Solving inverse-PDE problems with physics-aware neural networks
(2020)Abstract
We propose a novel composite framework that enables finding unknown fields in
the context of inverse problems for partial differential equations (PDEs). We
blend the high expressibility of deep neural networks as universal function
estimators with the accuracy and reliability of existing numerical algorithms
for partial differential equations. Our design brings together techniques of
computational mathematics, machine learning and pattern recognition under one
umbrella to seamlessly incorporate any domain-specific knowledge and insights
through modeling. The network is explicitly aware of the governing physics
through a hard-coded PDE solver stage; this subsequently focuses the
computational load to only the discovery of the hidden fields. In addition,
techniques of pattern recognition and surface reconstruction are used to
further represent the unknown fields in a straightforward fashion. Most
importantly, our inverse-PDE solver allows effortless integration of
domain-specific knowledge about the physics of underlying fields, such as
symmetries and proper basis functions. We call this approach Blended
Inverse-PDE Networks (hereby dubbed BIPDE-Nets) and demonstrate its
applicability on recovering the variable diffusion coefficient in Poisson
problems in one and two spatial dimensions. We also show that this approach is
robust to noise.
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