Systems and methods for training neural network models for control of high-dimensional physical systems

Abstract

An embodiment of the present disclosure provides a method for training a neural network model for controlling the behavior of a system represented by a partial differential equation (PDE). The method includes collecting digital representations of time series data indicative of measurements of the system's behavior at different time instances. The method further includes training a neural network model having an autoencoder architecture, the autoencoder architecture including an encoder for encoding the digital representations into a latent space, a linear predictor for propagating the digital representations into the latent space, and a decoder for decoding the digital representations to minimize a prediction error between an output of a neural network model that decodes a measurement of the behavior at a particular time instance and a measurement of the behavior collected at a subsequent time instance, and a loss function including a residual element of the PDE whose eigenvalues ​​depend on parameters of the linear predictor.

Description

[Technical Field]

[0001] The present disclosure relates generally to system modeling, prediction and control, and more particularly to systems and methods for training neural network models for control of high-dimensional physical systems. [Background technology]

[0002] Control theory in control systems engineering is a subfield of mathematical processing that deals with the control of continuously operating dynamic systems in engineered processes and machines. The objective is to develop control policies to control such systems with control actions in an optimal manner without delay or overshoot and to ensure control stability. Summary of the Invention [Problem to be solved by the invention]

[0003] Traditionally, some methods for 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 various complexities. Examples of such systems can include production lines, automobile engines, robots, numerically controlled machining, motors, satellites, and generators.

[0004] Furthermore, a model of the system's dynamics, or a model of the system, describes the system's dynamics using differential equations. However, in some situations, the model of the system may be nonlinear and difficult to design, difficult to use in real time, or inaccurate. Examples of such cases are prevalent in certain applications such as robotics, building control, e.g., heating, ventilation, and air conditioning (HVAC) systems, smart grids, factory automation, transportation, self-tuning machines, and traffic networks. In addition, even when a nonlinear model may be available, designing an optimal controller for control of the system can be an inherently difficult task.

[0005] Furthermore, in the absence of an accurate model of a dynamic system, some control methods utilize operational data generated by the dynamic system to construct feedback control policies that stabilize system dynamics or embed quantifiable control-related performance. Typically, various types of methods for controlling a system that utilize operational data may be used. In some embodiments, a control method may first build a model of the system and then utilize the model to design a controller. However, such control methods result in a black-box design of the control policy, which directly maps the system state to control commands. However, such control policies are not designed with the physical characteristics of the system in mind.

[0006] In another embodiment, a 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 the model-building step may require a large amount of data. In addition, the controller is calculated from an estimated model, for example, according to the certainty equivalence principle, but in practice, the model estimated from the data may not capture the physical characteristics of the system's dynamics. Therefore, some control techniques for a system may not be usable with the model built for the system.

[0007] Therefore, there is a need for a method and system for optimally controlling the system to address the above-mentioned problems. [Means for solving the problem]

[0008] The present disclosure provides a computer-implemented method and system for training neural network models for the control of high-dimensional physical systems.

[0009] An objective of some embodiments is to train a neural network model such that the trained neural network model can be utilized to control the behavior of a system having nonlinear dynamics described by a partial differential equation (PDE). The neural network model has an autoencoder architecture including an encoder, a linear predictor, and a decoder. The linear predictor may be based on a Koopman operator. Such a linear predictor may be a reduced-order model.

[0010] Another objective of some embodiments is to generate a model of the dynamics of a system that captures the physics of the system's behavior. In this way, the embodiments simplify the process of designing a model of the system while retaining the benefits of having a model of the system when designing a control application.

[0011] Some embodiments introduce an operator-theoretic perspective on dynamical systems, complementing traditional geometric perspectives. In this framework, a Koopman operator is defined that acts on an observation function (an observable) in an appropriate function space. When acted upon by the Koopman operator, the evolution of the observable is linear, but the function space may be infinite-dimensional. As a result, approximating the Koopman operator and finding its eigenfunctions is key to linearizing the nonlinear dynamics of a system.

[0012] Accordingly, one embodiment discloses a computer-implemented method for training a neural network model for controlling the operation of a system having nonlinear dynamics represented by a partial differential equation (PDE). The computer-implemented method includes collecting digital representations of time series data indicative of measurements of the system's operation at different time instances. The computer-implemented method further includes training a neural network model having an autoencoder architecture, the autoencoder architecture including: an encoder configured to encode the digital representations into a latent space; a linear predictor configured to propagate the encoded digital representations into the latent space using a linear transformation determined by values ​​of parameters of the linear predictor; and a decoder configured to decode the linearly transformed encoded digital representations to minimize a loss function including a prediction error between an output of a neural network model that decodes a measurement of the operation at a particular time instance and a measurement of the operation collected at a subsequent time instance, and a residual element of the PDE whose eigenvalues ​​depend on the parameters of the linear predictor. The linear predictor may be a reduced-order model represented by a Koopman operator, which may be nonlinear and high-dimensional. Such models can be useful for accurately representing systems with nonlinear dynamics. Linear predictors can be designed to meet desired properties, such as linearity and reduced order.

[0013] In some embodiments, the method further includes controlling the system using a linear control law including a control matrix formed by values ​​of parameters of a linear predictor, the control matrix being a finite-dimensional linear system that can be utilized to linearly transform the encoded digital representation to minimize a loss function.

[0014] In some embodiments, the method further includes performing an eigendecomposition into Lie operators. The residual elements of the PDE are based on the Lie operators. A square matrix is ​​used to approximate the Lie operators associated with the Koopman operator generator. The eigendecomposition may be based on determining the eigenvalues ​​of the residual elements.

[0015] In some embodiments, the digital representation of the time series data is obtained using computational fluid dynamics (CFD) simulation or experimentation. CFD simulation and experimentation are high-fidelity calculations for obtaining the digital representation of the time series data. CFD simulation or experimentation allows for improved accuracy and speed of complex simulation scenarios, such as transonic or turbulent fluid flows, in various applications of systems, such as heating, ventilation, and air conditioning (HVAC) applications, to describe airflow.

[0016] In some embodiments, the linear predictor represented by the Koopman operator is based on a reduced-order model. Advantageously, the reduced-order model is represented by a Koopman operator that allows it to meet desired properties, such as linearity and reduced order.

[0017] In some embodiments, the method further includes approximating the Koopman operator using a data-driven approximation technique. The data-driven approximation technique is generated using numerical or experimental snapshots. The data-driven approximation technique can be a dynamic mode decomposition (DMD) approximation technique. DMD can utilize snapshots of system state measurements, and the DMD algorithm can search for a linear operator that approximately drives the system state.

