Machine Learning-Based Calibration for Ultrasonic Flow Metering Using Transit-Time and Temperature Measurements

Ultrasonic flow meters measure fluid velocity by transmitting and receiving ultrasonic waves through the fluid. As the waves travel through a moving fluid, their speed is altered by the component of the fluid’s velocity along the wave path. By precisely measuring the transit times of ultrasonic pulses traveling in both the upstream and downstream directions, the average velocity of the fluid within the pipe can be determined, which subsequently allows for the calculation of the volumetric flow rate.

In a typical setup for a DN-25 pipe, two ultrasonic transducers are mounted on the outer surface of the pipe at a known distance $L$ apart along the axial direction. These transducers can function interchangeably as both transmitters and receivers. A short burst of ultrasonic waves is transmitted by one transducer, propagates through the pipe wall, into the fluid, crosses the fluid stream, goes through the opposite pipe wall, and is finally received by the other transducer. This process is then reversed, with the second transducer transmitting and the first receiving.

The key point of ultrasonic flow measurement is accurately determining the time of flight (ToF)–time difference Δt of theultrasonic pulses. Let $t_1$ be the time taken for the ultrasonic pulse to travel from the upstream transducer to the downstream transducer, and $t_2$ be the time taken for the pulse to travel from the downstream transducer to the upstream transducer.

When the fluid is stationary (zero flow), the transit times in both directions are equal $\left(t_{10}=t_{20}\right)$. However, when the fluid is flowing with an average velocity $v$ along the axis of the pipe, the effective speed of the ultrasonic wave in the downstream direction is increased to $c+v \cos (\theta)$, and in the upstream direction, it is decreased to $c-v \cos (\theta)$, where $c$ is the speed of sound in the fluid and $\theta$ is the angle between the ultrasonic path and the pipe axis. The path length of the ultrasonic beam through the fluid is typically $D / \sin (\theta)$, where $D$ is the inner diameter of the pipe. However, for simplicity in this initial explanation, let's consider the axial distance $L$ between the transducers and assume the angle $\theta$ is such that its cosine component along the axial direction is relevant. Therefore, the transit times can be expressed as:

$t_1=\frac{L}{c+v \cos (\theta)}$

$t_2=\frac{L}{c-v \cos (\theta)}$

By rearranging these equations, one can solve for the fluid velocity $v$ :

$v=\frac{L}{2 \cos (\theta)}\left(\frac{1}{t_1}-\frac{1}{t_2}\right)=\frac{L\left(t_2-t_1\right)}{2 \cos (\theta) t_1 t_2}$    (1)

For a given pipe geometry ( $L$ and $\theta$ are fixed) and assuming the speed of sound c remains relatively constant, the fluid velocity $v$ is directly proportional to the difference in the reciprocal of the transit times or, for small velocity differences, approximately proportional to the difference in transit times $\left(t_2-t_1\right)$. Accurate measurement of $t_1$ and $t_2$ is thus crucial for precise flow rate determination. Eq. (1) is theoretically sound; however, in practical applications, it exhibits significant sensitivity to small variations in $t_1$ and $t_2$. Given that these transit times are typically of the order of nanoseconds, even minimal measurement errors or numerical inaccuracies can result in significant deviations in the estimated velocity. As a result, the accuracy and reliability of the flow meter are adversely affected To address this limitation, a calibration factor is typically introduced into the velocity formulation:

$v=\kappa\left(t_1, t_2\right) \frac{\Delta t}{t_1 t_2}$        (2)

where, $\kappa\left(t_1, t_2\right)$ is a calibration parameter, which can be determined through empirical methods or advanced techniques, such as artificial intelligence-based algorithms. It is important to note that the neural network is not applied to raw ultrasonic waveforms or direct ToF estimation; instead, it operates on precomputed measurement-level features to determine the calibration factor.

Figure 1. Illustration of the experimental setup using DN-25

Figure 1 illustrates the typical ultrasonic flow measurement setup for a DN-25 pipe. Two ultrasonic transducers are mounted on opposite sides of the pipe wall at a distance $L$ apart. This configuration is used to measure the ToF of the ultrasonic pulses in both directions, which is critical to determining the velocity of the fluid.

