Predicting Wave Reflection and Transmission in Heterogeneous Media via Fourier Operator-Based Transformer Modeling

TL;DR

This paper introduces a Fourier operator-based transformer model trained on high-fidelity simulations to predict electromagnetic wave behavior at material interfaces, achieving accurate results with errors below 10%.

Contribution

It presents a novel ML surrogate model combining Fourier transforms and vision transformers to efficiently simulate wave interactions in heterogeneous media.

Findings

Model predicts wave reflection and transmission with less than 10% error.

Fourier transforms in latent space improve spectral alignment with data.

Prediction error grows linearly over time, with sharp increase at interfaces.

Abstract

We develop a machine learning (ML) surrogate model to approximate solutions to Maxwell's equations in one dimension, focusing on scenarios involving a material interface that reflects and transmits electro-magnetic waves. Derived from high-fidelity Finite Volume (FV) simulations, our training data includes variations of the initial conditions, as well as variations in one material's speed of light, allowing for the model to learn a range of wave-material interaction behaviors. The ML model autoregressively learns both the physical and frequency embeddings in a vision transformer-based framework. By incorporating Fourier transforms in the latent space, the wave number spectra of the solutions aligns closely with the simulation data. Prediction errors exhibit an approximately linear growth over time with a sharp increase at the material interface. Test results show that the ML solution has adequate relative errors below $10\%$ in over $75$ time step rollouts, despite the presence of the discontinuity and unknown material properties.