Test-Time Control Over Accuracy-Cost Trade-Offs in Neural Physics Simulators via Recurrent Depth

Accuracy-cost trade-offs are a fundamental aspect of scientific computing. Classical numerical methods inherently offer such a trade-off: increasing resolution, order, or precision typically yields more accurate solutions at higher computational cost. Inspired by adaptive-compute language models, we introduce the Recurrent-Depth Simulator (RDS), an architecture-agnostic, plug-and-play framework that enables explicit test-time control over accuracy-cost trade-offs. By setting the number of recurrent steps $K$, users can generate fast, less-accurate simulations for exploratory runs or real-time control loops, or increase $K$ for more-accurate simulations in critical applications or offline studies. We validate RDS on several fluid-dynamics benchmarks, including Burgers, Korteweg-De Vries, and Kuramoto Sivashinsky, and demonstrate 1) physically faithful simulations over long horizons, even in low compute settings; 2) superior accuracy-cost trade-offs compared to alternative adaptive-compute models, including Deep Equilibrium and diffusion-based models. We further validate the recurrent-depth simulator on the challenging task generating three-dimensional turbulent compressible Navier-Stokes simulations, where we demonstrate a 0.8B parameter model with a single recurrent-depth Fourier layer attains lower mean-squared error than a 1.6B parameter counterpart with six Fourier layers, while matching computational resources and utilizing 13.5\% less memory during training.