17 research outputs found
Positivity-preserving discontinuous spectral element methods for compressible multi-species flows
We introduce a novel positivity-preserving, parameter-free numerical
stabilisation approach for high-order discontinuous spectral element
approximations of compressible multi-species flows. The underlying
stabilisation method is the adaptive entropy filtering approach (Dzanic and
Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative
formulation of the multi-species flow equations. We show that the
straightforward enforcement of entropy constraints in the filter yields poor
results around species interfaces and propose an adaptive, parameter-free
switch for the entropy bounds based on the convergence properties of the
pressure field which drastically improves its performance for multi-species
flows. The proposed approach is shown in a variety of numerical experiments
applied to the multi-species Euler and Navier--Stokes equations computed on
unstructured grids, ranging from shock-fluid interaction problems to
three-dimensional viscous flow instabilities. We demonstrate that the approach
can retain the high-order accuracy of the underlying numerical scheme even at
smooth extrema, ensure the positivity of the species density and pressure in
the vicinity of shocks and contact discontinuities, and accurately predict
small-scale flow features with minimal numerical dissipation.Comment: Submitted for revie
Probing optimisation in physics-informed neural networks
A novel comparison is presented of the effect of optimiser choice on the
accuracy of physics-informed neural networks (PINNs). To give insight into why
some optimisers are better, a new approach is proposed that tracks the training
trajectory curvature and can be evaluated on the fly at a low computational
cost. The linear advection equation is studied for several advective
velocities, and we show that the optimiser choice substantially impacts PINNs
model performance and accuracy. Furthermore, using the curvature measure, we
found a negative correlation between the convergence error and the curvature in
the optimiser local reference frame. It is concluded that, in this case, larger
local curvature values result in better solutions. Consequently, optimisation
of PINNs is made more difficult as minima are in highly curved regions.Comment: Accepted at the ICLR 2023 Workshop on Physics for Machine Learnin