimensional model (e.g., variables of the PDE) so that the reduced-order model can be used in real time for predicting and controlling system 204. Furthermore, the generation of a reduced-order model for controlling system 204 will be described in detail with reference to Figure 5. In step 406, the apparatus 302 predicts and controls system 204 in real time using the reduced-order model. 【0137】 The generation of order reduction models for nonlinear operators such as 110 is further described in Figure 5. 【0138】 Figure 5 shows a block diagram 500 illustrating the generation of a reduced-order model according to some embodiments of the present disclosure. The nonlinear operator 110 is the reduced-order model. Block diagram 500 depicts an architecture including a digital representation 116 of time-series data and a neural network 106. The autoencoder 108 of the neural network 106 includes an encoder 216, a decoder 218, and a nonlinear operator 220. Block diagram 500 further depicts the output 502 of the neural network 106. 【0139】 A snapshot of a CFD simulation or experiment is the data required for an autoencoder, such as the autoencoder 108, which is a neural network model as shown in Figure 6. The latent space is governed by a nonlinear ODE, which is learned based on both the data snapshot and model information using DSC equations such as Equation 14. 【0140】 Furthermore, for a given time-dependent differential equation (e.g., ODE or PDE), there may be a set of feasible initial conditions. Some embodiments define these feasible initial conditions as those that can be classified into domains of system dynamics f. 【0141】 Typically, the domain of a function is the set of inputs it accepts. More precisely, if we assume a function f:X→Y, then the domain of f is X. The domain can be part of the function's definition rather than a property of the function. In such cases, both X and Y are subsets of R, and the function f can be graphed in Cartesian coordinates. In such cases, the domain is represented on the x-axis of the graph as a projection of the graph of the function onto the x-axis. 【0142】 In some embodiments, the generation of collocation points 118 is based on a subset of initial and boundary conditions having a structure that reduces the complexity of solving the PDE, and the function space of a system that satisfies the subset of initial and boundary conditions. The collocation points 118 can be samples extracted from the domain of system dynamics f such that, in the case of the PDE, the collocation points 118 can satisfy the initial and boundary conditions. 【0143】 In some embodiments, the subset of initial and boundary conditions may include at least one of sinusoidal, harmonic, periodic, or exponential functions. For example, if the boundary conditions for system dynamics f are periodic, then the collocation points 118 should also be periodic. If the boundary conditions are Dirichlet, i.e., if system dynamics f is equal to a specific value at its boundary point, then the collocation points 118 should also be equal to such a value at the corresponding boundary point. Advantageously, the collocation points 118 can be estimated at a significantly lower computational cost compared to the calculation of snapshots. Snapshots can be generated by either a simulator or experiment, while the collocation points 118 can be generated simply by sampling from the realizable function space. 【0144】 Furthermore, a function space is a set of functions between two definite sets. Often, a domain and / or codomain can have additometry that can be inherited by a function space. For example, a set of functions from any set X to a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, a function space can inherit a topological or metric structure. 【0145】 The autoencoder 108 can receive a digital representation 116 of the time-series data and collocation points 118 projected onto the differential equation. The encoder 216 encodes the digital representation into latent space. The nonlinear predictor 220 can propagate the encoded digital representation into latent space using a transformation determined by the parameter values ​​of the nonlinear operator 220. Furthermore, the decoder 218 can decode the transformed encoded digital representation. The output 502 of the transformed encoded digital representation may be a reconstructed snapshot or a decoded and linearly transformed encoded digital representation. 【0146】 The basic neural network implemented for the autoencoder 108 architecture is shown in Figure 6. 【0147】 Figure 6 shows a schematic diagram 600 of a neural network 106 according to some embodiments of the present disclosure. The neural network 106 may be a network or circuit of artificial neural networks composed of artificial neurons or nodes. Thus, the neural network 106 is an artificial neural network used to solve artificial intelligence (Al) problems. The connections of biological neurons are modeled in the artificial neural network as weights between nodes. Positive weights reflect excitatory connections, and negative weight values ​​signify inhibitory connections. All inputs 602 of the neural network 106 can be modified by the weights and summed up. Such activity is called linear combination. Finally, an activation function controls the amplitude of the output 604 of the neural network 106. For example, the acceptable range of the output 604 is typically from 0 to 1, or can be from -1 to 1. Artificial networks may be used for predictive modeling, adaptive control, and applications where the artificial network can be trained via training datasets. Self-learning derived from experience may occur within the network, which can lead to the drawing of conclusions from a complex and seemingly unrelated set of information. The block architecture of the autoencoder 106 is shown in Figures 7A, 7B, and 7C. 