DETAILED ACTION
This nonfinal action is in response to the amendment and remarks filed 05/12/2026, and subsequent request for continued examination filed 06/01/2026, for application 18/052,092.
Claims 1, 11, and 20 have been amended. Claims 8 and 18 are canceled.
Claims 1-2, 4-7, 9-12, 14-17, and 19-20 remain pending in the application. Claims 1, 11, and 20 are independent claims.
Notice of Pre-AIA or AIA Status
The present application, filed on or after March 16, 2013, is being examined under the first inventor to file provisions of the AIA .
Continued Examination Under 37 CFR 1.114
A request for continued examination under 37 CFR 1.114, including the fee set forth in 37 CFR 1.17(e), was filed in this application after final rejection. Since this application is eligible for continued examination under 37 CFR 1.114, and the fee set forth in 37 CFR 1.17(e) has been timely paid, the finality of the previous Office action has been withdrawn pursuant to 37 CFR 1.114. Applicant's submission filed on 05/12/2026 has been entered.
Claim Rejections - 35 USC § 103
In the event the determination of the status of the application as subject to AIA 35 U.S.C. 102 and 103 (or as subject to pre-AIA 35 U.S.C. 102 and 103) is incorrect, any correction of the statutory basis (i.e., changing from AIA to pre-AIA ) for the rejection will not be considered a new ground of rejection if the prior art relied upon, and the rationale supporting the rejection, would be the same under either status.
The following is a quotation of 35 U.S.C. 103 which forms the basis for all obviousness rejections set forth in this Office action:
A patent for a claimed invention may not be obtained, notwithstanding that the claimed invention is not identically disclosed as set forth in section 102, if the differences between the claimed invention and the prior art are such that the claimed invention as a whole would have been obvious before the effective filing date of the claimed invention to a person having ordinary skill in the art to which the claimed invention pertains. Patentability shall not be negated by the manner in which the invention was made.
Claims 1-2, 5-7, 9-12, 15-17, and 19-20 are rejected under 35 U.S.C. 103 as being unpatentable over Gin et al., (“Deep Learning Models for Global Coordinate Transformations that Linearize PDEs”, available arXiv 7 Nov 2019), hereinafter Gin, further in view of Klus et al., (“Data-driven approximation of the Koopman generator: Model reduction, system identification, and control”, available arXiv 13 Feb 2020), hereinafter Klus, and Raissi et al. (“Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”, available online 3 November 2018), hereinafter Raissi.
Regarding claim 1, Gin teaches A computer-implemented method of training a neural network model for modeling an operation of a system having non-linear dynamics, wherein the non-linear dynamics are represented by partial differential equations (PDEs) (“We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a nonlinear PDE into a linear PDE… We demonstrate our method on a number of examples, including the heat equation and Burgers equation, as well as the substantially more challenging Kuramoto-Sivashinsky equation, showing that our method provides a robust architecture for discovering interpretable, linearizing transforms for nonlinear PDEs” [Gin Abstract]), comprising:
collecting a digital representation of time series data indicative of the operation of the system at different instances of time; (“We demonstrate our method on a number of examples, including the heat equation and Burgers equation, as well as the substantially more challenging Kuramoto-Sivashinsky equation, showing that our method provides a robust architecture for discovering interpretable, linearizing transforms for nonlinear PDEs” [Gin Abstract]; “The data for training the neural networks is created by performing numerical simulations of the given PDE. The initial conditions used and discretization details are described for each example below” [Gin page 8 Building Networks for Time-Stepping Dynamics]; “The training data consists of 8000 trajectories from the heat equation…The trajectories consist of 50 equally spaced time steps with Δt = 0:0025” [Gin page 9 Heat Equation]; “In all cases, the training data consists of 120,000 trajectories from Burgers’ equation, each with 51 equally spaced time steps with Δt = 0:002” [Gin page 12 Data]; “The data set mirrors the data used for Burgers’ equation. The training data consists of 120,000 trajectories, each with 51 equally spaced times steps” [Gin page 18 Kuramoto-Sivashinsky Equation]; Via simulations, a collection of training data (i.e., digital representation) is formed for a given PDE (e.g., heat equation, Burgers’ equation, Kuramoto-Sivashinsky equation) that represents the operation of a system via trajectories taken at equally spaced time steps (i.e., sequential data taken at different instances of time, thereby time series data)) and
