A phase change heat transfer prediction method based on hard constraint DeepONet

Abstract

The application provides a phase change heat transfer prediction method based on a hard constraint DeepONet, and belongs to the field of metallurgical temperature prediction, and comprises the following steps: inputting an initial field characteristic vector, normalized space-time coordinates and normalized physical time into a physical information deep operator network to obtain an unconstrained prediction residual, wherein the physical information deep operator network comprises a branch network, a trunk network and a fusion layer; adding a hard constraint mapping to an output end of the physical information deep operator network to obtain a final normalized predicted temperature field; and calculating a partial differential equation residual based on a thermodynamic law according to the final normalized predicted temperature field to update network parameters of the physical information deep operator network. The method of the application can ensure prediction accuracy, and the time consumption of reasoning calculation is far lower than that of a traditional numerical simulation method, so that the prediction and update of a one-dimensional temperature field and a phase interface can be completed within a millisecond level time, and there is no boundary and initial loss.

Description

Technical Field

[0001] This invention belongs to the field of metallurgical temperature prediction, specifically involving a phase change heat transfer prediction method based on hard-constrained DeepONet. Background Technology

[0002] In industrial processes such as continuous casting in metallurgy, the real-time and accurate prediction of one-dimensional phase change heat transfer processes (such as the solidification of molten steel) is a key technology for controlling billet quality and avoiding internal defects. This type of physical process can be abstracted as a Stefan problem with a moving boundary. Its essential difficulty lies in the fact that the solid-liquid interface migrates over time, causing nonlinear abrupt changes in thermophysical parameters at the interface, accompanied by intense absorption or release of latent heat. Therefore, how to accurately and in real-time predict the one-dimensional temperature field and interface movement has remained a challenging technical problem in the field of industrial thermal control.

[0003] Currently, there are two main technical approaches for predicting one-dimensional phase change heat transfer processes:

[0004] One type is numerical simulation techniques based on the Finite Volume Method (FVM) or the Finite Element Method (FEM). While these techniques can achieve high simulation accuracy by using fine mesh generation and enthalpy methods to handle phase transition interfaces, they require iteratively solving large-scale algebraic equations, resulting in extremely high computational costs and making it difficult to meet the stringent requirements of millisecond-level real-time control for prediction speed.

[0005] Another type is the fast prediction technique based on Physics-Informed Neural Networks (PINN), which has been introduced in recent years. This technique constructs the partial differential equation of heat conduction, initial conditions, and boundary conditions as constraints in the loss function, and trains a multilayer perceptron using algorithms such as stochastic gradient descent. This enables rapid solution of the parameterized partial differential equation during the inference phase, thus eliminating the dependence on grids in traditional numerical methods. However, in practical one-dimensional phase change heat transfer prediction, this type of technique has revealed several fundamental flaws that lead to distorted prediction results, severely restricting its reliable deployment in industrial fields.

[0006] The one-dimensional phase change heat transfer prediction method based on PINN has the following main technical shortcomings:

[0007] (1) Inaccurate boundary temperature prediction: This method uses a soft constraint mechanism to weight and balance various residuals. During training, it is difficult to ensure that the Dirichlet condition is strictly satisfied at the wall boundary, resulting in non-physical drift and discontinuity in the predicted boundary temperature. This directly causes the calculation deviation of the phase interface position, making the prediction of key process parameters such as the thickness of the solidified shell unreliable.

[0008] (2) Initial time prediction failure: At the spacetime origin (the boundary between the initial time and the wall), the initial high temperature field and the instantaneously applied low temperature boundary condition form a singular conflict. Ordinary network structures are prone to gradient oscillations or explosions at this point, causing the temperature distribution in the very early solidification stage to deviate completely from the true physical value, resulting in the initial phase transition behavior being unable to be accurately captured.

[0009] (3) Long-term prediction of energy non-conservation: If the correction term in the boundary condition handling strategy is not constructed reasonably, it will introduce a false volume heat source near the boundary layer, which will destroy the overall energy conservation of the system. As the prediction time increases, the temperature field will deviate from the actual physical law, making this technology completely unreliable for long process such as continuous casting.

[0010] (4) Insufficient generalization ability under varying operating conditions: This technology is usually trained on a single fixed operating condition. Once the casting temperature, cooling intensity or material properties change, it is necessary to retrain at a high cost. This lack of operator mapping ability makes it impossible for the technology to adapt to the multivariable and variable operating conditions common in industrial production environments online, and it is difficult to form a general heat conduction prediction method that can be engineered. Summary of the Invention

[0011] To address the shortcomings of existing technologies, this application proposes a phase change heat transfer prediction method based on hard-constrained DeepONet. The aim is to overcome the deficiencies of existing Physical Information Neural Networks (PINNs) in solving one-dimensional Stefan phase change heat transfer problems, such as boundary soft constraint failure, spatiotemporal singularity oscillations, and training instability caused by cross-order-of-magnitude physical property parameters. By coupling a structured analytical mapping layer consisting of physical basis terms and distance suppression terms at the operator network output, the model is forced to strictly satisfy initial and boundary conditions in mathematical definition, thus solving the problems of boundary leakage and spurious heat sources. Simultaneously, a self-similar solution based on the hyperbolic tangent function is used to simulate and smooth the singularity of the spatiotemporal origin, and a completely dimensionless partial differential equation residual driving mechanism is employed. This aims to achieve a real-time phase change heat transfer numerical solution acceleration scheme that combines high physical fidelity with millisecond-level inference speed.

