Methods for determining a confidence interval for an estimated position of an object and methods for training a physics-informed neural network for this purpose.

Abstract

The invention relates to a method for determining a confidence interval for an estimated position of an object and a method for training a physics-informed neural network for this purpose. The invention further relates to a computer program, a device, and a storage medium for this purpose.

Description

[0001] The invention relates to a method for determining a confidence interval for an estimated position of an object and a method for training a physics-informed neural network for this purpose. The invention further relates to a computer program, a device, and a storage medium for this purpose. State of the art

[0002] Vehicle localization, whether in urban environments or on open highways, is a critical function for various applications such as navigation, autonomous driving, and traffic management. The process can involve determining a vehicle's precise location and orientation within a given environment, typically through the use of a combination of technologies and methodologies. Technologies used in vehicle localization include the following: GNSS (Global Navigation Satellite Systems) such as GPS, GLONASS, and Galileo provide geospatial positioning with global coverage. These systems can be fundamental for open-air environments but may be less reliable in urban environments due to signal clustering from buildings or other structures.Inertial navigation systems (INS) use accelerometers and gyroscopes to calculate a vehicle's position and orientation based on initial location inputs and subsequent movements. INS can operate independently of external signals, making them useful in environments without GNSS, such as tunnels. Odometrics can also be used and involves measuring the movement of vehicle wheels. While relatively simple, odometrics can accumulate errors over time but is useful in conjunction with other methods to improve accuracy. Furthermore, lidar and vision systems can be used, which gather detailed environmental data to determine the vehicle's position relative to known features. These are particularly critical for autonomous vehicles, which require a high degree of accuracy and environmental awareness.Wireless communication can also be used, such as mobile networks and DSRC (Dedicated Short Range Communication), which support localization through techniques such as TDoA (Time of Arrival) and RSSI (Received Signal Strength Indication).

[0003] Vehicle localization is fraught with challenges that can affect the accuracy and reliability of localization methods. Most known positioning processes provide only an estimate of the position without specifying the uncertainty of that estimate. Urban canyons created by tall buildings, as well as rural or natural obstacles such as hills and trees, can interfere with satellite signals used for GNSS. Weather conditions such as fog, rain, and snow can also affect sensor performance. Multipath interference can occur when signals are reflected off surfaces before reaching the receiver, creating multiple paths that confuse the system, especially in densely populated areas. Sensors can drift or become less accurate without regular calibration.For example, wheel slip can lead to errors in odometry data, and inertial sensors could accumulate time errors that can drift significantly if not corrected. Road changes, construction zones, and dynamic traffic conditions require real-time data updates, posing a problem for systems that rely on pre-mapped routes or older data. Disclosure of the invention

[0004] According to aspects of the invention, a method with the features of claim 1, a method with the features of claim 7, a computer program with the features of claim 8, a data processing device with the features of claim 9, and a computer-readable storage medium with the features of claim 10 are provided. Further features and details of the invention are disclosed in the respective dependent claims, in the description, and in the drawings. Features and details described in the context of the inventive methods also correspond in each individual case to the inventive computer program, the inventive data processing device, and the inventive computer-readable storage medium, and vice versa.

[0005] According to one aspect of the invention, a method for determining a confidence interval for an estimated position of an object is provided, wherein the method comprises the following steps, the steps of which can be repeated and / or performed in a specific order.

[0006] In a first possible step, channel parameters and noise power in a receiver are estimated using a channel string method based on observations at the receiver. The channel parameters and noise power can be specific to wireless communication between the receiver and the object. The channel parameters can describe properties of a communication channel between the receiver and the object. They can provide information about how a transmitted signal propagates from a particular transmitter to a particular receiver and can reflect a combined effect of scattering, fading, and power attenuation with distance. The channel parameters are, in particular, complex numbers defined by a magnitude and phase. Furthermore, a tapped delay line model is assumed; that is, each channel tap is also associated with a delay.In summary, a channel estimate is a list of tuples, where each tuple is defined by a delay, a phase, and a magnitude. The noise power can typically be estimated based on known reference signals; here, the received signal power and the noise power are estimated. This can yield a signal-to-noise power ratio.

