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

2026-02-02

AI Summary

This work proposes a data-driven surrogate modeling framework for partial differential equations (PDEs) that addresses instability in long-time integration caused by violations of conservation laws or physical bounds—such as non-negativity or concentration constraints in [0,1]. Inspired by the lattice Boltzmann method, the approach introduces a local transport operator that inherently enforces discrete conservation through state exchange among neighboring cells. By parameterizing the transport process within a capacity-constrained feasible set, the model structurally respects bound constraints without requiring post-hoc clipping. Coupled with an autoregressive architecture and a discrete velocity representation, the method enables large time steps and captures long-range transport dynamics. Demonstrated on shallow water waves, traffic flow, and spinodal decomposition phase-field simulations, the framework significantly enhances rollout stability, physical consistency, and fidelity in preserving both pointwise and statistical characteristics of fine-scale structures.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

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.

Problem

• conservation laws
• bounded PDE
• data-driven simulation
• long-horizon stability
• physical constraints

Innovation

• conservative transport operators
• capacity-constrained learning
• PDE surrogates
• boundedness enforcement
• discrete conservation