[0012] In a first aspect, the present invention provides a phase change heat transfer prediction method based on hard-constrained DeepONet, comprising:

[0013] The physical domain data of one-dimensional continuous casting or phase transformation process is obtained, and the physical domain data is preprocessed by spatiotemporal discretization to obtain normalized spatiotemporal coordinates, normalized spatial domain, normalized physical time and normalized cold wall temperature.

[0014] In the preprocessed spatial domain, m discrete temperature sensor sampling points are set up. The normalized initial temperature field distribution containing Gaussian random perturbation is collected through the temperature sensor sampling points. The m normalized initial temperature field distributions are used to construct an initial field feature vector with dimension m.

[0015] The initial field feature vector, normalized spatiotemporal coordinates, and normalized physical time are input into the physical information deep operator network to obtain the unconstrained prediction residual. The physical information deep operator network includes a branch network, a backbone network, and a fusion layer. The branch network is used to extract features from the initial field feature vector to obtain a high-dimensional implicit feature vector. The backbone network is used to extract features from the spatiotemporal coordinate query points to obtain a continuous basis feature vector. The high-dimensional implicit feature vector and the continuous basis features are passed through the fusion layer to obtain the unconstrained prediction residual.

[0016] Hard constraint mapping is added to the output of the physical information deep operator network to obtain the final normalized predicted temperature field.

[0017] Based on the final normalized predicted temperature field, the network parameters of the physical information deep operator network are updated by calculating the residuals of the partial differential equations based on the laws of thermodynamics.

[0018] Both the branch network and the backbone network are composed of multilayer perceptrons. The fusion layer is used to perform element-wise inner product calculation on the high-dimensional implicit feature vector and the continuous basis features to obtain the fused features.

[0019] The unconstrained prediction residual is expressed as follows:

[0020] ;

[0021] in, For unconstrained prediction residuals, Let k be the k-th high-dimensional implicit eigenvector. Let be the eigenvectors of the k-th continuous basis. Here, p is the bias term, x is the normalized spatiotemporal coordinate, and t is the normalized physical time.

[0022] The output of the physical information deep operator network is given a hard constraint mapping to obtain the final normalized predicted temperature field, including:

[0023] Based on the normalized cold wall temperature and the initial field eigenvector, the physical basis term for simulating the thermal diffusion wave penetration process is constructed using the hyperbolic tangent function;

[0024] On the physical boundary of the residual output by the backbone network, a composite distance suppression term is constructed. The composite distance suppression term is an analytical distance decay function, which is zero on the physical boundary.

[0025] Based on the unconstrained prediction residual, the physical basis term and the composite distance suppression term are nonlinearly combined to obtain the final normalized predicted temperature field.

[0026] Based on the normalized cold wall temperature and the initial field eigenvector, the physical basis term for simulating the thermal diffusion wave penetration process is constructed using the hyperbolic tangent function, as expressed below:

[0027] ;

[0028] in, For physical basis terms, This is the normalized cold wall temperature. For the initial field eigenvectors, To control the adjustable hyperparameters for the steepness of the thermal diffusion wavefront, To prevent values ​​with a denominator of zero, x represents the normalized spacetime coordinates, and t represents the normalized physical time. ( ) is the hyperbolic tangent function.

[0029] The composite distance suppression term is expressed as follows:

[0030] ;

[0031] in, This is a composite distance suppression term. To control the attenuation coefficient of the spatial boundary suppression intensity, The attenuation coefficient of the time boundary suppression intensity. ( ) is the hyperbolic tangent function.

[0032] The method based on unconstrained prediction residuals nonlinearly combines the physical basis term and the composite distance suppression term to obtain the final normalized predicted temperature field, as expressed below:

[0033] ;

[0034] in, To achieve the final normalized predicted temperature field, For physical basis terms, This is a composite distance suppression term. This represents the unconstrained prediction residuals.

[0035] The process of updating the network parameters of the physical information deep operator network by calculating the residuals of partial differential equations based on thermodynamic laws, according to the final normalized predicted temperature field, includes:

[0036] A smooth liquid phase volume fraction was constructed using the hyperbolic tangent function;

[0037] Based on the smooth liquid phase volume fraction, the equivalent density, equivalent specific heat capacity, and equivalent thermal conductivity of the sampling points are dynamically calculated using a linear mixing model.

[0038] The equivalent density, equivalent specific heat capacity, and equivalent thermal conductivity of the sampling points are normalized to obtain dimensionless equivalent density, dimensionless equivalent specific heat capacity, and dimensionless equivalent thermal conductivity.

[0039] The residuals of the partial differential equations are obtained using the effective heat capacity enthalpy method based on the smooth liquid phase volume fraction, dimensionless equivalent density, dimensionless equivalent specific heat capacity, and dimensionless equivalent thermal conductivity.

[0040] A loss function is constructed based on the residuals of the partial differential equations, and the network parameters of the physical information deep operator network are updated using the loss function.