[0007] In the next possible step, a line of sight from the receiver to the object is detected. The line of sight can refer to direct visual contact between a sender and a receiver. It can be a connecting line between the respective sender and receiver and can enable signal transmission without obstacles. There are various approaches to detecting whether the line of sight is blocked. One approach is machine learning-based detection. Here, a neural network can be trained to detect whether the line of sight is blocked based on the channel impulse response (CIR). This may require a suitable training dataset for the neural network. A second approach would be CIR-based detection. Based on the CIR, the strongest path can be compared with other detected paths.If the strongest path is not significantly stronger, it can be assumed that this path is not a line-of-sight path. This approach can be threshold-based detection. Choosing the right threshold may require specific optimization.

[0008] In a possible next step, the confidence interval for an estimated object position is determined if a detection result indicates that the line of sight from the receiver to the object is unobstructed. This requirement may be necessary to ensure that the position estimation is not further disturbed. Determining the confidence interval for the estimated position is specifically performed using a physics-informed neural network that receives a defined confidence level for the estimated position, along with the estimated channel parameters and noise power, as input and models an estimate of the object's position.

[0009] Physics-informed neural networks (PINNs) are a type of machine learning model specifically designed to solve problems directly involving differential equations. PINNs can integrate known physical laws into their learning process. These laws can be expressed as differential equations. Integrating these laws ensures that the solutions provided by the neural network not only fit the available data but also adhere to fundamental physical principles. A PINN can incorporate a standard neural network architecture, such as a feedforward, convolutional, or recurrent neural network, tailored to approximate the solutions to the applicable differential equations of the given problem.The physics-informed neural network can thus model the estimation of the object's position using partial differential equations that describe the process of estimating the object's position. Unlike traditional neural networks that rely solely on data-driven loss functions such as mean squared error on observed data, the loss function in a PINN can include terms that enforce physical laws. Specifically, it can have components that measure how well the network's predictions satisfy the differential equations in its domain, initial and boundary conditions, and additional constraints derived from physical laws.Training data for a PINN can include observations or measurements from experiments, boundary conditions or initial states, and sparse or noisy data, as the physical constraints can help guide the network toward plausible solutions, even when data is scarce. During training, the PINN can adjust its weights not only to minimize the discrepancy between its predictions and the actual data, but also to reduce the remainder of the differential equations—the difference between the left-hand and right-hand sides of the equations when the network's predictions are applied. PINNs can be able to generalize well from limited data due to the embedded physical laws that provide additional information not captured in the data alone.The solutions adhere to physical principles in particular, making them more interpretable and trustworthy for scientific and technical applications.

[0010] In one example, a Fisher information matrix of the estimated channel parameters and noise power is calculated, and the determination involves the following steps: - Inverting the Fisher information matrix to obtain a covariance matrix, - Extracting at least one component from the obtained covariance matrix that is specific for detecting the line of sight, wherein the at least one component is in particular an input on a diagonal of the inverted Fisher information matrix, - Calculating the confidence interval for the estimated position using the physics-informed neural network with the extracted at least one component and the desired confidence level.

[0011] The Fisher information matrix for a set of parameters θ is based, in particular, on a set of observations X and can be defined as the variance of the score or the expected value of the outer product of the score. The score can be the gradient (vector of partial derivatives) of the log-likelihood function with respect to the parameters θ. Mathematically, the Fisher information matrix I(θ) is given by I(θ)=E[(∂ log f(X;θ)∂θ)(∂ log f(X;θ)∂θ)T] where f(X; θ) is the probability density function of X, parameterized by θ.

