FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates

TL;DR

FluxNet introduces a novel approach for learning local transport operators that inherently conserve quantities and respect bounds, enabling stable, long-horizon PDE simulations across various physical models.

Contribution

The paper proposes a capacity-constrained, conservation-guaranteeing transport operator framework inspired by lattice Boltzmann methods, improving stability and physical fidelity in PDE surrogates.

Findings

Enhanced stability in long-horizon rollouts for conservation laws.

Improved physical consistency over baselines in shallow water and traffic flow.

Accelerated simulations in phase-field models with preserved microstructure.

Abstract

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.