  The article is aimed to address a mutually boosting use of asymptotic
analysis and machine learning, for fast stiffness design of configurations
infilled with smoothly-varying graded microstructures. The discussion is
conducted in the context of an improved asymptotic-homogenisation topology
optimisation (AHTO plus) framework. It is demonstrated that on one hand,
machine learning can be employed to represent the key but implicit
inter-relationships revealed from asymptotic analysis, and the evaluations of
the homogenised quantities, as well as the sensitivities of the design
variables, become quite efficient. On the other hand, the use of asymptotic
analysis identifies a computational routine for data acquisition, thus the
training data here are inexhaustible in theory. Key issues regarding
integration of the two methods, such as ensuring the positive definiteness of
the homogenised elasticity tensor represented with neural networks, are also
discussed. The accuracies and the efficiencies of the present scheme are
numerically demonstrated. For two-dimensional optimisation, it takes the
present algorithm roughly 300 seconds on a standard desktop computer, and this
qualifies the present scheme as one of the most efficient algorithms used for
the compliance optimisation of configurations infilled with complex
microstructures.
