Sample-efficient Surrogate Model for Frequency Response of Linear PDEs using Self-Attentive Complex Polynomials
This work addresses the costly design process for physical systems governed by linear PDEs, such as antennas, by providing a sample-efficient surrogate model, though it is incremental as it builds on existing neural network and search techniques.
The paper tackled the problem of slow and expensive explicit simulation of linear PDEs for physical system design by proving a parametric form in the Fourier domain (CZP framework) and applying it to antenna design, resulting in a surrogate model that outperformed baselines by 10-25% in test loss and achieved 33% greater success in finding verifiable designs.
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.