FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates
This work addresses the challenge of ensuring physical constraints like conservation and bounds in data-driven PDE surrogates, which is crucial for reliable simulations in fields like fluid dynamics and materials science, though it appears incremental as it builds on existing autoregressive and lattice Boltzmann-inspired approaches.
The authors tackled the problem of unstable long-horizon rollouts in data-driven PDE simulation by introducing FluxNet, a framework that learns capacity-constrained local transport operators to enforce conservation and state bounds structurally, resulting in improved rollout stability and physical consistency in experiments on shallow-water equations and traffic flow, and enabling large time-steps with preserved microstructure evolution in spinodal decomposition.
Autoregressive learning of time-stepping operators offers an effective approach to data-driven PDE simulation on grids. For conservation laws, however, long-horizon rollouts are often destabilized when learned updates violate global conservation and, in many applications, additional state bounds such as nonnegative mass and densities or concentrations constrained to [0,1]. Enforcing these coupled constraints via direct next-state regression remains difficult. We introduce a framework for learning conservative transport operators on regular grids, inspired by lattice Boltzmann-style discrete-velocity transport representations. Instead of predicting the next state, the model outputs local transport operators that update cells through neighborhood exchanges, guaranteeing discrete conservation by construction. For bounded quantities, we parameterize transport within a capacity-constrained feasible set, enforcing bounds structurally rather than by post-hoc clipping. We validate FluxNet on 1D convection-diffusion, 2D shallow water equations, 1D traffic flow, and 2D spinodal decomposition. Experiments on shallow-water equations and traffic flow show improved rollout stability and physical consistency over strong baselines. On phase-field spinodal decomposition, the method enables large time-steps with long-range transport, accelerating simulation while preserving microstructure evolution in both pointwise and statistical measures.