【0148】 Figure 7A shows Figure 700A illustrating the input of a digital representation in an encoder 216 (such as an autoencoder 108) of a neural network 106 according to some embodiments of the present disclosure. Figure 700A includes the encoder 216, a snapshot 702, a colocation point 118, and the final layer 704 of the encoder 216. 【0149】 The input to encoder 216 can be either a snapshot 702 or a colocation point 118. Snapshot 702 may be, for example, a digital representation 116 of time-series data. Encoder 216 takes the value of either snapshot 702 or colocation point 118. Encoder 216 outputs to latent space or nonlinear operator 220 through its final layer 704. Digital representations 116 of time-series data showing measurements of the system 204's behavior at different time instances can be collected. Furthermore, for training a neural network 106 (such as autoencoder 108) with an autoencoder architecture, encoder 216 can encode the digital representation into latent space. The encoding process is model reduction. 【0150】 Figure 7B shows Figure 700B illustrating the propagation of the encoded digital representation into the latent space by the nonlinear operator 220 of the neural network 106 according to some embodiments of the present disclosure. Figure 700B shows encoder 2 16 The final layer 704, Non This includes linear operators 220 and the last iteration 706 of the nonlinear operators 220 or latent space model. The nonlinear operators 220 are presented as NODEs, and h θ This is shown. 【0151】 The nonlinear operator 220 is configured to propagate the encoded digital representation into latent space using a linear transformation determined by the parameter values ​​of the nonlinear operator 220. The output of the last iteration 706 of the nonlinear operator 220 is sent to the decoder 218 of the neural network 106. The process of propagating the encoded digital representation into latent space is called reduced-order model propagation or time integration. 【0152】 Figure 7C shows Figure 700C illustrating the decoding of the transformed encoded digital representation by the decoder 218 of the neural network 106 according to some embodiment of the present disclosure. Figure 700C includes the decoder 218, the last iteration 706 of the nonlinear operator 220, and the output 708 of the decoder 218. 【0153】 Decoder 218 forwards the input and yields output 708. Decoder 218 is configured to decode the transformed encoded digital representation to produce output 708. Output 708 is the decoded and linearly transformed encoded digital representation, such as the reconstructed snapshot shown in Figure 5. The decoding process is the reconstruction of the snapshot. 【0154】 【number】 【0155】 An exemplary scenario for the real-time implementation of the device 302 for controlling the operation of system 204 is further described in Figure 8. 【0156】 Figure 8 shows an exemplary Figure 800 of a real-time implementation of a device 302 for controlling the operation of system 204 according to some embodiments of the present disclosure. Exemplary Figure 800 includes a room 802, a door 804, a window 806, a ventilation unit 808, and a set of sensors 810. 【0157】 In the exemplary scenario, system 204 is an air conditioning system. Exemplary Figure 800 shows a room 802 having a door 804 and at least one window 806. The temperature and airflow in room 802 are controlled by device 302 via the air conditioning system through a ventilation unit 808. A set of sensors 810, such as sensors 810a and 810b, is placed in room 802. At least one airflow sensor, such as sensor 810a, is used to measure the velocity of the airflow at a given point in room 802, and at least one temperature sensor, such as sensor 810b, is used to measure the temperature of the room. Note that other types of settings are also possible, for example, a room with multiple HVAC units, or a house with multiple rooms. 【0158】 Another exemplary scenario for the real-time implementation of the device 302 for imaging greenhouse gas leaks is further described in Figure 9. 【0159】 Figure 9 shows an exemplary Figure 900 of a real-time implementation of a device 302 for reconstructing the distribution of gases from a camera. For example, the device 302 is implemented to image greenhouse gas leaks 902. Exemplary Figure 900 may include a mid-infrared light source 904, a single-pixel camera 906, and a neural network 106. The single-pixel camera 906 may include two objective lenses, such as lens 908 and lens 912. The single-pixel camera 906 may further include a digital micromirror device (DMD) 910 and a photodetector 914. 【0160】 The DMD910 receives light from the gas scene and projects it onto the photodetector 914 through the lens 912. The DMD910 changes the orientation of its mirror at a high frequency, thereby obtaining a series of measurements 920 of the gas scene at all time instances. 【0161】 Next, a series of measurements 920 are compared to measurements 918 using a DMD configuration of the gas reconstruction generated from a trained neural network 106. The gas distribution reconstruction 916 is refined to minimize the difference between the series of measurements 920 from the mid-infrared (MIR) sensor and the gas reconstruction measurements 918. 【0162】 A system 204, such as an air conditioning system, can be described by a physics-based model called the Boussinesque equation, as illustrated in Figure 4. However, the Boussinesque equation involves infinite dimensions in order to solve the Boussinesque equation for controlling the air conditioning system. Data assimilation may be further added to the ODE model. The model reproduces the dynamics of the air conditioning system (e.g., airflow dynamics) in an optimal manner. Furthermore, in some embodiments, the airflow dynamics model correlates airflow values ​​(e.g., airflow velocity) with the temperature of the air-conditioned room during operation of the air conditioning system. In addition, the device 302 optimally controls the air conditioning system to generate airflow in a controlled manner. 