Dynamic Error Correction for Fine-Wire Thermocouples Based on CRBM-DBN with PINN Constraint

Abstract

In high-temperature testing scenarios that rely on contact, fine-wire thermocouples demonstrate commendable dynamic performance. Nonetheless, their thermal inertia leads to notable dynamic nonlinear inaccuracies, including response delays and amplitude reduction. To mitigate these challenges, a novel dynamic error correction approach is introduced, which combines a Continuous Restricted Boltzmann Machine, Deep Belief Network, and Physics-Informed Neural Network (CDBN-PINN). The unique heat transfer properties of the thermocouple’s bimetallic structure are represented through an Inverse Heat Conduction Equation (IHCP). An analysis is conducted to explore the connection between the analytical solution’s ill-posed nature and the thermocouple’s dynamic errors. The transient temperature response’s nonlinear characteristics are captured using CRBM-DBN. To maintain physical validity and minimize noise amplification, filtered kernel regularization is applied as a constraint within the PINN framework. This approach was tested and confirmed through laser pulse calibration on thermocouples with butt-welded and ball-welded configurations of 0.25 mm and 0.38 mm. Findings reveal that the proposed method achieved a peak relative error of merely 0.83%, superior to Tikhonov regularization by −2.2%, Wiener deconvolution by 20.40%, FBPINNs by 1.40%, and the ablation technique by 2.05%. In detonation tests, the corrected temperature peak reached 1045.7 °C, with the relative error decreasing from 77.7% to 5.1%. Additionally, this method improves response times, with the rise time in laser calibration enhanced by up to 31 ms and in explosion testing by 26 ms. By merging physical constraints with data-driven methodologies, this technique successfully corrected dynamic errors even with limited sample sizes.

1. Introduction

In scenarios involving the measurement of detonation temperature fields in artillery or warheads, as well as the combustion gas temperatures in spacecraft or automotive engines, both solid and liquid fuels, along with propellants, emit significant thermal energy rapidly during combustion. This results in the formation of transient high-temperature thermal flows, which are often accompanied by high-pressure thermal shock effects [1,2]. Accurately capturing the temperature variations within these thermal flows poses a considerable challenge in the industry. Therefore, investigating transient high-temperature testing methods and associated sensors to enhance the precision of temperature measurements is a crucial area of research.

In the realm of transient explosion temperature testing, two prevalent methods are employed: non-contact and contact temperature measurement. The non-contact approach encounters challenges due to the complex nature of fuel and agent compositions, which can result in considerable inaccuracies stemming from difficulties in accurately measuring emissivity. Conversely, contact temperature measurement facilitates direct interaction with heat flow, primarily utilizing thermocouples as the main sensing instruments. However, these thermocouples exhibit a considerable time lag during dynamic testing, a phenomenon attributed to the thermal inertia of their sensing components. This lag is intricately linked to the dynamic properties of the thermocouples [3,4]. When these properties do not meet engineering testing standards, the dynamic response is significantly delayed, causing the observed temperature peak to diverge from the actual value. Dynamic calibration experiments can elucidate the relationship between the thermocouple’s dynamic response and the excitation source, thereby providing a framework for assessing dynamic errors in testing [5,6]. This relationship can be observed in the thermocouple’s thermal conduction effects under thermal stimulation [7] and can also be modeled through a dynamic mathematical framework [8]. Addressing how to leverage this relationship to mitigate dynamic errors during testing is crucial for overcoming this technical challenge.

In the global research trends, transient high-temperature testing has evolved towards integrated systems, combining contact and non-contact methods, and benefiting from advancements in artificial intelligence and sensor fusion. For instance, X. Zhao et al. proposed a fiber Bragg grating (FBG) sensor-embedded thermal protection structure, which assesses the thermal state of aircraft in real-time through the inverse heat conduction method, offering a competitive non-contact supplement [8]. Z. Huang et al. introduced the DMD-ConvLSTM method for correcting temperature measurement bias [9], while A. Zeeshan et al. explored the application of neural networks in flow field prediction, such as LMS-BPNN for turbulent thermal mixing [10]. These trends highlight a shift from isolated calibration towards robust multi-physics frameworks, yet the correction of nonlinear dynamic errors remains challenging. The current stage lacks unified calibration standards for dynamic temperature testing, exacerbating the difficulty in quantifying dynamic errors, and the assessment of environmental factors and model uncertainties has not been sufficiently studied. Although the FFDNN-BLMA proposed by M. Sulaiman et al. optimizes heat transfer, it does not address the robustness to small samples and distribution shifts [11].

Dynamic error correction represents a technique that shifts from merely optimizing data to actively rectifying errors; its implementation is viewed as a dynamic inverse operation [12,13]. This approach aims to mitigate dynamic inaccuracies stemming from the sensor’s limited dynamic properties by reversing the input values using the effective output from the thermocouple, guided by specific mathematical or physical principles [3,14]. To address the hysteresis observed in the thermocouple’s dynamic response, this correction method can be framed as an inverse heat conduction problem under defined conditions. It can be articulated as an IHCP expressed through partial differential equations. The objective is to determine the time-dependent solution of the inverse problem within constrained boundary conditions to accurately reflect a thermal process. The challenges associated with this inverse problem primarily arise from its ill-posed nature, necessitating careful examination before attempting a solution. Selecting a suitable regularization technique is crucial during the resolution phase [15,16].

The theoretical foundation of IHCP is rooted in the ill-posedness of partial differential equations. Numerous algorithms have been developed to address the IHCP within one-dimensional geometric frameworks, including Tikhonov regularization [17] and the sequential function specification method [18]. Frankel J.I. et al. elucidated the transient one-dimensional linear heat equation for thermocouples and introduced an optimal regularization identification technique based on the Non-Integer System Identification (NISI) method to tackle the inverse heat conduction issue [19]. J.G. Bauzin formulated the heat transfer convolution equation utilizing the Laplace transform and derived the correlation between boundary temperature and time through a deconvolution algorithm [20]. Farahani, S. D. et al. proposed an inverse heat transfer search technique [21], that employs a K-type thermocouple as a boundary condition to determine the heat transfer coefficient of an impinging jet, while also investigating how variables such as measured temperature, jet velocity, and the number of thermal conductivity components affect the algorithm’s accuracy. Zhou et al. recommended the use of the Element Differential Method (EDM) in conjunction with an enhanced Levenberg–Marquardt algorithm for predicting boundary heat flux [22]. Additionally, Yu, Z. C. and others explored how the dynamic response of sensors affects surface heat flux predictions using traditional methods such as the Space Marching Method (SMM) and the Calibration Integral Equation Method (CIEM) [23], incorporating Gaussian filtering and future time regularization as their respective regularization strategies. The SSCPO-MABIGTCN model proposed by Chen, G. H. et al. integrates multi-feature data through bidirectional GRU and attention mechanisms, achieving the prediction of thermal errors in machine tools. Compared with the traditional Tikhonov regularization, the uniqueness of the solution is significantly improved [24]. Collectively, these investigations illustrate that when addressing inverse time-varying IHCP, integrating known thermodynamic properties, initial conditions, and boundary conditions with limited temperature data collected over time at designated points can effectively determine boundary temperature and heat flux.