training the neural network model having an autoencoder architecture (“We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a nonlinear PDE into a linear PDE” [Gin Abstract]; “In this work, we use the universal approximation properties of neural networks to find such linearizing coordinate transformations. The network architecture that we use is shown in Figure 4. The input of the the network uk is the state vector at time tk and the output is the state vector at time tk+1” [Gin page 6 Building Networks for Time-stepping Dynamics]; see network architecture in Figure 4 [Gin page 7]) to minimize a loss function (“The loss function used to train the network is the sum of five different losses. They are depicted in Figure 5” [Gin page 7 Building Networks for Time-stepping Dynamics]; see depiction of loss functions in Figure 5 [Gin page 8]), wherein the autoencoder architecture includes
an encoder configured to receive the digital representation of time series data, wherein the encoder is configured to encode the digital representation into a latent space (“The network consists of three parts: (i) the encoder φ, (ii) the linear dynamics K, and (iii) the decoder φ-1…The encoder consists of the outer encoder χ+I and the inner encoder ψ. The outer encoder performs a coordinate transformation into a space in which the dynamics are linear. The inner encoder diagonalizes the system and/or reduces the dimensionality” [Gin page 6 Building Networks for Time-stepping Dynamics]; see χ outer encoder and ψ inner encoder in Figure 4 [Gin page 7]; The encoder is configured to reduce dimensionality of the input, i.e., encode higher-dimensional input data into a lower-dimensional latent space),
a linear predictor configured to propagate the encoded digital representation into the latent space with linear transformation determined by values of parameters of the linear predictor (“The network consists of three parts: (i) the encoder φ, (ii) the linear dynamics K, and (iii) the decoder φ-1” [Gin page 6 Building Networks for Time-stepping Dynamics]; “r. The resulting dynamics are given by a Koopman operator matrix K” [Gin Abstract]; see deep autoencoder in Figure 1 – “Figure 1. A deep autoencoder is used to find coordinate transformations to linearize PDEs. The encoder finds a set of intrinsic coordinates for which the dynamics are linear. Then the dynamics are given by a matrix K. The decoder transforms back to the original coordinates. Multiple time step prediction can be performed by repeated multiplication by the matrix K in the intrinsic coordinates” [Gin page 2]; see network architecture in Figure 4 – K Linear takes output vk from encoder (after ψ inner encoder which reduces dimensionality, i.e., encodes data into latent space) and advances state forward in time to vk+1; The Koopman operator matrix K (i.e., linear predictor [see instant specification ¶ 0009]) is used to advance forward in time (i.e., propagate) encoder output via its parameters representing linearized dynamics of the system (i.e., linear transformation)), and
a decoder configured to decode the linearly transformed encoded digital representation (“The network consists of three parts: (i) the encoder φ, (ii) the linear dynamics K, and (iii) the decoder φ-1… The inner decoder ψ-1 and the outer decoder ζ+I are the inverses of the inner and outer encoder, respectively.” [Gin page 6 Building Networks for Time-stepping Dynamics]; see deep autoencoder in Figure 1 – “Figure 1. A deep autoencoder is used to find coordinate transformations to linearize PDEs…The decoder transforms back to the original coordinates” [Gin page 2]; see ψ-1 inner decoder and ζ outer decoder (after K Linear) in Figure 4 [Gin page 7])
wherein the loss function includes
a prediction error between
outputs of the neural network model, wherein the outputs of the neural network model are decoded measurements of the operation at an instant of time. and
measurements of the operation at a subsequent instance of time (“Loss 2: prediction loss. The output of the network should accurately predict the state uk+1 when given the state at the previous time uk. The loss is given by
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” [Gin page 7 Building Networks for Time-stepping Dynamics]; see Loss 2: Prediction in Figure 5 [Gin page 8]; The prediction loss (i.e., error) term calculates loss between the predicted state uk+1 output by the network (φ-1(Kφ(uk))) when using uk (i.e., operation at an instant of time) as input, and the actual state uk+1 (i.e., operation at a subsequent instance of time)), and
a residual factor of the PDEs having eigenvalues dependent on the parameters of the linear predictor (“Loss 3: linearity loss. The dynamics on the intrinsic coordinates should be linear. Therefore, we enforce a prediction loss within these coordinates:
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” [Gin page 8 Building Networks for Time-stepping Dynamics]; see Loss 3: Linearity in Figure 5 [Gin page 8]; “Because of its linearity, the behavior of the Koopman operator is completely determined by its eigenvalues and eigenfunctions. We use deep learning in order to approximate the Koopman eigenfunctions, which satisfy
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“ [Gin page 6 Building Networks for Time-stepping Dynamics]; “The first PDE that we consider is the one-dimensional heat equation:
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…For the heat equation, the discrete-time eigenvalues are
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…because the high-frequency waves decay faster than the low-frequency waves, we expect the 21×21 matrix K to have the eigenvalues
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, and therefore the eigenvalues satisfy
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” [Gin pages 9-10 Heat Equation]; The linearity loss term (i.e., residual factor) for the PDE (e.g., heat equation) is based on the Koopman operator matrix (i.e., linear predictor) K having linear dynamics, wherein behavior of the Koopman operator and its parameters is determined by its associated eigenfunctions and eigenvalues),
wherein the residual factor of the PDE is based on a Lie operator ([Gin page 8 Building Networks for Time-stepping Dynamics] and see Loss 3: Linearity in Figure 5 [Gin page 8] and [Gin page 6 Building Networks for Time-stepping Dynamics] and [Gin pages 9-10 Heat Equation], as detailed above; The linearity loss term (i.e., residual factor) for the PDE (e.g., heat equation) is based on the Koopman operator, which is in turn based on the Koopman generator which generates the operator over time. As per the instant specification (“an infinitesimal generator L of the Koopman operator family may be defined as: (equation 6)…The generator L is sometimes referred to as a Lie operator” [¶ 0065-0067]), the examiner has interpreted the term Lie operator as encompassing a Koopman generator (i.e., generator of a Koopman operator), and thereby related to the Koopman operator family).
However, Gin does not expressly teach controlling an operation of a system based on the determined predictions or using time series data that is indicative of measurements of the operation of a system to train the neural network model, or performing eigen-decomposition to a Lie operator.
In the same field of endeavor, Klus teaches a method of applying Koopman operator theory to high-dimensional systems (“We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition)… Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies” [Klus Abstract]) which further recites controlling an operation of a system based on the determined predictions (“we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies” [Klus Abstract]; “The predictive capabilities of the Koopman operator have also raised interest in the control community, where the aim is to determine a system input u such that the non-autonomous control system ˙x = b(x, u) behaves in a desired way, which results in the following control problem: [equation 14]… In order to achieve a feedback behavior, problem (14) is embedded into a model predictive control (MPC) [52] scheme, where it has to be solved repeatedly over a relatively short horizon while the system (the plant) is running at the same time. The first part [t0, t0 +h] of the optimal control u is then applied to the plant, and (14) has to be solved again on a shifted horizon [t0 + h, te + h]” [Klus pages 22-23 Control]) and uses time series data that is indicative of measurements of the operation of a system (“We now assume that we have m measurements of the states of the system, given by {xl} m l=1, and the corresponding time derivatives, given by {x˙l} m l=1. The derivatives might also be estimated from data, cf. [5]” [Klus page 6 Deterministic dynamical systems]; The method utilizes measurements of the states of a system at different instances of time (i.e., time series data)).
It would have been obvious to one of ordinary skill in the art before the effective filing date of the claimed invention to have incorporated controlling an operation of a system and using time series data that is indicative of measurements of the operation of a system as taught by Klus into Gin because they are both directed towards applying Koopman operator theory to high-dimensional systems. Gin already teaches that although the disclosed demonstrations (e.g., heat equation, Burgers’ equation, Kuramoto-Sivashinsky equation) are performed using simulated data, the disclosed data-driven approach can also be applied to experimental data (e.g., snapshots) (“Note that our approach is completely data driven - no knowledge of the underlying equations is needed. Therefore, it can be used for experimental data for which the governing equations are unknown” [Gin page 9 Building Networks for Time-stepping Dynamics]). Gin also already teaches that the disclosed approach would have clear applications to control systems (“The ability to embed nonlinear systems in a linear framework is particularly useful for estimation and control, where a wealth of techniques exist for linear systems. Therefore, it will likely be fruitful to extend these approaches to include inputs and control” [Gin page 21 Conclusion]). Therefore, a person of ordinary skill in the art would recognize the value of incorporating the teachings of Klus to enable further applications of the approach of Gin to training on measurement data of a real-time system for purposes of predictive control strategy.