[0041] The volume fraction of the smooth liquid phase is calculated using the following formula:

[0042] ;

[0043] in, The normalized temperature. To control the smoothing hyperparameter of the transition width of the paste region, tanh() represents the volume fraction of the smooth liquid phase, and tanh() is the hyperbolic tangent function.

[0044] Based on the smoothed liquid phase volume fraction, a linear mixing model is used to dynamically calculate the equivalent density, equivalent specific heat capacity, and equivalent thermal conductivity of the sampling points. The calculation formulas are as follows:

[0045] ;

[0046] in, The equivalent density of the sampling points. The equivalent specific heat capacity of the sampling point. The equivalent thermal conductivity at the sampling point is... For smooth liquid phase volume fraction, The density of the pure liquid phase is given. The specific heat capacity of the pure liquid phase. The thermal conductivity of the pure liquid phase is . The density of the pure solid phase. The specific heat capacity of a pure solid phase, It is the equivalent thermal conductivity of the pure solid phase.

[0047] The residual of the partial differential equation is expressed as follows:

[0048] ;

[0049] in, For the residuals of the partial differential equation, For dimensionless equivalent density, For dimensionless equivalent specific heat capacity, The dimensionless equivalent thermal conductivity The normalized temperature. For Fourier numbers, For Stefan numbers, denoted as the smoothed liquid volume fraction, x as the normalized spatiotemporal coordinate, and t as the normalized physical time.

[0050] Beneficial effects:

[0051] This application proposes a phase change heat transfer prediction method based on hard-constraint DeepONet, which achieves the following technical effects in practical applications:

[0052] (1) Effectively improve the accuracy of temperature prediction at the boundary and initial stages:

[0053] Through a hard-constraint mapping mechanism, the predicted temperature field is directly ensured to strictly match the actual physical settings at the boundary and at the initial moment during inference calculations. This makes the calculated values ​​of the solidified shell thickness and the phase interface advancement position during one-dimensional phase transitions more reliable, providing accurate temperature data support for setting initial solidification control parameters in processes such as continuous casting.

[0054] (2) Ensuring energy conservation and stability during long-term continuous prediction processes:

[0055] By structurally eliminating the generation of spurious heat sources, this method avoids overall temperature field drift during long-term simulations. In processes such as continuous casting of steel that last for tens of minutes or even hours, this method can ensure that the temperature prediction results always follow the true physical laws, guaranteeing that the prediction of the final solidification position does not become distorted over time, and reducing the risk of billet quality defects caused by prediction deviations.

[0056] (3) Enhance on-site adaptability under multiple working conditions:

[0057] This method possesses the capability for one-time operator mapping to different casting temperatures, cooling intensities, and material properties. Once trained, it can be directly applied to various production conditions without requiring retraining or manual model adjustments due to changes in conditions. In continuous casting scenarios involving alternating production of multiple steel grades and specifications, it can quickly adapt to changes in process parameters, significantly reducing debugging costs and response delays caused by switching prediction models.

[0058] (4) Meets the rapid response requirements for industrial-grade real-time temperature field prediction:

[0059] This method, while ensuring prediction accuracy, consumes significantly less time for inference calculations than traditional numerical simulation methods, enabling the prediction and updating of the one-dimensional temperature field and phase interface within milliseconds. This real-time prediction capability can be embedded in online applications such as the dynamic control system for secondary cooling water in continuous casting, providing timely and accurate temperature field information for real-time regulation of cooling water volume and optimization of the billet solidification process, effectively supporting online closed-loop control of billet quality. Attached Figure Description

[0060] Figure 1 Flowchart of a phase change heat transfer prediction method based on hard constraint DeepONet according to an embodiment of the present invention;

[0061] Figure 2 The overall architecture and hard constraints of this invention are illustrated in the following schematic diagram.

[0062] Figure 3 Smooth diffusion characteristics of the physical basis term over time in an embodiment of the present invention;

[0063] Figure 4 Spatiotemporal boundary suppression characteristic diagram of composite distance term in an embodiment of the present invention;

[0064] Figure 5 A schematic diagram of the global temperature error curve over time according to an embodiment of the present invention;

[0065] Figure 6 A schematic diagram of the interface position prediction and truth curve over time according to an embodiment of the present invention;

[0066] Figure 7 A comparison chart of the evolution of the absolute prediction error over time and the tracking of the solid-liquid phase transition interface position at an example initial temperature in this embodiment of the invention;

[0067] Figure 8 A schematic diagram illustrating the characteristics of smooth liquid phase volume fraction as a function of temperature based on the hyperbolic tangent function in an embodiment of the present invention. Detailed Implementation

[0068] The specific implementation methods of this application will be further described in detail below with reference to the accompanying drawings and embodiments.

