Underwater pulse discharge plasma channel time-varying resistance prediction method based on PINN

Abstract

The invention discloses an underwater pulse discharge plasma channel time-varying resistance prediction method based on a PINN, and belongs to the technical field of calculation. The method comprises the following steps: firstly, collecting discharge experiment data under various initial conditions, and performing preprocessing according to model training requirements; and secondly, a PINN neural network is constructed according to the requirement of an underwater pulse discharge actual task, a discharge equivalent loop and the Kirchhoff's law are combined to improve a PINN loss function, and the PINN loss function is made to be suitable for prediction of the plasma channel time-varying resistance. Thirdly, minimizing a loss function, optimizing model parameters, and training a PINN neural network to find optimal parameters; and finally, a corresponding time-varying resistance value can be obtained by inputting underwater pulse discharge time data. According to the method, the physical law and the neural network are combined for predicting the time-varying resistance of the underwater pulse discharge plasma channel for the first time, and the mapping relation of the time-varying resistance can be effectively constructed by combining physical information and the neural network.

Description

Technical Field

[0001] This invention belongs to the field of high voltage pulse power and physical information neural network (PINN) data simulation and calculation technology, and relates to a method for predicting the time-varying resistance of underwater pulsed discharge plasma channels based on PINN, and in particular, a method for calculating the resistance of time-varying plasma channels during underwater pulsed discharge based on PINN. Background Technology

[0002] In underwater pulsed discharge, energy is released into the electrode gap through a charging capacitor. Due to the large influx of energy in a short time, the electrode gap in the water is broken down, forming a plasma channel. The current increases sharply, causing the liquid within the channel to vaporize and expand instantaneously, generating a high-speed propagating shock wave. Subsequently, the bubble contracts, oscillates, and even collapses, generating a secondary pressure pulse. This phenomenon of converting electrical energy into shock wave mechanical energy through a liquid medium is called the "hydraulic-electric effect." This powerful shock wave effect has many advantages, including high energy conversion efficiency, wide frequency band, high low-frequency energy, long propagation distance, and controllable parameters and direction. Besides generating powerful mechanical shock waves, the underwater pulsed discharge process is also accompanied by rich physicochemical effects, producing ultraviolet light and various free radicals (OH, H, and O) and molecularly active species (H₂O₂, O₂, H₂, and O₃) in water. Underwater pulsed discharge has already been widely used in engineering fields such as energy extraction, environmental protection, rock breaking, and underwater sound sources. In addition, due to its high dielectric constant, pulse breakdown field strength, and self-recovery characteristics, water is widely used as an energy storage and insulation medium for miniaturized pulse power system devices, such as water capacitors, water switches, and water resistors. Underwater pulse discharge has a wide range of applications and has great potential in engineering applications.

[0003] In underwater pulsed discharge, the amount of deposited energy is related to the energy conversion efficiency and the magnitude of the generated shock wave, and the calculation of deposited energy is based on the plasma channel resistance. Therefore, constructing an accurate prediction system for the time-varying resistance of the plasma channel in underwater pulsed discharge, and then optimizing the discharge energy parameters, is a key technical bottleneck to overcome the constraints of "control-optimization-efficiency". Achieving the prediction of the time-varying resistance of the plasma channel is crucial for discharge parameters and improving shock wave efficiency.

[0004] Due to the extreme environment, it is difficult to directly implant measurement devices, and such methods can interfere with the discharge process. Traditional methods for predicting the resistance of underwater pulsed discharge plasma channels have many limitations, relying on accurate physical modeling, simplification assumptions, and plasma parameters (such as temperature, density, and conductivity models) that are difficult to obtain precisely. However, in actual discharge processes, there are complex and variable physicochemical reactions, which are often nonlinear. Traditional physical models struggle to effectively model these complex processes, often resulting in some degree of bias and poor applicability and generalization ability of the models.

[0005] Purely data-driven methods leverage the nonlinear fitting capabilities of machine learning and deep learning models to construct a latent nonlinear mapping model between input discharge parameters and the output data to be predicted, thereby modeling the time-varying resistance of underwater pulsed discharge plasma channels and indirectly calculating the time-varying resistance by bypassing the physical model. However, the effectiveness of data-driven prediction methods depends on high-quality data. Underwater discharge experiments are costly, and data acquisition is limited by accuracy, frequency response, and noise, resulting in a small amount of data with poor quality. This leads to severely insufficient model generalization ability and robustness. Furthermore, the failure to effectively integrate with the physical model results in poor interpretability and physical meaning.

