Variationally mimetic operator network approach to transient viscous flows

Abstract

The Variationally Mimetic Operator Network (VarMiON) approach is a machine learning technique, originally developed to predict the solution of elliptic differential problems, that combines operator networks with a structure inherited from the variational formulation of the equations. We investigate the capabilities of this method in the context of viscous flows, by extending its formulation to vector-valued unknown fields and with a particular emphasis on the space-time approximation context necessary to deal with transient flows. As a first step, we restrict attention to the regime of low-to-moderate Reynolds numbers, in which the Navier--Stokes equations can be linearized to give the time-dependent Stokes problem for incompressible fluids. The details of the method as well as its performance are illustrated in three paradigmatic flow geometries where we obtain a very good agreement between the VarMiON predictions and reference finite-element solutions.

Figures (11)

• Figure 1: Schematic of a generic DeepONet. The output prediction $\hat{\mathcal{N}}$ is given as a dot product between the output of the trunk (starting from some space input) and the output of the branch network (starting from some input data such as a forcing term). This structure is used to predict the outcome of an operator $\mathcal{N}: f ,k\mapsto u(x)$.
• Figure 2: VarMiON scheme for the time-dependent Stokes equation. The network prediction $\hat{\mathcal{N}}$ is given as a dot product between the outcome of the trunks and the outcome of the branches starting from the input data $(\rho, \mu,\hat{\boldsymbol{F}},\hat{\boldsymbol{U}}_0,\hat{\boldsymbol{P}}_0, \hat{\boldsymbol{G}})$.
• Figure 3: The meshes used for the FEM solution of (a) the cavity flow, (b) the flow past a cylinder, and (c) the contraction flow.
• Figure 4: The loss function computed on the training set (red line) and the validation one (blue line), respectively. On the left, the behaviour of the losses, on the right, their semi-log. The panels are related to the cavity flow, the flow past a cylinder, and the contraction flow, on the top (a)-(b), center (c)-(d) and bottom (e)-(f), respectively.
• Figure 5: The probability density of the relative $L_2$-error computed on the testing set. The panels are related to the cavity flow, the flow past a cylinder, and the contraction flow, on the left (a), center (b) and right (c), respectively.