[0018] In some embodiments, the method further comprises approximating the Koopman operator using deep learning techniques, which result in a linear embedding of the nonlinear dynamics of the system. Deep learning techniques for approximating the Koopman operator can be successful in long-term dynamic prediction of the system and control of the system.

[0019] In some embodiments, the method further includes generating collocation points associated with a function space of the system based on the PDE, the digital representation of the time series data, and the linearly transformed encoded digital representation. The method further includes training a neural network model based on the generated collocation points. The collocation points may be samples drawn from a domain of the function space of the system such that, in the case of a PDE, the collocation points also satisfy boundary conditions or other constraints associated with the system. Advantageously, generating the collocation points is computationally less expensive than computing snapshots in a CFD calculation.

[0020] In some embodiments, the method further includes generating control commands for controlling the system based on at least one of model-based control and estimation techniques or optimization-based control and estimation techniques. Such techniques can be advantageous for controlling dynamic systems. For example, model-based control and estimation techniques enable a model-based design framework in which system dynamics and constraints can be directly considered.

[0021] In some embodiments, the method further includes generating control commands for controlling the system based on data-driven control and estimation techniques, the goal of which is to design a control policy for the system from data and to control the system using the data-driven control policy.

[0022] Another embodiment discloses a training system for training a neural network model for controlling the operation of a system having nonlinear dynamics represented by a partial differential equation (PDE). The training system includes at least one processor and a memory having instructions stored thereon that, when executed by the at least one processor, cause the training system to collect digital representations of time series data indicative of measurements of the system's operation at different time instances. The at least one processor further causes the training system to train a neural network model having an autoencoder architecture, the autoencoder architecture including: an encoder configured to encode the digital representations into a latent space; a linear predictor configured to propagate the encoded digital representations into the latent space using a linear transformation determined by values ​​of parameters of the linear predictor; and a decoder configured to decode the linearly transformed encoded digital representations to minimize a loss function including a prediction error between an output of a neural network model that decodes a measurement of the operation at a particular time instance and a measurement of the operation collected at a subsequent time instance, and a residual element of the PDE having an eigenvalue that depends on the parameters of the linear predictor.

[0023] Yet another embodiment discloses a non-transitory computer-readable storage medium having embodied thereon a program executable by a processor for performing a method for training a neural network model for controlling 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 indicative of measurements of the system's operation at different time instances. The method further includes training the neural network model having an autoencoder architecture including an encoder configured to encode the digital representations into a latent space, a linear predictor configured to propagate the encoded digital representations into the latent space using a linear transformation determined by values ​​of parameters of the linear predictor, and a decoder configured to decode the linearly transformed encoded digital representations to minimize a loss function including a prediction error between an output of a neural network model that decodes a measurement of the operation at a particular time instance and a measurement of the operation collected at a subsequent time instance, and a residual element of the PDE having an eigenvalue that depends on the parameters of the linear predictor.

[0024] The present disclosure will be further described in the following detailed description with reference to the figures in which like reference numerals represent like parts throughout the several views of the drawings, by way of non-limiting examples of exemplary embodiments of the present disclosure. The drawings shown are not necessarily to scale, with emphasis generally being placed upon illustrating the principles of embodiments of the present disclosure. [Brief explanation of the drawings]

[0025] [Figure 1A] FIG. 1 is a block diagram illustrating two stages for training a neural network model in an offline stage to be used in an online stage to control the operation of a system, according to one embodiment of the present disclosure. [Figure 1B] FIG. 1 is a schematic diagram illustrating the architecture of a Koopman operator, according to some embodiments of the present disclosure. [Figure 2A] FIG. 1 is a schematic diagram illustrating principles used to control the operation of a system according to some embodiments of the present disclosure. [Figure 2B] FIG. 1 is a schematic diagram illustrating an example method for approximating a Koopman operator, according to some embodiments of the present disclosure. [Figure 2C] FIG. 1 is a schematic diagram illustrating an autoencoder architecture for a neural network model, according to some embodiments of the present disclosure. [Figure 3] FIG. 2 is a block diagram illustrating an apparatus for controlling the operation of a system according to some embodiments of the present disclosure. [Figure 4] 1 is a flowchart illustrating principles for controlling the operation of a system according to some embodiments of the present disclosure. [Figure 5] FIG. 1 is a block diagram illustrating generation of a reduced-order model in accordance with some embodiments of the present disclosure. [Figure 6] FIG. 1 is a schematic diagram illustrating a neural network model, according to some embodiments of the present disclosure. [Figure 7A] FIG. 1 illustrates an input of a digital representation in an encoder of a neural network model, according to some embodiments of the present disclosure. [Figure 7B] FIG. 1 illustrates the propagation of an encoded digital representation into a latent space by a linear predictor of a neural network model, in accordance with some embodiments of the present disclosure. [Figure 7C] FIG. 1 illustrates decoding of a linearly transformed encoded digital representation by a neural network model decoder, according to some embodiments of the present disclosure. [Figure 8] FIG. 1 illustrates an example of a real-time implementation of an apparatus for controlling the operation of a system, according to some embodiments of the present disclosure. [Figure 9] 1 is a flowchart illustrating a method for training a neural network model, according to some embodiments of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0026] In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. However, it will be apparent to those skilled in the art that the present disclosure may be practiced without these specific details. In other instances, devices and methods are shown only in block diagram form to avoid obscuring the disclosure. Various changes may be made in the function and arrangement of elements without departing from the spirit and scope of the disclosed subject matter as set forth in the appended claims.

[0027] As used in this specification and claims, the words "for example," "for example," and "such as," as well as "comprises," "has," "includes," and other verb forms thereof, when used in conjunction with a list of one or more components or other items, should be construed as open-ended. This means that the list should not be considered to exclude other additional components or items. The term "based on" means based at least in part on. Furthermore, it should be understood that the terms and terminology used herein are for descriptive purposes and should not be considered limiting. Any headings used within this specification are for convenience only and do not have any legal or limiting effect.

[0028] In the following description, specific details are provided to provide a thorough understanding of the embodiments. However, those skilled in the art will understand that the embodiments may be practiced without these specific details. For example, systems, processes, and other elements of the disclosed subject matter may be shown as components in block diagram form in order to avoid obscuring the embodiments in unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail in order to avoid obscuring the embodiments. Furthermore, like reference numbers and names in the various drawings indicate like elements.

[0029] In describing embodiments of the present disclosure, the following definitions are applicable throughout the disclosure: "Control system" or "controller" may refer to a device or set of devices for managing, commanding, directing, or regulating the behavior of other devices or systems. A control system may be implemented by software or hardware and may include one or several modules. A control system that includes a feedback loop may be implemented using a microprocessor. A control system may be an embedded system.