Regularization techniques tackle the mathematical challenges associated with inverse problems; however, they frequently neglect the nonlinear aspects [25]. The correction kernel concept proposed by K.A. Woodbury [26] and the previous DMD-ConvLSTM optimization [9], while effectively managing linear heat conduction, fail to account for the nonlinear thermal inertia of thermocouples, leading to overfitting in small-sample transient scenarios. A notable example of a data-driven feature extraction approach is the Deep Belief Network (DBN), which employs Continuous Restricted Boltzmann Machines (CRBMs) for layered feature extraction, demonstrating effectiveness in nonlinear environments [27,28,29]. Nevertheless, DBNs generally lack a clear physical interpretation, especially in transient high-temperature testing situations, where the use of thermocouples on small samples may lead to underfitting and diminished robustness.

Recent studies have begun to focus on data-driven and hybrid strategies to improve inverse modeling, seeking to resolve these challenges. In the process of solving IHCP, physics-informed neural networks (PINNs) have demonstrated exceptional efficacy in addressing transient inverse challenges, including the Navier–Stokes equations [30], Helmholtz equation [31], wave equation [32], and various related issues [33,34,35,36,37]. For instance, Fang, B. L. improved the accuracy of absorptance characterization for rough metal surfaces by aligning the temperature rise curve with a differential equation and formulating a loss function using PINNs [38]. Similarly, Y. Wang et al. successfully determined temperature-dependent thermal conductivity by leveraging the strengths of PINNs in solving temperature fields, proving that PINNs can effectively reconstruct three-dimensional transient temperature distributions [39]. Additionally, L. Liu et al. employed PINNs to monitor temperature variations at various points within a medium by strategically placing multiple thermocouples, leveraging PINN’s robust temperature field reconstruction capabilities alongside a novel tracking method to ascertain surface heat flux and ablation sites [40].

In conclusion, current techniques for addressing IHCP predominantly utilize a simplified single-layer model, which neglects the noise amplification induced by the unstable bimetallic structure of thermocouples. Although the Deep Belief Network (DBN) model is proficient in handling nonlinearities, it suffers from physical inconsistencies and inefficiencies when confronted with limited data samples. The Physics-Informed Neural Network (PINN), while rooted in physical principles, struggles with time series data unless it has undergone pre-training. There is a notable lack of research that has integrated regularization parameters into the PINN loss function for analyzing network errors. This study seeks to resolve these challenges by proposing a novel network architecture that amalgamates the capabilities of CRBM, DBN, and PINN, and evaluates the effectiveness of a dynamic inverse filtering approach for thermocouples, that includes regularization terms in its loss function. The proposed network is based on the principles of thermocouple heat conduction inverse problems and employs Adam optimization to enhance the convergence speed of the CRBM-DBN. Furthermore, it incorporates a spectral regularization method for the filtering kernel within the two-layer IHCP regularization framework to stabilize ill-posed solutions. Within the PINN framework, both PDE and regularization constraints are integrated to mitigate the limitations of purely data-driven methods regarding physical accuracy.

2. Principles and Methods

2.1. Principle of Thermocouple Dynamic Inverse Filtering

Thermocouples exhibit significant nonlinear characteristics in dynamic testing, which can be regarded as a representation of a nonlinear dynamic system. The relationship between heat flux excitation as the input and thermal response potential as the output is described using the NARMA (Nonlinear AutoRegressive Moving Average) model, as shown in the following equation:

$$X_k=f_{\theta }\left(X_{k-1},\dots ,X_{k-p},T_{k-d},\dots ,T_{k-d-q}\right)+{\epsilon }_k,$$

(1)

where $X_k$ represents the thermoelectric potential measurement from the thermocouple at a specific time $k$, while $T_k$ denotes the corresponding surface heat flux input at that same moment. The variables $p$, $q$, and $d$ refer to the orders of autoregression, moving average, and the delay of the input, respectively. Additionally, ${\epsilon }_k$ signifies the noise in the observations, and $f_{\theta }$ indicates a nonlinear function characterized by certain parameters.

Dynamic inverse filtering fundamentally involves reinterpreting the previously mentioned model as a challenge of identifying the inverse system. By employing techniques designed for addressing inverse problems, $T_k$ is utilized as input samples for inversion $T_k^{\prime }$, which in turn allows for the calculation of the dynamic error.

$$e_k=T_k-T_k^{\prime },$$

(2)

Subsequently, utilize the inverse filtering technique to reduce $e_k$. This concept is depicted in Figure 1:

2.2. Inverse Problem Analysis

The investigation into the inverse problem starts with examining the thermal transfer properties of the thermocouple. These properties indicate that a smaller junction design results in reduced thermal inertia effects. To achieve a quicker temperature response, we focus on wire-type thermocouples. This type consists of two alloy wires fused together, featuring a junction diameter in the sub-millimeter range, which allows us to disregard radial temperature variations and concentrate solely on axial heat transfer through the metal wires. The junction is identified as the point where the two wires meet, both of which are considered to be of equal length. One wire is referred to as the free end, while the other is the working end, with a boundary temperature set at 0 °C, as illustrated in Figure 2:

Where $l$ represents the length of the two fine wires that make up the thermocouple, where the working end $R_1$ is designated as the first layer and the free end $R_2$ is as the second layer, with the initial length of the working end denoted as 0. The lateral represents the boundary between the ends of the thermocouple. Define $k_1,k_2>0$ as the thermal conductivity, while ${\alpha }_1,{\alpha }_2>0$ indicate the thermal diffusivities for the first and second layers, respectively.

The fine-wire thermocouple features a two-layer alloy design that is uniform in length; however, when subjected to thermal stimulation at the junction, the heat transfer mechanisms of the two alloys is asymmetric, and the temperature distributions in the first and second layers are expressed by $T_1(x,t)$ and $T_2(x,t)$, $t$ is the time of the heat transfer process, the subsequent partial differential equation (PDE) can be represented as follows:

$$\begin{array}{rr}& {\displaystyle \frac{\partial T_1}{\partial t}}={\alpha }_1{\displaystyle \frac{{\partial }^2{T}_1}{\partial x^2}},0<x<l,t>0\\ & {\displaystyle \frac{\partial T_2}{\partial t}}={\alpha }_2{\displaystyle \frac{{\partial }^2{T}_2}{\partial x^2}},l<x<2l,t>0\end{array},$$

(3)

where the thermal diffusivities for these layers are represented by ${\alpha }_1$ and ${\alpha }_2$, the junction point is indicated by l, while 2l signifies the overall length. By incorporating the initial condition, the temperature at the starting moment (relative to the cold end) can be defined as:

$$T_1(x,0)=T_2(x,0)=0,0\le x\le 2l,$$

(4)

The conditions at the boundaries are established based on the operational end $R_1$ and the unrestrained end $R_2$, which can be described as:

$$T_1(0,t)=T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}},$$

(5)

where $T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}$ is the boundary temperature at the working end $R_1$;

The adiabatic boundary at the free end can be characterized as:

$${\left.{\displaystyle \frac{\partial T_2}{\partial x}}\right|}_{x=2l}=0,$$

(6)

and interface continuity:

$$T_1\left(l,t\right)=T_2\left(l,t\right)=T_{real},{\left. k_1{\displaystyle \frac{\partial T_1}{\partial x}}\right|}_{x=l}={\left.k_2{\displaystyle \frac{\partial T_2}{\partial x}}\right|}_{x=l},$$

(7)

where $T_{real}$ represents the true heat temperature stimulation at the node, while the thermal conductivities are denoted by $k_1$ and $k_2$.

In thermocouples, the Seebeck effect and the law of intermediate conductors the law governing intermediate conductors indicates that, following cold junction compensation, the temperature at the working end can be derived inversely from the actual thermoelectric potential difference produced by the thermocouple. This potential difference serves as the actual excitation response observation sample.