[0006] Physical Information Neural Networks (PINNs) are an emerging deep learning method that embeds physical laws as constraints into model training. The key lies in using prior physical knowledge to guide the learning process, significantly improving interpretability and physical meaning while maintaining strong nonlinear fitting capabilities. This effectively enhances the model's accuracy and generalization ability when dealing with complex scientific and engineering problems. The potential of introducing physical laws as constraints to improve model generalization and reliability in data-scarce scenarios is applicable to constructing time-varying resistance models for underwater pulsed discharge plasma channels. This invention is the first to apply this to the prediction of the specific, critical, and difficult-to-measure physical quantity of time-varying resistance in underwater pulsed discharge plasma channels. However, due to the unique nature of experiments during underwater pulsed discharge, it is impossible to actually measure the time-varying resistance value of the plasma channel, making it difficult to construct a training dataset. Therefore, this invention proposes using an unsupervised learning method to construct a PINN-based time-varying resistance model for underwater pulsed discharge plasma channels, completing the calculation of the time-varying resistance and overcoming the technical bottleneck. Summary of the Invention

[0007] To overcome the shortcomings of the aforementioned technologies, this invention provides a prediction method based on a PINN network that can effectively predict the time-varying resistance of underwater pulsed discharge plasma channels. By analyzing the equivalent circuit model of underwater pulsed discharge and applying Kirchhoff's loop voltage law as a physical prior to the neural network, the method replaces the constraints of empirical formulas, ensuring physical consistency throughout the modeling process.

[0008] To achieve the above objectives, the technical solution of the present invention is as follows:

[0009] A method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN. The prediction method includes the following steps:

[0010] Step 1: Based on the principle of underwater pulse discharge, an underwater pulse discharge experimental platform is built. The experimental platform mainly consists of three parts: a charging module, a discharging module, and a measurement module, which are used to conduct underwater pulse discharge experiments.

[0011] Step Two: Based on the underwater pulse discharge experimental platform built in Step One, conduct experiments with different discharge experimental parameters and obtain discharge data; specifically:

[0012] During the underwater pulsed discharge experiment, the loop current *i* and the voltage *u* between sampling points were recorded in real time using current and voltage sensors to obtain the underwater pulsed discharge loop current timing data, which served as the discharge data. Sampling points were defined based on the experimental reaction time. The final collected discharge data mainly included current *i*, voltage *u*, and sampling point time data *t*.

[0013] Step 3: Preprocess the discharge data obtained in Step 2, including data analysis, cleaning, smoothing and denoising, and format standardization; specifically:

[0014] Step 3.1: Analyze the numerical distribution of current and voltage in the discharge data, the overall distribution pattern, the trend of change, and the characteristics of data points.

[0015] Step 3.2: Check if there are any data in the discharge data that do not conform to the experimental phenomenon, and handle missing and outlier values ​​in continuous discharge data.

[0016] Step 3.3: Use a denoising algorithm to smooth and denoise the acquired underwater pulse discharge circuit current time series data to avoid the impact of noise generated during sensor data acquisition on model training.

[0017] Step 3.4: Normalize the format of the smoothed and denoised discharge data to obtain standardized data that conforms to the model input.

[0018] Step 4: The preprocessed discharge data from Step 3, mainly including current i and sampling time data t, is used as the model input, and current i is part of the loss function. The dataset is then used to construct training and testing sets proportionally. The training set is used to train the PINN neural network model, and the testing set is used to test the performance of the PINN neural network model.

[0019] Step 5: After constructing the dataset, based on the characteristics of underwater pulsed discharge, construct a suitable PINN neural network model, including an input layer, hidden layers, an output layer, activation functions, and a Dropout layer; construct the nonlinear mapping of the neural network model. Specifically:

[0020] The PINN neural network model is a multilayer perceptron (MLP) with a total of 7 layers, including an input layer, 5 hidden layers, and an output layer. Each hidden layer contains 50 neurons, and a tanh activation function is used in the middle of each layer. Dropout is set to 0.2 to prevent the model from overfitting.

