IEEE Computational Intelligence Magazine - May 2021 - 24

released at 2 m height, at an initial velocity of
8 m/s (simulating a human throwing a basketball-sized ball with his arm). We vary the
launch angles to simulate different projectiles.
In (15), R is the resistance coefficient which is
related to the air density and object velocity,
and g is the gravity. For simplicity, it is assumed
that the ball does not spin much so the Magnus
effect is neglected. Then the air resistance or
drag effect is modelled by R = CV, where
V = ^x t2 + y t2h , C = 0.5tC d A/m. t is the air density. C d is
the drag coefficient, A = rr 2 is the cross-sectional area of the ball,
and m is its mass. The radius and mass of a basketball are typically
r = 0.12 (m) and m = 0.6 (kg), and the drag coefficient C d is
around 0.54 [53].
Figure 6a shows the tNES optimized projectiles of the flying ball for various launch angles a 0 = 15, 30, 45, 75, between
the time domain t e 60, 2 s@ under the effect of Earth gravity
g = 9.8 ^m/s 2h and air t = 1.2 ^kg/m 2h . Moreover, tNES is
applied to solve the projectiles on another planet such as Mars
( g = 3.7, t = 0 -4.5 s, no drag effect) or on the Moon ( g = 1.6,
t = 0 -10 s, no drag effect), since the same physics laws apply
elsewhere in the universe. In Figure 6b, solutions from 20 SGD
and 40 tNES optimization runs for one particular scenario
( g = 1.6, a 0 = 45) are compared against the ground truth. The
SGD produces three different groups of projectiles, two of
which are poorly optimized solutions due to being trapped in
some bad local minima. There are also deviations within the
groups, because the optimization runs are yet to fully converge
into the minima. In contrast, the quality of tNES solutions are
consistent, so they all overlap onto an indistinguishable projectile. Their median mean squared errors against the ground truth
solution are 2.6e2 for SGD and 6.9e-7 for tNES. A similar pattern can be observed in other scenarios as well.
The convergence and optimized loss results given by tNES,
xNES and SGD are summarized in Figure 7.We show the convergence history for selected scenarios, and compare the optimized loss for all scenarios with a " letter value " plot [54].
Different from previous examples, for most of the problems here
the xNES cannot completely avoid being trapped in a bad local
minimum. Nevertheless, it offers a better chance to find better
minima when compared to SGD. At the same time, tNES gives
the best results by leveraging the experiential priors. These priors come from solving a related problem scenario in the absence
of drag, a scenario with different gravitational conditions but a
similar launch angle, or for one with different launch angle but
similar gravity. By transferring and reusing information from
such relevant sources, tNES avoids being trapped in a bad local
minimum. These results demonstrate that transfer neuroevolution indeed solves differential equations faster and better.

Classical numerical schemes may be able to solve
this differential equation faster, but are notoriously
prone to failure at high velocity values caused by
inappropriate meshing, which gives rise to unphysical
oscillations in the solution. Being mesh-free, the PINN
approach avoids this issue.
Figure 5c compares the convergence trends of different optimizers, derived from 30 independent runs for each problem. We
first compare the xNES with SGD. At lower v, SGD descends
very quickly, but starts levelling off even though the loss values
are far from the optimum. In contrast, xNES starts to catch up
with SGD given a sufficient number of evaluations, and it eventually converges (at least 2 orders of magnitude) better. As the
problem gets more difficult when v increases, both xNES and
SGD spend a significant amount of early evaluations on a plateau
before they can find a right path to descend. This is when the
SGD fails to find a good solution, because on some occasions
they could not find a right path to descend from the early plateau. Overall, xNES converges better than SGD in both speed
and accuracy when v increases. This simple example demonstrates the promise of neuroevolution in solving problems where
SGD gets trapped. We further demonstrate the benefit of transfer
neuroevolution in Figure 5c. For each problem, source distributions were obtained by solving the same equation with v - 0.5
using xNES; here, v is the velocity in the target problem. Then
30 independent tNES runs are performed. The proposed transfer
method successfully speeds up the convergence (quickly escapes
from the plateau) and also helps to achieve a better optimized
solution by exploiting experiential priors. In this example, the
optimized loss achieved by neuroevolution (tNES & xNES) are
at least 2 orders of magnitude lower than SGD. Furthermore,
under a limited computational budget of 50k evaluations, tNES
significantly outperforms xNES (as per the Mann-Whitney rank
test) across all the problems.
B. 2D Projectile Motion Equations

In our second example, transfer neuroevolution is applied to
solve the physics of projectile motion. Assuming a ball is thrown
into the air at specific launch angle a 0, initial velocity vel 0 and
location ^x 0, y 0h, the entire projectile of the flying ball can be
predicted by solving the following ordinary differential equations:

x tt = - R $ x t, t e 60, T @, (15a)
y tt = - R $ y t - g, t e 60, T @, (15b)

subject to the initial conditions x = x 0, x t = vel 0 cos ^a 0 r/180 h,
t = 0, and y = y 0, y t = vel 0 sin ^a 0 r/180 h, t = 0. The solution
to the above differential equations gives the horizontal and vertical position x ^ t h and y ^ t h of the flying ball, within the time
domain t e 60, T @ . Now, let us assume the following initial
conditions: ^x 0, y 0h = ^0, 2 mh, vel 0 = 8 m/s, i.e., the ball is
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IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2021

C. Model Equations of Traveling Waves

In this section, transfer neuroevolution is applied to solve for
several transient PDEs which describe traveling wave(s). The
first model equation-Burgers' equation-can be viewed as a

IEEE Computational Intelligence Magazine - May 2021

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