[0030] An "air conditioning system" or heating, ventilation, and air conditioning (HVAC) system can 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 mechanics, and / or heat transfer. Air conditioning systems span a very broad collection of systems, ranging from systems that provide only outside air to building occupants, to systems that control only the building's temperature, to systems that control both temperature and humidity.

[0031] A "central processing unit" (CPU) or "processor" can refer to a computer or a component of a computer that reads and executes software instructions. Additionally, a processor can be "at least one processor" or "one or more processors."

[0032] 1A shows a two-stage block diagram 100A for training a neural network model in an offline stage for use in an online stage for controlling the operation of a system, according to one embodiment of the present disclosure. Block diagram 100A includes two stages, such as offline stage 102 and online stage 104. Block diagram 100A illustrates the control and estimation of a large-scale system, such as a system with nonlinear dynamics represented by partial differential equations (PDEs), using a two-stage apparatus, i.e., offline stage 102 and online stage 104.

[0033] The offline stage 102 (or Stage I) may include a neural network model 106. The neural network model 106 has an autoencoder architecture. The neural network model 106 comprises an autoencoder 108 including an encoder and a decoder. The neural network model 106 further comprises a linear predictor 110. The offline stage 102 may further include a computational fluid dynamics (CFD) simulation or experiment module 112, differential equations 114 for representing the nonlinear dynamics of the system, and a digital representation 116 of time series data indicative of collocation points 118. The online stage 104 (or Stage II) may include a data assimilation module 120 and a control unit 122 for controlling the system.

[0034] In the offline stage 102, a linear predictor 110 may be derived by performing offline tasks for system control and estimation. In some embodiments, the linear predictor 110 may be based on a reduced-order model. The reduced-order model may be represented by a Koopman operator. Such a reduced-order model may also be referred to as a latent space model. In general, the dimensions of the latent space may be equal to, larger than, or smaller than the input. Further details of the architecture of the Koopman operator for representing the linear predictor 110 are shown, for example, in FIG. 1B.

[0035] Typically, latent space models can be nonlinear and high-dimensional models. The present disclosure enables the design of latent space models that meet desired properties, such as linearity and reduced order. Furthermore, data for the development of latent space models can be generated by performing high-fidelity CFD simulations and experiments using the CFD simulation or experimentation module 112.

[0036] Generally, CFD refers to a branch of fluid mechanics that utilizes numerical analysis and data structures to analyze and solve problems that may involve fluid flow. For example, a computer may be used to perform the calculations necessary to simulate the free-stream flow of a fluid and the interaction of the fluid (such as a liquid or gas) with surfaces defined by boundary conditions. Furthermore, to describe airflow within a system, several software programs have been designed to improve the accuracy and speed of complex simulation scenarios associated with transonic or turbulent flows that may occur in system applications, such as HVAC applications. Furthermore, initial validation of such software is typically performed using equipment such as a wind tunnel. Additionally, previously performed analytical or empirical analyses of specific airflow problems associated with a system may be used for comparison in CFD simulations.

[0037] In some embodiments, the digital representation 116 of the time series data is obtained using a CFD simulation or experiment module 112. The CFD simulation or experiment module 112 may output a data set, such as the digital representation 116 of the time series data, which may be utilized to develop a latent space model (or linear predictor 110). The latent space model may be constructed for several trajectories generated by the CFD simulation. In an exemplary scenario, an HVAC system may be installed in a room. The room may have various scenarios, such as a window may be open, a door may be closed, etc. CFD simulations may be performed for a room with a closed window, a room with an open window, a room with one, two, or multiple 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 the offline stage 102.

[0038] In some embodiments, 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 linearly transformed encoded digital representation (such as the output of the linear predictor 110). The neural network model 106 may be trained based on the generated collocation points 118. Specifically, the neural network model 106 may be trained based on the difference between the prediction of the latent space model and a dataset, such as the digital representation 116 of the time series data plus a physical information portion, i.e., differential equations 114 for representing the nonlinear dynamics of the system that generate the collocation points 118.

[0039] Additionally, the output of the neural network model 106 may be utilized by a data assimilation module 120 of the online stage 104. The data assimilation module 120 may output, for example, a reconstructed model of temperature and velocity within an area, such as a room, associated with a system, such as an HVAC system. The reconstructed model of temperature and velocity may be utilized by a control unit 122. The control unit 122 may generate control commands for controlling the operation (e.g., airflow) of the system, such as an HVAC system.

[0040] The data assimilation module 120 utilizes the process of data assimilation, which refers to assimilating accurate information from sensors with potentially inaccurate model information. For example, a room may be equipped with sensors to monitor specific sensory data. Examples of sensory data implemented in a room for HVAC applications may include, but are not limited to, thermocouple readings, thermal camera measurements, speed sensor data, and humidity sensor data. The information from the sensors may be assimilated by the data assimilation module 120.

[0041] Typically, data assimilation refers to the mathematical field that may seek to optimally combine predictions (usually in the form of numerical models) with observations related to a system. Data assimilation may be used for a variety of goals, such as to find optimal state estimates of a system, to determine initial conditions for a numerical predictive model of a system, to interpolate scarce observational data using knowledge of the system being observed, to identify numerical parameters of a model from observed experimental data, etc. Depending on the goal, a variety of solution methods may be used.

[0042] It should be noted that the offline stage 102 and the online stage 104 are examples of developing a simplified, robust neural network model 106, which can be used for estimation and control of a system with nonlinear dynamics by the control unit 122. Typically, system estimation and control involves estimating values ​​of parameters of the linear predictor 110 based on measured empirical data, which may have a random component. The parameters describe the underlying physical setting, such that the parameter values ​​can affect the distribution of the measured data. Furthermore, an estimator, such as the control unit 122, attempts to approximate the unknown parameters using the measurements. Generally, two approaches are considered for approximation. The first approach is a probabilistic approach, which may assume that the measured data is random and that the probability distribution depends on the parameters of interest. The second approach is a set membership approach, which may assume that the measured data vector belongs to a set that depends on the parameter vector. In the present disclosure, a probabilistic approach for approximation may be adopted.

[0043] It should be noted that by incorporating knowledge of the physical information portions or differential equations associated with the system, the need for large training data sets, such as the digital representation of time series data 116, to identify the latent space model may be reduced. Furthermore, because the neural network model 106 performs operator learning, it allows the neural network model 106 to make predictions beyond the training range and may further be used for compressed sensing, estimation, and control of the system.

[0044] The linear predictor 110 of the neural network model 106 can be represented by a Koopman operator, the architecture of which is further illustrated in Figure 1B.

[0045] 1B shows a schematic diagram 100B of the architecture of the Koopman operator according to some embodiments of the present disclosure. The schematic diagram 100B shows the Koopman operator in a finite-dimensional space represented by a matrix K that leads to a finite-dimensional linear system.

