BalanceNO: Discovering What to Conserve for Coupled PDE Forecasting

Accurate autoregressive prediction of coupled partial differential equations is essential for fast multiphysics simulation, where spatial transport coexists with interactions that redistribute conserved quantities among fields. Existing surrogates face a critical dilemma: unconstrained regression suffers from severe physical drift over long rollouts, while hard-coded conservation laws are overly rigid, failing to capture the dynamic redistribution of quantities among coupled fields. We introduce BalanceNO, an operator neural framework that reformulates coupled-field evolution as a transport–interaction balance governed by a linear invariant subspace identified from trajectory data. BalanceNO discovers and validates exact balance constraints solely from training trajectories. To strictly enforce them during updates, it integrates telescoping spatial fluxes with nullspace-projected local interactions—preserving field-coupled invariants without freezing individual field totals. By embedding selective conservation directly into its core discrete update, BalanceNO bypasses the need for complete governing equations, seamlessly accommodating weighted conservation laws and partially conservative dynamics. Evaluated across complex spatiotemporal systems combining local transport, global nonlocal interactions, and strongly coupled fields in heterogeneous, three-dimensional, and interface-evolving media, BalanceNO delivers accurate long-horizon rollouts alongside roundoff-level invariant preservation. The code for the experiments can be found in https://anonymous.4open.science/r/BNO-0BB3.