  Linear Partial Differential Equations (PDEs) govern the spatial-temporal
dynamics of physical systems that are essential to building modern technology.
When working with linear PDEs, designing a physical system for a specific
outcome is difficult and costly due to slow and expensive explicit simulation
of PDEs and the highly nonlinear relationship between a system's configuration
and its behavior. In this work, we prove a parametric form that certain
physical quantities in the Fourier domain must obey in linear PDEs, named the
CZP (Constant-Zeros-Poles) framework. Applying CZP to antenna design, an
industrial application using linear PDEs (i.e., Maxwell's equations), we derive
a sample-efficient parametric surrogate model that directly predicts its
scattering coefficients without explicit numerical PDE simulation. Combined
with a novel image-based antenna representation and an attention-based neural
network architecture, CZP outperforms baselines by 10% to 25% in terms of test
loss and also is able to find 2D antenna designs verifiable by commercial
software with $33\%$ greater success than baselines, when coupled with
sequential search techniques like reinforcement learning.