[0012] Each element of the Fisher information matrix can provide a measure of how much information the observations X provide about the interactions and relationships between the parameters in θ. One application of the Fisher information matrix is ​​setting a Cramér-Rao lower bound, which provides a lower bound for the variance of any unbiased estimator of a parameter vector. The inverse of the Fisher information matrix can specify this lower bound. In contexts where parameters of a distribution need to be estimated from data, the Fisher information matrix can help understand the achievable accuracy of these estimates.

[0013] In another example, the observations are based on a result of acquisition by at least one sensor, in particular a measurement of a travel time and / or an arrival angle between the receiver and the object, and are provided in particular as in-phase / quadrature data.

[0014] In-phase / quadrature data, or I / Q data, consists of two distinct components of a signal. The in-phase (I) component is the component of the signal that is in phase with a reference signal. The quadrature (Q) component is the component of the signal that is 90° out of phase with the reference signal. Together, these components can be used to represent complex signals, with the I component representing the real part and the Q component representing the imaginary part of a complex number. I / Q data allows a signal to be represented in a two-dimensional space called the IQ plane or complex plane. Any value in this plane can correspond to a unique state of the signal (amplitude and phase), thus enabling efficient encoding of information.I / Q data can be crucial for modulation techniques such as QAM (quadrature amplitude modulation), PSK (phase-shift keying), and FSK (frequency-shift keying). These techniques modulate data onto a carrier wave by changing the amplitude, phase, or frequency of the carrier. I / Q components can also help determine how a signal deviates due to noise or interference, which aids in error correction methods to improve communication reliability. Using I / Q data can allow for more efficient bandwidth utilization and ensures robustness against interference and noise. It can enable the simultaneous transmission of multiple signals and supports advanced modulation methods necessary for high-data-rate communication.

[0015] In another example, the receiver and / or the object is a vehicle. The receiver and / or the object can also be an infrastructure system in a road environment. In yet another example, the wireless communication between the receiver and the object is based on a unilateral or bilateral transmission of radio signals. The communication between the receiver and the object can be V2X (vehicle-to-everything) communication. Applying the method according to the invention to vehicles can be advantageous because position estimation can play a crucial role in this context. This can be particularly true for at least partially automated vehicles.

[0016] In another example, the channel estimation method is a channel state information estimator designed to estimate the state of a communication channel in wireless communication. Channel state information, or CSI, specifically describes properties of the communication channel between a transmitter and a receiver. This can include information about how a transmitted signal propagates through the environment, including effects such as fading, delay, and path loss, which are influenced by various factors, including the physical environment and the movement of objects. CSI can be represented as a matrix containing amplitude and phase information of the channel at different frequencies or time intervals. As a first step, a signal probe can be performed, and the corresponding response can be analyzed.A transmitter can therefore send a known signal (often called a pilot signal) over the channel. The receiver can then measure how this signal is affected by the channel. The received signal may contain distortions caused by the channel's characteristics (e.g., fading, delay distribution). The CSI estimator can then analyze these distortions to determine the channel's effect on the signal. Several techniques can be used for CSI estimation, including the following. One technique is least-squares (LS) estimation, which calculates the CSI by directly comparing the received pilot signals with the original transmitted pilot signals. Another technique is minimum mean square error (MMSE) estimation, which improves upon LS by incorporating noise estimates, making it more robust against noise but computationally more complex.Another technique would be pilot-based estimation, which uses specific symbols inserted into the data stream to assist in the estimation process.

[0017] Another aspect of the invention is a method for training a physics-informed neural network to determine a confidence interval for an estimated position of an object, wherein the method comprises the following steps, the steps of which can be repeated and / or performed in a specific order.

[0018] As a first possible step, a desired confidence level and baseline data are provided, where the baseline data is specific to an actual position of the object. The confidence level can be expressed as a percentage (usually 95%, but other values ​​such as 90% or 99% can also be used) and can reflect a confidence that the confidence interval contains the true parameter value. For example, a 95% confidence interval can imply that if multiple samples of the same population are taken and intervals calculated, approximately 95% of these will contain the true parameter. The baseline data can include a true position determination error variance and / or a bias and / or a true channel impulse response. The baseline can be obtained using simulations or measurements.High bandwidth can be used here to generate high-resolution CIR estimates, as higher bandwidth can provide better temporal resolution.