Klus further teaches performing eigen-decomposition to a Lie operator (“The purpose of this study is to present a general framework to compute a matrix approximation of the Koopman generator, both for deterministic and stochastic systems, and to explore a range of applications…1. We reformulate standard EDMD in such a way that it can be used to approximate the generator of the Koopman operator—as well as its eigenvalues, eigenfunctions, and modes—from data without resorting to trajectory integration” [Klus page 2 Introduction]; The disclosed method determines eigenvalues and eigenfunctions (i.e., performs eigen-decomposition) with respect to a Koopman generator (i.e., Lie operator [see instant specification ¶ 0065-0067]))
It would have been obvious to one of ordinary skill in the art before the effective filing date of the claimed invention to have incorporated performing eigen-decomposition to a Lie operator as taught by Klus into Gin because they are both directed towards applying Koopman operator theory to high-dimensional systems. Incorporating the teachings of Klus into Gin by modifying the Koopman operator matrix to represent a continuous-time Koopman generator would improve applicability of the combination to control systems, via removing drawbacks of large lag times and increasing suitability for time optimization approaches (“Regardless of the approach, a drawback of Koopman operator based surrogate models is that the control freedom is limited by the finite lag time. While larger lag times are often beneficial for the approximation of the dynamics, this is counterproductive for control, as the control frequency is strongly limited. This issue is overcome by the generator approach (15) since we can choose arbitrary time steps here, and results on mixed integer optimal control problems (see, e.g., [57]) suggest that fast switches allow for solutions of any desired accuracy. Moreover, the continuous-time generator model is much better suited for switching time optimization approaches” [Klus page 24 Control]).
However, the combination of Gin and Klus does not expressly teach generating collocation points associated with a function space of the system, wherein the collocation points are generated based on the PDEs, and the encoder recev[ing] the collocation points projected into differential equations, and the residual factor of the PDEs depend[ing] on the collocation points.
In the same field of endeavor, Raissi teaches a method of applying neural network architectures to modeling nonlinear partial differential equations (“We introduce physics-informed neural networks– neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models.” [Raissi Abstract]) that generat[es] collocation points associated with a function space of the system, wherein the collocation points are generated based on the PDEs, (“This example aims to highlight the ability of our method to handle periodic boundary conditions, complex-valued solutions, as well as different types of nonlinearities in the governing partial differential equations. The nonlinear Schrödinger equation along with periodic boundary conditions is given by
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where h(t,x) is the complex valued solution. Let us define f(t,x) to be given by
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…and {tif, xif}Nfi=1 represents the collocation points on f(t, x)” [Raissi page 689 Example (Schrodinger equation)])
has an encoder that receiv[es] the collocation points projected into differential equations, (“Let us define f(t,x) to be given by
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, and proceed by placing a complex-valued neural network prior on h(t,x)…this will result in the complex-valued (multi-output) physic-informed neural network f(t,x)… and {tif, xif}Nfi=1 represents the collocation points on f(t, x)” [Raissi page 689 Example (Schrodinger equation)]; The neural network prior (i.e., encoder) receives coordinate variables (t, x) representing collocation points and projects them into differential equations via computation of partial derivatives (e.g. h_t and h_xx in f(t,x)))
and has a physics-informed residual factor of the PDEs in the loss function that depends on the collocation points (“The shared parameters of the neural networks h(t, x)and f(t, x)can be learned by minimizing the mean squared error loss
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…where
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…… and {tif, xif}Nfi=1 represents the collocation points on f(t, x)…MSEf penalizes the Schrodinger equation not being satisfied on the collocation points” [Raissi page 689 Example (Schrodinger equation)]).