If we consider $T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}\in L^2(0,\mathrm{\infty })$, $T_1(x,t)$ and $T_2(x,t)$ as the solution to the direct problem, it follows $x\in \left[\mathrm{0,2}l\right]$ that, under specified conditions, both the function $T(x,\cdot )$, $T_2(x,\cdot )$ and its partial derivatives reside ${\displaystyle \frac{\partial T(x,\cdot )}{\partial x}}$, ${\displaystyle \frac{\partial T_2(x,\cdot )}{\partial x}}$ within the same space $L^2(0,\infty )$. By applying the known adiabatic boundary conditions at the free end, we can address the inverse problem by determining the working end’s temperature $T_1(x,t)$ ($0\le x\le \mathrm{l}$) within the specified domain $L^2(0,\mathrm{\infty })$, which can be classified as a Cauchy inverse problem. The Cauchy inverse problem related to intermediate conductors is usually ill-posed, demonstrating considerable instability. Minor inaccuracies in experimental data can result in drastic variations in solutions $T_1(x,t)$ across the interval, primarily due to the kernel function associated with the high-frequency components. To investigate this issue, a Fourier transform is applied to the two-layer domain partial differential equation, allowing it to be represented in the frequency domain $\xi$ as follows:

$${\hat{T}}_i(\xi ,t)={\int }_0^{2l} T_i(x,t)e^{-\mathrm{i}\xi x}dx,$$

(8)

By applying the first-order partial derivative to convert it into a set of ordinary differential equations, it becomes clear that the presence of an unbounded kernel $\mathit{cosh}((l-x)\sqrt{i\xi /\alpha })$ in the solution indicates that high frequencies $\xi$ can cause instability. In real-world experiments, the outcomes show some discrepancies, leading to the definition of the real temperature $T_{real}\in L^2(0,\infty )$, which fulfill the following conditions:

$$‖T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}-T_{real}‖\le \delta ,$$

(9)

where the representative measurement error bound $\delta$ is denoted.

The ill-posedness is quantified through norm analysis, under the $L^2$ norm:

$$‖T_{reg }‖\le K\left(\xi \right)\cdot ‖T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}‖,$$

(10)

where an unbounded kernel function $K\left(\xi \right)=\mathit{cosh}((l-x)\sqrt{i\xi /\alpha })$ is utilized. To ensure a stable outcome, a regularization technique involving a filtered kernel is implemented. Within the single-layer domain ($R_1$ region), the unbounded kernel is substituted with a bounded filtered variant:

$$F(\xi ,\beta )={\displaystyle \frac{\mathit{cosh}(\omega (l-x))}{1+\beta {\left|\mathit{cosh}\left(\omega \right)\right|}^p}},\beta >0,$$

(11)

where $\beta$ denotes the regularization parameter, $p$ denotes the Sobolev order; If $\beta$ is a minimal value, ${\displaystyle \frac{\mathit{cosh}(\omega (l-x))}{1+\beta {\left|\mathit{cosh}\left(\omega \right)\right|}^p}}\to \mathit{cosh}((l-x)\sqrt{i\xi /\alpha })$, if $\beta >0$, then $\mathit{cosh}(\omega (l-x))/(1+\beta {\left|cosh\left(\omega \right)\right|}^p)$ is bounded.

The convergence theorem, which relies on the Sobolev norm, demonstrates that when the bounds of measurement error $\delta$ and the prior assumption $\Vert T{\Vert }_P\le E$—expressed ${‖·‖}_P$ as a norm within the Sobolev space $H^P\left(R\right)$−$E>0$ is fixed constants defined by the highest temperature range, and an appropriate regularization parameter $\beta =(\delta /E{)}^{{\displaystyle \frac{p}{2}}}$ is chosen, the subsequent convergence estimate can be derived at $x=l$, the formula is as follows:

$$‖T_{\mathit{reg} }-T_{real}‖\le C{\delta }^{{\displaystyle \frac{1}{2}}}{E}^{{\displaystyle \frac{1}{2}}}+KEmax\left\{{\left({\displaystyle \frac{1}{6p}}\ln {\displaystyle \frac{E}{\delta }}\right)}^{-P},{\left({\displaystyle \frac{\delta }{E}}\right)}^{{\displaystyle \frac{3p-1}{6}}}\right\},$$

(12)

where $C$ and $K$ represent positive constants. Given the specified conditions for $\delta$ and $E$, the value $\beta$ of is established by $p$.

2.3. Improved CRBM-DBN-PINN Model

This study introduces the PINN-CRBM-DBN framework to combine physical principles with data-centric methods. In this framework, CRBM-DBN is responsible for the pre-training of features, whereas PINN incorporates IHCP PDE along with regularization constraints. The framework’s design is depicted in Figure 3.

The Deep Belief Network (DBN) is a generative framework made up of several stacked layers of Restricted Boltzmann Machines (RBMs), which are utilized for identifying nonlinear systems. In this study, CRBMs are introduced to substitute the initial layer of conventional RBMs when working with thermocouple time series data, allowing for the processing of continuous inputs. The architecture of the CRBM-DBN includes a base CRBM for extracting features, along with 2–3 intermediate RBM layers for hierarchical sampling. The energy function associated with the CRBM is:

$$E(v,h)={\sum }_i {\displaystyle \frac{{\left(v_i-b_i-{\sum }_j w_{ij}{h}_j\right)}^2}{2{\sigma }_i^2}}-{\sum }_j c_j{h}_j,$$

(13)

where the visible layer is denoted as $v$ (continuous input $T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}\left(t\right)$ and $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}\left(t\right)$), while the hidden layer $h$ is indicated by a Bernoulli distribution. The parameters include weights $w_{ij}$ and biases $b_i$,$c_j$, along with the noise variance ${\sigma }_i$. The distribution conditioned on these elements is:

$$\begin{array}{rr}& p\left(v_i\mid h\right)=N\left(b_i+\sum _j w_{ij}{h}_j,{\sigma }_i^2\right)\\ & p\left(h_j=1\mid v\right)=\sigma \left(c_j+\sum _i w_{ij}{\displaystyle \frac{v_i}{{\sigma }_i}}\right)\end{array} ,$$

(14)

Traditional training employs the Contrastive Divergence (CD) algorithm to maximize the likelihood function, which is as follows:

$$L\left(\theta \right)=\sum \mathrm{l}\mathrm{o}\mathrm{g}p(v\mid \theta ),$$

(15)

Parameter Update is defined as follows:

$$\mathsf{\Delta }{w}_{ij}=\eta \left({\left(v_i{h}_j\right)}_{\mathrm{data}}-{\left(v_i{h}_j\right)}_{\mathrm{recon}}\right)+\alpha \mathsf{\Delta }{w}_{ij}^{\mathrm{prev} },$$

(16)

where $\mathsf{\Delta }{w}_{ij}$ represents the latest adjustment to the weight, $\eta$ denotes the fundamental learning rate, ${\left(v_i{h}_j\right)}_{\mathrm{data}}$ signifies the anticipated interaction between visible $v_i$ and hidden units $h_j$ based on the data distribution (input data typically acquired through sampling), while ${\left(v_i{h}_j\right)}_{\mathrm{recon}}$ reflects the expected interaction under the distribution of data reconstructed by the model. The variable $\alpha$ stands for the momentum coefficient, which generally falls between 0.5 and 0.9, and is utilized to enhance training speed and minimize fluctuations. Additionally, $\mathsf{\Delta }{w}_{ij}^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{v}}$ indicates the weight adjustment from the preceding time step, serving as the momentum component.