[0046] The Koopman operator is defined as a basis for describing latent space models. The Koopman operator can be based on a Hamiltonian system to formulate the Koopman operator in discrete time. In certain cases, a continuous time formulation can be considered to formulate the Koopman operator.

[0047]

number

[0048] Hamiltonian systems can be used to describe the evolution equations of physical systems, such as systems with nonlinear dynamics. The advantage of Hamiltonian systems is that they provide significant insight into the dynamics of the system, even when the initial value problem can be solved analytically.

[0049]

number

[0050]

number

[0051] Equation 5 may also be utilized in dynamical systems with continuous spectra. Thus, the transformation of a dynamical system from a state-space representation to a Koopman representation trades nonlinear, finite-dimensional dynamics for linear, infinite-dimensional dynamics. The advantage of such a tradeoff is that linear differential equations can be solved using the spectral representation. In practical scenarios, a sufficiently large but finite sum of modes is used to approximate the Koopman spectral solution.

[0052]

number

[0053]

number

[0054]

number

[0055]

number

[0056]

number

[0057]

number

[0058]

number

[0059]

number

[0060]

number

[0061]

number

[0062] Thus, the Koopman eigenfunctions are important criteria, based on which any observable, such as the ORE, can be expressed. The Koopman eigenfunctions themselves are given by DSC. Upon transformation, it is observed that a finite-dimensional nonlinear dynamical system defined by the function f and an infinite-dimensional linear dynamical system defined by the Koopman equation are two equivalent representations of the same fundamental behavior. Furthermore, the observable g and the associated Koopman modal expansion can be successfully linked to the original evolution defined by the function f. Importantly, the Koopman operator captures everything about the nonlinear dynamical system, while the eigenfunctions define the nonlinear changes in coordinates in which the system becomes linear.

[0063] Note that if an observable function g is restricted to an invariant subspace spanned by the eigenfunctions of the Koopman operator, then this observable function g can induce a linear operator K that is finite-dimensional and can evolve the eigenobservable functions on this subspace. Such a subspace is represented in Figure 1B.

[0064] Furthermore, although asymptotic methods can be used to approximate certain eigenfunctions for simple mechanics (e.g., polynomial nonlinear mechanics), analytical procedures for determining eigenpairs of the Koopman operator generally do not exist. Several computational methods, for example, the Dynamic Mode Decomposition (DMD) technique, may be used to approximate the eigenfunctions of the Koopman operator. Details of the DMD technique are further provided, for example, in FIG. 2B.

[0065] 2A shows a schematic overview 200A of principles used to control the operation of a system according to some embodiments of the present disclosure. The schematic overview 200A shows a controller 202 and a system 204. The system 204 may be a system with nonlinear dynamics. The controller 202 may include a linear predictor 206. The linear predictor 206 may be the same as the linear predictor 110 of FIG. 1A. The controller 202 may further include a control unit 208 in communication with the linear predictor 206. The control unit 208 is similar to the control unit 122 of FIG. 1A.

[0066] The controller 202 may be configured to control continuously operating dynamic systems, such as systems 204 in engineering processes and machines. Hereinafter, the terms “controller” and “device” may be used interchangeably and may have the same meaning. Hereinafter, the terms “continuously operating dynamic system” and “system” may be used interchangeably and may have the same meaning. Examples of systems 204 may include, but are not limited to, HVAC systems, light detection and ranging (LIDAR) systems, condensing units, production lines, self-tuning machines, smart grids, automobile engines, robots, numerically controlled machining, motors, satellites, generators, and transportation networks. The controller 202 or control unit 208 may be configured to develop control policies, such as estimations and control commands, to optimally control the system 204 using control actions without delay or overshoot in the system 204 and ensure control stability.

[0067] In some embodiments, the control unit 208 may be configured to generate control commands for controlling the system 204 based on at least one of a model-based control and estimation technique or an optimization-based control and estimation technique, such as a model predictive control (MPC) technique. Model-based control and estimation techniques may be advantageous for controlling dynamic systems such as the system 204. For example, the MPC technique may enable a model-based design framework in which the dynamics and constraints of the system 204 may be directly considered. The MPC technique may develop control commands for controlling the system 204 based on a linear predictor 206 or a model of a latent space model. The linear predictor 206 of the system 204 refers to the dynamics of the system 204 being described using linear differential equations.

[0068] In some embodiments, control unit 208 may be configured to generate control commands for controlling system 204 based on data-driven control and estimation techniques. The underlying control and estimation techniques may utilize operational data generated by system 204 to develop feedback control policies that stabilize system 204. For example, states of system 204 measured during operation of system 204 may be provided as feedback for controlling system 204.

[0069] Typically, the use of operational data to design control policies or control commands is referred to as data-driven control and estimation techniques. Data-driven control and estimation techniques may be utilized to design control policies from data, which may then be used to control the system 204. Furthermore, in contrast to such data-driven control and estimation techniques, some embodiments may use operational data to design models, such as the linear predictor 206. Data-driven models, such as the linear predictor 206, may be used to control the system 204 using various model-based control methods. Furthermore, data-driven control and estimation techniques may be utilized to determine an actual model of the system 204 from the data, i.e., a model that can be used to estimate the behavior of the system 204 having nonlinear dynamics. In some examples, the model of the system 204 may be determined from data that may capture the dynamics of the system 204 using differential equations. Furthermore, physics-based PDE-accurate models may be learned from the operational data.

[0070] Furthermore, to simplify the computation of model generation, an ordinary linear differential equation (ODE) for the linear predictor 206 may be formulated to describe the dynamics of the system 204. In some embodiments, the ODE may be formulated using model reduction techniques. For example, the ODE may be a reduced-order PDE using, for example, an appropriate orthogonal decomposition and Galerkin projection or DMD. Furthermore, the ODE may be a portion of the PDE that describes, for example, boundary conditions. However, in some embodiments, the ODE may not be able to reproduce the actual dynamics of the system 204 (i.e., the dynamics described by the PDE) in the case of uncertainty conditions. Examples of uncertainty conditions may include cases where the boundary conditions of the PDE may be changing over time or where one of the coefficients included in the PDE may be changing.

[0071] Further, an example of a data-driven control and estimation technique is shown in FIG. 2B. FIG. 2B shows a schematic diagram 200B illustrating an exemplary method for approximating a Koopman operator according to some embodiments of the present disclosure. In some embodiments, the Koopman operator may be approximated using a data-driven approximation technique. The data-driven approximation technique may be generated using numerical or experimental snapshots. As the data-driven approximation technique, for example, a dynamic mode decomposition (DMD) approximation technique may be used. The schematic diagram 200B includes a snapshot 210, an algorithm step 212, a set of modes 214, a predictive reconstruction 216, and a shifted value 218 of the snapshot 210.

