Equation discovery method for weakly nonlinear dynamical systems based on mixed time scales
By constructing hybrid time-scale training data and dynamic symbol networks with phase space structures, the problem of identifying the dynamics of wind-induced vibrations of bridges in existing technologies has been solved. Stable and interpretable dynamic equations for identifying wind-induced vibrations of bridges have been achieved, which is applicable to the nonlinear self-excited vibration control of long-span bridges.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HARBIN INST OF TECH
- Filing Date
- 2026-03-25
- Publication Date
- 2026-06-05
AI Technical Summary
Existing technologies are insufficient to effectively identify the nonlinear dynamic mechanisms of wind-induced vibrations in bridges, especially dynamic problems with multi-timescale characteristics such as vortex-induced vibrations and flutter. Traditional methods rely on prior assumptions and experience, making them difficult to adapt to complex nonlinear dynamic processes.
By constructing hybrid time-scale training data based on phase space structure, using dynamic symbolic networks for recursive learning, and combining sparse regularization terms and loss functions, the governing equations of wind-induced vibration of bridges are identified, and the dynamic equations are discovered using a hybrid time-scale and symbolic network approach.
It improves the stability and interpretability of the dynamic equation identification for wind-induced vibration of bridges, effectively characterizes local rapid vibration features and overall slow evolution features, and is suitable for nonlinear self-excited vibration control of long-span bridges.
Smart Images
Figure CN122153218A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the interdisciplinary fields of bridge wind engineering, nonlinear dynamics, and artificial intelligence, and in particular to a method for discovering equations of weakly nonlinear dynamic systems based on mixed time scales. Specifically, it relates to a method for identifying aerodynamic self-excited vibration control equations based on phase space structural characteristics, mixed time scale data organization, and a dynamic symbol network based on this reconstructed data. This method is applicable to the discovery of dynamic equations for bridge flutter, vortex-induced vibration, and other nonlinear self-excited vibration processes. Background Technology
[0002] Bridges are crucial infrastructure in integrated transportation systems, playing a vital supporting role in economic operation and public travel. As bridges continue to evolve towards longer spans, lighter weight, and greater flexibility, their sensitivity to wind loads has significantly increased, making wind-induced vibration a more prominent issue. In long-span bridges, vortex-induced vibration and flutter are two typical wind-induced vibration phenomena. Vortex-induced vibration often occurs at lower wind speeds and typically affects the bridge's performance and comfort; flutter, on the other hand, occurs more frequently at higher wind speeds, exhibits significant self-excited coupling characteristics, and in severe cases, can threaten the structural safety and overall stability of the bridge.
[0003] Currently, research on the dynamic mechanism of wind-induced vibration in bridges mainly relies on theoretical analysis, semi-empirical modeling, and parameter identification methods based on prior structures. These methods typically require pre-assuming the functional form of aerodynamic or state equation terms, and then fitting parameters using experimental or response data. While they have certain applicability under specific conditions, they generally suffer from problems such as strong reliance on human experience and prior assumptions, insufficient model transferability, and difficulty in adapting to complex nonlinear dynamic processes.
[0004] In recent years, the development of machine learning, especially symbolic regression and neural symbolic methods, has provided new ideas for discovering equations in complex dynamic systems. Most existing data-driven equation identification methods primarily focus on regression learning for time-series samples, failing to adequately utilize the system's phase space structural features and trajectory organization patterns. Therefore, they struggle to simultaneously characterize the local rapid vibration features and overall slow evolution characteristics of wind-induced vibrations in bridges. Especially for dynamic problems with nonlinear, multi-time-scale characteristics such as vortex-induced vibration and flutter, relying solely on single-time-scale and single-trajectory recursive learning is insufficient to fully reveal the true evolutionary laws of the system. Therefore, it is necessary to propose a dynamic equation identification method that can combine phase space structural information, mixed time-scale features, and the interpretable expressive power of symbolic networks. Summary of the Invention
[0005] The purpose of this invention is to address the aforementioned deficiencies in the prior art by providing a method for discovering equations of weakly nonlinear dynamical systems based on mixed time scales.
