128 research outputs found
Local, Smooth, and Consistent Jacobi Set Simplification
The relation between two Morse functions defined on a common domain can be
studied in terms of their Jacobi set. The Jacobi set contains points in the
domain where the gradients of the functions are aligned. Both the Jacobi set
itself as well as the segmentation of the domain it induces have shown to be
useful in various applications. Unfortunately, in practice functions often
contain noise and discretization artifacts causing their Jacobi set to become
unmanageably large and complex. While there exist techniques to simplify Jacobi
sets, these are unsuitable for most applications as they lack fine-grained
control over the process and heavily restrict the type of simplifications
possible.
In this paper, we introduce a new framework that generalizes critical point
cancellations in scalar functions to Jacobi sets in two dimensions. We focus on
simplifications that can be realized by smooth approximations of the
corresponding functions and show how this implies simultaneously simplifying
contiguous subsets of the Jacobi set. These extended cancellations form the
atomic operations in our framework, and we introduce an algorithm to
successively cancel subsets of the Jacobi set with minimal modifications
according to some user-defined metric. We prove that the algorithm is correct
and terminates only once no more local, smooth and consistent simplifications
are possible. We disprove a previous claim on the minimal Jacobi set for
manifolds with arbitrary genus and show that for simply connected domains, our
algorithm reduces a given Jacobi set to its simplest configuration.Comment: 24 pages, 19 figure
Single Model Uncertainty Estimation via Stochastic Data Centering
We are interested in estimating the uncertainties of deep neural networks,
which play an important role in many scientific and engineering problems. In
this paper, we present a striking new finding that an ensemble of neural
networks with the same weight initialization, trained on datasets that are
shifted by a constant bias gives rise to slightly inconsistent trained models,
where the differences in predictions are a strong indicator of epistemic
uncertainties. Using the neural tangent kernel (NTK), we demonstrate that this
phenomena occurs in part because the NTK is not shift-invariant. Since this is
achieved via a trivial input transformation, we show that it can therefore be
approximated using just a single neural network -- using a technique that we
call UQ -- that estimates uncertainty around prediction by
marginalizing out the effect of the biases. We show that UQ's
uncertainty estimates are superior to many of the current methods on a variety
of benchmarks -- outlier rejection, calibration under distribution shift, and
sequential design optimization of black box functions
Accelerating Flow Simulations using Online Dynamic Mode Decomposition
We develop an on-the-fly reduced-order model (ROM) integrated with a flow
simulation, gradually replacing a corresponding full-order model (FOM) of a
physics solver. Unlike offline methods requiring a separate FOM-only simulation
prior to model reduction, our approach constructs a ROM dynamically during the
simulation, replacing the FOM when deemed credible. Dynamic mode decomposition
(DMD) is employed for online ROM construction, with a single snapshot vector
used for rank-1 updates in each iteration. Demonstrated on a flow over a
cylinder with Re = 100, our hybrid FOM/ROM simulation is verified in terms of
the Strouhal number, resulting in a 4.4 times speedup compared to the FOM
solver.Comment: Presented at Machine Learning and the Physical Sciences Workshop,
NeurIPS 202
Improved Surrogates in Inertial Confinement Fusion with Manifold and Cycle Consistencies
Neural networks have become very popular in surrogate modeling because of
their ability to characterize arbitrary, high dimensional functions in a data
driven fashion. This paper advocates for the training of surrogates that are
consistent with the physical manifold -- i.e., predictions are always
physically meaningful, and are cyclically consistent -- i.e., when the
predictions of the surrogate, when passed through an independently trained
inverse model give back the original input parameters. We find that these two
consistencies lead to surrogates that are superior in terms of predictive
performance, more resilient to sampling artifacts, and tend to be more data
efficient. Using Inertial Confinement Fusion (ICF) as a test bed problem, we
model a 1D semi-analytic numerical simulator and demonstrate the effectiveness
of our approach. Code and data are available at
https://github.com/rushilanirudh/macc/Comment: 10 pages, 6 figure
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