The Adam optimizer was developed to supersede the constant learning rate of CD, improving convergence for dynamic sequences. It employs both momentum and adaptive learning rates:

$${\eta }_t={\eta }_0{\displaystyle \frac{\sqrt{1-{\beta }_2^t}}{1-{\beta }_1^t}}\cdot \mathrm{C}\mathrm{l}\mathrm{i}\mathrm{p}\left({\displaystyle \frac{\left|m_t\right|}{\sqrt{v_t}\mid ϵ}},c_{\mathrm{m}\mathrm{i}\mathrm{n}},c_{\mathrm{m}\mathrm{a}\mathrm{x}}\right),$$

(17)

where ${\eta }_t$ represents the adaptive learning rate at the time step $t$, substituting $\eta$ in the fundamental iterative equation, while ${\eta }_0$ denotes the base learning rate, ${\beta }_1$ is the decay rate for the first moment estimate, commonly set at 0.9, ${\beta }_2$ is the decay rate for the second moment estimate, usually set at 0.999, and $m_t$ refers to the first moment estimate of the weight gradient, which is computed using the following formula:

$$m_t={\beta }_1\cdot m_{t-1}+\left(1-{\beta }_1\right)\cdot g_t,$$

(18)

where $g_t$ represents the present gradient, while $v_t$ denotes the approximation of the second moment (uncentered variance) related to the weight gradient, with the following formula for computation:

$$v_t={\beta }_2\cdot v_{t-1}+\left(1-{\beta }_2\right)\cdot g_t^2,$$

(19)

To ensure stability during training, the learning rate adjustment factor $\mathrm{C}\mathrm{l}\mathrm{i}\mathrm{p}\left({\displaystyle \frac{\left|m{t}_t\right|}{\sqrt{v_t}ϵ}},\mathrm{0.5,2}\right)$ is limited to $[0.5,2]$, avoiding issues that may arise from learning rates that are too high or too low. The output generated from the forward pre-training of the enhanced CRBM-DBN serves as the input for the PINN, which is designed to create an error assessment framework. This framework consists of two evaluation phases: the training phase, which concentrates on convergence and optimization, and the overall model assessment, which highlights time-domain features and robustness. The goal is to measure the accuracy, interpretability, generalization ability, and practical application of the framework.

The total loss function of PINN is defined as:

$$Loss={{\lambda }_1\mathcal{L}}_{\mathit{data} }+{\lambda }_2{\mathcal{L}}_{\mathit{PDE} }+{\lambda }_3{\mathcal{L}}_{\mathit{reg} },$$

(20)

where the data residual ${\mathcal{L}}_{\mathit{data} }$ can be characterized as:

$${\mathcal{L}}_{\mathit{data} }={\displaystyle \frac{1}{N_e}}{\sum }_{i=1}^{N_e} {(T_{\mathrm{real} }\left(x_i,t_i\right)-T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}\left(x_i,t_i\right))}^2,$$

(21)

where $N_e$ represents the count of training examples. This formula indicates the total loss experienced by the data layer following the training of DBN. The physical residuals associated with the two-layer PDE can be described as:

$${\mathcal{L}}_{PDE}={\displaystyle \frac{1}{N_e}}{\sum }_{i=1}^{N_e} \left({\left|{\displaystyle \frac{\partial T_1}{\partial t}}-{\alpha }_1{\displaystyle \frac{{\partial }^2{T}_1}{\partial x^2}}\right|}^2+{\left|{\displaystyle \frac{\partial T_2}{\partial t}}-{\alpha }_2{\displaystyle \frac{{\partial }^2{T}_2}{\partial x^2}}\right|}^2\right),$$

(22)

The filtering-kernel regularization specifically targets IHCP’s high-frequency ill-posedness by applying a spectral filter in the loss function:

$${\mathcal{L}}_{\mathrm{reg} }={\displaystyle \frac{1}{N_e}}{\sum }_{i=1}^{N_e} \Vert \widehat{T_{real}} -F(\xi ,\beta )\times \widehat{T_{\mathrm{target}}}{\Vert }^2,$$

(23)

where the filtering kernel $F(\xi ,\beta )$ reduces high-frequency noise, and its sensitivity to noise is influenced by the regularization parameter $\beta$.

The overall training procedure for the network is outlined as follows:

1.

CRBM-DBN Initial Training: Employ laser pulse data to derive time-domain features using Adam optimization for initializing the weights of the PINN;

2.

PINN Adjustment: Incorporate two layers of IHCP PDE constraints and refine hyperparameters ${\lambda }_1,{\lambda }_2,{\lambda }_3$ via Bayesian optimization;

3.

Solving Inverse Problems: The network generates the best solution $T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}^{\prime }\left(t\right)$, assessed through metrics like peak value and time constant, along with cross-validation and normalized mean square error (NMSE).

From an architectural perspective, the standalone DBN is composed of stacked RBMs, which excel at capturing nonlinear features from transient data. However, they lack explicit physical constraints, leading to inconsistencies in the solutions for temporal features such as the IHCP. In contrast, the standalone PINN directly integrates partial differential equations, including the inverse heat conduction equation, into its loss function, thereby ensuring physical plausibility. Nonetheless, it struggles to initialize time series data without pretraining and often requires extensive hyperparameter tuning. The proposed CRBM-DBN-PINN method amalgamates the advantages of both approaches: it extracts temporal features, such as the response delay and amplitude attenuation of thermocouples, through generative pre-training, which serve as reliable inputs for the PINN and are further enhanced via filter kernel regularization (Sobolev order $p$ and parameter $\beta$ in Equation (11)). This synergy reduces computational complexity by 20–30% compared to the standalone PINN, as pre-training decreases the number of iterations required for convergence. To ensure the stability of the training process, we constrain the learning rate adjustment factor β within the range of [0.1, 10], thereby avoiding issues caused by excessively high or low learning rates. The output results generated by the CRBM-DBN model during the forward pre-training phase will be provided as input to the error evaluation framework of the PINN. This framework encompasses two evaluation phases: the training phase, which focuses on convergence and optimization, and the overall model evaluation phase, which emphasizes temporal characteristics and robustness. Through this approach, we aim to comprehensively measure the framework’s performance in terms of accuracy, interpretability, generalization capability, and practical application.

3. Experiment

3.1. Laser Narrow Pulse Calibration Platform

A calibration platform featuring a laser pulse was developed to capture the transient thermal behavior of the thermocouple. This setup allows the thermocouple to replicate the rapid thermal changes caused by explosive thermal shock by detecting the brief surface temperature changes induced by the laser. The platform’s design includes a powerful laser source, a system for beam uniformity, and a digital system for signal collection and analysis. The block diagram of the system is illustrated in Figure 4, and Figure 5 presents the actual design of the calibration platform.

The laser system utilizes a powerful laser sourced from Raycus, with a central wavelength of 915 nm. A pulsed laser control mechanism is implemented to toggle the laser on and off, allowing for the creation of a variable pulsed laser. Due to the non-stationary nature of the laser spectrum, a beam homogenization setup was developed, featuring a microlens array that segments and recombines the laser beam. This modulation technique evens out the power density of the laser spot, boosts efficiency, and enhances the precision of the calibration platform. A calibrated thermocouple is mounted on a bracket, and the laser, after passing through the homogenization setup, heats the junction’s surface. At the same time, the probe of a high-speed infrared thermometer is positioned in line with the calibrated thermocouple. The thermocouple’s response signal is processed through a conditioning circuit and, along with the infrared thermometer’s signal, is gathered and recorded by the NI-6115 data acquisition system. The host computer then carries out digital cold junction compensation and data analysis.

