Stochastic goal-oriented error estimation with memory

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

“…neurons , depth , r e g = i n t ( hyperparams [ 0 ] ) , np . i n t ( hyperparams [ 1 ] ) , hyperparams [ 2 ] model = S e q u e n t i a l ( ) 4 model . add (LSTM( neurons , i n p u t s h a p e =(None , np .…”

“…The inner-loop optimization problem depends on the category within which the regression model falls. 1 1. For regression models in Category 1, we compute optimal model parameters θ f as the solution to the minimization problem minimize θ f µ∈Dtrain n∈Ttrain…”

“…where x : [0, T ] × D → R K are the generalized coordinates and x ref : D → R K denotes the reference state (typically chosen to be either x ref (µ) = x 0 (µ) or x ref (µ) = 0). The generalized coordinates are computed by performing a projection process performed on the FOM ODE (1). Namely, the approximation (11) is substituted in the FOM ODE (1), and the resulting residual is enforced to be orthogonal to range of the test-basis matrix.…”

“…The residual norm is informative of the error, as it typically appears in a posteriori error bounds and is usually strongly correlated with the error; it is often used within greedy methods for snapshot collection [7,6,19,48,49] and when using ROMs for PDE-constrained optimization within a trust-region framework [52,50,51]. In contrast, dual-weighted residuals are derived from a Taylor-series approximation of the error, and are typically used for goal-oriented error estimation and mesh adaptation [28,1,46,47]. Due to their practical utility, error indicators have been largely successful in quantifying and controlling errors through mesh adaptation for static problems (i.e., problems without time evolution).…”

“…As such, this technique can be considered a regression approach, where the regression model is provided by a GP, the features are the system parameters, and the response is the approximate-solution error. Recently, researchers have pursued improvements over this technique by (1) considering a wider class of modern machine-learning regression methods than simply Gaussian processes (which do not scale well to high-dimensional feature spaces), and (2) considering more informative features than the system parameters alone. First, the reduced-order model error surrogates (ROMES) method [12] was proposed in the context of static problems.…”