90 research outputs found
Fundamental solutions of an extended hydrodynamic model in two dimensions: derivation, theory and applications
The inability of the Navier-Stokes-Fourier equations to capture rarefaction
effects motivates us to adopt the extended hydrodynamic equations. In the
present work, a hydrodynamic model comprised of the conservation laws closed
with the recently propounded coupled constitutive relations (CCR) -- referred
to as the CCR model -- adequate for describing moderately rarefied gas is
utilized. A numerical framework based on the method of fundamental solutions is
developed and employed to solve the CCR model in two dimensions. To this end,
the fundamental solutions of the linearized CCR model are derived in two
dimensions. The significance of deriving the two-dimensional fundamental
solutions is that they cannot be deduced from their three-dimensional
counterparts that do exist in literature. As applications, the developed
numerical framework based on the derived fundamental solutions is used to
simulate (i) a rarefied gas flow confined between two coaxial cylinders with
evaporating walls and (ii) a temperature-driven rarefied gas flow between two
non-coaxial cylinders. The results for both problems have been validated
against those obtained with the other classical approaches. Through this, it is
shown that the method of fundamental solutions is an efficient tool for
addressing two-dimensional multiphase microscale gas flow problems at a low
computational cost. Moreover, the findings also show that the CCR model solved
with the method of fundamental solutions depicts rarefaction effects, like
transpiration flows and thermal stress, generally well.Comment: 14 figure
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Coupled constitutive relations: a second law based higher order closure for hydrodynamics
In the classical framework, the Navier-Stokes-Fourier equations are obtained
through the linear uncoupled thermodynamic force-flux relations which guarantee
the non-negativity of the entropy production. However, the conventional
thermodynamic description is only valid when the Knudsen number is sufficiently
small. Here, it is shown that the range of validity of the
Navier-Stokes-Fourier equations can be extended by incorporating the nonlinear
coupling among the thermodynamic forces and fluxes. The resulting system of
conservation laws closed with the coupled constitutive relations is able to
describe many interesting rarefaction effects, such as Knudsen paradox,
transpiration flows, thermal stress, heat flux without temperature gradients,
etc., which can not be predicted by the classical Navier-Stokes-Fourier
equations. For this system of equations, a set of phenomenological boundary
conditions, which respect the second law of thermodynamics, is also derived.
Some of the benchmark problems in fluid mechanics are studied to show the
applicability of the derived equations and boundary conditions.Comment: 20 pages, 6 figures, Proceedings of the Royal Society A (Open access
article
Skill-Mix: a Flexible and Expandable Family of Evaluations for AI models
With LLMs shifting their role from statistical modeling of language to
serving as general-purpose AI agents, how should LLM evaluations change?
Arguably, a key ability of an AI agent is to flexibly combine, as needed, the
basic skills it has learned. The capability to combine skills plays an
important role in (human) pedagogy and also in a paper on emergence phenomena
(Arora & Goyal, 2023).
This work introduces Skill-Mix, a new evaluation to measure ability to
combine skills. Using a list of skills the evaluator repeatedly picks
random subsets of skills and asks the LLM to produce text combining that
subset of skills. Since the number of subsets grows like , for even modest
this evaluation will, with high probability, require the LLM to produce
text significantly different from any text in the training set. The paper
develops a methodology for (a) designing and administering such an evaluation,
and (b) automatic grading (plus spot-checking by humans) of the results using
GPT-4 as well as the open LLaMA-2 70B model.
Administering a version of to popular chatbots gave results that, while
generally in line with prior expectations, contained surprises. Sizeable
differences exist among model capabilities that are not captured by their
ranking on popular LLM leaderboards ("cramming for the leaderboard").
Furthermore, simple probability calculations indicate that GPT-4's reasonable
performance on is suggestive of going beyond "stochastic parrot" behavior
(Bender et al., 2021), i.e., it combines skills in ways that it had not seen
during training.
We sketch how the methodology can lead to a Skill-Mix based eco-system of
open evaluations for AI capabilities of future models
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