Electrode type fluid electric field discharge heat exchange prediction method based on gaussian process regression

Abstract

The present application relates to heat generating device technical field, and further to an electrode type fluid electric field discharge heat exchange prediction method based on Gaussian process regression. The method comprises: step 1: synchronously collecting each parameter sequence on the electrode type fluid electric field discharge heat exchange device; generating a discharge heat exchange segment according to a discharge starting event slice, calculating a target heat exchange index, and establishing a sample set corresponding to the input feature set and the target heat exchange index; step 2: performing joint training of the variational input distortion mapping and the sparse Gaussian process on the sample set established in step 1 to obtain a joint modeling state of the variational input distortion mapping and the sparse Gaussian process model; step 3: obtaining the prediction value and the prediction interval of the target heat exchange index and outputting. The present application can fully solve the problems of uneven feature distribution, non-stationary time series features and high large-scale calculation complexity, realize high-precision prediction and uncertainty quantification, and improve the prediction stability.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of heat generating devices, and particularly relates to an electrode type fluid electric field discharge heat exchange prediction method based on Gaussian process regression. BACKGROUND

[0002] The electrode type fluid electric field discharge heat exchange technology is a process of generating a strong electric field in a fluid by a high-voltage electrode and triggering local discharge. The process can form a micro-scale discharge area with high energy density in a very short time, so that the fluid is rapidly heated and accompanied by heat exchange effect. This method is different from the traditional heating methods relying on heat conduction, convection or radiation heat transfer. It shows higher energy transfer efficiency in instantaneous high-power release and local strong turbulence. In recent years, with the growing demand for high-efficiency energy-saving heat exchange in industry, the electrode type fluid discharge heat exchange method has gradually attracted attention in research and application. In water treatment, industrial heating and new energy conversion systems, this kind of technology is tried to be applied to improve energy efficiency and rapid heat response performance.

[0003] In the prior art, the research on the electrode type fluid electric field discharge heat exchange process mainly focuses on experimental observation and physical modeling. In the aspect of experimental observation, researchers usually use high-speed voltage and current recorders, optical detectors and acoustic sensors to capture the transient information of the discharge process, and try to calculate the actual heat exchange effect by energy balance method. However, such experimental observation is often limited by sampling accuracy, measurement noise and sensor response speed, and it is difficult to obtain stable and reusable prediction results under complex working conditions. In the aspect of physical modeling, researchers try to construct analytical equations to describe the heat exchange process based on the coupling model of arc discharge, plasma heat transfer and fluid mechanics. However, due to the high nonlinearity and strong randomness of the discharge phenomenon, the analytical equation usually needs a lot of simplifying assumptions, such as homogenization of flow field or neglect of micro-scale disturbance, which leads to obvious difference between the prediction accuracy of the model and the actual measurement.

[0004] In existing research, some researchers have tried to use traditional statistical learning methods, such as multivariate linear regression, support vector machine or random forest, to establish a data-driven prediction relationship between input features and output indicators of discharge heat exchange. To some extent, this kind of method improves the fitting ability of the model to complex data, but it depends on fixed feature space and preset kernel function form, and has insufficient flexibility for nonlinear distribution. In addition, when facing the distribution drift and local sparse area of input features, these methods are prone to overfitting or unstable prediction, which limits their popularization and application in actual industrial environment. SUMMARY

[0005] The main objective of this invention is to provide a method for predicting heat transfer in an electrode-type fluid electric field discharge based on Gaussian process regression. This method involves simultaneously acquiring multi-source sequences and generating discharge heat transfer segments on an electrode-type fluid electric field discharge heat transfer device. A correspondence between the input feature set and the target heat transfer index is constructed. Joint training is performed using variational input distortion mapping combined with a sparse Gaussian process to obtain a reusable joint modeling state. During prediction, the sparse Gaussian process model is input with features processed by dimension-wise monotonically distorting and recalibration, outputting the predicted value and prediction interval of the target heat transfer index. This invention effectively addresses the problems of uneven feature distribution, non-stationary temporal features, and high computational complexity on a large scale, achieving high-precision prediction and uncertainty quantification. It offers benefits such as improved prediction stability, enhanced model generalization ability, reduced computational costs, and improved reliability for industrial applications.

[0006] To solve the above problems, the technical solution of the present invention is implemented as follows:

[0007] A method for predicting electric field discharge heat transfer in electrode-type fluids based on Gaussian process regression, comprising:

[0008] Step 1: Simultaneously acquire voltage, current, optical emission, acoustic emission, inlet temperature, outlet temperature, and volumetric flow rate sequences on the electrode-type fluid electric field discharge heat exchanger; generate discharge heat exchange segments according to the discharge initiation event slices, calculate the target heat exchange index, and establish a sample set that corresponds one-to-one with the input feature set and the target heat exchange index.

[0009] Step 2: Perform joint training of variational input distortion mapping and sparse Gaussian process on the sample set established in step 1 to obtain the joint finalized state of the variational input distortion mapping and sparse Gaussian process model;

[0010] Step 3: For the heat transfer segment to be predicted, generate the input feature set in the manner of Step 1; call the joint shaping state obtained in Step 2: first, perform monotonic distortion and recalibration on the input feature set dimension by dimension using variational input distortion mapping to form distortion features; input the distortion features into the sparse Gaussian process model to obtain the predicted value and prediction interval of the target heat transfer index and output it.

[0011] Furthermore, in step 1, the operation of simultaneously acquiring voltage, current, optical emission, acoustic emission, inlet temperature, outlet temperature, and volumetric flow rate sequences on the electrode-type fluid electric field discharge heat exchanger is specifically configured with a sampling frequency of 100,000 times per second. The continuous sequence is sliced ​​based on the following event detection rules: when the first-order difference of the current sequence shows 5 consecutive incremental sampling points within a 0.5-millisecond time window, and the optical emission sequence shows a peak value within the corresponding time window, this moment is defined as the start of discharge. Taking the defined discharge start time as zero, the process extends forward by 2 milliseconds and backward by 8 milliseconds to form a discharge heat exchange segment with a length of 10 milliseconds.

[0012] Furthermore, step 1 further includes processing the voltage and current sequences acquired in step 1 using zero-phase bandpass filtering to retain frequency components from 1000 Hz to 50000 Hz; calculating short-time energy and performing peak deduplication on the optical and acoustic emission sequences, with the deduplication interval set to 50 microseconds; performing a moving average on the temperature and volumetric flow rate sequences, with a window length set to 0.2 milliseconds; and obtaining the target heat transfer index for the current segment based on the measured values ​​of the inlet temperature, outlet temperature, and volumetric flow rate sequences within a 10-millisecond segment using a standard heat transfer calculation process, with the unit being energy per unit time.

[0013] Furthermore, the input feature set generated in step 1 includes: voltage peak value, current peak value, current rise time duration, optical peak count, acoustic peak count, time interval between optical peak and acoustic peak, average temperature difference within the segment, average volumetric flow rate, and peak position time coordinates that are cross-correlated with voltage and current within the segment; this input feature set is then matched one-to-one with the target heat transfer index to form the sample set of the current sampling batch.

