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Physics-informed neural networks (PINNs) have emerged as a popular approach for solving forward, inverse, and parametric problems involving partial differential equations. However, their performance is often limited by ill-conditioning in optimization. To address this, time-stepping-oriented neural network (TSONN) reformulate the optimization process into a sequence of well-conditioned sub-problems, offering improved robustness and efficiency for complex scenarios. This paper presents a solver for laminar flow around aerofoils based on TSONN, validated across various test cases. Specifically, the solver achieves mean relative errors of approximately 4.1% for lift coefficients and 2.2% for drag coefficients. Furthermore, this paper extends the solver to parametric problems involving flow conditions and aerofoils shapes, covering nearly all laminar flow scenarios in engineering. The parametric solver solves all laminar flows within the parameter space in just 4.6 day, at approximately 40 times the computational cost of solving a single flow. The model training involves hundreds of millions of flow conditions and aerofoil shapes, ultimately yielding a surrogate model with strong generalization capability that does not require labelled data. The surrogate model achieves average errors of 4.4% for lift coefficients and 1.7% for drag coefficients, highlighting its high generalizability and cost-effectiveness for high-dimensional parametric problems and surrogate modelling.
Cao et al. (Thu,) studied this question.