[0072] The DMD approximation technique may be utilized, for example, to approximate the Koopman operator of a fluid on a cylinder. The DMD approximation technique is a dimensionality reduction algorithm. Typically, given a time series of data, the DMD approximation technique calculates a set of modes 214. Each mode in the set of modes 214 may be associated with a fixed oscillation frequency and a decay or growth rate. For linear systems, the set of modes 214 and fixed oscillation frequencies may be analogous to normal modes of the system, but more generally, the set of modes 214 and fixed oscillation frequencies may be analogous to approximations of the eigenvalues ​​of a composite operator (called the Koopman operator).

[0073] Furthermore, due to the inherent temporal behavior associated with each mode of mode set 214, DMD approximation techniques differ from other dimensionality reduction methods, such as principal component analysis, which may compute orthogonal modes that lack predetermined temporal behavior. Because mode set 214 is not orthogonal, DMD approximation technique-based representations may not be as concise as representations produced by principal component analysis. However, because each mode of mode set 214 is associated with a time-dependent damped (or driven) sinusoidal behavior, DMD approximation techniques may be more physically meaningful than principal component analysis.

[0074]

number

[0075]

number

[0076] Optionally, matrix A may be further reduced by dropping one or more modes of mode set 214. Such an eigendecomposition of matrix A may provide the DMD eigenmodes represented in mode set 214. Matrix A may be used to reconstruct data corresponding to predictive reconstruction 216. Predictive reconstruction 216 may be output by data assimilation module 120 of FIG. 1A. For example, predictive reconstruction 216 may include data related to room temperature and velocity reconstruction in the case of an HVAC system.

[0077] Typically, DMD approximation techniques utilize computational methods to approximate the Koopman operator from data. Advantageously, DMD approximation techniques have a simple formulation in terms of linear regression. Accordingly, several methodological innovations have been introduced, such as using sparsity-promoting optimization to identify the set of modes 214, using randomized linear algebra to enhance the DMD approximation technique, utilizing extended DMD approximation techniques to include linear measurements, using higher-dimensional DMDs operating on delay coordinates to generate more complex models of the linear predictor 206, using multi-resolution DMD approximation techniques with multi-scale systems exhibiting transient or intermittent dynamics, and extending the DMD approximation algorithm to account for the natural dynamics and operation of the system. DMD approximation techniques may also include total least-squares DMD, forward-backward DMD, and variable projection, which may improve the DMD's performance relative to noise sensitivity. Such methods may be utilized in a variety of applications, such as fluid dynamics and heat transfer, epidemiology, neuroscience, finance, plasma physics, robotics, and video processing.

[0078] In some embodiments, the Koopman operator may be approximated using deep learning techniques. In certain scenarios, DMD approximation techniques may be unable to represent the Koopman eigenfunctions. In such cases, deep learning techniques, such as neural network models, may be utilized to approximate the Koopman operator, resulting in a linear embedding of the nonlinear dynamics of the system 204. Deep learning techniques may be successful in long-term dynamic prediction and fluid control for HVAC systems. Deep learning techniques may also be extended to consider uncertainty, PDE modeling, and optimal control of the system 204. Examples of neural network model architectures may include, but are not limited to, neural ODEs for dictionary learning and graphical neural networks utilized to learn synthetic Koopman operators.

[0079] An example of the use of deep learning techniques (or neural network models) to approximate the Koopman operator is further provided in Figure 2C.

[0080] 2C shows a schematic diagram 200C of an autoencoder architecture for a neural network model according to some embodiments of the present disclosure. The deep neural network model may be utilized to learn a linear basis and a Koopman operator using data from snapshot 210. Schematic diagram 200C includes autoencoder 108. Autoencoder 108 includes encoder 220, decoder 222, and linear predictor 224. Linear predictor 224 may be the same as linear predictor 110 of FIG. 1A. Schematic diagram 200C further includes linear predictor 226 and linear predictor 228.

[0081]

number

[0082] Furthermore, within the latent space of an autoencoder 108, such as a linear predictor 224, the dynamics of the system 204 are constrained to be linear. Thus, in some embodiments, a square matrix "K" is used to drive the evolution of the dynamics of the system 204. In general, there is no invariant finite-dimensional Koopman subspace that captures the evolution of all measurements of the system 204; in such cases, the square matrix K can only approximate the underlying truly linear operator.

[0083]

number

[0084] 3 shows a block diagram 300 of an apparatus 302 for controlling the operation of a system according to some embodiments of the present disclosure. The block diagram 300 may include the apparatus 302. The apparatus 302 may include an input interface 304, a processor 306, a memory 308, and storage 310. The storage 310 may further include a model 310a, a controller 310b, an update module 310c, and a control command module 310d. The apparatus 302 may further include a network interface controller 312 and an output interface 314. The block diagram 300 may further include a network 316, a state trajectory 318, and an actuator 322 associated with the system 204.

[0085] 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 multiple input interfaces and multiple output interfaces. The input interface 304 is configured to receive a state trajectory 318 of the system 204. The input interface 304 includes a network interface controller (NIC) 312 adapted to connect the device 302 to a network 316 via a bus. Additionally, the device 302 receives the state trajectory 318 of the system 204 via the network 316, either wirelessly or wired.

[0086] The state trajectory 318 may be a plurality of states of the system 204 that define the actual behavior of the dynamics of the system 204. For example, the state trajectory 318 may serve as a reference continuous state space for controlling the system 204. In some embodiments, the state trajectory 318 may be received from real-time measurements of some portion of the states of the system 204. In some other embodiments, the state trajectory 318 may be simulated using a PDE that describes the dynamics of the system 204. In some embodiments, a shape may be determined for the received state trajectory 318 as a function of time. The shape of the state trajectory 318 may represent the actual pattern of behavior of the system 204.

[0087] The apparatus 302 further includes a 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 coupled to one or more input / output devices via a bus. Furthermore, the stored instructions implement methods for controlling the operation of the system 204.

[0088] The memory 308 may be further expanded to include a storage 310. The storage 310 may be configured to store a model 310a, a controller 310b, an update module 310c, and a control command module 310d.

[0089] The controller 310b may be configured to store instructions that, when executed by the processor 306, execute one or more modules in the storage 310. Furthermore, the controller 310b manages each module of the storage 310 to control the system 204.