[0014] Furthermore, the variational input distortion mapping in step 2 includes feature-level distortion mapping and temporal-level distortion mapping. The construction process of feature-level distortion mapping includes: performing quantile normalization on each dimension of the input feature set to map the values ​​to the interval between 0 and 1; constructing a monotonic piecewise cubic spline for each dimension in the interval between 0 and 1, with the number of internal nodes set to 12. The horizontal coordinates of the nodes are placed according to the quantile points, and the vertical coordinates of the nodes are initialized according to the rule that the starting point is equal to 0, the ending point is equal to 1, and the vertical coordinates of the intermediate nodes are generated by positive incremental accumulation, forming a strictly monotonic initial spline; listing the vertical coordinate of each dimension of the spline as an optimizable control variable, automatically adjusting the vertical coordinate by improving the variational optimization objective, and ensuring the strict monotonicity of the spline by applying a positive domain mapping to the difference of the vertical coordinates of adjacent nodes and then performing forward accumulation.

[0015] Furthermore, the construction process of the time-series-level warp mapping includes: extracting a set of time axis anchor point sequences within each discharge heat transfer segment, which includes the current rising edge start point, optical peak apex, acoustic peak apex, temperature difference peak apex, and segment end point; dividing the 10-millisecond time axis into 5 segments based on the extracted anchor point sequences, and establishing an affine scaling mapping on each segment; using the scaling ratio and start-end alignment offset of each segment as optimizable control variables, and incorporating them together with the control variables of feature-level warp into the same variational optimization objective; reparameterizing the time coordinates of the entire segment using the time-series-level warp mapping to obtain the warped time axis, and then resampling all subsequent time-related features on the generated warped time axis.

[0016] Furthermore, after the variational input distortion mapping process, the maximum and minimum distance point selection is performed on the distorted input feature space: first, the sample with the largest Euclidean norm is selected as the first induced point; then, the sample is iteratively scanned, and in each round, the minimum distance from each candidate sample to the set of selected induced points is calculated, and the sample with the maximum and minimum distance is added to the set of induced points; the process terminates when the coverage radius is less than 0.15 or the number of induced points reaches 256; the coverage radius is calculated based on the standardized Euclidean distance metric, where the standardization coefficient is the scale obtained when performing quantile normalization in step 1.

[0017] Furthermore, in step 2, the sparse Gaussian process model uses a half-integer stationary kernel and an anisotropic length scale; a Gaussian approximate posterior is established for the function values ​​on the induced point set, described by multidimensional mean and multidimensional variance; the hyperparameters of the kernel, the induced point posterior parameters, and all control quantities of the variational input distortion mapping used in step 2, which includes feature-level distortion mapping and temporal-level distortion mapping, are incorporated into a single variational optimization objective.

[0018] Furthermore, in step 2, a mini-batch learning process is adopted, and each round of optimization strictly follows the following three stages in sequence: In the first stage, the posterior parameters of the induced points are updated using the natural gradient method; in the second stage, the hyperparameters of the kernel are updated using the quasi-Newton method; in the third stage, all control variables of the variational input distortion map, which includes feature-level distortion maps and temporal-level distortion maps, used in step 2 are updated using adaptive moment estimation. After each round of optimization, the improvement of the variational optimization objective is calculated. If the improvement is less than 1%, the learning step size of the third stage is reduced to 0.5 of the original and the next round continues. Training stops when the improvement is less than 1% for 10 consecutive rounds or the total number of rounds reaches 500. The training result is the joint finalized state of the variational input distortion map and the sparse Gaussian process model, in which the set of induced points, the hyperparameters of the kernel, the posterior parameters of the induced points, and the control variables of the variational input distortion map are stored together for inference.

[0019] The electrode-type fluid electric field discharge heat transfer prediction method based on Gaussian process regression of the present invention has the following beneficial effects:

[0020] First, by simultaneously acquiring voltage, current, optical emission, acoustic emission, inlet temperature, outlet temperature, and volumetric flow rate sequences on an electrode-type fluid electric field discharge heat exchange device, and generating discharge heat exchange segments according to the discharge initiation event slices, this invention can capture complete dynamic features within a very short time range, achieving refined characterization of the discharge process.

[0021] Secondly, based on establishing the correspondence between the input feature set and the target heat transfer index, this invention employs variational input distortion mapping to perform monotonically distorting and recalibrating the input features dimension by dimension. Simultaneously, it combines temporal-level distortion mapping to resample key anchor points within segments, effectively solving the problems of uneven input feature distribution and non-stationarity of temporal features, enabling the model to maintain stable expressive power under different operating conditions. Building upon this, this invention establishes a prediction model through a sparse Gaussian process, selects the induced point set using a maximum-minimum distance point selection strategy, and, under a joint shaping state, incorporates the kernel hyperparameter, induced point posterior parameters, and distortion mapping control quantity into the optimization objective, thereby significantly reducing computational complexity and ensuring the model's scalability and real-time performance under large-scale data.

[0022] Ultimately, during the prediction process, by loading the joint morphological state and transforming the input feature set into a distorted feature input sparse Gaussian process model, this invention can not only output the predicted value of the target heat transfer index, but also provide the prediction interval, thus providing reliable uncertainty quantification for system operation.

[0023] In summary, this invention has beneficial effects in improving prediction accuracy, enhancing model generalization ability, improving computational efficiency, and providing interpretability of results, and can better meet the requirements of high reliability and high real-time performance for electrode-type fluid electric field discharge heat transfer processes in industrial applications. Attached Figure Description

[0024] Figure 1 A schematic diagram of the method flow for predicting electric field discharge heat transfer of electrode fluid based on Gaussian process regression, provided in an embodiment of the present invention;

[0025] Figure 2 This is a schematic diagram of the feature-level variational input distortion mapping principle provided in an embodiment of the present invention;

[0026] Figure 3 This is a schematic diagram comparing the feature space before and after distortion, provided for an embodiment of the present invention. Detailed Implementation

[0027] Example 1:

[0028] The device synchronously acquires voltage, current, optical emission, acoustic emission, inlet temperature, outlet temperature, and volumetric flow rate sequences at a frequency of 100,000 times per second. Event detection is performed on the continuous sequences. If five consecutive incremental sampling points appear in the first-order difference of the current sequence within a 0.5-millisecond window, and the optical emission sequence simultaneously shows a peak within the same window, the discharge start time is detected and recorded as time 0. A discharge heat transfer segment of 10 milliseconds is constructed by extending forward 2 milliseconds and backward 8 milliseconds from time 0 as the zero point.

[0029] After zero-phase bandpass filtering and moving average processing, the input feature set of nine items was extracted from the signal within the segment: the voltage peak value was taken as the maximum value of the voltage sequence within the segment, resulting in a voltage peak value of 1100. The current peak value was taken as the maximum value of the current sequence within the segment, resulting in a current peak value of 75. The current rise time was calculated based on the time span between the 10% and 90% thresholds, resulting in a current rise time of 0.12 milliseconds. The optical peak count was calculated by removing duplicates over 50 microseconds, resulting in an optical peak count of 3. The acoustic peak count was calculated using the same method, resulting in an acoustic peak count of 2. The time interval between the optical and acoustic peaks was calculated between the main peak vertices, resulting in a time interval of 0.08 milliseconds. The average temperature difference within the segment was calculated by averaging the difference between the inlet and outlet temperature sequences within the segment, resulting in an average temperature difference of 4.8. The average volumetric flow rate was calculated by averaging the volumetric flow rate sequence within the segment, resulting in an average volumetric flow rate of 0.0012. The peak time coordinates of the voltage and current cross-correlation within the segment were obtained by using the peak position of the similarity metric, resulting in a peak time coordinate of 0.01 milliseconds.