It would have been obvious to one of ordinary skill in the art before the effective filing date of the claimed invention to have incorporated generating collocation points associated with a function space of the system, wherein the collocation points are generated based on the PDEs, and the encoder receiv[ing] the collocation points projected into differential equations, and the residual factor of the PDEs depend[ing] on the collocation points as taught by Raissi into the combination of Gin and Klus because both Gin and Raissi are directed towards applying neural network architectures to modeling nonlinear partial differential equations. Incorporating the teachings of Raissi would incorporate known strengths of physics-informed neural networks by encoding the structure of underlying physical laws of the system through collocation points, thereby obtaining an overall more accurate and data-efficient architecture (“To further analyze the performance of our method, we have performed the following systematic studies to quantify its predictive accuracy for different number of training and collocation points... The general trend shows increased prediction accuracy as the total number of training data Nuis increased, given a sufficient number of col-location points Nf. This observation highlights a key strength of physics-informed neural networks: by encoding the structure of the underlying physical law through the collocation points Nf, one can obtain a more accurate and data-efficient learning algorithm” [Raissi pages 699-700 Continuous time models]).
Regarding claim 2, the combination of Gin, Klus, and Raissi teaches the limitations of parent claim 1, and Klus further teaches controlling the system by using a linear control law including a control matrix formed by the values of the parameters of the linear predictor (“In what follows, we will focus mainly on the generator of the Koopman operator and its properties and applications…The purpose of this study is to present a general framework to compute a matrix approximation of the Koopman generator, both for deterministic and stochastic systems, and to explore a range of applications” [Klus page 2 Introduction]; “The Koopman lifting technique [22, 23] uses the infinitesimal generator L for system identification…. First, the Koopman operator for a fixed lag time τ is estimated from trajectory data with the aid of standard EDMD. Then an approximation of the generator is obtained by taking the matrix logarithm, i.e.,
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where K^τ is the matrix representation of the Koopman operator with respect to the chosen basis ψ (and lag time τ). The last step is to estimate the governing equations in the same way as illustrated in Example 3.3 for gEDMD” [Klus page 16 Koopman lifting technique]; “Since the real-time requirements in MPC are often very hard to satisfy, a promising approach is to replace the system dynamics by a surrogate model, and one possibility is to use the Koopman operator or its generator for prediction. Introducing the variable
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, we obtain a linear system via the approximation L of the generator:
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” [Klus pages 23-24 Control]; Via the Koopman lifting technique, a Koopman generator matrix approximation (i.e., control matrix) can be formed from the Koopman operator matrix representation (e.g., linear predictor of Gin) and its parameters, wherein the Koopman generator can then be used for predictive control of the system via linear approximation
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(i.e., linear control law)).
It would have been obvious to one of ordinary skill in the art before the effective filing date of the claimed invention to have incorporated controlling the system by using a linear control law including a control matrix formed by the values of the parameters of the linear predictor as taught by Klus into Gin because they are both directed towards applying Koopman operator theory to high-dimensional systems. Incorporating the teachings of Klus into Gin to determine an approximation of the Koopman generator would improve applicability of the combination to control systems, via removing drawbacks of large lag times and increasing suitability for time optimization approaches (“Regardless of the approach, a drawback of Koopman operator based surrogate models is that the control freedom is limited by the finite lag time. While larger lag times are often beneficial for the approximation of the dynamics, this is counterproductive for control, as the control frequency is strongly limited. This issue is overcome by the generator approach (15) since we can choose arbitrary time steps here, and results on mixed integer optimal control problems (see, e.g., [57]) suggest that fast switches allow for solutions of any desired accuracy. Moreover, the continuous-time generator model is much better suited for switching time optimization approaches” [Klus page 24 Control]).
Regarding claim 5, the combination of Gin, Klus, and Raissi teaches the limitations of parent claim 1, and Gin further teaches wherein the linear predictor is based on a reduced-order model, wherein the reduced-order model is represented by a Koopman operator (“Although the Koopman operator acts on an infinite-dimensional space, we can obtain a finite-dimensional approximation by considering the space spanned by finitely many Koopman eigenfunctions. Acting on this space, the Koopman operator is just a matrix. Therefore, Koopman operator theory provides an approach to find an intrinsic coordinate system in which the dynamical system has linear dynamics” [Gin page 6 Building Networks for Time-stepping Dynamics]; By linearizing nonlinear dynamical systems, the Koopman operator (i.e., linear predictor) performs model order reduction by reducing high-dimensional (and possibly infinite-dimensional) dynamical systems to their key dynamical features)
Regarding claim 6, the combination of Gin, Klus, and Raissi teaches the limitations of parent claim 5, and Gin further teaches approximating the Koopman operator by use of a data-driven approximation technique (“We use deep learning in order to approximate the Koopman eigenfunctions” [Gin page 6 Building Networks for Time-stepping Dynamics]; “The data for training the neural networks is created by performing numerical simulations of the given PDE…Note that our approach is completely data driven – no knowledge of the underlying equations is needed. Therefore, it can be used for experimental data for which the governing equations are unknown” [Gin page 9 Building Networks for Time-stepping Dynamics]; Training the neural network model (i.e., autoencoder architecture), including approximating parameters of Koopman operator matrix K, is data-driven). Klus further teaches a data-driven approximation technique that is generated using numerical or experimental snapshots (“As a more complex example, we derive a coarse-grained model from molecular dynamics simulations of alanine dipeptide, which has been used as a test case in numerous previous studies. The data set is the same as in reference [51] and comprises one million snapshots of Langevin dynamics saved every 1 ps” [Klus page 21 Example 2: Alanine dipeptide]).