[0021] Time-varying resistor R pl As a time-dependent variable, a nonlinear mapping between time-varying resistance and time is established through the model's input layer, hidden layer, and output layer. The preprocessed sampling time data t is used as the input to the PINN neural network model, and the output is the time-varying resistance R at the corresponding time step. pl size.

[0022] Step Six: After completing the construction of the PINN neural network model in Step Five, improve the loss function L of the PINN neural network model based on physical laws. PINN This makes it suitable for time-varying resistance prediction; the loss function is defined as shown in formula (1):

[0023] (1)

[0024] This includes physical prior loss and regularization terms. Regularization terms prevent overfitting by limiting the size of model parameters. Indicates physical prior loss; Represents the regularization coefficient; This represents the learnable parameters in the model; Indicates to Calculate the 2-norm;

[0025] Step 6.1, according to Kirchhoff's laws, The first specific representation is:

[0026] (2)

[0027] In the formula, N represents the number of data points; The total inductance in the discharge circuit is represented by ; i represents the current in the circuit; and t represents the sampling time data. R represents the inherent resistance of the circuit; pl This represents the time-varying resistance predicted by the model; Indicates the total capacitance of the circuit;

[0028] Formula (2) indicates that, based on the dataset constructed in step four, N time-step data t are taken as the input to the PINN neural network model to obtain the time-varying resistance R over N time steps. pl Then, calculate the value of formula (2), take the average value, and update the model parameters by optimizing the loss function.

[0029] Step 6.2, in the loss function This term is derived from Kirchhoff's Voltage Law, which states that in any closed loop, the algebraic sum of the voltage drops across all components is equal to the algebraic sum of the electromotive forces. That is, starting from a point, looping around the point back to that point, the algebraic sum of the voltages across all segments is always equal to zero. ;

[0030] Therefore, according to the equivalent circuit of underwater pulse discharge and Kirchhoff's voltage law, we have:

[0031] (3)

[0032] in, This represents the voltage across the capacitor. This represents the voltage across the resistor. This represents the voltage across the inductor;

[0033] The voltage and current relationship of a capacitor is as follows:

[0034] (4)

[0035] The points after sorting are:

[0036] (5)

[0037] in, The initial voltage across the capacitor is represented by , i represents the current in the circuit, and C represents the capacitance value.

[0038] The current-voltage relationship of the resistor in the underwater pulse discharge equivalent circuit is as follows:

[0039] (6)

[0040] Where i represents the current in the loop, R0+R pl R represents the total resistance of the circuit. pl R0 represents the time-varying resistance of the plasma channel, and R0 represents the inherent resistance of the circuit.

[0041] The current-voltage relationship of the inductor in the underwater pulse discharge equivalent circuit is as follows:

[0042] (7)

[0043] Where L represents the total inductance in the discharge circuit;

[0044] According to Kirchhoff's laws, the voltage relationship of the equivalent circuit of underwater pulse discharge is expressed as:

[0045] (8)

[0046] Because the above formula contains integral terms, it makes calculation difficult. Therefore, differentiating both sides of the above equation with respect to t yields:

[0047] (9)

[0048] The total inductance in the discharge circuit is represented by ; i represents the current in the circuit; and t represents the sampling time data. R represents the inherent resistance of the circuit; pl This represents the time-varying resistance predicted by the model; Indicates the total capacitance of the circuit;

[0049] The loss function of the underwater pulsed discharge PINN model is constructed based on formula (9) to guide the model in learning how to establish a nonlinear mapping between time and the time-varying resistance of the corresponding plasma channel.

[0050] Step 7: After constructing the dataset, model, and loss function through the above steps, train the PINN neural network using the AdamW optimization algorithm based on momentum and weight decay.

[0051] Step 8: Based on Step 7, further optimize the model by minimizing the loss function using the gradient-based L-BFGS algorithm;

[0052] Step 9: Determine the loss function L PINN If the number of iterations is less than the threshold or the training has reached the specified number of iterations, return to step four to continue training the model. If the requirement has been met, continue to step nine. The threshold in step nine is 0.01 and the number of iterations is 5000.