[0030] The target heat transfer parameters are calculated using a heat balance heat transfer process. Let the mass flow rate be denoted by [symbol missing]. Specific heat capacity at constant pressure is indicated by the symbol. The temperature difference within the segment is symbolic. Volumetric flow rate is a symbol Fluid density is symbol The relationship between mass flow rate, volumetric flow rate, and density is as follows: ;in Indicates mass flow rate. Indicates fluid density, This represents the volumetric flow rate. The target heat transfer parameter is indicated by the symbol... ,have ,in Indicates the target heat exchange index, This represents the specific heat capacity at constant pressure. This represents the temperature difference within the segment. Using water as the medium, [the following is taken]: Substitute have to This value is used to compare with the predicted value.

[0031] The variational input warp mapping maps the nine entries to intervals according to the quantile rules of the training phase. The normalized value is obtained. The symbols Indicates the first The normalized input of the dimension. For this fragment, give... Each dimension contains Cubic polynomial interpolation is used within the spline segments. Let the locally normalized coordinates be of sign . The interpolation coefficients are of the sign of The distorted output is a symbol. ,according to ,in express In the local coordinates of the segment, This represents the coefficient of the segment. Indicates the first The distortion characteristics of the dimension. The joint fixed state stores coefficients for each dimension and each segment. The calculation of the segments and coefficients that this segment falls into is given as follows (only the actual hit segments and coefficients are listed for each dimension): 1st dimension interval Local coordinates ,coefficient ,calculate The second dimension interval , The third dimension interval , The fourth dimension interval , The 5th dimension interval , The 6th dimension interval , The 7th dimension interval , 8th dimension interval , The 9th dimension interval , This yields the distorted feature vector. .

[0032] A half-integer-order stationary kernel is employed, and an anisotropic length scale is set. Let the input be a symbolic string. , representing the distortion feature; the set of induced points is a symbol , represents the position; kernel function is a symbol , representing similarity; signal variance is a sign-valued expression. , representing the magnitude of the kernel function at zero distance; the anisotropic length scale is denoted by . , representing the scale parameter for each dimension. A half-integer order is used. The kernel function has ;in This represents the distance weighted by the anisotropic length scale. Indicates another input. Take Take the length scale .

[0033] The number of induced points set is taken The representative storage location is:

[0034] ;

[0035] ;

[0036] .

[0037] The above construction makes the 4th to 9th dimensions and Since the distance is the same, it is mainly contributed by the first three dimensions, which facilitates the explanation of the calculation process. Calculate the distance and kernel value between the input and the induced point: ; Obtained by the same method Let the induced point kernel vector be denoted by the symbol . The kernel matrix between induced points is symbolic. Its diagonal element is The off-diagonal elements are calculated using the kernel function described above, resulting in... The posterior mean vector of the induced variable is denoted as symbolic. , representing the mean of the function values ​​at the induced location; the posterior covariance matrix is ​​denoted by the symbol . , representing the uncertainty of the induced variable. The joint type state gives... The predicted values ​​and predicted variances are expressed in a closed-form sparse variational approximation. The zero-distance kernel value is denoted as symbol […]. Let the inverse of the kernel matrix be denoted by the symbol . The expression for the predicted value is: ,in This represents the predicted value of the target heat transfer index. The expression for the prediction variance is: ,in This represents the prediction variance. (Number substitution) Calculated Using interval coverage probability Given the prediction interval, let the standard normal quantile coefficient be denoted as . ,Pick ,have .

[0038] The predicted value for this segment is 18893.80, and the prediction interval is [missing value]. The control value calculated based on the heat exchange balance is 24192. This control value is higher than the upper limit of the prediction interval, indicating that the current input is in the edge region covered by the induction point, or that there is a strong instantaneous increase in the combination of volumetric flow rate and temperature difference. This difference suggests adopting a conservative energy allocation strategy in process control; the uncertainty in the edge region can be reduced by adding new fragment samples and retraining within subsequent maintenance windows to update the joint shaping state.

[0039] Example 2:

[0040] A method for predicting electric field discharge heat transfer in electrode-type fluids based on Gaussian process regression, comprising:

[0041] Step 1: Simultaneously acquire voltage, current, optical emission, acoustic emission, inlet temperature, outlet temperature, and volumetric flow rate sequences on the electrode-type fluid electric field discharge heat exchanger; generate discharge heat exchange segments according to the discharge initiation event slices, calculate the target heat exchange index, and establish a sample set that corresponds one-to-one with the input feature set and the target heat exchange index.

[0042] First, voltage sensors, current sensors, optical emission sensors, acoustic emission sensors, inlet temperature sensors, outlet temperature sensors, and volumetric flow rate sensors are configured on the electrode-type fluid electric field discharge heat exchanger. All sensors are driven by the same time reference, with a sampling frequency set to 100,000 times per second. The time reference is provided by an integrated clock generator, and a unified timestamp is recorded for each sample to ensure strict alignment of the voltage, current, optical emission, acoustic emission, inlet temperature, outlet temperature, and volumetric flow rate sequences on the time axis. To ensure complete acquisition of high-frequency information, the analog signal is limited to less than half the sampling frequency by an anti-aliasing filter before entering the analog-to-digital conversion link, and an amplitude calibration procedure is used to ensure that the range of each channel covers the full amplitude variation range of the discharge process. This setup preserves rapid changes related to the discharge in subsequent feature extraction while maintaining time consistency between channels and avoiding cross-channel offsets caused by time drift.

[0043] During continuous acquisition, event detection is performed on the current sequence. First, adjacent sample points are differentiated on the current sequence to form a discrete change sequence representing an upward trend. A sliding window of 0.5 milliseconds is advanced along time. When five consecutive increasing sample points appear within any window, the upward trend of the current is determined to meet the trigger condition. Simultaneously, local peak vertices are detected in the optical emission sequence aligned with this window. The determination of local peak vertices combines short-time energy and neighborhood comparison: first, short-time energy is calculated with an analysis step size of 50 microseconds; then, the position with the maximum energy within three adjacent analysis step sizes is selected as a peak vertices candidate. A threshold is determined by static baseline plus dynamic noise estimation; candidates exceeding the threshold are confirmed as peak vertices. When both of the above conditions are met simultaneously, the corresponding time point is defined as the discharge start time. Taking the discharge start time as zero, a signal segment of 10 milliseconds is extracted by extending forward by 2 milliseconds and backward by 8 milliseconds to form the discharge heat transfer segment. This length covers the key stages of discharge formation, development, and initial decay, which is beneficial for observing the sequential relationship between electrical excitation, radiation response, acoustic disturbance, and heat transfer within a single segment, thereby improving the correspondence between subsequent input characteristics and target heat transfer indicators.

[0044] Preprocessing is performed on the data within the discharge heat transfer segment to improve the stability of the input features. Zero-phase bandpass filtering is applied to the voltage and current sequences, retaining frequency components from 1000 Hz to 50000 Hz. Zero-phase processing achieves waveform phase preservation through symmetrical forward and backward filtering, ensuring that the peak time position and rising edge shape do not shift, facilitating accurate extraction of the peak time coordinates correlated with the current rising edge duration and voltage-current cross-correlation. After calculating the short-time energy of the optical and acoustic emission sequences, peak deduplication is performed at a 50-microsecond interval to avoid repeated counting of high-frequency, fragmented fluctuations corresponding to a single discharge. The inlet temperature, outlet temperature, and volumetric flow rate sequences are averaged with a window length of 0.2 milliseconds to suppress instantaneous measurement jitter in temperature and volumetric flow rate, making the mean within the segment closer to the true steady-state change trend. These processing steps are designed around temporal positioning accuracy and statistical stability, enabling the preservation of the discharge microstructure characteristics while reducing the sensitivity of subsequent feature calculations to noise.

