  In this paper, we introduce LInK, a novel framework that integrates
contrastive learning of performance and design space with optimization
techniques for solving complex inverse problems in engineering design with
discrete and continuous variables. We focus on the path synthesis problem for
planar linkage mechanisms. By leveraging a multi-modal and
transformation-invariant contrastive learning framework, LInK learns a joint
representation that captures complex physics and design representations of
mechanisms, enabling rapid retrieval from a vast dataset of over 10 million
mechanisms. This approach improves precision through the warm start of a
hierarchical unconstrained nonlinear optimization algorithm, combining the
robustness of traditional optimization with the speed and adaptability of
modern deep learning methods. Our results on an existing benchmark demonstrate
that LInK outperforms existing methods with 28 times less error compared to a
state-of-the-art approach while taking 20 times less time on an existing
benchmark. Moreover, we introduce a significantly more challenging benchmark,
named LINK-ABC, which involves synthesizing linkages that trace the
trajectories of English capital alphabets - an inverse design benchmark task
that existing methods struggle with due to large non-linearities and tiny
feasible space. Our results demonstrate that LInK not only advances the field
of mechanism design but also broadens the applicability of contrastive learning
and optimization to other areas of engineering.
