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

arXiv 2602.01941 ↗First posted 2 Feb 2026cond-mat.mtrl-sci cs.CE cs.LGcited by 0

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Abstract (arXiv)

Autoregressive learning of time-stepping operators provides an effective approach to data-driven partial differential equation (PDE) simulation, yet for conservation laws, they face a fundamental challenge: learned updates may violate global conservation over long rollouts. For the important subclass of mass-conservation-type equations, the problem is compounded by inherent physical bounds (e.g., nonnegativity or concentrations in [0,1]) whose violation further destabilizes predictions. We introduce FluxNet, which learns cumulative transport amounts representing the total conserved quantity redistributed between each cell and a configurable neighborhood over the full surrogate interval. A conservative update guarantees exact discrete conservation by construction; modular capacity-constrained transport heads (L, U, and D) enforce lower bounds, upper bounds, or near-zero dual-bound violations through architectural design. Unlike flux-rate surrogates that require temporal integration and thus inherit CFL constraints, FluxNet involves no such integration; configurable transport neighborhoods enable large-timestep prediction at full spatial resolution. Ghost cells extend the framework to non-periodic boundaries. Experiments on four benchmarks (1D convection--diffusion, 2D shallow water, 1D traffic flow, 2D Cahn--Hilliard) demonstrate exact conservation, structural bound preservation, architecture modularity, and superior stability over flux-rate surrogates at large temporal strides. The code is publicly available at: https://github.com/Lan-zs/FluxNet.

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References · 21

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
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  11. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
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  1. 2022Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditionscited by 33
  2. 2021One-shot learning for solution operators of partial differential equationscited by 27
  3. 2021Composing Partial Differential Equations with Physics-Aware Neural Networkscited by 21
  4. 2024Graph Neural PDE Solvers with Conservation and Similarity-Equivariancecited by 15
  5. 2021Finite Volume Neural Network: Modeling Subsurface Contaminant Transportcited by 14
  6. 2023Exactly conservative physics-informed neural networks and deep operator networks for dynamical systemscited by 13
  7. 2024Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Lawscited by 10
  8. 2025Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation lawscited by 3
  9. 2025Learning Under Laws: A Constraint-Projected Neural PDE Solver that Eliminates Hallucinationscited by 2

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