  Reaction-diffusion (Turing) systems are fundamental to the formation of
spatial patterns in nature and engineering. These systems are governed by a set
of non-linear partial differential equations containing parameters that
determine the rate of constituent diffusion and reaction. Critically, these
parameters, such as diffusion coefficient, heavily influence the mode and type
of the final pattern, and quantitative characterization and knowledge of these
parameters can aid in bio-mimetic design or understanding of real-world
systems. However, the use of numerical methods to infer these parameters can be
difficult and computationally expensive. Typically, adjoint solvers may be
used, but they are frequently unstable for very non-linear systems.
Alternatively, massive amounts of iterative forward simulations are used to
find the best match, but this is extremely effortful. Recently,
physics-informed neural networks have been proposed as a means for data-driven
discovery of partial differential equations, and have seen success in various
applications. Thus, we investigate the use of physics-informed neural networks
as a tool to infer key parameters in reaction-diffusion systems in the
steady-state for scientific discovery or design. Our proof-of-concept results
show that the method is able to infer parameters for different pattern modes
and types with errors of less than 10\%. In addition, the stochastic nature of
this method can be exploited to provide multiple parameter alternatives to the
desired pattern, highlighting the versatility of this method for bio-mimetic
design. This work thus demonstrates the utility of physics-informed neural
networks for inverse parameter inference of reaction-diffusion systems to
enhance scientific discovery and design.