[0053] Step 10: Through the optimization in Steps 8 and 9, obtain the optimal parameters of the model. ,Will By incorporating the PINN neural network model, a method for predicting the time-varying resistance of underwater pulsed discharge plasma channels based on PINN is provided, thereby realizing the prediction of the time-varying resistance of underwater pulsed discharges. Based on this prediction, the energy deposited in the plasma channel can be calculated using the obtained identification resistance data, thus obtaining the energy deposition efficiency and calculating the magnitude of the released shock wave. This helps to optimize discharge parameters, achieve discharge parameter tuning, and optimize energy conversion efficiency.

[0054] This invention constructs a PINN model that synchronizes time-varying resistance with time as input. By embedding the circuit equations describing the discharge process as mandatory physical constraints into the loss function, the network is guided to learn a solution that both fits experimental observations and strictly follows physical laws under limited data, thereby reversing the time-varying resistance that cannot be directly measured.

[0055] The innovative design features of this invention are:

[0056] The core idea behind using PINN to predict the time-varying resistance of underwater pulsed discharge plasma channels is to move away from treating the plasma channel as a "black box" that needs to be learned from scratch. Instead, the governing equations describing its physical evolution are embedded directly into the loss function of the neural network as soft constraints. This allows for training a physically consistent and more generalizable model using less and coarser data. The core approach involves constructing a PINN model that takes time as input and synchronously outputs the time-varying resistance. By embedding the circuit equations describing the discharge process as mandatory physical constraints into the loss function, the network is guided to learn a solution that fits experimental observations and strictly adheres to physical laws, thus retrieving the time-varying resistance that cannot be directly measured.

[0057] The beneficial effects of this invention are:

[0058] (1) The invention can make good use of the physical priors of the circuit equations, while maintaining the powerful nonlinear learning ability of the neural network model, overcoming the difficulty of traditional methods in capturing the complex nonlinear relationships of the fit, while also maintaining good physical knowledge and interpretability, and effectively completing the modeling of the mapping relationship between time and time-varying resistance.

[0059] (2) This invention improves the loss function of traditional PINN by embedding Kirchhoff's voltage law into the PINN model, replacing the traditional empirical formula, so that the model still maintains physical consistency during calculation. At the same time, it can also predict the time-varying resistance of plasma channels without time-stamp values, and has high robustness and generalization.

[0060] (3) The invention is highly practical and can theoretically be applied to underwater pulse discharge time-varying resistance prediction under various initial discharge parameters. Attached Figure Description

[0061] Figure 1Flowchart for predicting the time-varying resistance of underwater pulsed discharge plasma channels using PINN;

[0062] Figure 2 The diagram shows the constructed prediction model based on PINN time-varying resistance. Detailed Implementation

[0063] The following will describe several exemplary embodiments of the present invention in detail with reference to the accompanying drawings. It should be noted that the embodiments shown in the drawings are for the purpose of more clearly illustrating the technical solutions of the present invention, and not for limiting its specific implementation. The fundamental purpose of providing these embodiments is to enable those skilled in the art to more deeply understand the essence of the present invention and to fully grasp the technical scope of the present invention accordingly.

[0064] This invention provides a method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN, the specific model structure of which is as follows: Figure 2 As shown.

[0065] The innovation of this invention lies in its approach. Unlike classic physics-based methods and pure machine learning methods, this invention establishes a nonlinear mapping between time and time-varying resistance through learning, using physical equations and deep learning models. By analyzing equivalent circuit characteristics and improving the loss function of PINN, a circuit voltage equation based on Kirchhoff's laws is embedded into the constructed physics-based neural network, achieving a fusion of physical information and the neural network. This is the first application of PINN in predicting the time-varying resistance of underwater pulsed discharge plasma channels. The specific implementation includes the following steps:

