Test-Time Accuracy-Cost Control in Neural 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. We introduce \textbf{Recurrent-Depth Simulator} (\textbf{RecurrSim}) an architecture-agnostic framework that enables explicit test-time control over accuracy-cost trade-offs in neural simulators without requiring retraining or architectural redesign. By setting the number of recurrent iterations $K$, 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 demonstrate RecurrSim's effectiveness across fluid dynamics benchmarks (Burgers, Korteweg-De Vries, Kuramoto-Sivashinsky), achieving physically faithful simulations over long horizons even in low-compute settings. On high-dimensional 3D compressible Navier-Stokes simulations with 262k points, a 0.8B parameter RecurrFNO outperforms 1.6B parameter baselines while using 13.5\% less training memory. RecurrSim consistently delivers superior accuracy-cost trade-offs compared to alternative adaptive-compute models, including Deep Equilibrium and diffusion-based approaches. We further validate broad architectural compatibility: RecurrViT reduces error accumulation by 90\% compared to standard Vision Transformers on Active Matter, while RecurrUPT matches UPT performance on ShapeNet-Car using 44\% fewer parameters.