Digital computers use
numbers based on flawed representations of real numbers, which may lead to
inaccuracies when simulating the motion of molecules, weather systems and
fluids, find scientists.
University College London – September 23,.
230219 -- The study, published today in Advanced Theory and Simulations,
shows that digital computers cannot reliably reproduce the behaviour of
'chaotic systems' which are widespread. This fundamental limitation could have
implications for high performance computation (HPC) and for applications of
machine learning to HPC.
Science and study co-author, said:
"Our work shows that the behaviour of the chaotic dynamical systems is
richer than any digital computer can capture. Chaos is more commonplace than
many people may realise and even for very simple chaotic systems, numbers used
by digital computers can lead to errors that are not obvious but can have a big
impact. Ultimately, computers can't simulate everything."
The team investigated the impact of
using floating-point arithmetic -- a method standardised by the IEEE and used
since the 1950s to approximate real numbers on digital computers.
Digital computers use only rational
numbers, ones that can be expressed as fractions. Moreover the denominator of
these fractions must be a power of two, such as 2, 4, 8, 16, etc. There are
infinitely more real numbers that cannot be expressed this way.
In the present work, the scientists used
all four billion of these single-precision floating-point numbers that range
from plus to minus infinity. The fact that the numbers are not distributed
uniformly may also contribute to some of the inaccuracies.
First author, Professor Bruce Boghosian
(Tufts University), said: "The four billion single-precision
floating-point numbers that digital computers use are spread unevenly, so there
are as many such numbers between 0.125 and 0.25, as there are between 0.25 and
0.5, as there are between 0.5 and 1.0.
It is amazing that they are able to
simulate real-world chaotic events as well as they do. But even so, we are now
aware that this simplification does not accurately represent the complexity of
chaotic dynamical systems, and this is a problem for such simulations on all
current and future digital computers.”
The study builds on the work of Edward
Lorenz of MIT whose weather simulations using a simple computer model in the
1960s showed that tiny rounding errors in the numbers fed into his computer led
to quite different forecasts, which is now known as the 'butterfly effect'.
The team compared the known mathematical
reality of a simple one-parameter chaotic system called the 'generalised
Bernoulli map' to what digital computers would predict if every one of the
available single-precision floating-point numbers were used.
They found that, for some values of the
parameter, the computer predictions are totally wrong, whilst for other choices
the calculations may appear correct, but deviate by up to 15%.
The authors say these pathological
results would persist even if double-precision floating-point numbers were
used, of which there are vastly more to draw on.
"We use the generalised Bernoulli
map as a mathematical representation for many other systems that change
chaotically over time, such as those seen across physics, biology and
chemistry," explained Professor Coveney. "These are being used to
predict important scenarios in climate change, in chemical reactions and in
nuclear reactors, for example, so it's imperative that computer-based
simulations are now carefully scrutinised."
The team say that their discovery has
implications for the field of artificial intelligence, when machine learning is
applied to data derived from computer simulations of chaotic dynamical systems,
and for those trying to model all kinds of natural processes.
More research is needed to examine the
extent to which the use of floating-point arithmetic is causing problems in
everyday computational science and modelling and, if errors are found, how to
correct them.
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