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Linear Partial Differential Equations (PDEs) govern the spatial-temporaldynamics of physical systems that are essential to building modern technology.When working with linear PDEs, designing a physical system for a specificoutcome is difficult and costly due to slow and expensive explicit simulationof PDEs and the highly nonlinear relationship between a system's configurationand its behavior. In this work, we prove a parametric form that certainphysical quantities in the Fourier domain must obey in linear PDEs, named theCZP (Constant-Zeros-Poles) framework. Applying CZP to antenna design, anindustrial application using linear PDEs (i.e., Maxwell's equations), we derivea sample-efficient parametric surrogate model that directly predicts itsscattering coefficients without explicit numerical PDE simulation. Combinedwith a novel image-based antenna representation and an attention-based neuralnetwork architecture, CZP outperforms baselines by 10% to 25% in terms of testloss and also is able to find 2D antenna designs verifiable by commercialsoftware with $33\%$ greater success than baselines, when coupled withsequential search techniques like reinforcement learning.

Sample-efficient Surrogate Model for Frequency Response of Linear PDEs using Self-Attentive Complex Polynomials