[0066] Step 1: Based on the principle of underwater pulse discharge, an underwater pulse discharge experimental platform was constructed. The platform mainly consists of three parts: a charging module, a discharging module, and a measurement module. The charging module comprises a high-voltage power supply, a current-limiting diode, and a charging switch to charge the main capacitor, which has a capacitance of 0.997 μF and a rated voltage of 35 kV. The discharging module consists of a discharge capacitor and a discharge switch. The main discharge switch is a triggered vacuum switch (TVS). The discharge capacitor is a cylindrical container with a diameter of 22 cm and a height of 23 cm, with observation windows of 9 cm diameter on both sides. Tap water with a conductivity of 182 μS / cm was used as the discharge liquid. The discharge electrodes are a combination of needle and plate electrodes. The anode uses a needle-shaped electrode with a tungsten-copper alloy tip, and the cathode uses a plate electrode made of stainless steel. The spacing between the needle and plate electrodes can be adjusted within the range of 0-20 mm. The measurement module consists of a high-voltage probe (EP-100k), a current sensor (Pearson Electronics 4997), an oscilloscope (DSO5054A), and a voltmeter (Q4-V). Furthermore, a piezoelectric pressure probe (model: P / N: W138A10 / M038CY006AC) is used to measure the shock wave pressure generated by the discharge, employing a pulse pressure signal detection method. Finally, the above experimental setup is used to conduct underwater pulse discharge experiments.

[0067] Step Two: Based on the underwater pulse discharge experimental platform built in Step One, different discharge experimental parameters are set according to the discharge requirements under different conditions to conduct experiments and obtain discharge data; specifically:

[0068] During the underwater pulsed discharge experiment, the loop current *i* and the voltage *u* between sampling points were recorded in real time using current and voltage sensors to obtain the underwater pulsed discharge loop current timing data, which served as the discharge data. Sampling points were defined based on the experimental reaction time. The final collected discharge data mainly included current *i*, voltage *u*, and sampling point time data *t*.

[0069] Step 3: Preprocess the discharge data obtained in Step 2, including data analysis, cleaning, smoothing and denoising, and format standardization; specifically:

[0070] Step 3.1: Analyze the numerical distribution of current and voltage in the discharge data, the overall distribution pattern, the trend of change, and the characteristics of data points.

[0071] Step 3.2: Check the discharge data for any data that does not conform to the experimental phenomena, and handle missing and outlier values ​​in the continuous discharge data. For missing values, use polynomial interpolation to supplement the data; for outliers, delete them directly or remove them using data fitting methods.

[0072] Step 3.3: A denoising algorithm is used to smooth and denoise the acquired underwater pulse discharge circuit current time series data to avoid the impact of noise generated during sensor data acquisition on model training. A high-order polynomial is used to fit the data points within the window, and then the value of this fitted polynomial at the center point is used as the output to remove noise present in the original data. This process is performed by sliding the fitting window across the entire data sequence. The expression for data smoothing and denoising is:

[0073] For a sliding window, assume there are 2m+1 data points within the window, with indices from -m to m (the center point index is 0). The coordinates of the data points within the window are (x, y), where x is the equally spaced position index and y is the data value, typically set to […-2, -1, 0, 1, 2…].

[0074] First, an nth-order polynomial (n < 2m+1) is used to fit 2m+1 points in the window:

[0075] (1)

[0076] In the formula This represents the fitted data value, where x represents the value of the data centered at x. i Sliding window, [ a 1 , a 2 , a 3 ,..., a n ] Represent the coefficients of the fitted polynomial, and determine the coefficient vector. [ a 1 , a 2 , a 3 ,..., a n ] This allows the polynomial (1) to fit the data points y within the window. The optimal solution is obtained by minimizing the mean square error between the data and the polynomial fit. [ a 1 , a 2 , a 3 ,..., a n ] Coefficient value:

[0077] (2)

[0078] In the formula, m represents half the window length, m=0 represents the midpoint of the window, 2m+1 represents the window size (odd number), and y i Let represent the actual data value, and n represent the nth-order polynomial. The minimum value of formula (2) is obtained by the least binary method, that is, the best-fit polynomial of the corresponding window is obtained, and the data denoising of the local window is completed. Then, the denoising of the overall data is completed by the sliding window.

[0079] Step 3.4: Normalize the format of the smoothed and denoised discharge data to obtain standardized data that conforms to the model input.

[0080] For discharge data where current exhibits alternating positive and negative values ​​and includes a zero value, a maximum value normalization method is used to maintain the zero-value current as zero. This avoids altering the fundamental trend of the data due to normalization, which could negatively impact the model's fitting performance. The expression for normalizing the data is as follows:

[0081] (3)

[0082] In the formula This represents time series data after normalization. This represents the original data point.

