IEEE Computational Intelligence Magazine - May 2021 - 25

simplified 1D version of the Navier-Stokes equations, for understanding the dynamics of fluids and other important physics [55].
The Burgers' equation has a general form:
	

with an initial condition u ^x, 0h = exp ^- ^2x h2h . Figure 8b
shows a tNES optimized solution for o = 0.01 (with o = 0.006
as source prior), illustrating the nonlinear waveform propagation with both compression and rarefaction effects. A steep gradient can be observed on one side. In Figure 8e, the mean
absolute residual (aggregated residual from differential equation
and initial condition) based on 20 independent tNES and SGD
optimization runs for the same problem are compared. The
overall results show that tNES performs better than SGD at the
initial stages (near t = 0) and near the steep gradient region,
suggesting a more accurate solution from tNES. The results
from multiple optimization runs for problems with o = 0.005,
0.01, 0.05, 0.1 are compared in Figure 8g. The optimized loss

u t + u $ u x = o $ u xx, (16)

which combines the nonlinear advection and diffusion effects.
The diffusion coefficient (or kinematic viscosity) o is an
important parameter for characterizing the fluid properties.
1) Linearized Burgers' Equation
We firstly demonstrate the efficacy of transfer neuroevolution
on a simpler version, linearized Burgers' equation (it can be
viewed as a transient extension of our first example's 1D steady
state convection-diffusion equation):
u t + c $ u x = o $ u xx, x e 6-1.5, 6.5@, t e 60, 5@, (17)

2) Nonlinear Burgers' Equation
Next, the full nonlinear Burgers' equation is considered:

10
0
-10
y (m)

-20
-30

g = 1.6, a0 = 15
g = 1.6, a0 = 45
g = 3.7, a0 = 60
g = 3.7, a0 = 75

-40
-50

g = 9.8, a0 = 15
g = 9.8, a0 = 45
g = 9.8, a0 = 60
g = 9.8, a0 = 75

-60
0

10

20

30

40
x (m)
(a)

50

60

SGD (ADAM)

70

80

t NES

30

30

20

20

10

10
y (m)

with an initial condition u ^x, 0 h =
10exp ^- ^2x h2h . In this setup, a single
waveform is propagated at constant
velocity c = 1. We vary the diffusion
coefficient o = 0, 0.001, 0.01, 0.1 to
form different problems for transfer
neuroevolution to solve; in all cases, the
experiential prior is drawn from the
optimized search distr ibution for
o = 0.02. At o = 0, there is no diffusion
effect so that the initial waveform
maintains its form, while at larger o the
waveform loses its magnitude over time
due to the diffusion effect.
Figure 8a presents a tNES solution for
the case o = 0.01 (with o = 0.02 as
source prior), where the propagation of
the wave can be seen. In Figure 8d, the
mean absolute residual (aggregated residual from differential equation and initial
condition) for this case based on 20 tNES
and SGD optimization runs are compared. On average, SGD gives higher
residual than tNES near the diffusive wave
region. Higher SGD residual is also
observed in capturing the initial condition. The distributions of optimized loss
given by tNES, xNES and SGD are compared in Figure 8g. The optimized loss
values given by SGD and xNES are in a
similar range. By leveraging the experiential priors, tNES gives (1-2 orders of magnitude) better optimized loss than SGD
and xNES.

y (m)

	

u t + u $ u x = o $ u xx, x e 6-1.5, 2@, t e 60, 2@ (18)

	

0
-10

Ground Truth
t NES

0

-10
Ground Truth
SGD (ADAM)

-20
0

10

20

-20

30 40
x (m)

50

60

0

10

20

30 40
x (m)

50

60

(b)

FIGURE 6 (a) The projectiles under the effect of Earth ( g = 9.8, t = 1.2), Mars ( g = 3.7, no drag
effect) and Moon ( g = 1.6, no drag effect) are respectively solved for t = 0-2 s, t = 0-4.5 s and
t = 0-10 s. (b) All individual solutions given by 20 SGD and 40 tNES independent optimization
runs for one particular scenario ( g = 1.6, a 0 = 45, t = 0-10 s) are plotted and compared against
the ground truth. The SGD runs lead to three different groups of projectiles, and deviations are
observed within the two main groups. By knowledge transfer from a relevant prior (source distribution obtained by solving the g = 3.7, at the same a 0), all tNES runs result in an indistinguishable projectile that overlaps almost exactly with the ground truth.

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