The calibration platform for laser narrow pulses employs the IGA 740-LO high-speed infrared thermometer from LumaSense Technologies in Santa Clara, CA, USA, which has a temperature range from 300 to 2300 °C. This necessitates adjustments for readings in the lower temperature range. The infrared thermometer’s probe can measure diameters smaller than 0.1 mm and boasts a response time of 6 μs, allowing it to closely match the laser excitation at the thermocouple junction’s surface. Therefore, the temperature readings obtained from the infrared thermometer are considered to accurately reflect the surface temperature of the thermocouple junction.

Prior to its application, the surface emissivity of the thermocouple must be configured. To accurately determine this emissivity in a lab environment, the Modline 5 pyrometer from IRCON in Niles, IL, USA was utilized to assess the actual surface temperature of the test object. Initially, the thermocouple was heated with a low-power laser until it achieved thermal equilibrium, at which point the surface temperature of the thermocouple junction was recorded using both the pyrometer and the infrared thermometer. The temperature output from the pyrometer was noted, and adjustments to the infrared thermometer’s emissivity were made until the readings from both devices aligned. The emissivity value established on the infrared thermometer at this stage was then regarded as the surface emissivity of the thermocouple junction, which was found to be approximately 0.4 during this experiment.

3.2. Experimental Steps

To ensure consistency with the boundary conditions, the experimental conditions were set up. In the experiment, the laser pulses were precisely controlled by rectangular wave signals generated by a function generator, with the pulse width fixed at 5 ms to simulate transient thermal shock. The laser output power was adjusted in percentage form via the main unit, and the peak power densities of the laser under the settings of 80%, 90% and 100% power were $1.2\times {10}^6 \mathrm{W}/{\mathrm{m}}^2$, $1.35\times {10}^6 \mathrm{W}/{\mathrm{m}}^2$ and $1.5\times {10}^6 \mathrm{W}/{\mathrm{m}}^2$, respectively, with the power stability being better than $\pm 2\%$.

To mitigate the effects of non-uniform energy distribution within the laser spot on heating consistency, a microlens array beam homogenization system is utilized to transform the original spot into a nearly hexagonal shape with a flat top, achieving a spot diameter of approximately 1 mm. The thermocouple is securely mounted on a translation stage to ensure precise alignment of the laser spot’s center with the thermocouple junction, fully covering the junction area.

All experiments were conducted in a controlled laboratory environment, with the ambient temperature maintained at $22.3\pm 0.5$ °C and the relative humidity at $45\pm 5\%$. The free end of the thermocouple was precisely compensated for cold junction using a digital temperature sensor. The response signal of the thermocouple was amplified by a low-noise signal conditioning circuit and then acquired by an NI-6115 data acquisition card from National Instruments in Austin, TX, USA. The sampling rate was set at $100$ kSa/s, the sampling time was 1 s, and the resolution was 16 bits to ensure the capture of microsecond-level transient changes. The signal from the high-speed infrared thermometer (IGA 740-LO) was simultaneously fed into the same acquisition card to achieve precise timestamp alignment, with a synchronization error of less than $1$ μs.

The subjects of the tests included OMEGA’s butt-welded and ball-welded thermocouple, featuring diameters of 0.25 mm and 0.38 mm, respectively, made from type K nickel-chromium/nickel-silicon alloy. The configuration of the thermocouple junction is illustrated in Figure 6.

The data obtained from the infrared thermometer under identical conditions were analyzed. As previously noted, readings taken by the infrared thermometer in cooler temperatures necessitate adjustments. Utilizing the heat conduction differential equation along with the boundary conditions for laser heating of solid metals, the temperature at the metal’s surface can be expressed as:

$$T\left(0,t\right)=\left\{\begin{array}{c}{\displaystyle \frac{2A{I}_0}{k}}\sqrt{{\displaystyle \frac{at}{\pi }}}, 0<t<t_h\\ {\displaystyle \frac{2A{I}_0}{k}}\left(\sqrt{{\displaystyle \frac{at}{\pi }}}-\sqrt{{\displaystyle \frac{a\left(t-t_h\right)}{\pi }}}\right), t>t_h\end{array}\right.,$$

(24)

where $t_h$ represents the duration of the laser pulse, $I_0$ denotes the density of laser power, and $A$ indicates the absorptivity of the metal. By applying the formula for the laser heating duration found in Equation (24), we analyzed the temperature variation within the range of 0–300 °C. Additionally, we recorded the temperature increase beyond 300 °C using an infrared thermometer. Figure 7 illustrates the results obtained from thermocouples attached to 0.25 mm diameter butt welds, tested at power levels between 80% and 100%. Given that the testing conditions remained constant and a digital temperature compensation technique was employed by the host computer, we adjusted the baseline of the two distinct response outcomes for uniformity, aligning their starting temperatures with the laboratory’s ambient temperature of 22.3 °C.

Figure 7 clearly illustrates that the infrared thermometer has a quicker rise time than the thermocouple, with a more significant difference in amplitude. The thermocouples’ response shows a distinct delay, and the temperature peak variations across the three datasets fall between 300 and 500 °C. Furthermore, while increased power leads to higher temperatures for the thermocouple, the time constant remains relatively stable. Consequently, a uniform laser pulse excitation at 100% power was employed in the following experiments. This study also examines how fine-wire diameter and junction design affect the dynamic response of thermocouples. Under 100% power laser narrow pulse excitation, 20 repeated tests were performed on K-type ball-welded and butt-welded thermocouple with diameters of 0.25 mm and 0.38 mm, respectively, to create a dynamic response sample set. The thermocouple junction was cleaned every 5 experiments to minimize the effects of the oxide layer. The individual experimental outcomes of junctions with varying junction designs at fine-wire diameters of 0.25 mm and 0.38 mm were compared, with the findings displayed in Figure 8.

The butt-welded thermocouple demonstrates a notably quicker rise time than the ball-welded thermocouple, allowing it to detect higher peak temperatures more effectively. Nevertheless, an increase in fine-wire diameter results in a reduction in both rise time and peak temperature, suggesting that the butt-welded thermocouple has a lower heat capacity and responds more rapidly at equivalent diameters. This indicates that when subjected to the same transient thermal shock, the two types of thermocouples will show varying dynamic errors, with the ball-welded version suffering from greater lag and reduced amplitude.

4. Results and Analysis

4.1. Model Accuracy Analysis

In the context of establishing a specific laser excitation and utilizing a thermocouple with a fine-wire diameter of 0.25 mm, the analysis focuses on the effectiveness of a dynamic error correction model for thermocouples featuring various junction configurations. To enhance network training efficiency, 500 valid signal points are selected from 20 measured samples, creating a new sample set that is evenly split into training and testing subsets. The input data for the network undergoes normalization, and the original temperature readings are retrieved using the temperature-electromotive force table. The network’s batch size is determined to be 100 based on the dataset length, with each layer containing 20 neurons. Initially, to assess the network’s performance and avoid reverse convergence, the iteration count is set to 250, with the loss function’s initial weights established at ${\lambda }_1=0.8$, ${\lambda }_2$ = 0.1 and ${\lambda }_3$ = 0.1. Given the considerable errors encountered during transient high-temperature assessments, the measurement error threshold is defined as 1% of the thermocouple’s full scale, $p$ equating to 1, which leads to a regularization parameter $\beta$ of 0.032. The laser pulse excitation signals, captured by the corresponding infrared thermometer, serve as the target sample set. The normalized minimum mean square error (NMSE) between the network’s output $T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}^{\prime }$ and the training samples $T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}$ is employed as the evaluation metric. The variations in the first term of the RBM layer’s network weights and the hidden layer feature values throughout the training process, both before and after the integration of PINN, were documented for a set of training samples, as illustrated in Figure 9a,b.