[0083] The above processing can transform time series data into the range [-1, 1] while preserving the original trend of the data.

[0084] The expression for normalizing time data is:

[0085] (4)

[0086] In the formula, This represents the data after normalization. Represents the original data. This represents the minimum value in the data. This represents the maximum value in the data.

[0087] Step 4: The preprocessed discharge data from Step 3, mainly including current i and sampling time data t, is used as the model input, and current i is part of the loss function. The dataset is then used to construct training and testing sets proportionally. The training set is used to train the PINN neural network model, and the testing set is used to test the performance of the PINN neural network model.

[0088] Step 5: After constructing the dataset, based on the characteristics of underwater pulsed discharge, construct a suitable PINN neural network model, including an input layer, hidden layers, an output layer, activation functions, and a Dropout layer; construct the nonlinear mapping of the neural network model. Specifically:

[0089] The PINN neural network model is a multilayer perceptron (MLP) with a total of 7 layers, including an input layer, 5 hidden layers, and an output layer. Each hidden layer contains 50 neurons, and a tanh activation function is used in the middle of each layer. Dropout is set to 0.2 to prevent overfitting. The specific results are shown in the figure below. Figure 2 As shown.

[0090] The constructed model can be represented as follows:

[0091] (5)

[0092] In the formula, h represents the output of the current layer of the model, f represents the mapping relationship, and X represents the input of the current layer. This represents the activation function, where W and b represent the learnable parameters. .

[0093] Time-varying resistor R pl As a time-dependent variable, a nonlinear mapping between time-varying resistance and time is established through the model's input layer, hidden layer, and output layer. The preprocessed sampling time data t is used as the input to the PINN neural network model, and the output is the time-varying resistance R at the corresponding time step. pl size.

[0094] Step Six: After completing the construction of the PINN neural network model in Step Five, improve the loss function L of the PINN neural network model based on physical laws. PINN This makes it suitable for time-varying resistance prediction; the loss function is defined as shown in formula (6):

[0095] (6)

[0096] Formula (4) includes physical prior loss and regularization term. The regularization term is used to prevent model overfitting by limiting the size of model parameters. Indicates physical prior loss; Represents the regularization coefficient; This represents the learnable parameters in the model; Indicates to Calculate the 2-norm;

[0097] Step 6.1, according to Kirchhoff's laws, The first specific representation is:

[0098] (7)

[0099] In the formula, N represents the number of data points; The total inductance in the discharge circuit is represented by ; i represents the current in the circuit; and t represents the sampling time data. R represents the inherent resistance of the circuit; pl This represents the time-varying resistance predicted by the model; Indicates the total capacitance of the circuit;

[0100] Formula (7) indicates that, based on the dataset constructed in step four, N time-step data t are taken as the input to the PINN neural network model to obtain the time-varying resistance R over N time steps. pl Next, calculate the value of formula (7), then take the average value, and update the model parameters by optimizing the loss function.

[0101] Step 6.2, in the loss function This term is derived from Kirchhoff's Voltage Law, which states that in any closed loop, the algebraic sum of the voltage drops across all components is equal to the algebraic sum of the electromotive forces. That is, starting from a point, looping around the point back to that point, the algebraic sum of the voltages across all segments is always equal to zero. ;

[0102] Therefore, according to the equivalent circuit of underwater pulse discharge and Kirchhoff's voltage law, we have:

[0103] (8)

[0104] in, This represents the voltage across the capacitor. This represents the voltage across the resistor. This represents the voltage across the inductor;

[0105] The voltage and current relationship of a capacitor is as follows:

[0106] (9)

[0107] The points after sorting are:

[0108] (10)

[0109] in, The initial voltage across the capacitor is represented by , i represents the current in the circuit, and C represents the capacitance value.

[0110] The current-voltage relationship of the resistor in the underwater pulse discharge equivalent circuit is as follows:

[0111] (11)

[0112] Where i represents the current in the loop, R0+R pl R represents the total resistance of the circuit. pl R0 represents the time-varying resistance of the plasma channel, and R0 represents the inherent resistance of the circuit.

