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

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.

Figures (9)

• Figure 1: Autoregressive modeling for forecasting long-range dynamics with a series of time snapshots of $\tilde{s}_t = [s_t, s_{t-1}, \dots, s_{t-m+1}]$ at each step rolling out for prediction of $s_{t+1}$.
• Figure 2: Dual-path architecture: Fourier transformer. Overlapping tokenizer is applied prior to the transformations into embeddings of $f_{t}$ and $e_{t}$. After the two embeddings are merged, an overlapping detokenizer is employed to reconstruct the spatial field, mapping the latent representation back to the original physical domain.
• Figure 3: Comparison between the simulated (top), ML predicted (middle), and residual (bottom) results for the electric field in Case 2, at $t=50$ (left column), and $t=100$ (right column).
• Figure 4: Simulated and predicted solutions for Case 1, at two time steps $t=100$ and $t=200$. The dotted line at $x_j=0.5$ is the material interface where the speed of light changes.
• Figure 5: Fourier domain content of the predicted solution for Case 1, at step $t=100$ (left), when the wave packet is crossing the material boundary, and $t=200$ (right) when it has moved away after reflecting/transmitting.