[0069] Example 1:

[0070] This embodiment provides a phase change heat transfer prediction method based on hard-constraint DeepONet, such as... Figure 1 As shown, it includes:

[0071] Step S1: Obtain physical domain data in one-dimensional continuous casting or phase transformation process, and perform spatiotemporal discretization preprocessing on physical domain data to obtain the preprocessed spatial domain and spatiotemporal coordinate query points;

[0072] In this embodiment, the macroscopic physical quantities of the one-dimensional Stefan phase change heat transfer problem are transformed into dimensionless data suitable for neural network processing. The specific implementation process includes:

[0073] 1. Obtain basic physical data: Obtain the total length L of the computational domain and the maximum prediction time for one-dimensional continuous casting or phase transformation processes. and Dirichlet boundary conditions (such as cold wall temperature) Far-field liquid temperature With the melting point temperature of the material At the same time, the initial temperature distribution field is obtained. ;

[0074] 2. Spatiotemporal coordinates and temperature normalization: To eliminate the adverse effects of differences in physical dimensions on the backpropagation of gradients in neural networks, the original physical space coordinates are normalized. and physical time Perform linear scaling to map to a normalized spatiotemporal computation domain:

[0075] ;

[0076] Simultaneously, the actual temperature T is linearly normalized to obtain the dimensionless normalized temperature. :

[0077] ;

[0078] In formula (2), T is a variable, which can be expressed as: , or , This refers to the normalized temperature, and is also a general term for the normalized temperatures of various types, making... This directly corresponds to the material's melting point, thus simplifying the subsequent calculations of the liquid phase fraction and latent heat term. , , Also obtained from the previous , , After normalization using formula (2), we obtain the result, where The spatial mean (or initial field eigenvector) of the initial temperature distribution field at the input of the branch network represents the global high-temperature benchmark under this operating condition. The value of the far-field liquid temperature;

[0079] Step S2: Set m discrete temperature sensor sampling points in the preprocessed spatial domain, collect the normalized initial temperature field distribution containing Gaussian random perturbation through the temperature sensor sampling points, and construct an initial field feature vector of dimension m using m normalized initial temperature field distributions.

[0080] In this embodiment, the initial field sensor sampling occurs in the normalized spatial domain. Inside, settings A set of discrete sensor points are used to collect a normalized initial temperature field distribution containing Gaussian random perturbations, constructing a dimensionless model. initial field eigenvectors , as input to subsequent branch networks;

[0081] Step S3: Input the initial field feature vector and the spatiotemporal coordinate query point into the physical information deep operator network to obtain the unconstrained prediction residual. The physical information deep operator network includes a branch network, a backbone network, and a fusion layer. The branch network is used to extract features from the initial field feature vector to obtain a high-dimensional implicit feature vector. The backbone network is used to extract features from the spatiotemporal coordinate query point to obtain a continuous basis feature vector. The high-dimensional implicit feature vector and the continuous basis features are passed through the fusion layer to obtain the unconstrained prediction residual.

[0082] In this embodiment, a Physical Information Deep Operator Network (PI-DeepONet) is constructed to perform dual-branch feature extraction;

[0083] like Figure 2 As shown, a dual-branch architecture is used to handle the initial field distribution and spatiotemporal coordinates separately, thereby endowing the model with powerful operator mapping capabilities. The specific implementation process includes:

[0084] 1. Branch Net Feature Extraction: Extract the discretized initial field feature vector obtained in step 1. Input branch network. The branch network is composed of a multi-layer perceptron (MLP), which undergoes layer-by-layer linear transformation and nonlinear activation (hyperbolic tangent). After the function, the output dimension is High-dimensional implicit feature vectors :

[0085] ;

[0086] in, This is a branch network.

[0087] 2. Trunk Net Feature Extraction: Querying normalized spatiotemporal coordinates... Input to the backbone network. The backbone network also uses a multilayer perceptron, responsible for extracting continuous basis features of spatiotemporal coordinates, and outputting the same dimension as the branch networks. 3D feature vector :

[0088] ;

[0089] in, It serves as the backbone network.

[0090] 3. Inner Product Combination and Unconstrained Residual Generation: This involves combining the feature vectors output by the branch network... The feature vector output by the backbone network Perform element-wise inner product calculations and add learnable scalar bias terms. This yields the basic output of PI-DeepONet, namely the unconstrained prediction residuals. :

[0091] ;

[0092] It should be noted that the data generated at this time... The results of a network's free fitting without any physical boundary constraints cannot guarantee that the extreme boundary conditions at the spacetime origin are met, and are highly susceptible to inducing spurious volumetric heat sources in subsequent calculations. Therefore, it is necessary to... The physical prior model layer in step S4 is sent for hard constraint mapping.

[0093] Step S4: Add hard constraint mapping to the output of the physical information deep operator network to obtain the final normalized predicted temperature field;

[0094] In this embodiment, a physical prior hypothesis (Ansatz) is constructed for hard constraint mapping;

[0095] Step S4 is the core innovation of this invention. Because the global high temperature in the initial state of a one-dimensional Stefan problem and the instantaneously applied low wall temperature will overlap at the spacetime origin... The occurrence of drastic physical transitions (Dirichlet boundary conflicts) at certain points easily leads to gradient explosion or the introduction of non-physical spurious volumetric heat sources when using traditional soft constraints. Therefore, this embodiment couples a hard constraint mapping architecture at the output of PI-DeepONet, with the specific implementation steps as follows:

[0096] (1) Calculate the physical basis terms ;

[0097] In this embodiment, in order to smoothly handle spatiotemporal singularities and satisfy the self-similar evolution characteristics of the heat conduction equation, this embodiment abandons the traditional linear interpolation basis and adopts the hyperbolic tangent function ( Construct the physical basis terms for simulating the thermal diffusion wave penetration process. For example... Figure 3 As shown ( , Its preferred mathematical expression is:

[0098] ;

[0099] in, For physical basis terms, To control the adjustable hyperparameters for the steepness of the thermal diffusion wavefront, To prevent values ​​with a denominator of zero. This is the normalized cold wall temperature (left boundary). The spatial mean of the initial temperature distribution field at the input of the branch network (i.e., the initial field eigenvector) represents the global high-temperature benchmark under this operating condition, and is also... The value of the far-field liquid temperature, Normalized spacetime coordinates (representing mapping to) (normalized spatial dimension of the interval) Normalized physical time (representing the time mapped to) (Normalized time dimension of the interval).