[0090] Additionally, in some embodiments, the update module 310c may be configured to update gains associated with the model of the system 204. The gains may be determined by reducing the error between the state of the system 204 estimated by the model 310a and the actual state of the system 204. In some embodiments, the actual state of the system 204 may be a measured state. In some other embodiments, the actual state of the system 204 may be a state estimated by a PDE that describes the dynamics of the system 204. In some embodiments, the update module 310c may update the gains using extremum searching. In some other embodiments, the update module 310c may update the gains using a Gaussian process-based optimization technique.

[0091] The control command module 310d may be configured to determine control commands based on the model 310a. The control command module 310d may control the operation of the system 204. In some embodiments, the operation of the system 204 may be subject to constraints. Furthermore, the control command module 310d uses predictive model-based control techniques to determine the control commands while enforcing the constraints. The constraints include state constraints in the continuous state space of the system 204 and control input constraints in the continuous control input space of the system 204.

[0092] The output interface 314 is configured to send control commands to the actuators 1220 of the system 204 to control the operation of the system 204. Some examples of the output interface 314 may include a control interface that issues the control commands to control the system 204.

[0093] Control of system 204 is further explained in Figure 4. Figure 4 shows a flowchart 400 of principles for controlling the operation of system 204, according to some embodiments of the present disclosure. Flowchart 400 may include steps 402, 404, and 406.

[0094] In some embodiments, the system 204 may be modeled from the laws of physics. For example, the dynamics of the system 204 may be represented by mathematical equations using the laws of physics.

[0095]

number

[0096]

number

[0097]

number

[0098] In some embodiments, such abstract mechanics may be obtained from the numerical discretization of nonlinear partial differential equations (PDEs), which typically require many n state dimensions.

[0099] In some embodiments, a high-dimensional physics-based model of system 204 needs to be solved to control the operation of system 204 in real time. For example, in the case of an HVAC system, the Boussinesq equation needs to be solved to control airflow dynamics and indoor temperature. In some embodiments, the high-dimensional physics-based model of system 204 includes a large number of equations and variables that can be complex to solve. For example, solving the high-dimensional physics-based model in real time requires more computational power. Therefore, the high-dimensional physics-based model of system 204 may be simplified.

[0100] In step 404, providing the apparatus 302 to efficiently control the system 204 generates a reduced-order model to reproduce the dynamics of the system 204. In some embodiments, the apparatus 302 may simplify the high-order physics-based model using model reduction techniques to generate the reduced-order model. In some embodiments, the model reduction techniques reduce the dimensionality of the high-order physics-based model (e.g., variables of a PDE) so that the reduced-order model can be used in real time for prediction and control of the system 204. Further details on generating a reduced-order model for controlling the system 204 are provided with reference to FIG. 5 . In step 406, the apparatus 302 uses the reduced-order model in real time for prediction and control of the system 204.

[0101] 5 illustrates a block diagram 500 showing generation of a reduced-order model according to some embodiments of the present disclosure. The linear predictor 110 is a reduced-order model. The block diagram 500 shows an architecture including a digital representation 116 of time-series data and an autoencoder 106. The autoencoder 106 includes an encoder 220, a decoder 222, and a linear predictor 224. The block diagram 500 further shows an output 502 of the autoencoder 106.

[0102] The CFD simulation or experiment snapshots 210 are the data required for an autoencoder, such as the neural network model autoencoder 106 illustrated in Figure 6. The latent space is governed by a linear ODE that will be trained based on both the data snapshots 210 and the model information, using a DSC equation such as Equation 12.

[0103] Furthermore, for a given time-dependent differential equation (e.g., an ODE or PDE), there may be a set of feasible initial conditions. Some embodiments define feasible initial conditions as those that can fall within the domain of the system dynamics f.

[0104] Typically, the domain of a function is the set of inputs accepted by the function. More precisely, given a function f:X → Y, the domain of f is X. The domain may also be part of the definition of the function rather than a property of it. In such cases, X and Y are both subsets of R, and the function f may be graphed in a Cartesian coordinate system. In such cases, the domain is represented on the x-axis of the graph as the projection of the function's graph onto the x-axis.

[0105] The collocation points 118 may be samples extracted from the domain of the system dynamics f such that, in the case of a PDE, the collocation points 118 may satisfy the boundary conditions. For example, if the boundary conditions of the system dynamics f are periodic, then the collocation points 118 should be periodic. If the boundary conditions are Dirichlet, i.e., if the system dynamics f is equal to a particular value at its boundary point, then the collocation points 118 should also be equal to such value at the corresponding boundary point. Advantageously, the collocation points 118 may also be much more computationally inexpensive to evaluate compared to computing the snapshots 210. While the snapshots 210 may be generated by a simulator or experiment, the collocation points 118 may be generated by simply sampling them from a feasible function space.

[0106] Furthermore, a function space is a set of functions between two definite sets. Often, the domain and / or codomain may have addition properties that can be inherited by the function space. For example, the 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, the function space may inherit a topological or metric structure.

[0107] The autoencoder 106 may receive digital representations of the time series data 116 and the collocation points 118 that are projected onto the differential equations. The encoder 220 encodes the digital representations into a latent space. The linear predictor 224 may propagate the encoded digital representation into the latent space using a linear transformation determined by the values ​​of parameters of the linear predictor 224. Furthermore, the decoder 222 may decode the linearly transformed encoded digital representation. The output 502 of the linearly transformed encoded digital representation may be a reconstructed snapshot or a decoded linearly transformed encoded digital representation.

[0108] A basic neural network model implemented for the autoencoder 106 architecture is illustrated in FIG. 6. FIG. 6 shows a schematic diagram 600 of a neural network model according to some embodiments of the present disclosure. A neural network can be a network or circuit of artificial neural networks composed of artificial neurons or nodes. Thus, a neural network is an artificial neural network used to solve artificial intelligence (AI) problems. The connections of biological neurons are modeled in an artificial neural network as weights between nodes. Positive weights reflect excitatory connections, while negative weight values ​​represent inhibitory connections. All inputs 602 of the neural network model can be summed, modified by the weights. Such activity is referred to as a linear combination. Finally, an activation function controls the amplitude of the output 604 of the neural network model. For example, the acceptable range of the output 604 is typically between 0 and 1, or can be between -1 and 1. Artificial networks can be used for predictive modeling, adaptive control, and applications, where they can be trained via a training dataset. Self-learning due to experience can occur within a network, allowing it to draw conclusions from complex and seemingly unrelated sets of information.

[0109] The architecture of the blocks of the autoencoder 106 is illustrated in Figures 7A, 7B and 7C.

[0110] 7A illustrates a diagram 700A showing the input of a digital representation in an encoder 220 of a neural network model (such as an autoencoder 106) according to some embodiments of the present disclosure. The diagram 700A includes the encoder 220, a snapshot 210, a collocation point 118, and a final layer 702 of the encoder 220.

