22-26 March 2021
Artificial neural network for bifurcating phenomena modelled by nonlinear parametrized PDEs
Tuesday session 2 (Zoom) (15:00–16:30 GMT)
View slides (pdf) (available under a CC BY 4.0 license)
This work aims to develop and investigate a computational framework to study parametrized partial differential equations (PDEs) which model nonlinear systems undergoing bifurcations. Bifurcation analysis, ie following the coexisting branches due to the non-uniqueness of the solution, as well as determining the bifurcation points themselves, are complex computational tasks [1] [2] [3]. The combination of reduced basis (RB) model reduction and artificial neural network (ANN) can potentially reduce the computational burden by several orders of magnitude and shed light on new strategies. Following the POD-NN approach [4], we analyzed two CFD applications where both physical and geometrical parameters were considered. We studied the Navier–Stokes equations for a viscous, steady, and incompressible flow: (i) in a planar straight channel with a narrow inlet of varying width, and (ii) in a triangular parametrized cavity [5]. All the simulations were performed within the open source software 
FEniCS and 
RBniCS [6] for the RB framework, integrated with 
PyTorch to construct the neural network.
References
- [1] F. Pichi, A. Quaini, and G. Rozza. A reduced order modeling technique to study bifurcating phenomena: Application to the Gross–Pitaevskii equation, SIAM Journal on Scientific Computing 42, B1115–B1135, 2020. [DOI: 10.1137/20M1313106]
- [2] F. Pichi and G. Rozza. Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations, Journal of Scientific Computing 81, 112–135, 2019. [DOI: 10.1007/s10915-019-01003-3]
- [3] F. Pichi, M. Strazzullo, F. Ballarin, and G. Rozza. Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction, arΧiv: 2010.13506, 2020.
- [4] J. S. Hesthaven and S. Ubbiali. Non-intrusive reduced order modeling of nonlinear problems using neural networks, Journal of Computational Physics 363, 55–78, 2018. [DOI: 10.1016/j.jcp.2018.02.037]
- [5] F. Pichi, F. Ballarin, G. Rozza, and J. S. Hesthaven. Artificial neural network for bifurcating phenomena modelled by nonlinear parametrized PDEs, in preparation, 2020.
- [6] RBniCS - reduced order modelling in FEniCS. www.rbnicsproject.org