4.1 Machine learning architectures and training of the calibration factor

This stage maps the measured transit times $\left(t_1, t_2\right)$ and temperature ( $T$ ) to two distinct target variables: the calibration factor (κ) and the time difference correction (Δt). For the field calculation of the velocity, using the measurements directly would result in inaccurate conclusions. To avoid such inaccuracies, the manufacturer provides Δt and κ values for a given set of time and temperature measurements. We were provided a dataset of 950 points, including measured initial and final time, temperature, and calculated Δt and κ. Δt and κ were calculated with the same input parameters. Therefore, we split the data set into two distinct sets for independent modeling. We have separately optimized each set to arrive at the highest possible accuracy.

Table 1. Performance metrics comparison for $\Delta \mathrm{t}$ and $\kappa$ predictions

To ensure statistical validity, we implemented a regression pipeline using Python’s scikit-learn library. We applied a Yeo-Johnson Power Transformation to stabilize variance and a RobustScaler to mitigate outliers in the timing data. Datasets were split into 80% training, 10% validation, and 10% testing. Training ran for up to 500 epochs with early stopping, and hyperparameters were tuned using GridSearchCV. In Table 1, we present the performance metrics for both models.

The Δt model’s high R² shows that manufacturer-supplied Δt values are produced using the input parameters, mainly dependent on $t_2$. In contrast, the κ model’s lower percentage errors make it practical for real-time applications.

4.2 Calibration factor numerical outputs

In this section, we provide details of the modeling for each prediction.

4.2.1 Learning curves

Learning curve is a diagnostic plot that shows how the model’s performance evolves as a function of the size of the training dataset. We display the accuracy (R²). The model is used to estimate both the training set (so far) and the validation set (20% of the total training set) as more data is used to train the model. The curve helps assess whether the model is overfitting (a large gap between training and validation performance) or underfitting (both scores are low). A smooth convergence of the training and validation curves indicates that the model generalizes well to unseen data and benefits from additional training samples.

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Figure 2. Learning curves for (a) $\Delta \mathrm{t}$ prediction and (b) $\kappa$ prediction

Figure 2 shows how R² changes with training size. In Figure 2(a), Δt prediction reaches near-perfect R² even with small datasets, showing excellent generalization. We are reaching the measurement accuracy level. We see minimal improvement with the increased training size. In Figure 2(b), κ prediction improves gradually with training size, reaching a validation R² of 0.914, indicating effective prediction. κ prediction efficiency is a more gradual increase. Efficiency can be improved with larger datasets. 1.777% mean error shows that even with this small set of data, the model can be used in real-life situations.

4.2.2 Shapley Additive Explanations value analysis

SHAP values are a model-agnostic method for interpreting machine learning predictions by assigning an importance value to each feature for every individual prediction. Rooted in cooperative game theory, SHAP treats each feature as a "player" in a game where the model's prediction is the "payout" and calculates how much each feature contributes to shifting the prediction from the baseline (the average prediction over all data points). In the SHAP summary plots used in this study, the horizontal axis shows the SHAP values in the same units as the model output ( $\Delta \mathrm{t}$ or $\kappa$ ). A positive (negative) SHAP value means the feature increases (decreases) the predicted output relative to the baseline. For instance, if $t_2$ has a SHAP value of +300 , it contributes to increasing the predicted $\Delta \mathrm{t}$ (or $\kappa$ ) by 300 units, while a SHAP value of - 100 indicates $t_2$ lowers the prediction by 100 units. This spread illustrates how the effect of $t_2$ on the prediction depends on its value and its interaction with other features in the neural network.

The color gradient in the plots provides the actual value of the feature for each data point. Red dots represent high feature values (e.g., large $t_2$ ), and blue dots indicate low feature values. This allows us to observe trends such as high $t_2$ values generally contributing positively to $\Delta \mathrm{t}$ predictions (red dots on the positive side) and low $t_2$ values often decrease the predictions (blue dots on the negative side). This combination of SHAP values and color mapping helps to disentangle how both the magnitude and direction of each feature influence the model's predictions, which is especially important for interpreting complex, nonlinear models like MLPs.

