539 research outputs found
An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs
To solve optimization problems with parabolic PDE constraints, often methods
working on the reduced objective functional are used. They are computationally
expensive due to the necessity of solving both the state equation and a
backward-in-time adjoint equation to evaluate the reduced gradient in each
iteration of the optimization method. In this study, we investigate the use of
the parallel-in-time method PFASST in the setting of PDE constrained
optimization. In order to develop an efficient fully time-parallel algorithm we
discuss different options for applying PFASST to adjoint gradient computation,
including the possibility of doing PFASST iterations on both the state and
adjoint equations simultaneously. We also explore the additional gains in
efficiency from reusing information from previous optimization iterations when
solving each equation. Numerical results for both a linear and a non-linear
reaction-diffusion optimal control problem demonstrate the parallel speedup and
efficiency of different approaches
Comparison of neural closure models for discretised PDEs
Neural closure models have recently been proposed as a method for efficiently approximating small scales in multiscale systems with neural networks. The choice of loss function and associated training procedure has a large effect on the accuracy and stability of the resulting neural closure model. In this work, we systematically compare three distinct procedures: “derivative fitting”, “trajectory fitting” with discretise-then-optimise, and “trajectory fitting” with optimise-then-discretise. Derivative fitting is conceptually the simplest and computationally the most efficient approach and is found to perform reasonably well on one of the test problems (Kuramoto-Sivashinsky) but poorly on the other (Burgers). Trajectory fitting is computationally more expensive but is more robust and is therefore the preferred approach. Of the two trajectory fitting procedures, the discretise-then-optimise approach produces more accurate models than the optimise-then-discretise approach. While the optimise-then-discretise approach can still produce accurate models, care must be taken in choosing the length of the trajectories used for training, in order to train the models on long-term behaviour while still producing reasonably accurate gradients during training. Two existing theorems are interpreted in a novel way that gives insight into the long-term accuracy of a neural closure model based on how accurate it is in the short term.<br/
A shallow physics-informed neural network for solving partial differential equations on surfaces
In this paper, we introduce a shallow (one-hidden-layer) physics-informed
neural network for solving partial differential equations on static and
evolving surfaces. For the static surface case, with the aid of level set
function, the surface normal and mean curvature used in the surface
differential expressions can be computed easily. So instead of imposing the
normal extension constraints used in literature, we write the surface
differential operators in the form of traditional Cartesian differential
operators and use them in the loss function directly. We perform a series of
performance study for the present methodology by solving Laplace-Beltrami
equation and surface diffusion equation on complex static surfaces. With just a
moderate number of neurons used in the hidden layer, we are able to attain
satisfactory prediction results. Then we extend the present methodology to
solve the advection-diffusion equation on an evolving surface with given
velocity. To track the surface, we additionally introduce a prescribed hidden
layer to enforce the topological structure of the surface and use the network
to learn the homeomorphism between the surface and the prescribed topology. The
proposed network structure is designed to track the surface and solve the
equation simultaneously. Again, the numerical results show comparable accuracy
as the static cases. As an application, we simulate the surfactant transport on
the droplet surface under shear flow and obtain some physically plausible
results
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