[0111] The input of the encoder 220 can be either a snapshot 210 or a collocation point 118. The snapshot 210 may be, for example, a digital representation 116 of time-series data. The encoder 220 obtains the values ​​of the snapshot 210 or the collocation point 118. The encoder 220 outputs to a latent space or a linear predictor 224 through the last layer 702 of the encoder 220. The digital representation 116 of time-series data representing measurements of the behavior of the system 204 at different time instances can be collected. Furthermore, for training a neural network model (such as the autoencoder 106) with an autoencoder architecture, the encoder 220 can encode the digital representation into a latent space. The encoding process is model reduction.

[0112] 7B illustrates a diagram 700B depicting the propagation of an encoded digital representation into a latent space by the neural network model linear predictor 224, according to some embodiments of the present disclosure. Diagram 700A includes the last layer 702 of the encoder 220, the linear predictor 224, and the last iteration 704 of the linear predictor 224, or latent space model.

[0113] The linear predictor 224 is configured to propagate the coded digital representation into the latent space using a linear transformation determined by the values ​​of the parameters of the linear predictor 224. The output of the final iteration 704 of the linear predictor 224 is passed to the neural network model decoder 222. The process of propagating the coded digital representation into the latent space is called reduced-order model propagation or time integration.

[0114] 7C illustrates a diagram 700C showing decoding of a linearly transformed encoded digital representation by a neural network model decoder 222, according to some embodiments of the present disclosure. Diagram 700C includes the decoder 222, a final iteration 704 of the linear predictor 224, and an output 706 of the decoder 222.

[0115] The decoder 222 forwards the input and provides an output 706. The decoder is configured to decode the linearly transformed coded digital representation to produce the output 706. The output 706 is a decoded linearly transformed coded digital representation, such as a reconstructed snapshot as described in Figure 5. The process of decoding is a reconstruction of the snapshot.

[0116]

number

[0117]

number

[0118]

number

[0119] Physics-informed neural networks (PINNs) can seamlessly integrate measurement data with governing physical laws by using automatic differentiation to penalize residuals of differential equations in a loss function. Such an approach alleviates the need for large amounts of data by assimilating knowledge of the equations into the training process.

[0120] In some embodiments, the system 204 may be controlled using a linear control law that includes a control matrix formed by the values ​​of the parameters of the linear predictor 224 .

[0121]

number

[0122]

number

[0123]

number

[0124] 8 shows an example diagram 800 for a real-time implementation of an apparatus 302 for controlling the operation of a system 204, according to some embodiments of the present disclosure. The example diagram 800 includes a room 802, a door 804, a window 806, a ventilation unit 808, and a set of sensors 810.

[0125] In an exemplary scenario, system 204 is an air conditioning system. Exemplary diagram 800 shows a room 802 having a door 804 and at least one window 806. The temperature and airflow of 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 disposed in room 802. At least one airflow sensor, such as sensor 810a, is used to measure the speed of airflow at a given point within room 802, and at least one temperature sensor, such as sensor 810b, is used to measure the room temperature. It should be noted that other types of setups are possible, such as rooms with multiple HVAC units or houses with multiple rooms.

[0126] A system 204, such as an air conditioning system, may be described by a physics-based model called the Boussinesq equation, as exemplarily shown in FIG. 4. However, the Boussinesq equation includes infinite dimensions for solving the Boussinesq equation to control the air conditioning system. The model includes an ODE. Data assimilation may be added to the ODE model. The model optimally reproduces the dynamics (e.g., airflow dynamics) of the air conditioning system. Furthermore, in some embodiments, the model of airflow dynamics relates airflow values ​​(e.g., airflow velocity) to the temperature of the air-conditioned room during operation of the air conditioning system. Furthermore, the device 302 optimally controls the air conditioning system to generate airflow in a regulated manner.

[0127] FIG. 9 illustrates a flowchart 900 showing a method for training a neural network model according to some embodiments of the present disclosure.

[0128] In step 902, digital representations 116 of time series data may be collected that indicate measurements of the operation of the system 204 at different time instances. Details of collecting digital representations 116 of time series data are further described, for example, in Figure 2B.

[0129] In step 904, the neural network model 106 may be trained. The neural network model 106 has an autoencoder architecture including an encoder 220 configured to encode the digital representation into a latent space, a linear predictor 224 configured to propagate the encoded digital representation into the latent space using a linear transformation determined by values ​​of parameters of the linear predictor 224, and a decoder 222 configured to decode the linearly transformed encoded digital representation to minimize a loss function including a prediction error between the output of the neural network model 106, which decodes a measurement of motion at a particular time instance and a measurement of motion collected at a subsequent time instance, and a residual element of a PDE having eigenvalues ​​that depend on the parameters of the linear predictor 224.

[0130] The foregoing description provides exemplary embodiments only and is not intended to limit the scope, applicability, or configuration of the present disclosure. Rather, the description of exemplary embodiments will provide those skilled in the art with an enabling description for implementing one or more exemplary embodiments. Various changes are contemplated that may be made in the function and arrangement of elements without departing from the spirit and scope of the disclosed subject matter as set forth in the appended claims.

[0131] In the above description, specific details are given to provide a thorough understanding of the embodiments. However, those skilled in the art will understand that the embodiments may be practiced without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in block diagram form in order to avoid obscuring the embodiments in unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail in order to avoid obscuring the embodiments. Furthermore, like reference numbers and names in the various drawings indicate like elements.

[0132] Also, particular embodiments may be described as a process that is depicted as a flowchart, flow diagram, data flow diagram, structure diagram, or block diagram. While a flowchart may describe operations as a sequential process, many of the operations may be performed in parallel or simultaneously. Additionally, the order of operations may be rearranged. A process may terminate when its operations are completed, but may include additional steps not described or included in the diagram. Moreover, not all operations in any specifically described process may be performed in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the end of the function may correspond to a return of the function to the calling function or the main function.

[0133] Furthermore, embodiments of the disclosed subject matter may be implemented, at least in part, either manually or automatically. Manual or automatic implementations may be performed or at least assisted by machines, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware, or microcode, program code or code segments to perform the necessary tasks may be stored on a machine-readable medium. A processor may perform the necessary tasks.

[0134] The various methods or processes outlined herein may be coded as software executable on one or more processors employing any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages ​​and / or programming or scripting tools, and compiled as executable machine language code or intermediate code that runs on a framework or virtual machine. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments.

[0135] Each embodiment described above is described as a process that is depicted as a flowchart, flow diagram, data flow diagram, structure diagram, or block diagram. While a flowchart depicts operations as a sequential process, many of the operations can be performed in parallel or simultaneously. Additionally, the order of operations may be rearranged. A process may terminate when its operations are completed, or may have additional steps not described or included in the diagram. Moreover, not all operations in any specifically described process may be performed in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the end of the function may correspond to a return of the function to a calling function or a main function.

