# Fast Temperature Field Extrapolation Under Non-Periodic Boundary Conditions
## 1 Introduction  
Engineering equipment is often faced with complex and variable operation conditions, such as anomalous thermal stimuli, force stimuli, and electromagnetic stimuli [1, 2, 3]. In particular, roughly 55% of the failures seen in equipment are caused by severe thermal stimuli, and the thermal field analysis of equipment is critical [4, 5]. It is impossible to obtain comprehensive and accurate thermal field data through experiments using intrusive means. Theoretically, numerical heat transfer theory can be used to simulate the temperature field by establishing a heat transfer model for any complex structure [6, 7, 8]. However, a full-order model such as this entails a large-scale system of algebraic equations, which requires huge computational resources [9, 10]. The tedious calculation process is a great obstacle to the need for rapid thermal analysis and optimization, so it is necessary to introduce a reduced-order model to accelerate their calculation. The proper orthogonal decomposition (POD) method is one of the most commonly used methods in the field of model order reducing [11, 12, 13], which is able to extract a set of low-dimensional optimal POD bases from the huge physical field of its target, afterwhich this set of optimal POD bases is used to approximate the original physical field. The method was originally proposed by [PERSON] [14] in 1901 and is now widely used in fields such as signal analysis, statistics, meteorology, and fluid mechanics [15; 16; 17]. Taking the field of fluid dynamics as an example, [PERSON] [18] first applied the POD method to turbulence research in 1967 when identifying coherent structures in the flow field, and he processed the spatial correlation matrix directly. The method he used was named the direct POD method. The dimension of the spatial correlation matrix is very large, being comparable to the scale of the number of meshes, which severely limits the application of the POD method. To address such problems, [PERSON] [19] proposed the snapshot POD method in 1987, replacing the spatial correlation matrix with the temporal correlation matrix when the spatial dimension was larger than the temporal dimension, which greatly reduced the computational cost. Since then, the POD method has been developed for use in many contexts. In the field of heat transfer, [PERSON] et al. [20] established a POD-Galerkin reduced-order model for the unsteady thermal conductivity problem of a two-dimensional flat plate, which reduced the degrees of freedom of the system from 1452 to 5. [PERSON] et al. [21] developed the boundary-independent POD reduced-order model. [PERSON] et al. [22] coupled the POD-Galerkin method with the Navier-Stokes equations for the unsteady flow and heat transfer problem of a three-dimensional T-tube. The results showed that their method was hundreds of times faster than the full-order model. [PERSON] et al. [23] combined the POD reduced-order method with cubic spline interpolation to study the cross-scale flow and heat transfer in an air-cooled island, which significantly reduced the computational cost. [PERSON] et al. [24] combined the POD reduced-order method with deep learning to construct the relationship of POD mode coefficients with different operating conditions for a steam generator. However, most of the studies mentioned above are multi-case problems, the purpose of which is to use the known solutions of parts of case studies for POD reduced-order problems to obtain the solutions for other scenarios [25; 26]. The method consumes huge resources for the computation of numerical samples and it involves repetitive computation in terms of the time scales. [PERSON] et al. [27; 28; 29] proposed a reduced-order extrapolation algorithm based on the POD method, which is a method that uses the known solution to quickly calculate the solution for future scenarios under the same operating conditions. It is of great importance to speed up the calculation of transient equations.