[0006] This invention is achieved through the following technical solution: This invention proposes a method for discovering weakly nonlinear dynamic system equations based on mixed time scales. This method targets the structural characteristics of the nonlinear dynamic system of the self-excited aerodynamic equations of long-span bridges in phase space. The method includes: Step 1: Construct an aerodynamic self-excited vibration dataset and represent it in phase space; obtain bridge wind-induced vibration response data based on wind tunnel tests or numerical simulations of bridge segment models; for the bridge torsional and vertical coupled vibration system, the state variables consist of torsional displacement, torsional velocity, vertical displacement, and vertical velocity; further, discretize the entire trajectory into a set of phase space points to characterize the evolution trajectory of the system in phase space; Step 2: Construct mixed time-scale training data based on phase space structure; Step 3: Constructing a dynamic symbolic network; After obtaining mixed time-scale training data, a dynamic symbolic network is constructed to identify the control equations for the wind-induced limit cycle vibration of the bridge; The dynamic symbolic network introduces a time recursive structure; Step 4: Construct the loss function and train the dynamic symbolic network; Step 5: Output the control equations.
[0007] Furthermore, in step one, for each set of dynamic responses, it is converted into a state vector form:
[0008] in For system state variables, The dynamic function to be discovered.
[0009] Furthermore, in step two, several center points are first selected from the set of points in phase space, and then local neighborhoods are constructed around each center point. Each local neighborhood is used to characterize the local dynamic characteristics of a certain region in phase space. Multiple local neighborhoods together reflect the overall evolution law of the system. In order to preserve temporal information, time index and trajectory number are further introduced when constructing neighborhood samples, so as to form training samples that take into account both local phase space structure and global temporal evolution.
[0010] Furthermore, in step three, let the dynamic function represented by the dynamic symbolic network be... ,in The network parameters are represented; by introducing symbolic operators into the dynamic symbolic network, the linear and nonlinear terms in the control equations are explicitly combined and expressed; during the time progression, for any current state, the output of the dynamic symbolic network serves as the basis for updating the state at the next moment; numerical integration is used for recursion, so that the network training process is consistent with the continuous evolution process of the dynamic system.
[0011] Furthermore, in step four, a loss function consisting of a data fitting term and a sparse regularization term is used; the loss function is expressed as:
[0012] in, This represents the regularization weight.
[0013] Furthermore, the data fitting term is:
[0014] in, This indicates the total number of samples in the current window.
[0015] Furthermore, the sparse regularization term is: .
[0016] Furthermore, in step five, after training is completed, the candidate basis functions in the dynamic symbolic network are screened and truncated according to the network parameter size, the importance ranking of the candidates, and the sparse constraint results, so as to obtain the explicit control equation expression of the wind-induced limit ring vibration system of the target bridge.
[0017] The present invention also proposes an electronic device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of the method for discovering weak nonlinear dynamic system equations based on mixed time scales.
[0018] The present invention also proposes a computer-readable storage medium for storing computer instructions, which, when executed by a processor, implement the steps of the method for discovering weakly nonlinear dynamic system equations based on mixed time scales.
[0019] The beneficial effects of this invention are: This invention differs from traditional methods that directly perform symbolic regression from time-domain sequences. Its core lies in first constructing mixed-time-scale training data based on the phase space structure, and then inputting this data into a dynamic symbolic network for recursive learning, thereby achieving a unified modeling from "phase space structure representation" to "governance equation recovery." This method exhibits stronger adaptability and interpretability for nonlinear, self-excited, and multi-time-scale coupled dynamic processes such as wind-induced weak nonlinear limit cycle vibrations of bridges. Attached Figure Description
[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.
[0021] Figure 1 A schematic diagram for constructing mixed time-scale data for weakly nonlinear dynamic systems based on phase space structure.
[0022] Figure 2 This is a schematic diagram of a dynamic symbol network structure.
[0023] Figure 3 This is a schematic diagram of the control equation recognition process based on a dynamic symbolic network trained using mixed time scales.
[0024] Figure 4 A schematic diagram showing the results of the wind-induced limit cycle vibration control equations for bridges. Detailed Implementation
[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0026] This invention proposes a method for discovering equations of weakly nonlinear dynamic systems based on mixed time scales. The method first constructs the state variables and phase space trajectories of the dynamic system based on the wind-induced vibration response of a bridge, and organizes training data based on the phase space structure. Then, a dynamic symbolic network is established to learn the local dynamic laws and overall evolution laws of the system during time recursion. Finally, by combining sparse constraints and parameter selection, the explicit control equations of the target dynamic system are obtained, and the method is validated on vortex-induced vibration and flutter data.