[0045] The target heat transfer index is expressed in units of energy per unit time, representing the heat transfer capacity corresponding to a discharge heat transfer segment. The calculation process follows the method of heat balance heat transfer. The temperature difference within the segment is calculated using the inlet and outlet temperature sequences of the discharge heat transfer segment, and the average volumetric flow rate within the segment is calculated using the volumetric flow rate sequence. Based on the density and specific heat capacity at constant pressure of the heated medium, the corresponding values ​​are read from the medium property table, and the average volumetric flow rate is converted into a mass flow rate. The mass flow rate is multiplied by the specific heat capacity at constant pressure and the temperature difference within the segment to obtain the target heat transfer index. In the example using water as the medium, the specific heat capacity at constant pressure can be taken as 4200 joules per kilogram per Kelvin, and the density as 1000 kilograms per cubic meter. If the average volumetric flow rate within the segment is 0.001 cubic meters per second and the temperature difference within the segment is 5 Kelvin, then the target heat transfer index is calculated to be 21000 watts. This definition directly reflects the scale of heat transfer per unit time, making it suitable for establishing a causal relationship with electrical excitation and radiative response, thereby serving subsequent model training and prediction.

[0046] The input feature set is extracted from the discharge heat transfer segment using uniform rules and includes the following entries: **Voltage Peak:** The maximum amplitude within the segment is taken from the voltage sequence after zero-phase bandpass. This feature reflects the electric field excitation intensity. **Current Peak:** The maximum amplitude within the segment is taken from the current sequence after zero-phase bandpass. This feature characterizes the conduction degree of the discharge channel. **Current Rise Time:** Two thresholds, 10% and 90% of the amplitude corresponding to the peak value, are determined on the current sequence, and the time interval between the current rising from the 10% threshold to the 90% threshold is recorded. This feature is related to the discharge formation rate and can distinguish between slow and fast breakdown. **Optical Peak Count:** The number of peak vertices is counted in the optical emission sequence after peak deduplication. This feature reflects the discreteness and activity of radiation events. **Acoustic Peak Count:** The number of peak vertices is counted in the acoustic emission sequence after peak deduplication. This feature corresponds to acoustic disturbances caused by bubble bursts, microscale discharge bursts, etc. **Time Interval Between Optical and Acoustic Peaks:** The main peak vertices of the optical and acoustic emission sequences are found within the segment, and the absolute value of the difference between their time positions is calculated. Selecting the primary peak apex can pinpoint the temporal sequence of radiation and pressure disturbances caused by the discharge. Smaller values ​​indicate more synchronized radiation and pressure events, typically corresponding to a more compact energy release process. Average temperature difference within a segment: This is the average difference between the inlet and outlet temperature sequences within the segment. This feature reflects the average level of thermal driving force. Average volumetric flow rate: This is the average volumetric flow rate sequence within the segment. This feature determines the fluid flux entering the heat exchange zone per unit time. Peak time coordinates of voltage-current cross-correlation within a segment: Using a sliding relative displacement method, a similarity metric is calculated between the voltage and current sequences. The position where the similarity metric reaches its peak is identified, and the relative time corresponding to this position is recorded as the peak time coordinate. This feature quantifies the phase relationship between voltage and current. A peak time coordinate close to zero indicates greater synchronization, typically corresponding to stable discharge; a peak time coordinate deviating from zero indicates a significant delay or lead, commonly seen during discharge initiation or extinction phases.

[0047] For each discharge heat transfer segment, a one-to-one correspondence is established between the input feature set and the corresponding segment's target heat transfer index, forming a sample record. Each sample record includes a segment timestamp, the specific values ​​of the nine entries in the input feature set, and the value of the target heat transfer index. All sample records are written into the sample set in chronological order. The sample set is used for subsequent joint training of variational input distortion mapping and sparse Gaussian process. To ensure the integrity of the sample set, a consistency check is performed before writing: verifying that the segment length is 10 milliseconds, all nine feature entries have been generated, the target heat transfer index has been calculated according to thermal balance, and adding records of sampling frequency and filter configuration to facilitate offline reproduction of experimental conditions.

[0048] In another implementation, without changing the sampling frequency, slice length, and event detection rules, an adaptive baseline update procedure can be added to the optical and acoustic emission sequences. Specifically, noise energy levels are continuously estimated within a discharge-free window, and the threshold is set to a fixed multiple of the noise energy level, thereby maintaining a stable peak detection rate despite changes in ambient light or background noise. Without changing the definitions of the nine feature entries, a robust calculation strategy for the average temperature difference and average volumetric flow rate within a segment can be added. Specifically, a truncated average within the percentile range is used to reduce the impact of extreme jitter on the mean, thereby maintaining a stable correspondence between the target heat transfer index and the input feature set under conditions of instantaneous disturbances in heat measurement. Without changing the definition of the peak time coordinate of voltage-current cross-correlation, amplitude normalization can be performed on the voltage and current sequences before similarity measurement calculation, making the similarity measurement focus more on waveform shape and time alignment, reducing the bias caused by amplitude differences.

[0049] Step 2: Perform joint training of variational input distortion mapping and sparse Gaussian process on the sample set established in step 1 to obtain the joint finalized state of the variational input distortion mapping and sparse Gaussian process model.

[0050] During implementation, the sample set obtained in step 1 is divided into training batches and validation batches according to time sequence. Each training batch contains continuous discharge heat transfer segment sample records, with the number of samples in a single batch set to 512 to 2048 to balance convergence speed and statistical stability. Each sample record contains the values ​​of the nine entries in the input feature set and the value of the target heat transfer index. Mini-batch organization allows the variational optimization objective to reflect the statistical state of recent segments in each round of calculation, enabling the training process to rapidly absorb new information while maintaining numerical stability. Quantile normalization is performed on each of the nine entries in the input feature set dimension by dimension, mapping the value of each dimension to the interval between 0 and 1. Based on this interval, a monotonic piecewise cubic spline is constructed for each dimension, with the number of internal nodes set to 12, and the x-coordinates of the nodes are placed at equal quantile points. The y-coordinates of the nodes are initialized by setting the starting point to 0, the ending point to 1, and accumulating the y-coordinates of intermediate nodes with positive increments. The difference between the y-coordinates of adjacent nodes is accumulated forward after positive domain mapping, thus maintaining strict monotonicity throughout the training period. Monotone piecewise cubic splines perform learnable nonlinear recalibration on each feature dimension, compressing the original dimensional differences and distribution skew to a uniform scale, making the subsequent induced point set coverage and kernel function measurement more consistent with the variation pattern within the segment. The reason for choosing monotone mapping is to maintain the order relationship of features and facilitate the interpretation of the consistency between the direction of input and output changes.