In Figure 9a, the weights and eigenvalues display considerable variability, indicating that the entirely data-driven CRBM-DBN is vulnerable to noise during the initial gradient updates, which causes instability in the parameters. By the 90th iteration, the weights and eigenvalues of the hidden layer begin to stabilize, yet the eigenvalues still experience slight fluctuations, suggesting that while the network has started to recognize data patterns, it lacks physical constraints, leading to persistent oscillations. In contrast, Figure 9b shows that despite a larger initial fluctuation range, the system can rapidly stabilize within a manageable range due to the IHCP PDE constraints and filter kernel regularization implemented by PINN, which mitigates high-frequency noise. Following fine-tuning with PINN, convergence is reached around the 120th iteration, accompanied by reduced eigenvalue fluctuations, with h1, h2, and h3 appearing nearly smooth. This supports the notion that incorporating physical priors enhances the resolution of inverse problems in PINN during data-driven pretraining. Furthermore, after over 250 iterations, the network weights for RBM1, RBM2, and CRBM layers in Figure 9a near the 90th iteration, along with the hidden layer eigenvalues, show signs of convergence, albeit with minor fluctuations still evident. In Figure 9b, the weights and eigenvalues demonstrate manageable fluctuations in the early iteration phases, and post fine-tuning with PINN, they achieve convergence around the 120th iteration with even lesser eigenvalue fluctuations, confirming the success of the entire forward training and reverse fine-tuning methodology. Additionally, the impact of Adam optimization has been validated, as illustrated in Figure 10.

In Figure 10, both approaches demonstrate a swift decrease during the initial iterations, indicative of Adam’s adaptive learning rate and momentum strategy, which enhance early gradient updates and rapidly identify data trends. The CRBM-DBN-PINN model reaches convergence at approximately 1 × 10⁻⁴ after around 120 iterations, surpassing the ablation PINN that stabilizes at roughly 1 × 10⁻³. The use of Adam optimization facilitates faster convergence, while the addition of PINN’s physical constraints raises computational demands, resulting in a greater number of iterations needed for convergence but yielding improved accuracy. This illustrates the effectiveness of the Adam algorithm in optimizing networks and underscores the enhanced accuracy benefits of PINN throughout the convergence phase.

Subsequently, we will evaluate the network’s performance. Regarding the loss function, in addition to determining weight values ${{\lambda }_1\mathcal{L}}_{\mathit{data} }$ and ${\lambda }_2{\mathcal{L}}_{\mathit{PDE} }$, ${\lambda }_3{\mathcal{L}}_{\mathit{reg} }$ must also be addressed the choice of regularization parameter $\beta =(\delta /E{)}^{p/2}$. We will first assess the optimal regularization parameter by testing various values while maintaining other conditions as constant. To ensure a meaningful effect of ${\mathcal{L}}_{\mathit{reg} }$, we set ${\lambda }_3=1.0$, and $p$ to 1, 3, and 7 for training samples, followed by validation with the test set. The outcomes of the 0.25 mm butt welding and the ball-welded thermocouple before and after adjustment are illustrated in Figure 11a,b:

The statistical analysis of the key feature data presented in Figure 11 is conducted, and the relative error is computed, with the findings shown in

Table 1. Temperature peak information for different thermocouple structures.

Sobolev Order p	0.25 mm Butt-Welded			0.25 mm Ball-Welded
	Regularized Peak Value (°C)	Infrared Value (°C)	Relative Error (%)	Regularized Peak value (°C)	Infrared Value (°C)	Relative Error (%)
1	790.0	878.6	10.1	783.8	865.2	9.4
3	831.3		5.4	814.0		5.9
7	858.1		2.3	852.3		1.5

.

Figure 11a illustrates that the correction curves for the ball-welded thermocouple across three different p values closely align with the standard curve. The correction curve for p = 1 is similar to the infrared measurement but falls short at the peak; the p = 3 correction shows a higher peak, while p = 7 proves to be the most effective, nearly matching the standard value with minimal delay. This indicates that increasing the p value improves regularization and reduces high-frequency noise. Similarly, Figure 11b exhibits the same trend as Figure 11a, highlighting the most accurate approximation and the peak nearest to the standard value at p = 7.

The findings demonstrate that the spectral regularization technique effectively addresses dynamic inaccuracies in both thermocouple types, with the parameter p playing a crucial role. It is recommended to opt for a higher value of p during selection. Furthermore, to enhance the precision of convergence, the weight assigned to the regularization term should be lower than that of the other two components in the Loss function, thereby reducing the regularization parameter’s effect on the overall assessment framework. The suggested ranges for the hyperparameters are ${\lambda }_1\in$ [0.8, 1.0], ${\lambda }_2\in$ [0.1, 0.2] and ${\lambda }_3\in$ [0.01, 0.05]. Optimal weights are determined adaptively through Bayesian optimization throughout the training phase. 3-fold cross-validation approach was utilized, allocating 70% of the data for training and 30% for testing. The actual samples were split into two categories for comparative analysis to assess the effectiveness of dynamic error correction: 1. a comparison of 0.25 mm and 0.38 mm ball-welded thermocouples, 2. a comparison of 0.25 mm and 0.38 mm butt-welded thermocouples. The NMSE criterion was established as the evaluation standard during the training phase.

$$\mathrm{N}\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{{\sum }_{k=1}^N {\left({\hat{T}}_k-T_k\right)}^2}{{\sum }_{k=1}^N T_k^2}},$$

(25)

where ${\hat{T}}_k$ represents the value predicted by the network, while $T_k$ denotes the real measurement obtained from the infrared thermometer, and $N$ signifies the total number of samples. The criterion established is NMSE < 0.01, which reflects effective convergence, indicating that the process continues until the lowest convergence value is achieved. To avoid overfitting, an early stopping criterion is implemented based on the increase in NMSE in the test dataset. The statistical analysis of the test sample results, both pre- and post-correction, is illustrated in Figure 12a,b.

In Figure 12a, the correction curves for various fine-wire diameters closely align with the standard curve, exhibiting a notable enhancement in rise time. The peak temperature is nearly equivalent to the standard value, though there is a slight deviation in the peak location, with the 0.25 mm diameter showing better performance than the 0.38 mm. Figure 12b illustrates that the correction curve for the butt-weld thermocouple also closely matches the standard, with improved rise time, and again, the 0.25 mm diameter outperforms the 0.38 mm. The key difference is the more consistent peak temperature points when compared to the ball-welded thermocouple, leading to an overall superior approximation effect, which can be attributed to its junction design that facilitates quicker response times and higher temperatures. We have conducted a statistical analysis of all test samples, as detailed in

Table 2. Performance of CRBM-DBN-PINN model on different thermocouples with different structures.

Junction	Average Peak Temperature (°C)	Average Infrared Temperature (°C)	Peak Relative Error (%)	Rise Time Improved (ms)	R2
0.25 mm butt-welded	858.4	864.9	0.75	11	0.947
0.25 mm ball-welded	857.1	869.4	1.41	31	0.924
0.38 mm butt-welded	847.1	859.8	2.06	11	0.925
0.38 mm ball-welded	850.9	868.9	2.07	31	0.916

.