[0100] In this embodiment, the constructed physical basis term is mainly a basis function simulating heat diffusion, and its main functional form is as follows: Figure 3 As shown, its expression is a self-similar solution of the traditional heat conduction equation. Using it as a basis, the physical law of heat diffusion according to the square root of time can be directly hard-coded into the model. The output is 0 at the left boundary (x=0), and the output of the left boundary is the normalized cold wall temperature. The output is 1 at the far field (x=1), and the output at the right boundary is... (Right now That is, at the initial time point (t=0) and the initial spatial point, the output of the network is masked, and the true value corresponding to the physical basis terms is used instead to achieve the purpose of hard constraint, so that the boundary and initial errors are 0.

[0101] Physical mechanism explanation: when time At time t, the basis function exhibits an extremely steep step response, perfectly locking the initial domain temperature except for the boundary. As time t gradually increases, the function shows a smooth nonlinear diffusion degradation, perfectly matching the heat conduction attenuation law in the real physical world. The second spatial derivative of this nonlinear basis is not zero, effectively avoiding the defect of the network losing its heat diffusion fitting ability caused by the linear basis.

[0102] (2) Calculate the composite distance suppression term ;

[0103] In this embodiment, to ensure that the residual output by the backbone network does not interfere with the known physical boundaries (spatial boundaries and initial time), this embodiment constructs an analytical distance decay function that is strictly zero at the boundaries. For example... Figure 4 As shown, the specific expression for the composite distance suppression term is:

[0104] ;

[0105] in, and These are the attenuation coefficients for controlling the suppression intensity of the spatial and temporal boundaries, respectively. It should be noted that whether or not hard constraints are applied to the left and right boundaries is optional; if only the left boundary is hard-constrained, then this coefficient should be removed from the equation. item, Figure 5 and Figure 6 The diagram shows the case where both the left and right boundaries are subject to hard constraints.

[0106] Physical mechanism explanation: When the query coordinates are located on the left wall Right boundary or initial time hour, The value of strictly decays to 0; within the computational domain, The value rapidly approaches 1. This is equivalent to adding a hard boundary constraint layer to the network residual. It forces the output to 0 at the time zero point and the spatial boundary, preventing the model prediction from taking effect at that point, and in conjunction with the physical basis term, outputs the true value at the boundary.

[0107] (3) Combined output of predicted temperature field without spurious heat sources:

[0108] The physical basis terms obtained in (1) (2) The distance suppression term obtained and the network unconstrained residuals obtained in step S3 A nonlinear combination is performed to generate the final normalized predicted temperature field. :

[0109] ;

[0110] Proof of the principle for eliminating spurious heat sources: due to the distance function The value is strictly zero at the boundary, therefore the influence of network prediction error on the boundary is completely masked, and the value at the boundary is entirely determined by the laws of real physics. Dominant. This architecture, from a mathematical definition, completely eliminates the non-physical residual fluctuations generated by ordinary PINN when fitting the boundary, thereby avoiding being misjudged as a false volume heat source in the backpropagation of partial differential equations, and significantly improving the physical fidelity of the model under extreme conditions;

[0111] Step S5: Based on the final normalized predicted temperature field, the residuals of the partial differential equations are calculated using thermodynamic laws, and the network parameters of the physical information deep operator network are updated.

[0112] After obtaining the hard-constrained predicted temperature field, this embodiment will calculate the residuals of the partial differential equation (PDE) based on the laws of thermodynamics to drive the network parameter update, specifically including:

[0113] (1) Property updates based on effective heat capacity enthalpy method and smooth liquid phase fraction:

[0114] To address the problem of automatic differential gradient breakage caused by discontinuities in physical properties at the phase transition interface, this embodiment introduces a method based on hyperbolic tangent (… ) continuous smooth liquid phase volume fraction Its mathematical construction is as follows:

[0115] ;

[0116] in, Normalized temperature; To control the smoothing hyperparameter of the transition width of the pasty region; refer to Figure 8 As shown, different Values ​​(e.g.) This determines the smoothness of the latent heat release process of phase change;

[0117] Based on calculations The equivalent density under the current collocation point is dynamically calculated using a linear mixture model. Equivalent specific heat capacity With equivalent thermal conductivity ;

[0118] ;

[0119] in, The equivalent density of the sampling points. The equivalent specific heat capacity of the sampling point. The equivalent thermal conductivity at the sampling point is... For smooth liquid phase volume fraction, The density of the pure liquid phase is given. The specific heat capacity of the pure liquid phase. The thermal conductivity of the pure liquid phase is . The density of the pure solid phase. The specific heat capacity of a pure solid phase. This represents the equivalent thermal conductivity of a pure solid phase. Where the subscript... and These represent the physical properties of the pure liquid phase and the pure solid phase, respectively.