[0051] Feature-level distortion mapping performs learnable monotonic recalibration on each of the nine entries in the input feature set, forming a representation with a uniform scale. Initialization and node placement: Quantile normalization is performed on each dimension, mapping values ​​to the interval 0 to 1. A monotonic piecewise cubic spline is constructed for this dimension on this interval, with 12 internal nodes. The x-coordinates of the nodes are placed at equal quantiles. The y-coordinates of the nodes are initialized by setting the starting point to 0, the ending point to 1, and accumulating the y-coordinates of intermediate nodes with positive increments. Monotonicity constraints and updates: The differences in the y-coordinates of adjacent nodes are positively valued and then forward-accumulated to ensure strict monotonicity during training updates. Only the y-coordinates of the nodes are adjusted during training iterations, without changing the x-coordinate positions, thus stabilizing the ordering relationship of each dimension. Design motivation and effects: Monotonic piecewise cubic splines can redistribute interval density without changing the numerical order. In actual data of discharge heat transfer segments, voltage and current peak values ​​typically exhibit a spike-like distribution, while the current rise time duration and cross-correlation peak time coordinates often show a long-tailed distribution. By using monotonic recalibration, concentrated regions are stretched and sparse regions are compressed, making the similarity measurement focus more on the intervals most sensitive to feature changes. For example, when the current rise time is short, small changes have a greater impact on the target heat transfer parameters. Monotonic recalibration will enhance the resolution within this range, thereby improving the effectiveness and interpretability of subsequent approximations.

[0052] Construction and Training of the Temporal-Level Twisted Map: The temporal-level twisted map performs piecewise affine scaling on the time coordinates of the discharge heat transfer segment, enabling higher temporal resolution for critical stages. Within each discharge heat transfer segment, five time axis anchor points are extracted: the current rise point, optical peak, acoustic peak, temperature difference peak, and segment end. The 10-millisecond time axis is divided into five segments. A scaling ratio and start-end alignment offset are set as learnable control variables for each segment, ensuring linear mapping within segments, continuity between segments at anchor points, and global strict increasing. The original time coordinates are converted into a twisted time axis via this mapping, and temporally relevant features are resampled on the twisted time axis. Discharge formation and energy release often concentrate in a very short time range after the discharge begins. Piecewise affine scaling allocates more of the limited time budget to this important segment, preserving the fine-grained sequential relationship between electrical excitation, radiation response, and acoustic disturbance. After this processing, the changes in the input feature set during critical stages are more clearly expressed, and the reasons for changes in the target heat transfer parameters are more easily captured by the approximate model.

[0053] After completing feature-level and temporal-level distortion mapping, a maximum-minimum distance selection process is performed in the distorted input feature space to construct an induced point set. The standardized Euclidean norm of each sample is calculated, and the sample with the largest norm is selected as the first induced point. The standardization coefficients are scaled using quantile normalization. In each round of scanning, unselected samples are scanned, and their minimum distance to the selected induced point set is calculated. The sample with the largest minimum distance is added to the induced point set. The process stops when the coverage radius is less than 0.15 or when the number of induced points reaches 256. The coverage radius is measured using standardized Euclidean distance to indicate the farthest uncovered area in the input space. The maximum-minimum distance strategy creates uniform coverage in the distorted space, approximating a larger input region with fewer representative points, reducing redundant coverage in dense regions, and enhancing representativeness in sparse regions, thereby improving the overall quality of the sparse approximation.

[0054] A Gaussian approximate posterior is established for the function values ​​on the induced point set, described by multidimensional mean and multidimensional variance. A half-integer stationary kernel is chosen as the kernel function, employing an anisotropic length scale. The reason for choosing a half-integer stationary kernel is that it exhibits strong discriminative power over short distances while maintaining overall smoothness, making it suitable for describing local structures with short current rise times and rapid radiation responses, and maintaining a stationary approximation during the slow changes in the later stages of discharge. The role of the anisotropic length scale is that different entries in the input feature set differ significantly in physical dimensions and range of variation; for example, the relative variation amplitude of voltage peaks and the relative variation amplitude of cross-correlation peak time coordinates are not on the same order of magnitude. Assigning an independent length scale to each dimension makes the sensitivity of the similarity measure to different dimensions more closely match its contribution to the target heat transfer index, thereby improving the sufficiency of the fit and the stability of the prediction.

[0055] The posterior parameters of the induced points, the hyperparameters of the kernel, the node ordinates of the feature-level distortion map, and the scaling ratio and start-end alignment offset of the temporal-level distortion map are all incorporated into a single variational optimization objective. This objective simultaneously considers data fitting terms and complexity constraints, ensuring that training both conforms to the statistical regularities of the sample set and suppresses overfitting.

[0056] Mini-batch learning process: Each training round executes three phases sequentially. Phase 1 uses the natural gradient method to update the posterior parameters of the induced points, enabling fast and stable convergence in directions with geometrically consistent information. Phase 2 uses a quasi-Newton method to update the hyperparameters of the kernel, improving the quality of the step direction using curvature information without significantly increasing computational burden. Phase 3 uses adaptive moment estimation to update the control variables of the feature-level and temporal-level distortion maps, allowing different control variables to obtain appropriate learning step sizes based on historical gradient magnitudes. Step size adjustment and stopping conditions: After each training round, the improvement of the variational optimization objective is calculated. When the improvement is less than 1%, the learning step size in Phase 3 is adjusted to half of the original value, and the next round continues. Training stops when the improvement is less than 1% for 10 consecutive rounds or when the total number of rounds reaches 500. This rule strikes a balance between numerical stability and training time, ensuring the repeatability of the finalized state. Validation strategy: After each round, the predicted values ​​and prediction intervals are evaluated using a validation batch, and the residual distribution is compared with the target heat transfer index. If the residual is significantly larger in a certain interval, the probability of samples appearing in that interval should be increased in subsequent small-batch organization, so that training can cover edge conditions more quickly.

[0057] After training, a joint shaping state is obtained. This joint shaping state includes the node x and y coordinates of the feature-level distortion map, the scaling ratio and start-end alignment offset of the time-series distortion map, the sample index of the induced point set, the posterior parameters of the induced points, the hyperparameters and anisotropic length scale of the kernel, the final variational optimization objective value, the training stopping epoch, and the coverage radius. The joint shaping state is written to a storage medium as a structured record for generating the prediction output in step 3. When a new discharge heat transfer segment enters the inference process, nine entries are first recalibrated according to the feature-level distortion map, then the time-series related features are resampled according to the time-series distortion map. The distorted input is then fed into a sparse Gaussian process to obtain the predicted value and prediction interval of the target heat transfer index.

[0058] Feature-level distortion mapping maintains monotonicity during node ordinate updates through positive conversion and forward accumulation, preventing reverse bending in later training stages. Temporal-level distortion mapping enforces continuity at anchor points and maintains a total distortion time axis length of 10 milliseconds through a unified total duration constraint, ensuring matching with segment length. The maximum-minimum distance strategy ensures coverage in the distortion space better aligns with the recalibrated distribution, making the similarity assessment of the half-integer-order stationary kernel consistent with actual differences, improving the approximation accuracy in boundary and sparse regions. Feature-level distortion mapping preserves order relationships, temporal-level distortion mapping highlights key stages, and the induced point set covers representative positions in the input space. Together, these three elements clearly define the correspondence between prediction results and input changes, facilitating traceability and optimization on the device side.

[0059] In another implementation, while keeping the number of entries unchanged, the number of internal nodes in the feature-level distortion map is set to 16 to accommodate more complex input distributions. Increasing the number of nodes refines the shape of the recalibrated curve and enhances resolution in multi-peak regions. While maintaining the anchor point set at five, the temperature difference peak is replaced with the moment of abrupt change in volumetric flow rate. When the instantaneous change in volumetric flow rate has a stronger impact on the target heat transfer index, this replacement can more accurately segment time periods, and the scaling ratio better matches the energy transport rhythm. Based on a coverage radius threshold of 0.15 for the induced point set, the threshold is set to 0.10 for data with significant cluster structure to improve the resolution of the sparse approximation at cluster boundaries. While maintaining the three-stage sequence in the alternating update process, a joint refinement round is inserted every five rounds, using a small step size to simultaneously fine-tune the posterior parameters of the induced points, the hyperparameters of the kernel, and the control variables of the distortion map, reducing slow oscillations caused by asynchronous updates of different variables.