【0163】 Figure 10 shows a flowchart 1000 illustrating a method for training a neural network 106 according to some embodiments of this disclosure. Flowchart 1000 may include steps 1002, 1004, and 1006. Fewer or more steps may be provided. In addition, one or more steps may be combined or separated without departing from the scope of the disclosure. 【0164】 In step 1002, the method may include collecting a digital representation 116 of the time-series data. The digital representation 116 of the time-series data represents an instance of the system's function space and corresponding measurements of the system's operating state at different time instances. The collection of the digital representation 116 of the time-series data by CFD simulation or experimental module 112 is further described, for example, in Figure 1A. 【0165】 In step 1004, the method may include generating collocation points 118 corresponding to a solution of the PDE representing the nonlinear dynamics of a set of initial and boundary conditions in the operating state of the system and constraints in the operating state of the system, which are unfolded from the boundary conditions according to the PDE. In some embodiments, the generation of collocation points 118 may be based on a subset of the set of initial and boundary conditions having a structure that reduces the complexity of solving the PDE. The generation of collocation points 118 may further be based on the function space of the system that satisfies the subset of initial and boundary conditions. Further details of the generation of collocation points 118 are provided, for example, in Figure 1D. 【0166】 【number】 【0167】 The above description provides only specific embodiments and is not intended to limit the scope, applicability, or configuration of the disclosure. Rather, the following description of specific embodiments will provide a description that enables the implementation of one or more specific embodiments. Various modifications are intended to be made to the function and configuration of the elements without departing from the spirit and scope of the subject matter disclosed in the appended claims. 【0168】 Specific details are provided in the following description to ensure a full understanding of the embodiments. However, the embodiments can be carried out without these specific details, as understood by those skilled in the art. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in block diagrams to avoid obscuring the embodiments with unnecessary details. In other examples, well-known processes, structures, and techniques may be shown without unnecessary details to avoid obscuring the embodiments. Furthermore, similar reference numbers and names in different drawings refer to similar elements. 【0169】 Furthermore, individual embodiments may be described as processes shown as flowcharts, flow diagrams, data flow diagrams, structural diagrams, or block diagrams. While flowcharts may describe operations as sequential processes, many operations can be performed in parallel or simultaneously. In addition, the order of operations may be reordered. A process may terminate when its operations are complete, but it may have additional steps that are not discussed or included in the diagrams. Moreover, not all operations in any specifically described process can occur in all embodiments. A process may correspond to a method, function, procedure, subroutine, subprogram, etc. If a process corresponds to a function, the termination of the function may correspond to returning the function to the calling function or the main function. 【0170】 Furthermore, embodiments of the disclosed subject matter may be implemented either manually or automatically, at least in part. Manual or automatic implementation may be performed, or at least assisted, through a machine, hardware, software, firmware, middleware, microcode, hardware description language, or any combination thereof. If implemented in software, firmware, middleware, or microcode, the program code or code segments for performing the required tasks may be stored in a machine-readable medium. A processor(s) may perform the required tasks. 【0171】 The various methods or processes outlined herein may be encoded as software executable on one or more processors using any one of a variety of operating systems or platforms. In addition, such software may be written using any of several suitable programming languages ​​and / or programming or scripting tools, and may be compiled as executable machine language code or intermediate code that runs on a framework or virtual machine. Typically, the functions of program modules may be combined or distributed as desired in various embodiments. 【0172】 Each embodiment is described as a process shown as a flowchart, flow diagram, data flow diagram, structure diagram, or block diagram. While flowcharts show operations as sequential processes, many operations can be performed in parallel or simultaneously. In addition, the order of operations may be reordered. A process may terminate when its operations are complete, but it may have additional steps that are not discussed or included in the diagram. Furthermore, not all operations in any process specifically described can occur in all embodiments. A process can correspond to a method, function, procedure, subroutine, subprogram, etc. If a process corresponds to a function, the termination of the function may correspond to returning the function to the calling function or the main function. 