  We use Gaussian stochastic weight averaging (SWAG) to assess the model-form
uncertainty associated with neural-network-based function approximation
relevant to fluid flows. SWAG approximates a posterior Gaussian distribution of
each weight, given training data, and a constant learning rate. Having access
to this distribution, it is able to create multiple models with various
combinations of sampled weights, which can be used to obtain ensemble
predictions. The average of such an ensemble can be regarded as the `mean
estimation', whereas its standard deviation can be used to construct
`confidence intervals', which enable us to perform uncertainty quantification
(UQ) with regard to the training process of neural networks. We utilize
representative neural-network-based function approximation tasks for the
following cases: (i) a two-dimensional circular-cylinder wake; (ii) the DayMET
dataset (maximum daily temperature in North America); (iii) a three-dimensional
square-cylinder wake; and (iv) urban flow, to assess the generalizability of
the present idea for a wide range of complex datasets. SWAG-based UQ can be
applied regardless of the network architecture, and therefore, we demonstrate
the applicability of the method for two types of neural networks: (i) global
field reconstruction from sparse sensors by combining convolutional neural
network (CNN) and multi-layer perceptron (MLP); and (ii) far-field state
estimation from sectional data with two-dimensional CNN. We find that SWAG can
obtain physically-interpretable confidence-interval estimates from the
perspective of model-form uncertainty. This capability supports its use for a
wide range of problems in science and engineering.