[0027] Specifically, in combination Figures 1-4 This invention proposes a method for discovering equations of weakly nonlinear dynamic systems based on mixed time scales. The method targets the structural characteristics of the nonlinear dynamic system of the self-excited aerodynamic equations of long-span bridges in phase space. The method includes: Step 1: Construct an aerodynamic self-excited vibration dataset and represent it in phase space; obtain bridge wind-induced vibration response data based on bridge segment model wind tunnel tests, numerical simulations or other vibration testing methods; for the bridge torsional and vertical coupled vibration system, the state variables consist of torsional displacement, torsional velocity, vertical displacement and vertical velocity; further, discretize the entire trajectory into a set of phase space points to characterize the evolution trajectory of the system in phase space; In step one, the system state variables consist of the bridge's wind-induced vibration response, and the dynamic structure of the limit cycle vibration is characterized by phase space trajectory. For each set of dynamic responses, it is converted into a state vector form:
[0028] in For system state variables, The dynamic function to be discovered.
[0029] Step 2: Construct mixed time-scale training data based on phase space structure; the training samples are not directly composed of the original time series point by point, but are constructed through the phase space center point and its local neighborhood, thereby simultaneously introducing local structural features and overall evolutionary information; In step two, considering the characteristics of aerodynamic self-excited vibration, which simultaneously exhibits local rapid vibration features and overall slow evolution features, this invention does not directly model the original time series point by point. Instead, it constructs hybrid time-scale training data based on the phase space structure. Specifically, firstly, several center points are selected from the phase space point set, and then local neighborhoods are constructed around each center point. Each local neighborhood is used to characterize the local dynamic characteristics within a certain region of the phase space. Multiple local neighborhoods collectively reflect the overall evolution law of the system. To preserve temporal information, time indexes and trajectory numbers are further introduced when constructing neighborhood samples, thereby forming training samples that take into account both the local phase space structure and the global time evolution.
[0030] Step 3: Constructing a dynamic symbolic network; After obtaining mixed time-scale training data, a dynamic symbolic network is constructed to identify the control equations for the wind-induced limit cycle vibration of bridges; The dynamic symbolic network introduces a time recursive structure on the basis of the traditional symbolic network, enabling the network not only to explicitly express candidate equation terms, but also to learn the evolutionary relationship of the dynamic system over time; The dynamic symbolic network extends from static symbolic expression to dynamic equation identification through the time recursive mechanism, making it suitable for the discovery of weakly nonlinear control equations; In step three, let the dynamic function represented by the dynamic symbolic network be... ,in The network parameters are represented; by introducing symbolic operators such as identities, multiplication, power functions and trigonometric functions into the dynamic symbolic network, the linear and nonlinear terms in the control equations are explicitly combined and expressed; during the time progression, for any current state, the output of the dynamic symbolic network serves as the basis for updating the state at the next moment; numerical integration is used for recursion, so that the network training process is consistent with the continuous evolution process of the dynamic system.
[0031] Step 4: Construct the loss function and train the dynamic symbolic network; the loss function is composed of local neighborhood data error, overall data error, length constraint error and sparse regularization term, so as to improve the simplicity and interpretability of the final equation while ensuring the fitting ability. In step four, in order to enable the dynamic symbolic network to learn both the dynamic changes in the local neighborhood and the evolution of the overall trajectory, a loss function consisting of a data fitting term and a sparse regularization term is adopted. For the Let there be a local neighborhood, and let the true state be... The predicted state is The data fitting loss in the local neighborhood can then be defined as:
[0032] in, This indicates the number of samples in the local neighborhood. This term is used to characterize the fast-timescale dynamic error within the local region. For all... After averaging the data across local neighborhoods, the overall data loss is obtained:
[0033] This term reflects the overall fitting error across all local neighborhoods, thus taking into account both local dynamic features and the overall slow evolution. To mitigate training instability caused by long-term recursion, the prediction length can be further constrained. After dividing the long trajectory into several restricted segments, the data loss under the preset length constraint can be expressed as:
[0034] in, This indicates the number of sub-segments obtained from the partitioning. Indicates the first Data error across sub-segments. During training, a sliding window mini-batch method is preferred for organizing samples; therefore, the window-level data loss, i.e., the data fitting term, can be written as:
[0035] in, This represents the total number of samples in the current window. In addition to the data fitting term, to enhance the simplicity and interpretability of the final control equation, a sparse regularization term is further added during training:
[0036] Therefore, the total loss function can be expressed as:
[0037] in, This represents the regularization weight. Preferably, the regularization weight can be adaptively adjusted according to the gradient ratio between the data term and the regularization term to balance the fitting accuracy and the sparsity of the equation. After training, the candidate symbol terms are filtered and truncated according to their parameters to obtain the final expression for the weakly nonlinear aerodynamic self-excited vibration control equations, represented by vortex oscillation and flutter.
[0038] Step 5: Output the control equations; through parameter filtering and threshold truncation, a more stable, concise, and physically meaningful control equation expression can be obtained.