Regarding claim 7, the combination of Gin, Klus, and Raissi teaches the limitations of parent claim 5, and Gin further teaches approximating the Koopman operator by use of a deep learning technique (“We use deep learning in order to approximate the Koopman eigenfunctions” [Gin page 6 Building Networks for Time-stepping Dynamics])
Regarding claim 8, the combination of Gin, Klus, and Raissi teaches the limitations of parent claim 1, and Gin further teaches generating collocation points associated with a function space of the system, wherein the generating is based on the PDEs, the digital representation of time series data and the linearly transformed encoded digital representation; (see Figure 1 – “Figure 1. A deep autoencoder is used to find coordinate transformations to linearize PDEs. The encoder finds a set of intrinsic coordinates for which the dynamics are linear. Then the dynamics are given by a matrix K. The decoder transforms back to the original coordinates.” [Gin page 2]; “Loss 1: autoencoder loss. We want an invertible transformation between the state space and intrinsic coordinates for which the dynamics are linear“ [Gin page 7 Building Networks for Time-stepping Dynamics]; The autoencoder architecture uses the input data (i.e., digital representation of time series data) to determine intrinsic coordinates (i.e., collocation points) within the state space (i.e., function space) of the system for which the associated PDE dynamics become linear) and
training the neural network model based on the generated collocation points. (“The loss function used to train the network is the sum of five different losses…Loss 1: autoencoder loss. We want an invertible transformation between the state space and intrinsic coordinates for which the dynamics are linear. The transformation into the intrinsic coordinates is given by the encoder φ and the transformation back into the state space is given by the decoder φ −1 . Therefore, we wish for the autoencoder φ−1 ◦ φ to reconstruct the inputs of the network as closely as possible. This loss is given by
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” [Gin page 7 Building Networks for Time-stepping Dynamics]).
Regarding claim 9, the combination of Gin, Klus, and Raissi teaches the limitations of parent claim 1, and Klus further teaches generating control commands to control the system based on at least one of: a model-based control and estimation technique or an optimization-based control and estimation technique (“The predictive capabilities of the Koopman operator have also raised interest in the control community, where the aim is to determine a system input u such that the non-autonomous control system ˙x = b(x, u) behaves in a desired way, which results in the following control problem: [equation 14]…In order to achieve a feedback behavior, problem (14) is embedded into a model predictive control (MPC) [52] scheme, where it has to be solved repeatedly over a relatively short horizon while the system (the plant) is running at the same time” [Klus pages 22-23 Control]; In light of the instant specification (“In some embodiments, the control unit 208 may be configured to generate the 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, for example, a model predictive control (MPC) technique” [¶ 0080]), the examiner has interpreted model-based and optimization-based control and estimation techniques as encompassing model predictive control (MPC) techniques. The disclosed method determines system input (i.e., generates control commands) to make the control system behave in a desired way using an MPC technique)