[0113] The current-voltage relationship of the inductor in the underwater pulse discharge equivalent circuit is as follows:

[0114] (12)

[0115] Where L represents the total inductance in the discharge circuit;

[0116] According to Kirchhoff's laws, the voltage relationship of the equivalent circuit of underwater pulse discharge is expressed as:

[0117] (13)

[0118] Because the above formula contains integral terms, it makes calculation difficult. Therefore, differentiating both sides of the above equation with respect to t yields:

[0119] (14)

[0120] The total inductance in the discharge circuit is represented by ; i represents the current in the circuit; and t represents the sampling time data. R represents the inherent resistance of the circuit; pl This represents the time-varying resistance predicted by the model; Indicates the total capacitance of the circuit;

[0121] The loss function of the underwater pulsed discharge PINN model is constructed based on formula (14) to guide the model in learning how to establish a nonlinear mapping between time data t and the corresponding time-varying resistance of the plasma channel.

[0122] Step 7: After constructing the dataset, model, and loss function through the above steps, train the PINN neural network using the AdamW optimization algorithm based on momentum and weight decay.

[0123] Step 8: Based on Step 7, further optimize the model by minimizing the loss function using the gradient-based L-BFGS algorithm;

[0124] Step 9: Determine the loss function L PINN If the threshold is less than 0.01 or the training has reached the specified number of iterations, return to step four to continue model training. If the requirement has been met, proceed to step nine, where the threshold is 0.01 and the number of iterations is 5000. The training process is as follows: Figure 1 As shown;

[0125] Step 10: Through the optimization in Steps 8 and 9, obtain the optimal parameters of the model. ,Will By incorporating the PINN neural network model, a method for predicting the time-varying resistance of underwater pulsed discharge plasma channels based on PINN is provided, thereby realizing the prediction of the time-varying resistance of underwater pulsed discharges. Based on this prediction, the energy deposited in the plasma channel can be calculated using the obtained identification resistance data, thus obtaining the energy deposition efficiency and calculating the magnitude of the released shock wave. This helps to optimize discharge parameters, achieve discharge parameter tuning, and optimize energy conversion efficiency.

[0126] This embodiment combines Kirchhoff's laws with neural networks, overcoming the limitations of traditional physical models that require precise models and assumptions, while also improving the physical meaning and interpretability of neural networks.

[0127] It should be noted that the embodiments described herein are all exemplary and are intended to illustrate the technical solutions of the present invention, rather than constituting specific limitations. For those skilled in the art, any modifications to the foregoing embodiments, or equivalent substitutions for some or all of the technical features therein, without departing from the essence of the present invention and the scope defined by the claims, should be covered within the protection scope of the present invention.

Claims

1. A method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN, characterized in that, The prediction method includes the following steps: Step 1: Construct an underwater pulse discharge experimental platform. The platform consists of three parts: a charging module, a discharging module, and a measurement module, and is used to conduct underwater pulse discharge experiments. Step 2: Based on the underwater pulse discharge experimental platform, set different discharge experimental parameters to conduct experiments and obtain discharge data, including current i, voltage u, and sampling point time data t; Step 3: Preprocess the discharge data obtained in Step 2 to construct a dataset, including data analysis, cleaning, smoothing and denoising, and format standardization; Step 4: Take the discharge data preprocessed in Step 3, including current i and sampling time data t, where time data t is used as the input to the model and current i is used as part of the loss function; divide the dataset into training set and test set. The training set is used to train the PINN neural network model, and the test set is used to test the performance of the PINN neural network model. Step 5: After constructing the dataset, based on the characteristics of underwater pulsed discharge, construct a suitable PINN neural network model, including an input layer, hidden layers, an output layer, activation functions, and a Dropout layer; construct the nonlinear mapping of the neural network model. ; Step Six: After completing the construction of the PINN neural network model in Step Five, improve the loss function L of the PINN neural network model based on physical laws. PINN This makes it suitable for time-varying resistance prediction; the loss function includes physical prior loss and regularization term, as shown in formula (1): (1) ； in, Indicates physical prior loss; Represents the regularization coefficient; This represents the learnable parameters in the model; Indicates to Calculate the 2-norm; Step 7: After constructing the dataset, model, and loss function through the above steps, train the PINN neural network using the AdamW optimization algorithm based on momentum and weight decay. Step 8: Based on Step 7, further optimize the model by minimizing the loss function using the gradient-based L-BFGS algorithm; Step 9: Determine the loss function L PINN Is the number of iterations less than the threshold or has the training reached the specified number of iterations? If not, return to step four to continue training the model. If the requirements have been met, proceed to step nine. Step 10: Through the optimization in Steps 8 and 9, obtain the optimal parameters of the model. ,Will Substituting into the PINN neural network model, the time-varying resistance of underwater pulsed discharge is realized. The prediction.