[0039] In step five, after training is complete, candidate basis functions in the dynamic symbolic network are screened and truncated based on network parameter size, candidate importance ranking, and sparse constraint results to obtain the explicit governing equation expression of the target bridge wind-induced limit cycle vibration system. Because this invention employs a hierarchical approach of "phase space structural modeling + hybrid time-scale training + dynamic symbolic network recognition," it effectively enhances the model's ability to express the weakly nonlinear dynamic processes of bridge wind-induced vibration and improves the stability and interpretability of the dynamic equation recognition.
[0040] Example The invention will be further described below with reference to the accompanying drawings. This invention proposes a method for discovering equations of weakly nonlinear dynamical systems based on mixed time scales, the method comprising: The first step is to construct a dataset of weakly nonlinear dynamic systems for training. This dataset can be obtained through wind tunnel tests of bridge segment models, numerical simulations, or other vibration tests. For each set of response data, state variables are first constructed, and then they are discretized and mapped to a set of phase space trajectory points.
[0041] The second step is to construct mixed time-scale training samples based on the phase space structure, such as... Figure 1 As shown, firstly, center points are selected from the phase space point set, then local neighborhoods are constructed around each center point, and time index and trajectory number information are introduced into the samples. The training data obtained in this way can simultaneously reflect the local rapid vibration characteristics and the overall slow evolution law.
[0042] The third step is to construct a dynamic symbol network, such as... Figure 2 As shown, the network adds a time recursion mechanism to the symbolic operator combination structure, enabling the network output to not only express candidate dynamic terms but also to be used for continuous updates of the system state. The candidate operators in the network can be set according to the characteristics of the target dynamic system.
[0043] The fourth step involves training the dynamic symbolic network using the mixed time-scale training samples constructed in the second step, such as... Figure 3 As shown. During training, a loss function consisting of a data fitting term and a sparse regularization term is used, and training stability is improved through methods such as sliding window and prediction length constraints. After training, the candidate parameters are filtered and truncated to obtain the explicit governing equations of the wind-induced limit ring vibration system of the target bridge, as shown. Figure 4 As shown.
[0044] In this embodiment, the present invention can be implemented using the PyTorch framework. The dynamic symbolic network can adopt an explicit operator combination structure to maintain the interpretability of the equation expression; the optimization algorithm can use Adam or AdamW; an early-stop mechanism can be set during training, and gradient pruning strategies can be introduced as needed to improve training stability.
[0045] This invention proposes a method for discovering equations of weakly nonlinear dynamic systems based on hybrid timescales. Addressing the issue that long-span bridges are prone to typical self-vibrations such as vortex-induced vibration and flutter under wind, and that existing equation modeling methods rely heavily on prior assumptions and struggle to adapt to complex nonlinear dynamic processes, the method described in this invention first constructs state variables and phase space trajectories based on the bridge's wind-induced vibration response. Then, based on the phase space structure, it selects center points and constructs local neighborhoods, introducing time indices and trajectory numbers to form hybrid timescale training data that considers both local rapid vibration characteristics and overall slow evolution characteristics. Furthermore, it constructs a dynamic symbol network, learning the evolution law of the dynamic system through time recursion, and uses a loss function composed of data fitting terms and sparse regularization terms for training. Finally, it filters and truncates candidate symbol terms to obtain the explicit control equations of bridge wind-induced vibration. This method effectively combines phase space structure information, multi-timescale features, and the interpretable expressive power of the symbol network, improving the stability and interpretability of identifying the dynamic equations of bridge wind-induced vibration.
[0046] The present invention also proposes an electronic device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of the method for discovering weak nonlinear dynamic system equations based on mixed time scales.
[0047] The present invention also proposes a computer-readable storage medium for storing computer instructions, which, when executed by a processor, implement the steps of the method for discovering weakly nonlinear dynamic system equations based on mixed time scales.
[0048] The memory in this application embodiment can be volatile memory or non-volatile memory, or it can include both volatile and non-volatile memory. The non-volatile memory can be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), or flash memory. The volatile memory can be random access memory (RAM), which is used as an external cache. By way of example, but not limitation, many forms of RAM are available, such as static random access memory (SRAM), dynamic random access memory (DRAM), synchronous dynamic random access memory (SDRAM), double data rate synchronous dynamic random access memory (DDRSDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous linked dynamic random access memory (SLDRAM), and direct rambus RAM (DR RAM). It should be noted that the memory used in the methods described in this invention is intended to include, but is not limited to, these and any other suitable types of memory.