[0120] (2) Calculate the PDE physical loss function:

[0121] In this embodiment, automatic differentiation technology is used to differentiate the predicted temperature field and dynamic physical property parameters. Based on the thermodynamic law of conservation of energy, a physical residual of the one-dimensional phase change heat conduction partial differential equation is constructed using an expression that separates sensible heat and latent heat. :

[0122] Because of physical time ,so and ;

[0123] Similarly, physical space Strictly following the chain rule and employing the effective heat capacity enthalpy method, the spatial second derivative and gradient term are expanded as follows:

[0124] ;

[0125] Substituting the above coordinate scaling relationship into the physical equation, we obtain the physical residual based on the normalized coordinate expansion:

[0126] ;

[0127] To completely eliminate the excessively large absolute values ​​of physical parameters in actual industrial scenarios (such as...) Product up to To mitigate the risk of gradient explosion (at magnitude of 1 / 3 oz), this invention further introduces dimensionless temperature. ( ) and solid-phase reference properties ( Divide both sides of the equation by the characteristic enthalpy coefficient. The process involves making the quantities completely dimensionless, and the dimensionless property coefficients and physical quantities involved are defined as follows:

[0128] ;

[0129] Solid thermal diffusivity :

[0130] Thermal diffusivity is the coefficient of thermal diffusivity in the solid phase of a material, used to characterize the rate at which heat diffuses within the material.

[0131] ;

[0132] Finally, the dimensionless partial differential equation residuals used for network training are extracted:

[0133] ;

[0134] in, The Fourier number represents the rate of heat diffusion. The Stefan number represents the relative intensity of sensible heat and latent heat. Far-field liquid temperature With the melting point temperature of the material The temperature difference between them This refers to the latent heat of phase transformation in the material.

[0135] This dimensionless residual formula not only employs a normalized density with a smooth transition, but also... It also fully accounts for the dimensionless spatial gradient of thermal conductivity caused by the movement of the phase transition interface. This eliminates mathematical errors caused by neglecting the derivative terms of variable physical properties; at the same time, it utilizes and The quasi-number converges the residual to This significantly improves the stability of backpropagation in deep learning frameworks;

[0136] (3) Loss function construction and optimization:

[0137] In this embodiment, since the hard constraint architecture in step S4 strictly guarantees that the initial and boundary conditions are met, the total loss function consists only of the physical equation residuals within the spatiotemporal domain:

[0138] ;

[0139] in, The number of coordinate points randomly sampled within the spatiotemporal domain. For loss function, Let be the residual of the partial differential equation at the i-th sampling point.

[0140] (4) Model hyperparameter adaptation and inference:

[0141] In this embodiment, the geometric boundary and physical attenuation hyperparameters involved are... ) and phase transition smoothing hyperparameter ( It can be preset based on material properties, or it can be used as learnable parameters of the model to be backpropagated and optimized synchronously in the Adam optimizer along with the network weights, thereby achieving adaptive physical manifold capture for different initial working conditions.

[0142] Once the model converges, inputting any given initial state will allow the model to output the full-time and spatiotemporal temperature field prediction results. The temperature field is updated to the real physical temperature field through inverse normalization. :

[0143] Physical space coordinate restoration:

[0144] ;

[0145] Physical Time Restoration:

[0146] ;

[0147] Physical temperature reduction:

[0148] ;

[0149] And by tracking The isotherm positions are determined, and the dynamic evolution curve of the solid-liquid phase transition interface over time is directly extracted. This is then combined with simulation algorithms for error analysis, such as... Figure 5 , Figure 6 and Figure 7 As shown. Figure 5 It is the change of prediction error over time. From this, we can see that the average error at the initial time position t=0 is 0, while Figure 6 The output is a comparison between the predicted and actual positions displayed on the screen. The maximum error is only 4.93mm. Figure 7 The prediction error varies with space. It can be seen that the error is 0 at the left boundary (cold wall) and the right boundary (far field), which achieves a good hard constraint effect. Secondly, the overall error is on the order of 10 to the power of 0, which is relatively small.

[0150] To address the technical problems in existing one-dimensional phase change heat transfer prediction technologies, such as boundary inaccuracies, initial-time prediction failures, long-term energy non-conservation, and insufficient generalization ability under varying operating conditions, this application provides a phase change heat transfer prediction method based on hard-constraint DeepONet. Its key features lie in its targeted solutions to the background technical problems from the following four aspects:

[0151] (1) To address the issues of inaccurate boundary temperature prediction and energy non-conservation: An analytical mapping layer consisting of physical basis terms and composite distance suppression terms is introduced at the output of the deep operator network, which forces the network-predicted temperature field to mathematically satisfy the Dirichlet boundary conditions and initial conditions. This hard constraint construction method eliminates the contradiction between the boundary conditions and the residuals of partial differential equations in traditional soft constraint methods, prevents boundary value drift, and fundamentally avoids the generation of false volume heat sources, thus ensuring the energy conservation of the system in the entire time and space domain.