Regarding claim 10, the combination of Gin, Klus, and Raissi teaches the limitations of parent claim 1, and Klus further teaches generating control commands to control the system based on a data-driven based control and estimation technique ([Klus pages 22-23 Control], as detailed above in claim 9; “Since the real-time requirements in MPC are often very hard to satisfy, a promising approach is to replace the system dynamics by a surrogate model, and one possibility is to use the Koopman operator or its generator for prediction” [Klus page 23 Control]; In light of the instant specification (“Typically, use of the operational data to design the control policies or the control commands is referred as the data-driven based control and estimation technique” [¶ 0082]), the examiner has interpreted a data-driven based control and estimation technique as encompassing any technique that uses operational data to control the system. To determine appropriate system input (i.e., control commands) to the control system, the disclosed method models (i.e., estimates) system dynamics via Koopman operator/generator approximation, which are further determined using input time series data (see [Klus page 6 Deterministic dynamical systems], as detailed above in claim 1) which measure states of the system (i.e., system operations over time))
Regarding claim 11, it is a system/apparatus claim that corresponds to the method of claim 1, which is already taught by the combination of Gin, Klus, and Raissi. Gin further teaches A training system, the training system comprising 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: perform the disclosed functions (“We demonstrate our method on a number of examples, including the heat equation and Burgers equation, as well as the substantially more challenging Kuramoto-Sivashinsky equation, showing that our method provides a robust architecture for discovering interpretable, linearizing transforms for nonlinear PDEs” [Gin Abstract]; “The training data consists of 8000 trajectories from the heat equation” [Gin page 9 Heat Equation]; “In all cases, the training data consists of 120,000 trajectories from Burgers’ equation” [Gin page 12 Data]; “The data set mirrors the data used for Burgers’ equation. The training data consists of 120,000 trajectories, each with 51 equally spaced times steps” [Gin page 18 Kuramoto-Sivashinsky Equation]; Training the autoencoder architecture on datasets comprising hundreds of thousands of trajectories inherently requires a computer with adequate processing (i.e., at least one processor) and storage (i.e., memory) capabilities to perform the disclosed functions). Consequently, claim 11 is rejected for the same reasons as claim 1.
Regarding claim 20, it is a product claim that corresponds to the method of claim 1, which is already taught by the combination of Gin, Klus, and Raissi. Gin further teaches A non-transitory computer readable storage medium embodied thereon a program executable by a processor for performing a method, the method comprising: the disclosed functions ([Gin Abstract] and [Gin page 9 Heat Equation] and [Gin page 12 Data] and [Gin page 18 Kuramoto-Sivashinsky Equation], as detailed above in claim 11; Training the autoencoder architecture on datasets comprising hundreds of thousands of trajectories inherently requires a computer with adequate processing and storage capabilities (i.e., a processor coupled to a storage medium) for performing the disclosed functions). Consequently, claim 20 is rejected for the same reasons as claim 1.
Regarding claims 12 and 15-19, they recite substantially similar limitations to those recited in claims 2 and 5-9, which are already taught by the combination of Gin, Klus, and Raissi. Consequently, claims 12 and 15-19 are rejected for the same reasons as claims 2-3 and 5-9.
Claims 4 and 14 are rejected under 35 U.S.C. 103 as being unpatentable over Gin, Klus, and Raissi as applied to claims 1 and 11 above, further in view of Brunton et al., (“Modern Koopman Theory for Dynamical Systems”, available arXiv 29 Oct 2021), hereinafter Brunton.
Regarding claim 4, the combination of Gin, Klus, and Raissi teaches the limitations of parent claim 1, and Klus further teaches wherein the digital representation of the time series data is obtained by use of simulation or experiments (“As a more complex example, we derive a coarse-grained model from molecular dynamics simulations of alanine dipeptide, which has been used as a test case in numerous previous studies. The data set is the same as in reference [51] and comprises one million snapshots of Langevin dynamics saved every 1 ps” [Klus page 21 Example 2: Alanine dipeptide]).
However, the combination does not explicitly teach wherein the digital representation of the time series data is obtained by use of computational fluid dynamics (CFD) simulation or experiments.