The SHAP summary plots show the mean absolute SHAP value for each feature, which reflects its overall importance in the model. These plots allow us to identify dominant predictors (e.g., $t_2$ for Δt prediction) and evaluate whether the model relies excessively on a single variable or distributes influence more evenly across features.

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Figure 3. Shapley Additive Explanations (SHAP) value analysis for (a) Δt prediction and (b) κ prediction

Figure 3(a) reveals that the $\Delta t$ prediction model relies almost exclusively on $t_2$ ( $>89 \%$ importance), with minimal contribution from $t_1$ or temperature. While this yields high accuracy for larger transit times, the model essentially learns a linear mapping of $t_2$ rather than the complex differential relationship required for low-flow precision. This overfitting to $t_2$ causes significant errors in the low-flow regime where $\Delta \mathrm{t}$. is small. Consequently, the pure $\Delta \mathrm{t}$ prediction model is deemed unsuitable for the final implementation. Instead, the proposed framework utilizes a more balanced $\kappa$ model (Figure 3(b)), which distributes importance more effectively across all inputs.

4.2.3 Actual vs. predicted values

Here we do a visual comparison of the actual and predicted values. In this plot, the horizontal axis represents the actual observed values from the dataset, while the vertical axis shows the corresponding values predicted by the model. A perfect model would produce predictions that fall exactly on the diagonal line (y = x), indicating that the predicted values match the actual values without error.

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Figure 4. Actual vs. predicted values for (a) Δt prediction and (b) κ prediction

Tight clustering of points along the diagonal line indicates strong agreement between the model’s predictions and the true values, with minimal error. The degree of scatter around the line reveals the residual variance. Figure 4 compares the actual and predicted outputs over 190 points. In Figure 4(a), the Δt prediction points cluster tightly along the diagonal, visualizing the high accuracy. In Figure 4(b), κ prediction shows slightly more spread, but remains closely aligned to the diagonal. For Δt prediction, points are densely concentrated along the diagonal, reflecting the model’s near-perfect fit (R² = 0.9999). For κ prediction, the points also align closely but show slightly more dispersion, consistent with its lower R² (0.914). With an RMSE value of 1.034, the model’s predictions are generally close to the actual values.

Although Figure 4 shows the overall trend, it is important to see the percent error to validate the results for the whole range. In Figure 5, we plot the percent error for both parameters. For κ, the results are mostly uniform and less than 10% (except for 2 points) error confirms that predicted values are a good approximation for the actual values, for a training set of 760 points. For Δt, on the other hand, we see a very good agreement for large values, while for small values of Δt, the error shoots up to 50%.

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Figure 5. % error vs. actual values for (a) Δt prediction and (b) κ prediction

4.3 Calculation of flow rate

The purpose of this study is to calculate the flow rate. With the predictions described in this section, the calculation of the flow rate using different methodologies is performed. The performance of the proposed ANN-based methodology for precise ToF estimation in ultrasonic flow metering was evaluated across a comprehensive spectrum of flow rates, ranging from 15 L/h to 4000 L/h.

First, we compare experimental measurements of the flow rate to flow rates determined by the conventional cross-correlation technique, a widely employed method in ultrasonic flow measurement. Table 2 presents a comparative analysis of the reference readings, the flow rates estimated by the cross-correlation technique, along with their corresponding percentage errors. DN-25 pipe is exposed to an assumed flow rate that moves the fluid through the pipe. We measured the actual flow rate using the total volume of the liquid displaced. Each assumed flow rate is repeated six times to arrive at the measured value. Since we use a single pipe, sequential flow rates are expected to behave in a similar way. The measured values are smoothed out using a power law interpolation. For completeness, we provide all the values in Table 2. The cross-correlation values for each data point are also given and based on the errors, the interpolation improves the accuracy of our measurements.

It is important to distinguish between the experimental reference values and the ANN-based predictions reported in Tables 2 and 3. The measured flow rates used as reference values were obtained by averaging six repeated experimental measurements at each nominal flow condition, ensuring stability and repeatability of the physical system. In contrast, the ANN-predicted values correspond to individual predictions generated on the test dataset, where each sample represents a single measurement consisting of upstream transit time (t₁), downstream transit time (t₂), and fluid temperature (T). The minimum reported error of 0.01% reflects the closest agreement between the ANN-predicted flow rate and the corresponding averaged experimental reference value for a specific test sample. Therefore, the reported errors represent point-wise prediction accuracy rather than statistical dispersion, and the consistently low errors across the dataset demonstrate the robustness and generalisation capability of the trained model.