2. The method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN according to claim 1, characterized in that, Step two specifically involves: During the underwater pulse discharge experiment, the loop current i and the voltage u between the discharge gaps at the sampling points were recorded in real time using current and voltage sensors to obtain the underwater pulse discharge loop current timing data, which was used as the discharge data. The sampling points are defined based on the experimental reaction time.

3. The method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN according to claim 1, characterized in that, Step three specifically includes: Step 3.1: Analyze the numerical distribution of current and voltage in the discharge data, the overall distribution pattern, the trend of change, and the characteristics of data points; Step 3.2: Check if there are any data in the discharge data that do not conform to the experimental phenomenon, and handle missing and outlier values ​​in continuous discharge data. Step 3.3: Use a denoising algorithm to smooth and denoise the acquired underwater pulse discharge circuit current timing data; Step 3.4: Normalize the format of the smoothed and denoised discharge data to obtain standardized data that conforms to the model input.

4. The method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN according to claim 1, characterized in that, The specific details of step five are as follows: The PINN neural network model consists of an input layer, 5 hidden layers, and an output layer, forming a total of 7 layers in a multilayer perceptron (MLP). Each hidden layer contains 50 neurons, and a tanh activation function is used in the middle of each layer. Dropout is set to 0.2 to prevent overfitting. Time-varying resistor R pl As a time-dependent variable, a nonlinear mapping between time-varying resistance and time is established through the model's input layer, hidden layer, and output layer. The preprocessed sampling time data t is used as the input to the PINN neural network model, and the output is the time-varying resistance R at the corresponding time step. pl size.

5. The method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN according to claim 1, characterized in that, In step six: Step 6.1, according to Kirchhoff's laws, The first specific representation is: (2) ； In the formula, N represents the number of data points; The total inductance in the discharge circuit is represented by ; i represents the current in the circuit; and t represents the sampling time data. R represents the inherent resistance of the circuit; pl This represents the time-varying resistance predicted by the model; Indicates the total capacitance of the circuit; Based on the dataset constructed in step four, N time-step data points t are taken as input to the PINN neural network model to obtain the time-varying resistance R over N time steps. pl Next, calculate the value of formula (2), then take the average value, and update the model parameters by optimizing the loss function; Step 6.2, in the loss function This term is derived from Kirchhoff's voltage law; The current-voltage relationship of the resistor in the equivalent circuit of underwater pulsed discharge is as follows: (6) ； Where i represents the current in the loop, R0+R pl R represents the total resistance of the circuit. pl R0 represents the time-varying resistance of the plasma channel, and R0 represents the inherent resistance of the circuit. The current-voltage relationship of the inductor in the equivalent circuit of underwater pulse discharge is as follows: (7) ； Where L represents the total inductance in the discharge circuit; The voltage relationship of the equivalent circuit of underwater pulsed discharge is expressed as: (8) ； Therefore, we get: (9) ； in, The total inductance in the discharge circuit is represented by ; i represents the current in the circuit; and t represents the sampling time data. R represents the inherent resistance of the circuit; pl This represents the time-varying resistance predicted by the model; Indicates the total capacitance of the circuit; The loss function of the underwater pulsed discharge PINN model is constructed based on formula (9) to guide the model in learning how to establish a nonlinear mapping between time and the time-varying resistance of the corresponding plasma channel.

6. The method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN according to claim 1, characterized in that, The threshold in step nine is 0.01, and the number of iterations is 5000.

7. The method for predicting the time-varying resistance of an underwater pulsed discharge plasma channel based on PINN according to claim 1, characterized in that, In step ten, the energy deposited in the plasma channel is calculated by identifying resistance data to obtain the energy deposition efficiency, and the magnitude of the released shock wave is calculated, thereby achieving discharge parameter optimization and improving energy conversion efficiency.

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