In the same field of endeavor, Brunton teaches a method of applying Koopman operator theory to high-dimensional systems (“Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems…In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications” [Brunton Abstract]) wherein the digital representation of the time series data is obtained by use of computational fluid dynamics (CFD) simulation or experiments (“Dynamic mode decomposition, originally introduced by Schmid [380, 379] in the fluid dynamics community, has rapidly become the standard algorithm to approximate the Koopman operator from data [366, 433, 227]... The DMD algorithm was originally developed to identify spatio-temporal coherent structures from high-dimensional time-series data, as are commonly found in fluid dynamics.” [Brunton page 21 Dynamic mode decomposition]; “The DMD algorithm is purely data-driven, and is thus equally applicable to experimental and numerical data” [Brunton page 25 Alternative optimizations for DMD]; “DMD originated in the fluid dynamics community [379], and has since been applied to a wide range of flow geometries (jets, cavity flow, wakes, channel flow, boundary layers, etc.), to study mixing, acoustics, and combustion, among other phenomena” [Brunton page 29 Fluid dynamics]; see Figures 3.2 to 3.4 – fluid dynamics examples [Brunton pages 30-32]; The disclosure teaches that Koopman operator approximation can be performed using computational processes (e.g., dynamic mode decomposition) performed on experimental, high-dimensional time-series data, such as that found in the field of fluid dynamics)
It would have been obvious to one of ordinary skill in the art before the effective filing date of the claimed invention to have incorporated obtaining the digital representation of the time series data by use of computational fluid dynamics (CFD) simulation or experiments as taught by Brunton into the combination of Gin, Klus, and Raissi because Gin, Klus, and Brunton are all directed towards methods of applying Koopman operator theory to high-dimensional systems. Given recognition in the art that dynamic mode decomposition (DMD), which is strongly correlated to Koopman operator theory (“The connection between DMD and the Koopman operator [366, 433, 227] has motivated several extensions for strongly nonlinear systems” [Brunton page 28 Nonlinear measurements and latent variables]; “several approaches have been proposed to extend DMD, including with nonlinear measurements…It is expected that neural network representations of dynamical systems, and Koopman embeddings in particular, will remain a growing area of interest in data-driven dynamics” [Brunton page 90 Discussion and outlook]), has applications to a wide variety of fields including fluid dynamics, (“Algorithms such as DMD [1, 2], EDMD [3, 4], SINDy [5], and their various kernel- [3, 6, 7], tensor- [8, 9, 10], or neural network-based [11, 12, 13] extensions and generalizations have been successfully applied to a plethora of different problems, including molecular and fluid dynamics” [Klus page 1 Introduction), a person of ordinary skill in the art would recognize the value of incorporating the teachings of Brunton to enable applicability of the combination to fluid dynamics simulations/experiments as a particular field of use.
Regarding claim 14, it recites substantially similar limitations to those recited in claims 4, which are already taught by the combination of Gin, Klus, Raissi, and Brunton. Consequently, claim 14 is rejected for the same reasons as claim 4.
Response to Amendment and Arguments
The amendment filed 05/12/2026 has been entered.
Applicant’s amendment to the claims with respect to resolving indefiniteness rejections under 35 U.S.C. 112(b) has been considered, and the rejections are consequently withdrawn.
The remarks filed 05/12/2026 have been fully considered.
Applicant’s remarks [Remarks pages 1-4] traversing the non-eligible subject matter rejections under 35 U.S.C. 101 set forth in the office action mailed 02/12/2026, in view of claims 1-2, 4-7, 9-12, 14-17, and 19-20 as amended, have been considered and are persuasive in part.
In concordance with the issues discussed during the telephonic interview held 05/06/2026 (see Examiner Interview Summary Record mailed 05/11/2026), the examiner at least agrees that the amended claims now recite a technical procedure of implementation of an autoencoder architecture, wherein the architecture comprises specific structural features (e.g., a particularly configured “residual factor’ within the loss function utilized to train the autoencoder architecture) that extend beyond what is well-understood, routine and conventional in the field of implementing physics-informed neural architectures. The claims are thereby interpretable as adequately reflecting the claimed improvements over conventional models as discussed in applicant’s argument (e.g., better enforcing consistency with underlying system dynamics to improve numerical stability and predictive accuracy).
Consequently, the rejections are withdrawn.
Applicant’s remarks [Remarks pages 4-6] traversing the obviousness rejections under 35 U.S.C. 103 set forth in the office action mailed 02/12/2026, in view of claims 1-2, 4-7, 9-12, 14-17, and 19-20 as amended, have been considered, but are moot because the new grounds of rejection set forth above does not rely on the reference(s) applied in the prior rejection of record for the subject matter being challenged in applicant' s argument.
Conclusion
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/V.M.B./
Examiner, Art Unit 2143
/JENNIFER N WELCH/Supervisory Patent Examiner, Art Unit 2143
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