233 research outputs found
Multifidelity Information Fusion Algorithms for High-Dimensional Systems and Massive Data sets
We develop a framework for multifidelity information fusion and predictive inference in high-dimensional input spaces and in the presence of massive data sets. Hence, we tackle simultaneously the “big N" problem for big data and the curse of dimensionality in multivariate parametric problems. The proposed methodology establishes a new paradigm for constructing response surfaces of high-dimensional stochastic dynamical systems, simultaneously accounting for multifidelity in physical models as well as multifidelity in probability space. Scaling to high dimensions is achieved by data-driven dimensionality reduction techniques based on hierarchical functional decompositions and a graph-theoretic approach for encoding custom autocorrelation structure in Gaussian process priors. Multifidelity information fusion is facilitated through stochastic autoregressive schemes and frequency-domain machine learning algorithms that scale linearly with the data. Taking together these new developments leads to linear complexity algorithms as demonstrated in benchmark problems involving deterministic and stochastic fields in up to 10⁵ input dimensions and 10⁵ training points on a standard desktop computer
The ROMES method for statistical modeling of reduced-order-model error
This work presents a technique for statistically modeling errors introduced
by reduced-order models. The method employs Gaussian-process regression to
construct a mapping from a small number of computationally inexpensive `error
indicators' to a distribution over the true error. The variance of this
distribution can be interpreted as the (epistemic) uncertainty introduced by
the reduced-order model. To model normed errors, the method employs existing
rigorous error bounds and residual norms as indicators; numerical experiments
show that the method leads to a near-optimal expected effectivity in contrast
to typical error bounds. To model errors in general outputs, the method uses
dual-weighted residuals---which are amenable to uncertainty control---as
indicators. Experiments illustrate that correcting the reduced-order-model
output with this surrogate can improve prediction accuracy by an order of
magnitude; this contrasts with existing `multifidelity correction' approaches,
which often fail for reduced-order models and suffer from the curse of
dimensionality. The proposed error surrogates also lead to a notion of
`probabilistic rigor', i.e., the surrogate bounds the error with specified
probability
Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)
Multifidelity simulation methodologies are often used in an attempt to
judiciously combine low-fidelity and high-fidelity simulation results in an
accuracy-increasing, cost-saving way. Candidates for this approach are
simulation methodologies for which there are fidelity differences connected
with significant computational cost differences. Physics-informed Neural
Networks (PINNs) are candidates for these types of approaches due to the
significant difference in training times required when different fidelities
(expressed in terms of architecture width and depth as well as optimization
criteria) are employed. In this paper, we propose a particular multifidelity
approach applied to PINNs that exploits low-rank structure. We demonstrate that
width, depth, and optimization criteria can be used as parameters related to
model fidelity, and show numerical justification of cost differences in
training due to fidelity parameter choices. We test our multifidelity scheme on
various canonical forward PDE models that have been presented in the emerging
PINNs literature
- …