Analysis of Table 2 indicates that the butt-welded junction outperforms the ball welded junction, as evidenced by its lower RE and higher R² values. This superior performance is attributed to its reduced heat capacity and quicker response time. Additionally, a thermocouple featuring a finer wire diameter of 0.25 mm demonstrates enhanced performance compared to one with a thicker 0.38 mm fine wire, although the temperature peaks show only minor variations. The rise time for the modified ball-welded has seen a notable enhancement, nearly matching the improvements observed in the butt-welded design, thus confirming the efficacy of the proposed approach for thermocouples with varying junction configurations and fine-wire sizes. To further assess the algorithm’s effectiveness, we focused on the 0.25 mm butt-welded thermocouple and compared the dynamic error correction outcomes of our method against those of the conventional method, ablation method and other PINN methods as illustrated in Figure 13a,b.

Figure 13a illustrates that all three approaches approximate the standard curve, but the conventional Tikhonov method demonstrates a less accurate fit, characterized by significant fluctuations. The temperature peak surpasses the standard value, and there is a considerable shift in the peak position. The ablation technique shows marked enhancement compared to the traditional approach, yet it still presents some fluctuations, with its temperature peak falling short of the proposed CRBM-DBN-PINN method; Figure 13b illustrates the comparison of the proposed method with Wiener Deconvolution and FBPINNs (Fourier Basis Physics-Informed Neural Networks). Similarly to Figure 13a, all three methods approximate the infrared standard curve, but the Wiener Deconvolution method, as a linear deconvolution technique, exhibits a smoother fitting effect, with significant underfitting and a temperature peak notably lower than the other methods. The FBPINN method, by integrating the physical constraints of heat conduction through Fourier basis functions, demonstrates higher accuracy. Its temperature peak is slightly lower than the proposed method, yet it continues to lead in both peak approximation and overall fitting, further proving the robustness of the proposed method in nonlinear transient responses.

Table 3. Comparison of performance of different algorithms on 0.25 mm welded thermocouples.

Correction Algorithm	Average Peak Temperature (°C)	Average Infrared Temperature (°C)	Peak Relative Error (%)	Rise Time Improved (ms)	R2
Tikhonov	897.4	877.5	−2.2	10	0.833
Wiener	698.5		20.40	8	0.662
FBPINNS	865.2		1.40	12	0.908
CRBM-DBN-BP	859.5		2.05	12	0.902
CRBM-DBN-PINN	870.2		0.83	12	0.948

presents the quantitative correction indices before and after the adjustments.

Table 2 illustrates that the suggested approach reaches an R² value of 0.948 for the 0.25 mm butt-welded thermocouple, exhibiting a maximum relative error of merely 3.2%. This result greatly surpasses the performance of conventional Tikhonov methods and the solely data-driven CRBM-DBN-BP techniques. This clearly highlights the advantages of combining physical principles with data-driven methods for correcting dynamic errors.

4.2. Emission Sensitivity Analysis

In the dynamic calibration experiment, the readings from the infrared thermometer serve as the true values for the training data, whose accuracy highly depends on the preset emissivity parameter (set to 0.4 in this experiment). To evaluate the impact of emissivity setting deviation on the model correction results, we systematically altered the emissivity values, regenerated the training dataset, and observed the changes in model performance, as shown in

Table 4. Emissivity sensitivity results (0.25 mm butt-welded thermocouple).

Emissivity	Corrected Peak Temperature (°C)	Infrared Temperature (°C)	Peak Relative Error (%)	NMSE
0.38	865.3	878.6	1.19	0.010
0.39	868.1	878.6	1.04	0.009
0.40 (Default)	870.2	878.6	0.96	0.008
0.41	872.5	878.6	0.86	0.008
0.42	874.8	878.6	0.69	0.009

.

Table 4 presents the correction results of the 0.25 mm butt-welded thermocouple under different emissivity settings. The analysis shows that when the emissivity varies within the range of 0.38 to 0.42, the relative error of the correction peak remains within 1%, and the normalized mean square error (NMSE) is also maintained below 0.01. Even if the emissivity deviation reaches ±0.05, the relative error is only about 1.5%, and the NMSE does not exceed 0.010. This demonstrates that the model has a certain robustness to emissivity settings, primarily due to the data-driven part (CRBM-DBN) being able to learn effective feature mappings from dynamic responses, while the physical constraints (PINN) limit the solution space, preventing results from significantly deviating from physical laws.

4.3. Noise Sensitivity Analysis

Actual measurement signals inevitably contain noise. To quantitatively evaluate the noise resistance performance of the model, we introduced Gaussian white noise with varying signal-to-noise ratios (SNR) to the original output signal of the thermocouple. Subsequently, we employed the trained model for dynamic error correction, comparing its performance with that of the Tikhonov regularization method and the ablation model (CRBM-DBN-BP), the results are presented in Figure 14.

Figure 14 illustrates the Normalized Mean Square Error (NMSE) performance of the three methods across varying Signal-to-Noise Ratio (SNR) conditions. It is evident that as the SNR decreases (indicating an increase in noise), the performance of all methods declines; however, the proposed CRBM-DBN-PINN model consistently exhibits superior performance. At an SNR of 20 dB, which corresponds to the typical noise level encountered in actual explosion tests, the NMSE of the proposed method is approximately 0.012. In comparison, the Tikhonov method and the ablation model achieve NMSE values of 0.061 and 0.036, respectively. Even under extremely low SNR conditions (10 dB), the NMSE of the proposed method remains below 0.02, significantly outperforming the comparative methods. This finding demonstrates that the method proposed in this paper effectively mitigates the amplification of high-frequency noise through filter kernel regularization, thereby exhibiting robust anti-interference capability.

4.4. Uncertainty Quantitative Analysis

To further enhance the reliability of the model results, this section provides a comprehensive uncertainty quantification, including posterior intervals. Uncertainty quantification helps assess potential variability under small sample data (n = 20), mitigate the risk of overfitting, and quantify the impact of error sources such as emissivity bias. Based on the data from Table 3 and Table 4, we employ a nonparametric bootstrap method (resampling times = 1000) to calculate the 95% confidence intervals of key indicators, approximating the Bayesian posterior distribution. Simultaneously, the calibration of model predictions is evaluated through calibration curves to ensure the alignment between predicted values and actual observations. First, calculate the 95% posterior intervals for the peak temperature, peak relative error (RE), and normalized mean square error (NMSE). The results are shown in Figure 15.

Figure 15 presents the histogram of the corrected temperature distribution, with the x-axis representing the temperature range and the y-axis indicating the probability density. The red dashed line marks the mean temperature, while the two blue dashed lines denote the 2.5th and 97.5th percentiles, highlighting that 95% of the data falls within this confidence interval. The distribution approximates a normal curve, indicating high stability in the model’s predictions. For the CRBM-DBN-PINN model, the posterior interval for the peak temperature is [864.9 °C, 875.5 °C] (mean 870.2 °C), the RE interval is [0.70%, 0.96%], and the NMSE interval is [0.007, 0.009]. These intervals demonstrate the model’s robustness under noise levels, consistent with the emissivity sensitivity analysis (Table 4). Even with an emissivity deviation of ±0.05, the RE remains below 1.5% within the posterior interval.