[0136] Furthermore, embodiments of the disclosed subject matter may be implemented, at least in part, either manually or automatically. Manual or automatic implementations may be performed or at least assisted by machines, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware, or microcode, program code or code segments to perform the necessary tasks may be stored on a machine-readable medium. A processor may perform the necessary tasks.

[0137] Many modifications and other embodiments of the disclosures described herein will come to mind to one skilled in the art to which these disclosures pertain having the benefit of the teachings presented in the foregoing description and the associated drawings. It is to be understood that the disclosure is not limited to the particular embodiments disclosed, and that modifications and other embodiments are intended to be included within the scope of the appended claims. Moreover, while the foregoing description and the associated drawings describe exemplary embodiments in the context of certain illustrative combinations of elements and / or functions, it is to be understood 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, combinations of elements and / or functions other than those expressly described above are also contemplated, for example, as may be set forth in some of the appended claims. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation.

Claims

1. A control method for controlling the operation of a system having nonlinear dynamics described by partial differential equations, the system being an air conditioning system including a ventilation unit, an airflow sensor, and a temperature sensor, comprising: The control method includes a computer-implemented method for training a neural network model for controlling the operation of the system having nonlinear dynamics described by partial differential equations, the computer-implemented method comprising: collecting digital representations of time series data indicative of measurements of the operation of the system at different instances of time; training the neural network model having an autoencoder architecture, the autoencoder architecture including: an encoder configured to encode the digital representation into a latent space; a linear predictor configured to propagate the encoded digital representation into the latent space using a linear transformation determined by values ​​of parameters of the linear predictor; and a decoder configured to decode the linearly transformed encoded digital representation to minimize a loss function including a prediction error between an output of the neural network model that decodes a measurement of the motion at a particular time instance and a measurement of the motion collected at a subsequent time instance, and a residual element of the partial differential equation having an eigenvalue that depends on the parameters of the linear predictor, the residual element of the partial differential equation being based on a Lie operator, the computer-implemented method further comprising performing an eigendecomposition into the Lie operators; The control method further includes controlling the system to control the temperature and airflow in a room via the ventilation unit using a linear control law including a control matrix formed by the values ​​of the parameters of the linear predictor.

2. The method of claim 1 , wherein the digital representation of the time series data is obtained using computational fluid dynamics simulation or experimentation.

3. The control method of claim 1 , wherein the linear predictor is based on a reduced order model represented by a Koopman operator.

4. The control method of claim 3 , further comprising approximating the Koopman operator using a data-driven approximation technique, wherein the data-driven approximation technique is generated using numerical or experimental snapshots.

5. The control method of claim 3 , further comprising approximating the Koopman operator using deep learning techniques.

6. generating collocation points associated with a function space of the system based on the partial differential equation, the digital representation of the time series data, and the linearly transformed encoded digital representation; and training the neural network model based on the generated collocation points.

7. The control method of claim 1 , further comprising generating control commands for controlling the system based on at least one of model-based control and estimation techniques or optimization-based control and estimation techniques.

8. The control method of claim 1 , further comprising generating control commands for controlling the system based on data-driven control and estimation techniques.

9. A training system for training a neural network model for controlling the operation of a system, the system being an air conditioning system having nonlinear dynamics described by partial differential equations and including a ventilation unit, an airflow sensor, and a temperature sensor, the training system comprising at least one processor and a memory storing instructions, the instructions, when executed by the at least one processor, causing the training system to: collecting digital representations of time series data indicative of measurements of the operation of the system at different instances of time; and training the neural network model having an autoencoder architecture, the autoencoder architecture including: an encoder configured to encode the digital representation into a latent space; a linear predictor configured to propagate the encoded digital representation into the latent space using a linear transformation determined by values ​​of parameters of the linear predictor; and a decoder configured to decode the linearly transformed encoded digital representation to minimize a loss function including a prediction error between an output of the neural network model that decodes a measurement of the motion at a particular time instance and a measurement of the motion collected at a subsequent time instance, and a residual element of the partial differential equation having an eigenvalue that depends on the parameters of the linear predictor, the residual element of the partial differential equation being based on a Lie operator, and the instructions further cause the training system to perform an eigendecomposition into the Lie operators; the at least one processor is further configured to control the system to control temperature and airflow in a room via the ventilation unit using a linear control law including a control matrix formed by the values ​​of the parameters of the linear predictor.

10. 10. The training system of claim 9, wherein the digital representation of the time series data is obtained using computational fluid dynamics simulation or experimentation.

11. The training system of claim 9 , wherein the linear predictor is based on a reduced order model represented by a Koopman operator.

12. the at least one processor is further configured to approximate the Koopman operator using a data-driven approximation technique, the data-driven approximation technique being generated using numerical or experimental snapshots; or The training system of claim 11 , wherein the at least one processor is further configured to approximate the Koopman operator using deep learning techniques.

13. The at least one processor further comprises: generating collocation points associated with a function space of the system based on the partial differential equation, the digital representation of the time series data, and the linearly transformed encoded digital representation; The training system of claim 9 , configured to train the neural network model based on the generated collocation points.

14. 10. The training system of claim 9, wherein the at least one processor is further configured to generate control commands for controlling the system based on at least one of model-based control and estimation techniques or optimization-based control and estimation techniques.

15. 1. A non-transitory computer-readable storage medium having embodied thereon a program executable by a processor for performing a method of training a neural network model for controlling the operation of a system having nonlinear dynamics described by partial differential equations, the system being an air conditioning system including a ventilation unit, an airflow sensor, and a temperature sensor, the method comprising: collecting digital representations of time series data indicative of measurements of the operation of the system at different instances of time; training the neural network model having an autoencoder architecture, the autoencoder architecture including: an encoder configured to encode the digital representation into a latent space; a linear predictor configured to propagate the encoded digital representation into the latent space using a linear transformation determined by values ​​of parameters of the linear predictor; and a decoder configured to decode the linearly transformed encoded digital representation to minimize a loss function including a prediction error between an output of the neural network model that decodes a measurement of the motion at a particular time instance and a measurement of the motion collected at a subsequent time instance, and a residual element of the partial differential equation having an eigenvalue that depends on the parameters of the linear predictor, the residual element of the partial differential equation being based on a Lie operator, the method further comprising performing an eigendecomposition into the Lie operators; The non-transitory computer-readable storage medium further includes instructions for generating control commands to control the system to control temperature and airflow in a room via the ventilation unit using a linear control law including a control matrix formed by the values ​​of the parameters of the linear predictor.

Images