COSMOS: Soft Mechanism Mixtures with Verifiable Routing for Long-Horizon PDE Forecasting

Anupam Rawat, Manikandan Padmanaban, Jagabondhu Hazra

Neural operators offer an efficient alternative to classical PDE solvers, but most learn a monolithic map per equation family and discretization. Real systems are compositional: transport, diffusion, wave, and reaction processes can act simultaneously. Existing mixture-of-experts operators typically use sparse top-$K$ routing, although concurrent physics is naturally a blend rather than a discrete choice. We propose COSMOS (Cooperative Operator Specialists with Mechanism-level Operator Soft-routing), a soft mechanism-mixture neural operator. Four process-biased specialists remain active at every step and are continuously mixed by a learned gate, with their features fused by a small network. Specialists share a coarse latent grid, while a zero-initialized full-resolution residual restores detail lost through the bottleneck. We also introduce an operator-splitting compositional benchmark with known mixture weights $w^\star$ per trajectory. Against family-tuned FNO under 20-step rollouts, an initial 3-seed evaluation suggested gains on diffusion--reaction, parity on Navier--Stokes, and weaker shallow-water performance. An 11-seed audit showed that the diffusion--reaction gain was…