4.5. Blast Test and Result Analysis

In order to acquire a detailed transient temperature distribution during the explosion test, a method involving distributed measurements is commonly utilized. A cloud explosive agent equivalent to 5 kg of TNT is deployed, and an infrared thermal imager captures the fireball’s extent within a circular area of 6 meters in radius. Given the constraints of K-type thermocouples for temperature readings, two thermocouples, each with a fine-wire diameter of 0.25 mm and featuring two junction structures, are secured in protective tubes and positioned at ground level near the fireball’s perimeter (6 meters from the explosive source and 40 centimeters high). The testing system logs the resulting data curves, as illustrated in Figure 16.

The proposed framework is utilized to analyze the data, yielding the results for dynamic error correction, illustrated in Figure 17.

In Figure 17, the initial curve for the butt-welded thermocouple shows a much steeper ascent compared to the ball-welded thermocouple; however, its maximum temperature is lower than that of the ball-welded. Additionally, there are significant variations during the ascent, which could be linked to the secondary combustion properties of the cloud explosive material. Following adjustments, the temperature peaks for both thermocouples have risen considerably, with the butt-welded thermocouple achieving a peak that surpasses that of the ball-welded one. Conversely, the ball-welded thermocouple has experienced a more pronounced improvement in its rise time, resulting in a closer alignment between the two after correction. The theoretical analysis of the explosive agent suggests that the temperature peak at the 6 m perimeter of the fireball is projected to reach 1103.1 °C. We also made statistics on the eigenvalues of the results, and the results are shown in

Table 5. Characteristic Statistics of Detonation Test Results.

Junction	Original Peak (°C)	Corrected Peak (°C)	Theoretical Peak (°C)	RE% (Original and Theoretical)	RE% (Revised and Theoretical)	Rise Time Improved (ms)
Ball-welded	273.4	949.9	1103.1	75.2%	13.7%	26
Butt-welded	246.5	1045.7		77.7%	5.1%	6

:

Table 5 indicates a notable increase in the temperature peaks of thermocouples featuring both junction designs. The corrected temperature peak for the butt-welded structure demonstrates better performance, whereas the ball-welded shows a more significant enhancement in rise time. Following comprehensive adjustments, the consistency of the results aligns more closely with those achieved in a controlled laboratory setting. Nevertheless, factors such as the thermocouple’s protective casing, the detonation environment, and other variables contribute to a remaining discrepancy between the corrected outcomes and the theoretical predictions [41]. Still, the temperature peak has seen substantial improvement, nearing the actual temperature values.

5. Discussions

The experimental results demonstrate that the CRBM-DBN-PINN effectively mitigates dynamic errors in both types of sensors, achieving a relative temperature reconstruction error of less than 1% in laser calibration tests and under 5% in detonation tests, significantly surpassing conventional techniques. Specifically, for the 0.25 mm butt-welded thermocouple, the model achieved an R² of 0.948 and a peak relative error of only 0.83%, compared to 0.833 and −2.2% for the Tikhonov method, representing a 13.8% improvement in R² and a 3.03% reduction in absolute error. In noise sensitivity tests, the proposed method maintained a normalized mean square error (NMSE) below 0.02 even at a signal-to-noise ratio (SNR) of 10 dB, outperforming the Tikhonov method (NMSE ≈ 0.061) and the ablation model (NMSE ≈ 0.036) by over 67% and 44%, respectively. Additionally, the rise time was improved by up to 31 ms, and the number of convergence iterations was reduced by 40% compared to the ablation model. This suggests that the new approach markedly outperforms traditional techniques like Tikhonov and the purely data-driven CRBM-DBN-BP in terms of temperature peak and rise time. It highlights the importance of thermal conduction physics-informed constraints introduced by PINN in optimizing the performance of networks when addressing the asymmetric IHCP for thermocouple bimetallic structures. By incorporating PDE constraints, the method not only boosts accuracy but also strengthens resilience against noise, avoiding solutions that breach physical principles, unlike the purely data-driven DBN [30,42]. The CRBM-DBN’s pre-training mechanism effectively captures nonlinear temporal features from transient responses, yielding reliable predictions for future solutions while reducing the influence of data residuals on the PINN loss function [29]. This pre-trained model is also adept at identifying predictive features across various time series. We integrated a filtering kernel regularization constraint into the PINN loss function, which, in contrast to conventional Tikhonov regularization methods [18,20,43], proves to be more effective in mitigating noise amplification and achieving notable dynamic error correction in small sample detonation tests. This methodology offers valuable insights for other transient tests involving limited samples. While this approach focuses on dynamic error correction for filamentary thermocouples, it can also be adapted for other fast-response thermocouple sensors or nonlinear materials with delayed temperature responses by modifying the loss function weights [38,39,40].

Despite some progress, the framework still has limitations. Firstly, although the filtering kernel aids in reducing the ill-posedness of the problem, the non-uniqueness of the solution in high-dimensional scenarios, such as 3D thermocouple arrays, may result in larger errors [12]. In comparison to purely data-driven independent Deep Belief Networks (DBNs), which can exhibit unbounded errors on sparse datasets [44], our hybrid approach effectively mitigates this issue by incorporating the partial differential equation constraints of Physics-Informed Neural Networks (PINNs) along with Sobolev norm bounds (Equation (12)), achieving a 25% reduction in mean absolute error (MSE) in simulations with 10% noise. While standalone PINNs can enhance physical accuracy, they encounter convergence issues when addressing discontinuous boundaries. However, our CRBM-DBN-based pretraining method reduces the number of iterations by 40% (Figure 10), although a risk of overfitting remains in extremely sparse data. Furthermore, PINNs face convergence challenges when managing discontinuous boundary conditions, necessitating further adjustments to hyperparameters [37,44]. Simultaneously, the CRBM-DBN, based on generative pretraining, is susceptible to overfitting, particularly when handling sparse data, which may compromise physical consistency. Overall, these challenges arise from the complexity of the multidimensional finite element method (IHCP) and the opacity of deep learning. Nevertheless, by integrating principal component analysis (PCA) techniques with sensitivity analysis of features, the applicability of this method can be broadened.

6. Conclusions

This study focuses on the challenge of dynamic inaccuracies in transient high-temperature readings due to the limited dynamic properties of thermocouples. We reformulated the correction of dynamic errors in thermocouples as an inverse heat conduction issue, introducing and validating a hybrid model that combines CRBM-DBN with PINN to tackle this inverse problem. A novel aspect of our approach is the incorporation of a filtering kernel regularization constraint within the PINN loss function, aimed at improving the convergence of errors. Experimental findings reveal that, as theoretically anticipated, the butt-welded and spherical-welded thermocouples exhibit distinct dynamic properties, with the butt-welded junction demonstrating a faster inherent response due to its lower heat capacity compared to the spherical-welded junction. For the same wire diameter, the original response signal from the spherical weld thermocouple shows greater time lag and amplitude reduction.

The CRBM-DBN-PINN effectively mitigates the dynamic errors in both types of sensors, achieving a relative temperature reconstruction error of less than 1% and under 5% in laser calibration and detonation tests, respectively, significantly surpassing conventional techniques. This approach is not only suitable for fine-wire thermocouples but also shows a degree of applicability for sensors exhibiting similar thermal hysteresis. This research not only delivers a precise method for correcting dynamic errors in thermocouples but also establishes a valuable research framework that integrates physical mechanisms with data-driven strategies for solving dynamic inverse problems in various sensor types. However, these findings present several limitations: the dataset size is restricted to only 20 samples derived from controlled laser tests, the assumption that the heat transfer process is primarily governed by conduction without considering convection effects, and the possible influence of external interference in real-world scenarios, such as noise variations in blast environments. Future research should aim to improve generalizability by increasing the dataset size and validating the model under actual operating conditions.