IEEE Computational Intelligence Magazine - May 2021 - 21

computation of gradient estimates and search
We augment neuroevolution with transfer optimization,
distribution update. The tNES algorithm also
updates the mixture model coefficients as per
boosting search efficiency by reusing experiential
(9) in the transfer mode, before entering the
priors from related source problem instances. This
next iteration. The optimization loop termicapability is enabled via a mixture model-based
nates once the stopping criteria, for example, a
target loss or maximum iterations, is met. The
adaptive transfer method.
algorithm outputs the best found pseudo-offspring (solution) for the target problem as well
as the optimized search distribution, which is added to the
across different problems derived from the same differential
solved problems library as an experiential prior for future use.
equation example is chosen. It is reckoned that the converThere are some tunable parameters in the tNES algorithm; see
gence of SGD (ADAM) is highly sensitive to the learning
Table I. The predetermined transfer plan schedules the transfer to
rate. Hence, a learning plan is adopted to reduce its learntake place after every Tt iterations, and until t max iterations.
ing rate by half on plateaus, until a minimum learning rate
of 1e-6 is reached. This strategy significantly improves
Although it is safe to specify a large t max since the adaptive design
the robustness and performance of SGD by preventing it
will auto-terminate the transfer once it is no longer beneficial,
from being easily trapped in a plateau and also speeding up
setting an appropriate t max can prevent unnecessary computations.
the convergence.
Other tunable parameters include the pseudo-offspring projection threshold, population size, learning rates and utility function.
Like its baseline xNES algorithm, tNES is sensitive to these
A. 1D Steady State Convection-Diffusion Equation
parameters, and the best settings may vary across problems.
The convection-diffusion equation describes a very common
phenomenon where a physical quantity, such as particles, energy, and temperature, are transferred inside a physical system due
V. Experimental Study
to two processes: convection and diffusion (also known as
This section contains empirical demonstrations that neuroevodirectional and non-directional transfer). As a motivating examlution and transfer neuroevolution are noteworthy approaches
ple, a simplified 1D steady state convection-diffusion equation
for solving differential equations. Results on several pedagogical
is solved:
examples representative of real world phenomena: A. 1D steady
state convection-diffusion equation, B. 2D projectile motion equations,
and C. model equations of traveling waves, are presented. In each

v $ u x = k $ u xx, x e 60, L@, (12)
example, some parameter in the differential equation or initial
condition is subject to change, forming multiple related probwith the boundary conditions u = 0, x = 0; u = 1, x = L . We
lems to be solved. We investigate physics-informed neural netare interested in the solution u ^ x h, the temperature in the spaworks to emulate the solution to the differential equations.
tial domain x e 60, L@ . The velocity v, diffusion coefficient k,
The neural network architecture is fixed beforehand, without
and domain boundary L are the problem specific constants.
carrying out exhaustive architecture search. The selected archiIn this special case, the ground truth solution can be analyticaltecture is nevertheless robust enough to produce good
ly derived as:
approximations for different problems derived from the same
1 - exp ^xv/kh
differential equation example. By altering the neural network

u ^xh =
, (13)
1 - exp ^Lv/kh
weights, the differential equation as well as the prescribed
initial and/or boundary conditions must be satisfied. The
resultant global optimization problem can often be very chalwhich is used to verify our optimized PINN solution. In our
lenging even if the differential equation looks simple on
example, a domain boundary of L = 5 and diffusion coefficient
the surface.
of k = 1 is used. We solve the equation for different velocity valThe configurations of the physics-informed neural netues, v = 2, 5, 8, 10. Classical numerical schemes may be able to
works and the optimization algorithms for different examples
solve this differential equation faster, but are notoriously prone
are given in Table II. We compare tNES (instantiation of transfer neuroevolution) and xNES (example of neuroevolution)
with the ADAM (state-of-the-art variant of SGD) algorithm
TABLE I Tuning parameters of tNES.
[51]. A similar setting for all 3 optimizers is used for fairness
" Tt, t max ,: transfer plan
of comparisons. For each loss (fitness) evaluation, the sum of
mean squared residuals from the differential equations as well
m: population size
as the initial and boundary conditions (2) is computed over
r: Mahalanobis distance threshold for pseudo-offspring projection
m randomly sampled collocation points. Similar to the netu k: utility function
work architecture, we do not exhaustively search for the best
lr = " h n, h A , h a ,: learning rate
optimization setting. A sufficiently robust setting that works

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IEEE Computational Intelligence Magazine - May 2021

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