[0019] In a possible next step, channel parameters and noise power are estimated in a receiver using a channel estimation method based on observations at the receiver. The channel parameters and noise power are particularly specific for wireless communication between the receiver and the object. Furthermore, a Fisher information matrix of the estimated channel parameters and noise power can be calculated.

[0020] In a possible next step, the Fisher information matrix of the estimated channel parameters and noise power is inverted to obtain a covariance matrix.

[0021] In a possible next step, at least one component is extracted from the obtained covariance matrix that is specific for detecting a line of sight between the receiver and the object.

[0022] In a possible next step, the confidence interval for the estimated position is calculated using the extracted at least one component and the provided desired confidence level.

[0023] In a possible next step, the size of the calculated confidence interval is set using the provided baseline data. Setting the size of the calculated confidence interval can indicate that, during the training process, the confidence interval should approach the baseline confidence interval based on the PINN-corrected FIM.

[0024] In another aspect of the invention, a computer program, in particular a computer program product, can be provided, comprising instructions which, when executed by a computer, cause the computer to perform the method according to the invention. The computer program according to the invention can thus have the same advantages as described in detail with reference to a method according to the invention.

[0025] In another aspect of the invention, a data processing device can be provided that is designed to carry out the method according to the invention. For example, the device can be a computer that executes the computer program according to the invention. The computer can include at least one processor that can be used to execute the computer program. Furthermore, a non-volatile data storage device can be provided in which the computer program can be stored and from which the computer program can be read by the processor for execution.

[0026] According to another aspect of the invention, a computer-readable storage medium can be provided that includes the computer program according to the invention and / or instructions which, when executed by a computer, cause the computer to perform the steps of the method according to the invention. The storage medium can be designed as a data storage device, such as a hard disk and / or non-volatile memory and / or a memory card and / or a solid-state drive. The storage medium can, for example, be integrated into the computer.

[0027] Furthermore, the method according to the invention can be implemented as a computer-implemented method. Alternatively or additionally, at least one of the disclosed method steps can be computer-implemented and / or automated.

[0028] Further advantages, features, and details of the invention will become apparent from the following description, in which embodiments of the invention are described in detail with reference to the drawings. In this context, the features mentioned in the claims and in the description can be essential to the invention, either individually or in any combination. The drawings show: Fig. 1 a method, computer program, storage medium and device according to embodiments of the invention, Fig. 2 a method according to embodiments of the invention, Fig. 3 a method according to embodiments of the invention.

[0029] Fig. Figure 1 shows a method 100, a computer program 20, a computer-readable storage medium 15 and a data processing device 10 according to embodiments of the invention.

[0030] Fig. Figure 1 shows in particular an embodiment of a method 100 for determining a confidence interval for an estimated position of an object 1. In a first step 101, channel parameters and noise power are estimated in a receiver 2 using a channel estimation method based on observations in the receiver 2, wherein the channel parameters and noise power are specific for wireless communication between the receiver 2 and the object 1. In a second step 102, a line of sight from the receiver 2 to the object 1 is detected. In a third step 103, the confidence interval for the estimated position is determined if a result of the detection step 102 indicates that the line of sight from the receiver 2 to the object 1 is unobstructed.Determination step 103 is performed using a physics-informed neural network that receives a defined confidence level for the estimated position and the estimated channel parameters and noise power as input and models an estimate of the position of object 1.

