  Physics-constrained neural networks are commonly employed to enhance
prediction robustness compared to purely data-driven models, achieved through
the inclusion of physical constraint losses during the model training process.
However, one of the major challenges of physics-constrained neural networks
consists of the training complexity especially for high-dimensional systems. In
fact, conventional physics-constrained models rely on singular-fidelity data
necessitating the assessment of physical constraints within high-dimensional
fields, which introduces computational difficulties. Furthermore, due to the
fixed input size of the neural networks, employing multi-fidelity training data
can also be cumbersome. In this paper, we propose the Multi-Scale
Physics-Constrained Neural Network (MSPCNN), which offers a novel methodology
for incorporating data with different levels of fidelity into a unified latent
space through a customised multi-fidelity autoencoder. Additionally, multiple
decoders are concurrently trained to map latent representations of inputs into
various fidelity physical spaces. As a result, during the training of
predictive models, physical constraints can be evaluated within low-fidelity
spaces, yielding a trade-off between training efficiency and accuracy. In
addition, unlike conventional methods, MSPCNN also manages to employ
multi-fidelity data to train the predictive model. We assess the performance of
MSPCNN in two fluid dynamics problems, namely a two-dimensional Burgers' system
and a shallow water system. Numerical results clearly demonstrate the
enhancement of prediction accuracy and noise robustness when introducing
physical constraints in low-fidelity fields. On the other hand, as expected,
the training complexity can be significantly reduced by computing physical
constraint loss in the low-fidelity field rather than the high-fidelity one.