In Table 3, we repeat a similar analysis using the proposed ANN-based method. In this method, we use equation 2 to calculate the velocity of fluid and calculate the flow rate accordingly. Although we have completed analyses using both Δt and κ, the inaccuracies at small values of Δt forced us to use sensor output rather than our findings. Therefore, for the calculation of the flow rate in Table 3, we have used κ value from ANN calculations, while actual Δt values from the sensor are used. All parameters used are given in Table 3. Since the data used in the prediction algorithm is not directly related to the experiment, we provide flow rate values close to the assumed flow rate for comparison with the cross-correlation method. Notably, the percentage errors associated with the ANN estimations are demonstrably lower than those obtained using the conventional cross-correlation technique at every measurement point. The error percentages for the cross-correlation method exhibit a significant variability, ranging from a minimum of 0.11% to a maximum of 5.70%, suggesting a sensitivity to specific flow regimes or signal characteristics that the ANN appears to mitigate. In contrast, ANN-based estimations yield a remarkably constrained error range, ranging from a mere 0.01% to 2.6%, underscoring the robustness and enhanced accuracy of the neural network approach in predicting flow rates from ultrasonic sensor data.

At the lower end of the flow rate range, the two methods behave equally well. For instance, at standard flow values of 15 L/h and 25, the cross-correlation technique yields error percentages of 0.76% and 0.94%, respectively. The ANN method has 0.92% and 0.63%. This enhanced sensitivity and accuracy at lower flow rates are of considerable importance in applications such as leak detection in water distribution systems, precise metering in low-flow chemical processes, and accurate fluid delivery in medical devices, where even small deviations from actual flow can have significant implications.

The enhanced accuracy afforded by the ANN is particularly evident at the higher end of the flow rate range. For instance, at standard flow values of 2200 L/h and 4000, the cross-correlation technique yields error percentages of 2.54% and 0.22%, respectively. The ANN method reduces these inaccuracies to 0.88% and 0.01%, representing a substantial improvement in the precision of the measurement under high flow conditions.

In the midrange, ANN shows a higher error range due to a small number of data points available at that range, causing incomplete prediction. With enough data points, ANN shows a much better overall result. This can be fundamentally attributed to its inherent capability to learn intricate and non-linear relationships. Unlike the cross-correlation technique, which relies on a direct linear relationship between transit time difference and flow velocity, the MLP can automatically learn complex, non-linear mappings between measured transit times, temperature, and the calibration factor, effectively compensating for flow-dependent nonlinearities and temperature effects.

Table 2. Experimental values of the flow rates

Table 3. Comparison of calculated and ANN-predicted calibration factor (κ) with corresponding flow rate estimations

This learning process enables the ANN to effectively compensate for a multitude of factors that can introduce errors in traditional ToF estimation methods, including variations in fluid properties (such as temperature, viscosity, and density), distortions in the flow profile (e.g., turbulent flow regimes), and the presence of noise and artifacts in the acquired ultrasonic signals.

The consistently low error percentages achieved by the ANN across the tested flow range highlight its potential for providing more reliable and accurate flow measurements in real-world industrial settings. This improvement in measurement accuracy can translate to significant benefits, including enhanced efficiency in industrial process control, more precise and fair billing in utility services, improved safety and control in critical fluid handling systems, and more accurate data for process optimization and monitoring.

While the results presented herein are promising, future research endeavors should focus on further validating the robustness and generalization capability of the trained ANN model under a broader spectrum of operating conditions. This includes evaluating its performance experimentally with different pipe sizes and materials, various fluid types exhibiting diverse acoustic properties, and in the presence of varying levels and types of noise interference. Investigating the sensitivity of the ANN to sensor variations and the potential for transfer learning across different ultrasonic sensor configurations would also be valuable steps toward practical industrial deployment. Additionally, exploring the interpretability of the learned features within the ANN could provide further insights into the underlying physical phenomena governing ultrasonic flow measurement and potentially lead to further improvements in the methodology.