[0031] Fig. Figure 2 shows in particular an embodiment of a method 200 for training a physics-informed neural network to determine a confidence interval for an estimated position of an object 1. In a first step 201, a desired confidence level and background truth data are provided, wherein the background truth data are specific for an actual position of the object 1. In a second step 202, channel parameters and a noise power in a receiver 2 are estimated using a channel estimation method based on observations in the receiver 2, wherein the channel parameters and the noise power are specific for wireless communication between the receiver 2 and the object 1, and a Fisher information matrix of the estimated channel parameters and the noise power is computed.In a third step 203, the Fisher information matrix of the estimated channel parameters and the noise power is inverted to obtain a covariance matrix. In a fourth step 204, at least one component specific for detecting a line of sight between receiver 2 and object 1 is extracted from the obtained covariance matrix. In a fifth step 205, the confidence interval for the estimated position is calculated using the extracted at least one component and the desired confidence level. In a sixth step 206, the size of the calculated confidence interval is set using the provided basic truth data.

[0032] Physics-informed neural networks (PINNs) are a type of universal function approximator that can incorporate knowledge of arbitrary physical laws applicable to a given dataset during the learning process, laws that can be described by partial differential equations (PDEs). They can overcome the limited data availability of some biological and technical systems, which renders most state-of-the-art machine learning techniques ineffective in these scenarios due to a lack of robustness. Prior knowledge of general physical laws can be used in training as a regulatory factor that limits the space of feasible solutions, thereby increasing the accuracy of the function approximation.In this way, embedding this pre-known information into a neural network can lead to an improvement in the information content of the available data, making it easier for the learning algorithm to grasp the correct solution and generalize well, even with a small amount of training examples.

[0033] Localization can be achieved through a number of technologies, such as multilateration of radio signals between (multiple) cell towers of the network and an object like a mobile phone or a vehicle, or through the use of GNSS. To locate a mobile phone or vehicle using multilateration of cell signals, the phone or vehicle can transmit at least one idle signal to contact nearby cell towers and does not require an active call.

[0034] Positioning methods are also applicable to automotive applications. Localization and / or positioning, originally used for navigation, can now be used for autonomous driving or advanced driver assistance functions. One example is issuing collision avoidance warnings. While the position estimate may be the most interesting value in this process, the uncertainty of this estimate may also be very important.

[0035] Most position estimation processes only output an estimate of the position. However, it can generally be useful to also have a good understanding of the uncertainty of the estimator. This is called uncertainty quantification. One aspect of the invention according to embodiments is estimating the confidence in the position estimator using prior knowledge of the physical processes and some statistical data.

[0036] One aspect of the invention, according to embodiments, is based on the use of physics-informed neural networks using observations and ground truth data to estimate the confidence in the estimator. The method is based, in particular, on a training process that can generate a corresponding model, i.e., the physics-informed neural network, adapted to the specific situation.

[0037] The process may therefore require the training of a physics-informed neural network and then the use of the physics-informed neural network in so-called operating mode.

[0038] With reference to Fig.Section 3 describes the training method 300 according to embodiments of the invention. Input for the training can include a desired confidence level (alpha) 301, observations in the receiver 302, and basic truth data 303. In a first step 304, channel parameters and noise power can be estimated using a channel estimation method based on observations in the receiver. The observations in the receiver can be composed of I / Q data. This step can be implemented using a CSI estimator based on pilot signals, such as least squares / minimal mean square error, but also with more advanced methods, such as neural networks. In a second step 305, it can be detected whether the line of sight is blocked, and the method can be terminated if this is the case.In a third step, the physics-informed neural network can be trained using the channel parameters and the baseline truth data. Here, the Fisher Information Matrix (FIM) of the estimated channel parameters can be considered the baseline truth parameter. This Fisher Information Matrix can then be inverted to obtain a covariance matrix. A line-of-sight covariance component can then be extracted. Specifically, the line-of-sight covariance components are entries on the diagonal of the inverted FIM. The confidence interval can then be calculated using the extracted covariance component and the provided desired confidence level. Afterward, the baseline truth data can be used to adjust the size of the confidence interval. The output can be the confidence interval of the line-of-sight path parameters, provided at the desired confidence level.

[0039] Confidence is related to the CRB (Cramer-Rao band), which in the simplest case is the trace of the inverted FIM. There are various methods for calculating the CRB. The variance of the line of sight (LOS) can also be calculated based on simulations and ray tracing.

