Traditional methods for uncertainty quantification (UQ) struggle with the curse of dimensionality when dealing with high-dimensional problems. One approach to address this challenge is to leverage the potent approximation capabilities of deep neural networks (DNNs). However, conventional DNNs often demand a substantial amount of high-fidelity (HF) training data to ensure precise predictions. Unfortunately, the availability of such data is limited due to computational or experimental constraints, primarily driven by associated costs. To mitigate these training expenses, multi-fidelity deep neural networks (MF-DNNs) were developed here, wherein a subnetwork was constructed to simultaneously capture both linear and non-linear correlations between HF- and low-fidelity (LF) data. The efficacy of MF-DNNs was initially demonstrated by accurately approximating diverse benchmark functions. Subsequently, the developed MF-DNNs were employed for the first time to simulate the aleatory uncertainty propagation in 1-, 32-, and 100-dimensional contexts, considering either uniform or Gaussian distributions of input uncertainties. The UQ results affirm that MF-DNNs adeptly predicted probability density distributions of quantities of interest (QoI) and their statistical moments without significant compromise of accuracy. Furthermore, MF-DNNs were applied to model the physical flow inside an aircraft propulsion system while accounting for aleatory uncertainties originating from experimental measurement errors. The distributions of isentropic Mach number were accurately predicted by MF-DNNs based on the 2D Euler flow field and few experimental data points. In conclusion, the proposed MF-DNN framework exhibits significant promise in addressing UQ and robust optimization challenges in practical engineering applications, particularly when dealing with multi-fidelity data sources. The main results were published in the Reliability Engineering & System Safety 245 (2024): 109975.
The above-mentioned MF-DNN-based framework was then applied to LS89 nozzle modified by fouling. Geometrical uncertainties significantly influence aerodynamic performance of gas turbines. One representative example is given by the airfoil shape modified by fouling deposition, as in turbine nozzle vanes, which generates high-dimensional input uncertainties. However, the traditional UQ approaches suffer from the curse of dimensionality phenomenon in predicting the influence of high-dimensional uncertainties. The prediction accuracy of MF-DNNs was first evaluated using a 15-dimensional benchmark function. An affordable turbomachinery UQ platform was then built based on the MF-DNN model, the sampling-, the parameterization- and the statistical processing modules. The impact of fouling deposition on LS89 nozzle vane flow was investigated using the proposed UQ framework. In detail, the MF-DNNs were fine-tuned based on bi-level numerical simulation results: the 2D Euler flow field as low-fidelity data and the 3D Reynolds-Averaged Navier-Stokes (RANS) flow field as high-fidelity data. The UQ results show that the total pressure loss of LS89 vane was increased by at most 17.1 % or reduced by at most 4.3 %, while the mean value of loss was increased by 3.4 % compared to the baseline. The main reason for relative changes in turbine nozzle performance is that the geometric uncertainties induced by fouling deposition significantly alter the intensity of shock waves near the throat area and trailing edge. The main work was published in the Journal of Turbomachinery 145, no. 11 (2023).
Besides, the emerging physics-informed neural networks (PINNs) approach was utilized for the first time to predict the flowfield of a compressor cascade. Different from conventional training methods, a new adaptive learning strategy that mitigates gradient imbalance through incorporating adaptive weights in conjunction with a dynamically adjusting learning rate was used during the training process to improve the convergence of PINNs. The performance of PINNs was assessed here by solving both the forward and inverse problems. In the forward problem, by encapsulating the physical relations among relevant variables, PINNs demonstrated their effectiveness in accurately forecasting the compressor’s flowfield. PINNs also showed obvious advantages over the traditional computational fluid dynamics (CFD) approaches, particularly in scenarios lacking complete boundary conditions, as is often the case in inverse engineering problems. PINNs successfully reconstructed the flowfield of the compressor cascade solely based on partial velocity vectors and near-wall pressure information. Furthermore, PINNs showed robust performance in the environment of various levels of aleatory uncertainties stemming from labelled data. This research provides evidence that PINNs can offer turbomachinery designers an additional and promising option alongside the current dominant CFD methods. This research work was published in the AIAA Journal (2024): 1-11.
Furthermore, the research work on the MF-DNN framework was presented by Dr. Li to a wider audience at the American Society of Mechanical Engineers (ASME) Turbo Expo Conference in June 2023 in Boston, United States. Dr. Li also showcased the MF-DNN model to the Rolls-Royce aerodynamics team in March 2023, and explored potential collaborations with industrial partners. The main results on PG-GNNs will be presented at the ASME Turbo Expo Conference in June 2024 in London, United Kingdom.