[0060] Step 3: For the heat transfer segment to be predicted, generate the input feature set in the manner of Step 1; call the joint shaping state obtained in Step 2: first, perform monotonic distortion and recalibration on the input feature set dimension by dimension using variational input distortion mapping to form distortion features; input the distortion features into the sparse Gaussian process model to obtain the predicted value and prediction interval of the target heat transfer index and output it.

[0061] During implementation, the discharge heat transfer segment to be predicted is subjected to synchronous acquisition, event detection, slicing, preprocessing, and feature extraction as described in step 1, forming an input feature set. The input feature set includes nine items: voltage peak value, current peak value, current rise time duration, optical peak count, acoustic peak count, time interval between optical and acoustic peaks, average temperature difference within the segment, average volumetric flow rate, and peak position time coordinates that are cross-correlated with voltage and current within the segment.

[0062] Peak value and rising edge determination: The voltage and current sequences after zero-phase bandpass filtering are determined by taking the maximum amplitude within the segment as the voltage peak and current peak, respectively. The current rising edge duration is calculated by crossing the thresholds at 10% and 90%. Zero-phase processing is used to ensure that the peak position and rising edge position do not shift in time, allowing comparisons of different segments and subsequent recalibration to be performed under a unified time reference. Optical and acoustic event counting: The number of peak vertices is counted at 50-microsecond deduplication intervals to obtain optical peak counts and acoustic peak counts. Deduplication merges the fragmented fluctuations generated by the same energy release into a single event, preventing excessive amplification of the proportion of high-frequency jitter in the statistics. Time interval between optical and acoustic peaks: The main peak vertices of the optical and acoustic emission sequences are selected within the segment, prioritizing peak vertices that simultaneously satisfy sufficient peak height and clear neighborhood contrast. When multiple candidates exist, the pair closest to the energy centers of the optical and acoustic peak vertices is selected. This selection can stably reflect the temporal relationship between radiation and pressure disturbances, avoiding random biases introduced by weak secondary peaks. Average temperature difference and average volumetric flow rate within a segment: The average temperature difference and average volumetric flow rate within a segment are obtained by averaging the results using a 0.2 ms moving average. Using a moving average suppresses instantaneous measurement jitter, making the average values ​​more closely reflect the slow changes in the actual heat transfer process. Cross-correlation peak time coordinates: A similarity metric is calculated between the voltage and current sequences, and the position where the similarity metric reaches its peak is recorded as the peak time coordinate. When multiple local peaks exist, the peak with the larger amplitude and closest to zero relative displacement is selected. This selection reflects the synchronization between excitation and response, providing more stable identification of the discharge formation stage.

[0063] The joint shaping state is loaded from the storage medium. This joint shaping state includes the node x and y coordinates of the feature-level warp mapping, the scaling ratio and start-end alignment offset of the temporal-level warp mapping, the sample index of the induced point set, the posterior parameters of the induced points, the hyperparameters and anisotropic length scale of the kernel, the final variational optimization objective value, the training stopping epoch, and the coverage radius. The consistency between the preprocessing configuration, sampling frequency, fragment length, and the joint shaping state record is verified. If inconsistencies exist, configuration alignment is performed before proceeding to the prediction process to ensure that the warp mapping and induced point set operate within the same metric space as in the training phase. Each of the nine entries is checked to ensure it falls within the quantile range of the training phase. A small number of out-of-bounds values ​​are truncated to maintain recalibration stability and avoid mapping distortion caused by extrapolation.

[0064] The nine entries in the input feature set are subjected to monotonic distortion and recalibration dimension-by-dimensional according to the feature-level distortion mapping in the joint shaping state, forming distorted features. The application of monotonic piecewise cubic splines: For each dimension, piecewise interpolation is performed based on the node's x-coordinate and y-coordinate to obtain the recalibrated values. The node's y-coordinate undergoes positiveization and forward accumulation during training to ensure the mapping is strictly increasing throughout the interval, thus maintaining consistency with the original size order and sensitive intervals. The role of recalibration: Mapping entries with different dimensions and distribution patterns to a unified scale, making the distorted features comparable across dimensions. For entries with a peak distribution, recalibration stretches the representation range near the peak, enhancing the discriminative power of small changes; for entries with a long-tail distribution, recalibration compresses the representation range at the tail, reducing the excessive influence of outliers on similarity metrics. Processing of time-related entries: For entries derived from time relationships, such as the time interval between optical and acoustic peaks and the time coordinates of cross-correlated peak positions, the shape is recalibrated using the distribution determined during the training phase, giving the distortion features higher resolution near key time relationships. This processing can more sensitively reflect the synchronicity of radiation and pressure perturbations and the alignment of excitation and response, thereby improving the consistent correspondence between predicted values ​​and target heat transfer parameters.

[0065] The distortion features are input into a sparse Gaussian process model, which outputs predicted values ​​and prediction intervals for the target heat transfer parameters. Similarity assessment: The sparse Gaussian process model, based on a half-integer stationary kernel and an anisotropic length scale, performs similarity assessment of distortion features on the induced point set. The anisotropic length scale makes the influence range of different items in the similarity assessment independent, providing appropriate sensitivity to differences in voltage peak values, current rise time duration, and cross-correlation peak time coordinates. Representative approximation: The induced point set is formed by selecting points with maximum and minimum distances, covering representative positions in the distortion space. During inference, the local structure of the input space is approximated using these representative positions, preserving key changes while controlling computational load. This approximation is validated during training by convergence of the target value in the joint shaped state and is directly reused in the inference stage. Prediction interval generation: The sparse Gaussian process model provides an uncertainty range along with the predicted values, presented as an interval. The interval width reflects the local sparsity of the input in the distortion space and the coverage of induced points. The interval is narrower near dense areas of the induced point set and wider near coverage boundaries. This range helps in selecting a prudent energy allocation strategy in process control, such as prioritizing conservative heating plans to avoid overshoot when the range is wide.

[0066] Dimensionally progressive monotonic distortion and recalibration redistribute interval density without altering the order of entry sizes, resulting in higher resolution for sensitive intervals. For discharge heat transfer segments, a slight reduction in the current rise time often leads to a significant increase in energy release rate. Dimensionally progressive monotonic distortion can allocate more representational power to a shorter time range, thus reflecting an enhanced effect of rapid release in the predicted values. The set of induced points obtained by selecting points with maximum and minimum distances forms a uniform coverage in the distortion space. This improves the comprehensiveness of the approximation without increasing the number of samples, making the prediction interval more consistent with the local density of the input distribution. When the input falls at the coverage boundary, the prediction interval widens appropriately, prompting process control to adopt a conservative strategy and reduce the probability of energy overshoot. The prediction interval simultaneously reflects data density and model confidence. In energy allocation strategies, when the intervals of continuous segments converge and remain narrow, it indicates stable operating conditions, which can increase the incentive for heating plans; when the interval suddenly increases, it indicates that the input features have fallen into a rare region, making a conservative strategy safer.