[0049] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. The computer program product includes one or more computer instructions. When the computer instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium accessible to a computer or a data storage device such as a server or data center that integrates one or more available media. The available media may be magnetic media (e.g., floppy disks, hard disks, magnetic tapes), optical media (e.g., high-density digital video discs (DVDs)), or semiconductor media (e.g., solid-state disks (SSDs)).
[0050] In implementation, each step of the above method can be completed by integrated logic circuits in the processor's hardware or by instructions in software. The steps of the method disclosed in the embodiments of this application can be directly implemented by a hardware processor, or by a combination of hardware and software modules in the processor. The software modules can reside in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory, and the processor reads information from the memory and, in conjunction with its hardware, completes the steps of the above method. To avoid repetition, detailed descriptions are omitted here.
[0051] It should be noted that the processor in the embodiments of this application can be an integrated circuit chip with signal processing capabilities. During implementation, each step of the above method embodiments can be completed by the integrated logic circuitry in the processor's hardware or by instructions in software form. The processor can be a general-purpose processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components. It can implement or execute the methods, steps, and logic block diagrams disclosed in the embodiments of this application. The general-purpose processor can be a microprocessor or any conventional processor. The steps of the methods disclosed in the embodiments of this application can be directly embodied as execution by a hardware decoding processor, or as a combination of hardware and software modules in the decoding processor. The software modules can be located in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory, and the processor reads the information in the memory and, in conjunction with its hardware, completes the steps of the above methods.
[0052] The above provides a detailed description of the proposed method for discovering weakly nonlinear dynamic system equations based on mixed time scales. Specific examples have been used to illustrate the principles and implementation methods of this invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of this invention. At the same time, those skilled in the art will recognize that there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.
Claims
1. A method for discovering equations of weakly nonlinear dynamical systems based on mixed time scales, characterized in that, The method addresses the structural characteristics of the nonlinear dynamic system of the self-excited aerodynamic equations of long-span bridges in phase space; the method includes: Step 1: Construct an aerodynamic self-excited vibration dataset and represent it in phase space; obtain bridge wind-induced vibration response data based on wind tunnel tests or numerical simulations of bridge segment models; for the bridge torsional and vertical coupled vibration system, the state variables consist of torsional displacement, torsional velocity, vertical displacement, and vertical velocity; further, discretize the entire trajectory into a set of phase space points to characterize the evolution trajectory of the system in phase space; Step 2: Construct mixed time-scale training data based on phase space structure; Step 3: Constructing a dynamic symbolic network; After obtaining mixed time-scale training data, a dynamic symbolic network is constructed to identify the control equations for the wind-induced limit cycle vibration of the bridge; The dynamic symbolic network introduces a time recursive structure; Step 4: Construct the loss function and train the dynamic symbolic network; Step 5: Output the control equations.
2. The method according to claim 1, characterized in that, In step one, for each set of dynamic responses, it is converted into a state vector form: in For system state variables, The dynamic function to be discovered.
3. The method according to claim 1, characterized in that, In step two, several center points are first selected from the set of points in phase space, and then local neighborhoods are constructed around each center point; each local neighborhood is used to characterize the local dynamic characteristics of a certain region in phase space. Multiple local neighborhoods collectively reflect the overall evolution of the system; to preserve temporal information, time indexes and trajectory numbers are further introduced when constructing neighborhood samples, thus forming training samples that take into account both local phase space structure and global temporal evolution.
4. The method according to claim 1, characterized in that, In step three, let the dynamic function represented by the dynamic symbolic network be... ,in It represents network parameters; by introducing symbolic operators into the dynamic symbolic network, it explicitly combines and expresses the linear and nonlinear terms in the governing equations; during time progression, for any current state, the output of the dynamic symbolic network serves as the basis for updating the state at the next time step; Numerical integration is used for recursion, so that the network training process is consistent with the continuous evolution process of the dynamic system.
5. The method according to claim 1, characterized in that, In step four, a loss function consisting of a data fitting term and a sparse regularization term is used; the loss function is expressed as: in, This represents the regularization weight.
6. The method according to claim 5, characterized in that, The data fitting term is: in, This indicates the total number of samples in the current window.
7. The method according to claim 6, characterized in that, The sparse regularization term is: 。 8. The method according to claim 1, characterized in that, In step five, after training is completed, candidate basis functions in the dynamic symbolic network are screened and truncated according to the network parameter size, candidate importance ranking, and sparse constraint results, so as to obtain the explicit control equation expression of the wind-induced limit ring vibration system of the target bridge.
9. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1-8.
10. A computer-readable storage medium for storing computer instructions, characterized in that, When the computer instructions are executed by the processor, they implement the steps of the method according to any one of claims 1-8.