【0173】 Furthermore, embodiments of the disclosed subject matter may be implemented either manually or automatically, at least in part. Manual or automatic implementation may be performed, or at least assisted, through a machine, hardware, software, firmware, middleware, microcode, hardware description language, or any combination thereof. If implemented in software, firmware, middleware, or microcode, the program code or code segments for performing the required tasks may be stored in a machine-readable medium. A processor(s) may perform the required tasks. 【0174】 A person skilled in the art who enjoys the benefit of the teachings presented in the above description and the accompanying drawings will likely conceive of numerous modifications and other embodiments of these disclosures. It should be understood that the disclosures are not limited to the specific embodiments disclosed, and that modifications and other embodiments are intended to be included within the scope of the appended claims. Furthermore, while the above description and the accompanying drawings describe exemplary embodiments in the context of several exemplary combinations of elements and / or functions, it should be recognized that various combinations of elements and / or functions may be provided by alternative embodiments without departing from the scope of the appended claims. In this regard, different combinations of elements and / or functions than those expressly described above are also intended, for example, which may be described in some of the appended claims. Specific terminology is used herein, but these terms are used only in a general and descriptive sense and are not intended to be limiting.

Claims

[Claim 1] A computer-implemented method for training a neural network for controlling the operation of systems in robotics, building control of heating, ventilation, and air conditioning systems, gas leak detection, smart grids, factory automation, transportation, self-tuning machines, and transportation networks, wherein the neural network includes nonlinear operators of the nonlinear dynamics of the system represented in latent space by parameterized ordinary differential equations (ODEs) having parameters determined by the training, and the method is: Collecting digital representations of time-series data showing instances of the function space of the system and corresponding measurements of the operating state of the system at different time instances, The method includes generating collocation points corresponding to a solution of a PDE representing the nonlinear dynamics of a set of initial and boundary conditions in the state of the operation of the system and constraints in the operation of the system that are unfolded from the boundary conditions according to the PDE, wherein the collocation points corresponding to the solution of the PDE are used to train the parameters of the nonlinear operator represented by the ODE, and the method is A computer-implemented method comprising training the neural network with training data including the collected time-series data and the colocation points to train the parameters of the nonlinear operator represented by the ODE, wherein the neural network has an autoencoder architecture including an encoder and a decoder, the encoder configured to encode each instance of the training data into a latent space, the nonlinear operator configured to propagate the encoded instances of the training data into the latent space using a transformation determined by the parameters of the nonlinear operator, and the decoder configured to decode the transformed encoded instances of the training data to minimize a hybrid loss function including a data-driven loss between the decoding of the neural network and the collected time-series data, and a physical information loss between the decoding of the neural network and the solution of the PDE at the colocation points. [Claim 2] The generation of the aforementioned collocation point is A subset of the set of initial and boundary conditions having a structure that reduces the complexity of solving the PDE, Based on the function space of the system that satisfies the subset of the initial and boundary conditions, The method according to claim 1, wherein the structure of the subset of the initial and boundary conditions includes at least one of a sine function, a harmonic function, a periodic function, or an exponential function. [Claim 3] The parameters of the nonlinear operator are determined based on a probabilistic approach, or The method according to claim 1, wherein the nonlinear operator is based on a continuous-time dynamic system. [Claim 4] The method according to claim 1, further comprising fine-tuning the parameters of the nonlinear operator in real time based on a set of expected measurements and the output of the neural network. [Claim 5] The method according to claim 1, further comprising generating estimation techniques and control commands for controlling the operation of the system. [Claim 6] The estimation techniques and the generation of control commands for controlling the operation of the system are based on model-based control and estimation techniques, or The estimation technique and the generation of control commands for controlling the operation of the system are based on optimization-based control and estimation techniques, or The method according to claim 5, wherein the estimation technique and the generation of control commands for controlling the operation of the system are based on data-driven control and estimation techniques. [Claim 7] A training system for training a neural network for