[0067] refer to Figure 2 ,like Figure 2The diagram illustrates the principle of the feature-level variational input distortion mapping of this invention. It uses a two-dimensional coordinate system to demonstrate the construction principle and optimization mechanism of the monotonically piecewise cubic spline mapping. The horizontal axis represents the original eigenvalues, which are mapped to the standard interval of 0 to 1 after quantile normalization. The vertical axis represents the distorted eigenvalues, also within the interval of 0 to 1. The core function of this mapping function is to independently perform a nonlinear transformation on each dimension of the input features to improve the distribution characteristics of the feature space, making it more suitable for Gaussian process modeling. The diagonal line drawn with dashed lines represents the identity mapping, i.e., the linear relationship where the distorted eigenvalues ​​equal the original eigenvalues. This diagonal line serves as a reference benchmark for comparing the transformation effect of the actual distortion mapping. The curve drawn with thick solid lines represents the monotonically piecewise cubic spline mapping function obtained after variational optimization. This curve is strictly monotonically increasing, with its starting point fixed at the origin (0,0) and its ending point fixed at the upper right corner (1,1), ensuring the reversibility of the mapping and the rationality of its physical meaning. The curve exhibits an overall S-shaped characteristic, relatively steep in the middle region and relatively gentle at both ends. This shape reflects the adaptive adjustment of the feature distribution by the variational optimization process. Ten solid black dots are marked along the distorted mapping curve; these dots represent the control nodes of the monotonically piecewise cubic spline. According to the design of this invention, 12 nodes are set inside the spline, plus the start and end points, for a total of 14 nodes. The figure shows the positions of 10 representative nodes. The horizontal coordinates of these nodes are placed according to the quantile rule, that is, the interval from 0 to 1 is divided into 13 equal segments, with the node horizontal coordinates corresponding to 0, 1 / 13, 2 / 13, ..., 12 / 13, and 1, respectively. This quantile placement strategy ensures that each node has the same sample coverage density in the original feature value space, avoiding local fitting accuracy differences caused by uneven node distribution. The vertical coordinate of the node is the control variable of the variational optimization, which follows a strictly monotonically increasing constraint during initialization. The initialization process uses a positive incremental accumulation method: the starting ordinate is set to 0, the ending ordinate to 1, and the increment of the ordinate of each intermediate node is set to 1 / 13. An initial ordinate sequence {0, 1 / 13, 2 / 13, ..., 12 / 13, 1} is generated through forward accumulation. This initialization strategy makes the initial spline approximately an identity mapping, providing a reasonable starting point for subsequent optimization. During training, the node ordinates are iteratively updated using the gradient information of the variational optimization objective. The rectangle in the upper left of the figure marks the control quantity optimization: the node ordinates are adjusted through variational optimization, illustrating the optimization objective. Monotonicity guarantee is a key technical challenge in feature-level warped mapping design. If the node ordinates are directly optimized without constraints, the optimization process may result in disordered ordinates, causing the mapping function to lose its monotonicity and thus invertibility. This invention uses a reparameterization technique to solve this problem. The rectangle in the lower right of the figure details the monotonicity guarantee mechanism: the difference between the ordinates of adjacent nodes is applied with a positive domain mapping and then accumulated forward. Specifically, a set of auxiliary variables is defined. The increments of the y-coordinates of adjacent nodes are represented by a positive domain mapping function (such as the softplus function or the exponential function), which is applied to each increment to ensure... This holds true. Then, the node's ordinate is calculated through forward accumulation: This needs to be ensured through normalization adjustments. This reparameterization method transforms a constrained optimization problem into an unconstrained one, allowing the optimization algorithm to freely update auxiliary variables. Meanwhile, the monotonicity of the node's ordinate is automatically guaranteed. A line in the figure marks the monotonically increasing characteristic of a certain segment of the spline, and the text label "strictly monotonic" emphasizes the importance of this property. The spline uses cubic polynomial interpolation between adjacent nodes, ensuring that the mapping function is continuous not only at the nodes but also in its first and second derivatives, i.e., it possesses... Continuity. This high-order smoothness ensures that the twisted mapping does not introduce artificial discontinuities or sharp changes, which is beneficial to the training stability of the Gaussian process model. The bottom of the figure indicates the number of internal nodes: 12, and the node x-coordinates: placed according to equipartition points, summarizing the key parameter settings for spline construction. By independently applying the above monotone piecewise cubic spline mapping to each dimension of the input feature set, this invention achieves feature-level nonlinear twisting transformation. This dimension-independent design ensures the physical independence of each dimension's features while allowing for differentiated nonlinear adjustments to each dimension based on its statistical characteristics. Compared to traditional global linear transformations (such as Principal Component Analysis (PCA)) or fixed nonlinear transformations (such as logarithmic transformations and Box-Cox transformations), the variational input twisted mapping, through end-to-end joint training, can automatically learn the optimal feature transformation method, significantly improving the Gaussian process regression model's ability to express complex nonlinear relationships.

[0068] like Figure 3The diagram shows a comparison of the feature space before and after distortion according to the present invention. This section visually reveals the effect of the distortion mapping on the geometric structure of the feature space and its influence on the linearization of the classification boundary by displaying the two-dimensional feature space distribution before and after the variational input distortion mapping process side by side. The diagram contains an independent two-dimensional coordinate system on each side. The left side is labeled as the feature space before distortion, and the right side is labeled as the feature space after distortion. The two are connected by a thick arrow in the center and the text "variational input distortion mapping," clearly indicating the directionality of the transformation. The horizontal axis of the feature space coordinate system on the left before distortion is labeled as Feature 1, and the vertical axis is labeled as Feature 2, representing the original feature space after basic preprocessing (such as quantile normalization). Multiple small black dots are scattered throughout the diagram, each dot representing the projection position of a training sample in this two-dimensional feature subspace. The distribution of these sample points exhibits obvious non-linear separation characteristics: the sample points in the upper half are roughly clustered in the higher Feature 2 value region, while the sample points in the lower half are clustered in the lower Feature 2 value region. However, there is significant overlap between the two classes of samples in the Feature 1 direction, and the distribution boundary exhibits a curved shape. The dashed curve in the figure is labeled as the category boundary (non-linear). This curve approximates the decision boundary separating the two classes of samples using a quadratic polynomial or cubic spline form. The non-linear characteristic of this boundary indicates that, in the original feature space, sample points corresponding to different discharge heat transfer states (such as low heat transfer efficiency and high heat transfer efficiency) cannot be completely separated by a simple linear hyperplane, requiring a complex non-linear classifier to achieve effective differentiation. This non-linear distribution poses a challenge to Gaussian process regression modeling. Although Gaussian processes can theoretically capture arbitrarily complex non-linear relationships through appropriate kernel functions, the training efficiency and prediction accuracy of the model are affected when the feature space distribution deviates significantly from the Gaussian assumption or presents a highly complex topological structure. Especially in the sparse Gaussian process framework, a limited number of induced points are insufficient to fully cover the complex region near the non-linear boundary, which may lead to distorted estimation of prediction uncertainty. Furthermore, the hyperparameter optimization of the kernel function often gets trapped in local optima on non-linear distributions, making it difficult to find the globally optimal combination of length scale and variance parameters. The horizontal axis of the distorted feature space coordinate system on the right is labeled as distorted feature 1, and the vertical axis is labeled as distorted feature 2, representing the feature space after variational input distortion mapping. The figure also shows the training sample points corresponding to the left side, but their spatial distribution has changed significantly. The most obvious change is that the distribution of sample points is more regular, and the degree of class separation is significantly improved. The separation of sample points in the upper and lower halves along the feature 2 direction has increased, while the overlapping area along the feature 1 direction has decreased. More importantly, the class boundary (linearization) drawn with a solid line in the figure has changed from a curved shape to an approximately straight line shape, and the annotation text specifically emphasizes this key characteristic of linearization. This straight line passes horizontally through the middle of the coordinate system, clearly separating the upper and lower class sample points, with only a small number of confusing samples that are difficult to classify near the boundary.