[0040] In operation, the trained model or PINN can be used with an input of a desired confidence level (alpha) and observations in the receiver and can provide a confidence interval of the line-of-sight path parameters with the desired confidence level.

[0041] For training the physics-informed neural network, an exemplary loss function can be an RMSE (Root Mean Square Error).

[0042] The above explanation of the embodiments describes the present invention in the context of examples. Of course, individual features of the embodiments can be freely combined with one another, as long as it is technically reasonable without departing from the scope of protection of the present invention.

Claims

[1] Method (100) for determining a confidence interval for an estimated position of an object (1) comprising the following steps: - Estimating (101) channel parameters and noise power in a receiver (2) using a channel estimation method based on observations in the receiver (2), wherein the channel parameters and noise power are specific for wireless communication between the receiver (2) and the object (1), - Detecting (102) a line of sight from the receiver (2) to the object (1), - Determining (103) the confidence interval for an estimated position of the object (1) if a result of the detection (102) indicates that the line of sight from the receiver (2) to the object (1) is not obstructed, wherein the determination (103) is performed using a physics-informed neural network that ◯ receives a defined confidence level for the estimated position and the estimated channel parameters and noise power as input and ◯ an estimate of the position of the object (1) is modeled. [2] Method (100) according to claim 1, characterized by , that a Fisher information matrix of the estimated channel parameters and noise power is calculated and the determination (103) includes the following steps: - Inverting the Fisher information matrix to obtain a covariance matrix, - Extracting at least one component from the obtained covariance matrix that is specific for detecting (102) the line of sight, - Calculating the confidence interval for the estimated position using the physics-informed neural network with the extracted at least one component and the desired confidence level. [3] Method (100) according to any one of the preceding claims, characterized bythat the observations are based on a result of a procurement by at least one sensor (3), in particular a measurement of a transit time and / or an arrival angle between the receiver (2) and the object (1), and are preferably provided as in-phase / quadrature data. [4] Method (100) according to any one of the preceding claims, characterized by , that the receiver (2) and / or the object (1) is a vehicle, wherein the wireless communication between the receiver (2) and the object (1) is based on a unilateral or bilateral transmission of radio signals. [5] Method (100) according to any one of the preceding claims, characterized by , that the physics-informed neural network models the estimation of the position of the object (1) using partial differential equations that describe the process of estimating the position of the object (1). [6] Method (100) according to any one of the preceding claims, characterized by, that the channel estimation method is a channel state information estimator designed to estimate the state of a communication channel in wireless communication. [7] Method (200) for training a physics-informed neural network to determine a confidence interval for an estimated position of an object (1), comprising the following steps: - Providing (201) a desired confidence level and background truth data, wherein the background truth data are specific for an actual position of the object (1), - Estimating (202) channel parameters and noise power in a receiver (2) using a channel estimation method based on observations in the receiver (2), wherein the channel parameters and noise power are specific for wireless communication between the receiver (2) and the object (1), and wherein a Fisher information matrix of the estimated channel parameters and noise power is calculated, - Inverting (203) the Fisher information matrix of the estimated channel parameters and noise power to obtain a covariance matrix, - Extracting (204) at least one component from the obtained covariance matrix that is specific for detecting a line of sight between the receiver (2) and the object (1), - Calculate (205) the confidence interval for the estimated position using the extracted at least one component and the desired confidence level. - Setting (206) a size of the calculated confidence interval by using the provided basic truth data. [8] Computer program (20) comprising instructions which, when the computer program (20) is executed by a computer (10), cause the computer (10) to execute the method (100) according to any of the preceding claims. [9] Data processing device (10) comprising means for carrying out the method (100) according to any one of claims 1 to 7. [10] Computer-readable storage medium (15) comprising instructions which, when executed by a computer (10), cause the computer (10) to execute the method (100) according to any one of claims 1 to 7.

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