controlling the operation of systems in robotics, building control of heating, ventilation, and HVAC systems, gas leak detection, smart grids, factory automation, transportation, self-tuning machines, and transportation networks, wherein the neural network includes nonlinear operators of the nonlinear dynamics of the system represented in latent space by parameterized ordinary differential equations (ODEs) having parameters determined by the training, the training system comprises at least one processor and a memory storing instructions, the instructions, when executed by the at least one processor, the training system Collecting digital representations of time-series data showing instances of the function space of the system and corresponding measurements of the operating state of the system at different time instances, The method includes generating collocation points corresponding to a solution of a PDE representing the nonlinear dynamics of a set of initial and boundary conditions in the state of the operation of the system and constraints in the operation of the system, which are unfolded from the boundary conditions according to the PDE, wherein the collocation points corresponding to the solution of the PDE are used to train the parameters of the nonlinear operator represented by the ODE, and the method includes, A training system comprising: training the neural network using training data including the collected time-series data and the colocation points to train the parameters of the nonlinear operator represented by the ODE, wherein the neural network has an autoencoder architecture including an encoder and a decoder, the encoder configured to encode each instance of the training data into a latent space, the nonlinear operator configured to propagate the encoded instances of the training data into the latent space using a transformation determined by the parameters of the nonlinear operator, and the decoder configured to decode the transformed encoded instances of the training data to minimize a hybrid loss function including a data-driven loss between the decoding of the neural network and the collected time-series data, and a physical information loss between the decoding of the neural network and the solution of the PDE at the colocation points. [Claim 8] The generation of the aforementioned collocation point is A subset of the set of initial and boundary conditions having a structure that reduces the complexity of solving the PDE, Based on the function space of the system that satisfies the subset of the initial and boundary conditions, The training system according to claim 7, wherein the structure of the subset of the initial and boundary conditions includes at least one of a sine function, a harmonic function, a periodic function, or an exponential function. [Claim 9] The training system according to claim 7, wherein the parameters of the nonlinear operator are determined based on a probabilistic approach. [Claim 10] The training system according to claim 7, wherein the nonlinear operator is based on a continuous-time dynamic system. [Claim 11] The training system according to claim 7, further configured to fine-tune the parameters of the nonlinear operator in real time based on a set of expected measurements and the output of the neural network. [Claim 12] The training system according to claim 7, further configured to generate estimation techniques and control commands for controlling the operation of the system. [Claim 13] A non-temporary computer-readable storage medium containing a processor-executable program for performing a method for controlling the operation of systems in robotics, building control of heating, ventilation, and air conditioning systems, gas leak detection, smart grids, factory automation, transportation, self-tuning machines, and transportation networks, wherein the neural network includes nonlinear operators of the nonlinear dynamics of the system represented in latent space by parameterized ordinary differential equations (ODEs) having parameters determined by training, and the method is Collecting digital representations of time-series data showing instances of the function space of the system and corresponding measurements of the operating state of the system at different time instances, The method includes generating collocation points corresponding to a solution of a PDE representing the nonlinear dynamics of a set of initial and boundary conditions in the state of the operation of the system and constraints in the operation of the system that are unfolded from the boundary conditions according to the PDE, wherein the collocation points corresponding to the solution of the PDE are used to train the parameters of the nonlinear operator represented by the ODE, and the method is A non-temporary computer-readable storage medium comprising training the neural network with training data including the collected time-series data and the colocation points to train the parameters of the nonlinear operator represented by the ODE, wherein the neural network has an autoencoder architecture including an encoder and a decoder, the encoder configured to encode each instance of the training data into a latent space, the nonlinear operator configured to propagate the encoded instances of the training data into the latent space using a transformation determined by the parameters of the nonlinear operator, and the decoder configured to decode the transformed encoded instances of the training data to minimize a hybrid loss function including a data-driven loss between the decoding of the neural network and the collected time-series data and a physical information loss between the decoding of the neural network and the solution of the PDE at the colocation points.