A hybrid method based on proper orthogonal decomposition and deep neural networks for flow and heat field reconstruction

Estimating the full state of physical systems, including thermal and flow status, from sparse measurements of limited sensors is a critical technology for perception and control. Neural networks have been used in recent studies to reconstruct the global field in a supervised learning paradigm. However, these studies encounter two major challenges: the lack of interpretability of black-box models and performance bottleneck caused by network structure and parameter optimization. This paper proposes a hybrid method based on proper orthogonal decomposition (POD) and deep neural networks (DNNs) to further enhance the interpretability and accuracy of flow and heat field reconstruction. The key idea is to leverage the inherent data modes extracted by POD that capture essential features in physical fields, and formulate the reconstruction problem as finding an optimal linear combination of dominant POD modes. To reduce the error introduced by underfitting and model structure, this paper estimates the coefficients of POD modes by establishing and solving a linear optimization problem that minimizes the gap between the recovered field and the exact measurements, rather than employing regression models. However, the underdetermined issue cased by the sparse measurements restricts the optimization problem to obtain a proper solution. To alleviate this problem, this paper presents to utilize the powerful non-linear approximation ability of DNNs to produce a reference field as auxiliary observations, which combines exact measurements to jointly constrain the optimization problem solving. Finally, the global physical field is reconstructed by superposing dominant POD modes weighted with the solved coefficients. By combining with POD technology, the proposed method can also improve the performance of neural networks on reconstruction problems with large-scale and irregular domains. The experiments conducted on the fluid and thermal benchmarks demonstrate that the proposed method can significantly boost neural network reconstruction performance and outperform existing POD-based methods.