[0069] The above-described embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for predicting electric field discharge heat transfer in electrode-type fluids based on Gaussian process regression, characterized in that, The method includes: Step 1: Simultaneously acquire voltage, current, optical emission, acoustic emission, inlet temperature, outlet temperature, and volumetric flow rate sequences on the electrode-type fluid electric field discharge heat exchanger; generate discharge heat exchange segments according to the discharge initiation event slices, calculate the target heat exchange index, and establish a sample set that corresponds one-to-one with the input feature set and the target heat exchange index. Step 2: Perform joint training of variational input distortion mapping and sparse Gaussian process on the sample set established in step 1 to obtain the joint finalized state of the variational input distortion mapping and sparse Gaussian process model; Step 3: For the heat transfer segment to be predicted, generate the input feature set in the manner of Step 1; call the joint shaping state obtained in Step 2: first, perform monotonic distortion and recalibration on the input feature set dimension by dimension using variational input distortion mapping to form distortion features; input the distortion features into the sparse Gaussian process model to obtain the predicted value and prediction interval of the target heat transfer index and output it. The variational input distortion mapping in step 2 includes feature-level distortion mapping and temporal-level distortion mapping. The construction process of feature-level distortion mapping includes: performing quantile normalization on each dimension of the input feature set to map the values ​​to the interval between 0 and 1; constructing a monotonic piecewise cubic spline for each dimension in the interval between 0 and 1, with the number of internal nodes set to 12. The horizontal coordinates of the nodes are placed according to the quantile points, and the vertical coordinates of the nodes are initialized according to the rule that the starting point is equal to 0, the ending point is equal to 1, and the vertical coordinates of the intermediate nodes are generated by positive incremental accumulation, forming a strictly monotonic initial spline; listing the vertical coordinate of each dimension of the spline as an optimizable control variable, automatically adjusting the vertical coordinate by improving the variational optimization objective, and ensuring the strict monotonicity of the spline by applying a positive domain mapping to the difference between the vertical coordinates of adjacent nodes and then performing forward accumulation.

2. The electrode-type fluid electric field discharge heat transfer prediction method based on Gaussian process regression as described in claim 1, characterized in that, In step 1, the operation of simultaneously acquiring voltage, current, optical emission, acoustic emission, inlet temperature, outlet temperature, and volumetric flow rate sequences on the electrode-type fluid electric field discharge heat exchanger is specifically configured with a sampling frequency of 100,000 times per second. The continuous sequence is sliced ​​based on the following event detection rules: when the first-order difference of the current sequence has 5 consecutive incremental sampling points within a 0.5-millisecond time window, and the optical emission sequence has a peak value within the corresponding time window, this moment is defined as the discharge start. Taking the defined discharge start time as zero, the process extends forward by 2 milliseconds and backward by 8 milliseconds to form a discharge heat exchange segment with a length of 10 milliseconds.

3. The electrode-type fluid electric field discharge heat transfer prediction method based on Gaussian process regression as described in claim 2, characterized in that, Step 1 further includes processing the voltage and current sequences acquired in Step 1 using zero-phase bandpass filtering to retain frequency components from 1000 Hz to 50000 Hz; calculating short-time energy and performing peak deduplication on the optical and acoustic emission sequences, with a deduplication interval set to 50 microseconds; performing a moving average on the temperature and volumetric flow rate sequences, with a window length set to 0.2 milliseconds; and obtaining the target heat transfer index for the current segment based on the measured values ​​of the inlet temperature, outlet temperature, and volumetric flow rate sequences within a 10-millisecond segment using a standard heat transfer calculation process, with the unit being energy per unit time.

4. The electrode-type fluid electric field discharge heat transfer prediction method based on Gaussian process regression as described in claim 3, characterized in that, The input feature set generated in step 1 includes: voltage peak value, current peak value, current rise time duration, optical peak count, acoustic peak count, time interval between optical peak and acoustic peak, average temperature difference within the segment, average volumetric flow rate, and peak position time coordinates that are cross-correlated with voltage and current within the segment. This input feature set is then matched one-to-one with the target heat transfer index to form the sample set for the current sampling batch.

5. The electrode-type fluid electric field discharge heat transfer prediction method based on Gaussian process regression as described in claim 4, characterized in that, The construction process of the time-series-level warp mapping includes: extracting a set of time axis anchor points within each discharge heat transfer segment, which includes the current rising edge start point, optical peak apex, acoustic peak apex, temperature difference peak apex, and segment end point; dividing the 10-millisecond time axis into 5 segments based on the extracted anchor point sequence, and establishing an affine scaling mapping on each segment; using the scaling ratio and start-end alignment offset of each segment as optimizable control variables, and incorporating them together with the control variables of the feature-level warp into the same variational optimization objective; reparameterizing the time coordinates of the entire segment using the time-series-level warp mapping to obtain the warped time axis, and then resampling all subsequent time-related features on the generated warped time axis.

6. The electrode-type fluid electric field discharge heat transfer prediction method based on Gaussian process regression as described in claim 5, characterized in that, After the variational input distortion mapping process, the maximum and minimum distance point selection is performed on the distorted input feature space: first, the sample with the largest Euclidean norm is selected as the first induced point; then, the sample is iteratively scanned, and in each round, the minimum distance from each candidate sample to the set of selected induced points is calculated, and the sample with the maximum and minimum distance is added to the set of induced points; the process terminates when the coverage radius is less than 0.15 or the number of induced points reaches 256; the coverage radius is calculated based on the standardized Euclidean distance metric, where the standardization coefficient is the scale obtained when performing quantile normalization in step 1.

7. The electrode-type fluid electric field discharge heat transfer prediction method based on Gaussian process regression as described in claim 6, characterized in that, In step 2, the sparse Gaussian process model uses a half-integer stationary kernel and an anisotropic length scale; a Gaussian approximate posterior is established for the function values ​​on the induced point set, described by multidimensional mean and multidimensional variance; the hyperparameters of the kernel, the induced point posterior parameters, and all control quantities of the variational input distortion mapping used in step 2, which includes feature-level distortion mapping and temporal-level distortion mapping, are incorporated into a single variational optimization objective.

8. The electrode-type fluid electric field discharge heat transfer prediction method based on Gaussian process regression as described in claim 7, characterized in that, In step 2, a mini-batch learning process is adopted, and the following three stages are strictly executed in sequence for each round of optimization: In the first stage, the posterior parameters of the induced points are updated using the natural gradient method; In the second stage, the hyperparameters of the kernel are updated using the quasi-Newton method. In the third stage, adaptive moment estimation is used to update all control variables of the variational input distortion map, which includes feature-level distortion maps and temporal-level distortion maps, used in step 2. After each round of optimization, the improvement of the variational optimization objective is calculated. If the improvement is less than 1%, the learning step size of the third stage is reduced to 0.5 of the original and the next round continues. Training stops when the improvement is less than 1% for 10 consecutive rounds or the total number of rounds reaches 500. The training result is the joint shaping state of the variational input distortion map and the sparse Gaussian process model, in which the set of induced points, the hyperparameters of the kernel, the posterior parameters of the induced points, and the control variables of the variational input distortion map are stored together for inference.

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