50 research outputs found
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
This work derives a residual-based a posteriori error estimator for reduced
models learned with non-intrusive model reduction from data of high-dimensional
systems governed by linear parabolic partial differential equations with
control inputs. It is shown that quantities that are necessary for the error
estimator can be either obtained exactly as the solutions of least-squares
problems in a non-intrusive way from data such as initial conditions, control
inputs, and high-dimensional solution trajectories or bounded in a
probabilistic sense. The computational procedure follows an offline/online
decomposition. In the offline (training) phase, the high-dimensional system is
judiciously solved in a black-box fashion to generate data and to set up the
error estimator. In the online phase, the estimator is used to bound the error
of the reduced-model predictions for new initial conditions and new control
inputs without recourse to the high-dimensional system. Numerical results
demonstrate the workflow of the proposed approach from data to reduced models
to certified predictions
Greedy construction of quadratic manifolds for nonlinear dimensionality reduction and nonlinear model reduction
Dimensionality reduction on quadratic manifolds augments linear
approximations with quadratic correction terms. Previous works rely on linear
approximations given by projections onto the first few leading principal
components of the training data; however, linear approximations in subspaces
spanned by the leading principal components alone can miss information that are
necessary for the quadratic correction terms to be efficient. In this work, we
propose a greedy method that constructs subspaces from leading as well as later
principal components so that the corresponding linear approximations can be
corrected most efficiently with quadratic terms.
Properties of the greedily constructed manifolds allow applying linear
algebra reformulations so that the greedy method scales to data points with
millions of dimensions. Numerical experiments demonstrate that an orders of
magnitude higher accuracy is achieved with the greedily constructed quadratic
manifolds compared to manifolds that are based on the leading principal
components alone.Comment: 25 pages, 14 figure
Dynamic data-driven model reduction: adapting reduced models from incomplete data
This work presents a data-driven online adaptive model reduction approach for systems that undergo dynamic changes. Classical model reduction constructs a reduced model of a large-scale system in an offline phase and then keeps the reduced model unchanged during the evaluations in an online phase; however, if the system changes online, the reduced model may fail to predict the behavior of the changed system. Rebuilding the reduced model from scratch is often too expensive in time-critical and real-time environments. We introduce a dynamic data-driven adaptation approach that adapts the reduced model from incomplete sensor data obtained from the system during the online computations. The updates to the reduced models are derived directly from the incomplete data, without recourse to the full model. Our adaptivity approach approximates the missing values in the incomplete sensor data with gappy proper orthogonal decomposition. These approximate data are then used to derive low-rank updates to the reduced basis and the reduced operators. In our numerical examples, incomplete data with 30–40 % known values are sufficient to recover the reduced model that would be obtained via rebuilding from scratch.United States. Air Force Office of Scientific Research (AFOSR MURI on multi-information sources of multi-physics systems, Award Number FA9550-15-1-0038)United States. Dept. of Energy (Applied Mathematics Program, Award DE-FG02 08ER2585)United States. Dept. of Energy (Applied Mathematics Program, Award DE-SC0009297
Context-aware controller inference for stabilizing dynamical systems from scarce data
This work introduces a data-driven control approach for stabilizing
high-dimensional dynamical systems from scarce data. The proposed context-aware
controller inference approach is based on the observation that controllers need
to act locally only on the unstable dynamics to stabilize systems. This means
it is sufficient to learn the unstable dynamics alone, which are typically
confined to much lower dimensional spaces than the high-dimensional state
spaces of all system dynamics and thus few data samples are sufficient to
identify them. Numerical experiments demonstrate that context-aware controller
inference learns stabilizing controllers from orders of magnitude fewer data
samples than traditional data-driven control techniques and variants of
reinforcement learning. The experiments further show that the low data
requirements of context-aware controller inference are especially beneficial in
data-scarce engineering problems with complex physics, for which learning
complete system dynamics is often intractable in terms of data and training
costs.Comment: 26 pages, 10 figure