Bridging Numerical PDE Solvers and Neural Emulators for Faster Simulation

This thesis challenges the conventional view of neural emulators as mere opaque alternatives to traditional numerical solvers for Partial Differential Equations (PDEs). Instead, it posits a strong, reciprocal relationship between these two paradigms, suggesting that neural architectures often mirror classical discretizations and their errors are amenable to similar spectral analysis. The core argument is that understanding this interplay allows for significant advancements in both fields, particularly in accelerating scientific and engineering simulations where the computational cost of traditional solvers is a bottleneck. The work disentangles the multiple roles a solver plays in the emulator learning pipeline and uses mode-wise Fourier analysis as a common language to interpret solver errors, architectural inductive biases, and training objectives. This approach culminates in three key contributions: APEBench, a benchmarking suite for autoregressive neural PDE emulators; an investigation into the effects of unconverged solvers on surrogate training; and an analysis of how numerical errors and architectural biases influence neural emulator superiority.