[0152] (2) To address the initial time prediction failure problem: A self-similar analytical basis term based on the hyperbolic tangent function was constructed to smooth the singular conflict between the initial high-temperature field at the spacetime origin and the instantaneous low-temperature boundary of the wall in the Stefan problem. The quasi-step evolution characteristics exhibited by this basis term in the very early time can accurately capture the physical penetration process of thermal diffusion waves into the material interior, effectively suppressing the gradient oscillation and numerical collapse of ordinary network structures near this singular point, and ensuring the reliability of temperature field prediction in the very early solidification stage.

[0153] (3) To address the problem of insufficient generalization ability under varying operating conditions: a dual-branch deep operator network architecture is adopted, enabling the model to learn operator mappings from different initial temperature distributions and different cooling boundary conditions to the complete temperature field in one go. After training, the model can directly perform millisecond-level inference prediction for any new operating condition without retraining, thereby improving the ability of heat conduction prediction technology from single-condition numerical fitting to multi-condition real-time generalization.

[0154] (4) To address the numerical stability problem of model training: a dimensionless physical loss function based on Stefan number and Fourier number was designed to uniformly scale the physical residual to the unit order of magnitude, effectively suppressing the risk of gradient runaway caused by the multiplication of physical property parameters such as density and specific heat across orders of magnitude. Combined with the smooth liquid phase volume fraction model, the numerical stability of automatic differential backpropagation during the tracking of complex phase change interfaces was greatly improved.

[0155] Example 2:

[0156] This embodiment proposes an electronic device, including: one or more processors, and a memory, wherein the memory is used to store instructions, and when the instructions are executed by the one or more processors, the one or more processors execute the phase change heat transfer prediction method based on hard constraint DeepONet.

[0157] The electronic device may be a mobile phone, computer, or tablet computer, etc., and includes a memory and a processor. The memory stores a computer program, which, when executed by the processor, implements a phase change heat transfer prediction method based on hard-constraint DeepONet as described in the embodiments. It is understood that the electronic device may also include input / output (I / O) interfaces and communication components.

[0158] The processor is used to execute all or part of the steps in the phase change heat transfer prediction method based on hard-constraint DeepONet as described in the above embodiments. The memory is used to store various types of data, which may include, for example, instructions for any application or method in the electronic device, as well as application-related data.

[0159] The processor can be implemented as an Application Specific Integrated Circuit (ASIC), Digital Signal Processor (DSP), Programmable Logic Device (PLD), Field Programmable Gate Array (FPGA), controller, microcontroller, microprocessor, or other electronic components, and is used to execute the phase change heat transfer prediction method based on hard constraint DeepONet described in the above embodiments.

[0160] Example 3:

[0161] This embodiment proposes a computer-readable storage medium that stores executable instructions. When these instructions are executed, if they are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium.

[0162] The computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the phase change heat transfer prediction method based on hard constraint DeepONet described in various embodiments of this application.

[0163] The aforementioned storage media include: flash memory, hard disks, multimedia cards, card-type memory (e.g., SD (Secure Digital Memory Card) or DX (Memory Data Register, MDR) memory), random access memory (RAM), static random-access memory (SRAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), programmable read-only memory (PROM), magnetic storage, disks, optical discs, servers, APP (Application) application stores, and other media capable of storing program verification codes. These media store computer programs, which, when executed by a processor, can implement the various steps of the aforementioned hard-constraint DeepONet-based phase change heat transfer prediction method.

[0164] Example 4:

[0165] This embodiment proposes a computer program product, including a computer program or instructions, which, when executed by a processor, implements the phase change heat transfer prediction method based on hard constraint DeepONet.

[0166] Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or part of the technical solution, can be embodied in the form of a computer program product.

[0167] The various embodiments in this application are described in a progressive manner. The same or similar parts between the various embodiments can be referred to each other. Each embodiment focuses on describing the differences from other embodiments.

[0168] The scope of protection of this application is not limited to the embodiments described above. Obviously, those skilled in the art can make various modifications and variations to this disclosure without departing from the scope and spirit of this disclosure. If such modifications and variations fall within the scope of equivalent technology of this disclosure, then the intent of this disclosure also includes such modifications and variations.

Claims

1. A phase change heat transfer prediction method based on hard-constraint DeepONet, characterized in that, include: The physical domain data of one-dimensional continuous casting or phase transformation process is obtained, and the physical domain data is preprocessed by spatiotemporal discretization to obtain normalized spatiotemporal coordinates, normalized spatial domain, normalized physical time and normalized cold wall temperature. In the preprocessed spatial domain, m discrete temperature sensor sampling points are set up. The normalized initial temperature field distribution containing Gaussian random perturbation is collected through the temperature sensor sampling points. The m normalized initial temperature field distributions are used to construct an initial field feature vector with dimension m. The initial field feature vector, normalized spatiotemporal coordinates, and normalized physical time are input into the physical information deep operator network to obtain the unconstrained prediction residual. The physical information deep operator network includes a branch network, a backbone network, and a fusion layer. The branch network is used to extract features from the initial field feature vector to obtain a high-dimensional implicit feature vector. The backbone network is used to extract features from the spatiotemporal coordinate query points to obtain a continuous basis feature vector. The high-dimensional implicit feature vector and the continuous basis features are passed through the fusion layer to obtain the unconstrained prediction residual. Hard constraint mapping is added to the output of the physical information deep operator network to obtain the final normalized predicted temperature field. Based on the final normalized predicted temperature field, the network parameters of the physical information deep operator network are updated by calculating the residuals of the partial differential equations based on the laws of thermodynamics.

2. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 1, characterized in that, Both the branch network and the backbone network are composed of multilayer perceptrons. The fusion layer is used to perform element-wise inner product calculation on the high-dimensional implicit feature vector and the continuous basis features to obtain the fused features.

3. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 1, characterized in that, The unconstrained prediction residual is expressed as follows: ； in, For unconstrained prediction residuals, Let k be the k-th high-dimensional implicit eigenvector. Let be the eigenvectors of the k-th continuous basis. Here, p is the bias term, x is the normalized spatiotemporal coordinate, and t is the normalized physical time.

4. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 1, characterized in that, The output of the physical information deep operator network is given a hard constraint mapping to obtain the final normalized predicted temperature field, including: Based on the normalized cold wall temperature and the initial field eigenvector, the physical basis term for simulating the thermal diffusion wave penetration process is constructed using the hyperbolic tangent function; On the physical boundary of the residual output by the backbone network, a composite distance suppression term is constructed. The composite distance suppression term is an analytical distance decay function, which is zero on the physical boundary. Based on the unconstrained prediction residual, the physical basis term and the composite distance suppression term are nonlinearly combined to obtain the final normalized predicted temperature field.

5. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 4, characterized in that, Based on the normalized cold wall temperature and the initial field eigenvector, the physical basis term for simulating the thermal diffusion wave penetration process is constructed using the hyperbolic tangent function, as expressed below: ； in, For physical basis terms, This is the normalized cold wall temperature. For the initial field eigenvectors, To control the adjustable hyperparameters for the steepness of the thermal diffusion wavefront, To prevent values ​​with a denominator of zero, x represents the normalized spacetime coordinates, and t represents the normalized physical time. ( ) is the hyperbolic tangent function.

6. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 4, characterized in that, The composite distance suppression term is expressed as follows: ； in, This is a composite distance suppression term. To control the attenuation coefficient of the spatial boundary suppression intensity, The attenuation coefficient of the time boundary suppression intensity. ( ) is the hyperbolic tangent function.

7. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 4, characterized in that, The method based on unconstrained prediction residuals nonlinearly combines the physical basis term and the composite distance suppression term to obtain the final normalized predicted temperature field, as expressed below: ； in, To achieve the final normalized predicted temperature field, For physical basis terms, This is a composite distance suppression term. This represents the unconstrained prediction residuals.

8. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 1, characterized in that, The process of updating the network parameters of the physical information deep operator network by calculating the residuals of partial differential equations based on thermodynamic laws, according to the final normalized predicted temperature field, includes: A smooth liquid phase volume fraction was constructed using the hyperbolic tangent function; Based on the smooth liquid phase volume fraction, the equivalent density, equivalent specific heat capacity, and equivalent thermal conductivity of the sampling points are dynamically calculated using a linear mixing model. The equivalent density, equivalent specific heat capacity, and equivalent thermal conductivity of the sampling points are normalized to obtain dimensionless equivalent density, dimensionless equivalent specific heat capacity, and dimensionless equivalent thermal conductivity. The residuals of the partial differential equations are obtained using the effective heat capacity enthalpy method based on the smooth liquid phase volume fraction, dimensionless equivalent density, dimensionless equivalent specific heat capacity, and dimensionless equivalent thermal conductivity. A loss function is constructed based on the residuals of the partial differential equations, and the network parameters of the physical information deep operator network are updated using the loss function.

9. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 8, characterized in that, The volume fraction of the smooth liquid phase is calculated using the following formula: ； in, The normalized temperature. To control the smoothing hyperparameter of the transition width of the paste region, tanh() represents the volume fraction of the smooth liquid phase, and tanh() is the hyperbolic tangent function. Based on the smoothed liquid phase volume fraction, a linear mixing model is used to dynamically calculate the equivalent density, equivalent specific heat capacity, and equivalent thermal conductivity of the sampling points. The calculation formulas are as follows: ； in, The equivalent density of the sampling points. The equivalent specific heat capacity of the sampling point. The equivalent thermal conductivity at the sampling point is... For smooth liquid phase volume fraction, The density of the pure liquid phase is given. The specific heat capacity of the pure liquid phase. The thermal conductivity of the pure liquid phase is . The density of the pure solid phase. The specific heat capacity of a pure solid phase, It is the equivalent thermal conductivity of the pure solid phase.

10. The phase change heat transfer prediction method based on hard-constraint DeepONet according to claim 8, characterized in that, The residual of the partial differential equation is expressed as follows: ； in, For the residuals of the partial differential equation, For dimensionless equivalent density, For dimensionless equivalent specific heat capacity, The dimensionless equivalent thermal conductivity The normalized temperature. For Fourier numbers, For Stefan numbers, denoted as the smoothed liquid volume fraction, x as the normalized spatiotemporal coordinate, and t as the normalized physical time.

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