LGNO: A Local-Global Neural Operator for Hyperbolic Conservation Laws

Technical Summary: LGNO: A Local–Global Neural Operator for Hyperbolic Conservation Laws

Problem Statement
Hyperbolic conservation laws govern phenomena in gas dynamics and plasma physics, characterized by solutions that evolve from smooth initial data into sharp gradients, shocks, and contact discontinuities in finite time. Accurately approximating these solutions is fundamentally challenging because successful methods must simultaneously propagate nonlinear waves, resolve localized discontinuities sharply, and remain robust over long-time integration.

Conventional high-order numerical schemes, such as WENO (Weighted Essentially Non-Oscillatory), achieve this by using local reconstruction and nonlinear stabilization. However, these methods incur a trade-off: preserving fine-scale structures requires higher-order reconstructions or finer meshes, which significantly increases computational cost. Conversely, reducing numerical dissipation to preserve structures often leads to spurious oscillations.

Neural operators, such as the Fourier Neural Operator (FNO), offer a data-driven alternative for learning solution operators. While FNOs efficiently capture long-range interactions and smooth dynamics via global spectral convolutions, they struggle with the nonsmooth structures inherent in conservation laws. Existing studies indicate that FNOs tend to smear localized features like shocks due to excessive numerical dissipation and spectral truncation, leading to accuracy deterioration as discontinuity structures become complex.

Methodology
The authors propose the Local–Global Neural Operator (LGNO), a neural architecture designed to learn a one-step discrete flow map ($G_{\Delta t}$) that couples global and local mechanisms to address the dual nature of conservation law solutions.

1. Architecture Design:

   • Global Branch: Built upon the FNO framework, this branch utilizes learnable Fourier multipliers on low-frequency modes to capture coherent large-scale dynamics and long-range interactions efficiently.
   • Local Multiresolution Branch: To resolve sharp localized features (shocks, contact discontinuities), the model incorporates a local branch inspired by U-Net's coarse-to-fine fusion. This branch applies local convolutions on a coarsened grid (via average pooling) and interpolates the result back to the original resolution. This design injects stencil-type locality—a key inductive bias in classical shock-capturing schemes—while maintaining a large effective receptive field at low computational cost.
   • Coupling: The global and local features are combined via a multiplicative interaction operator, allowing the local response to be modulated by global spectral features. The layers employ residual updates to refine latent representations.
   • Boundary Handling: For outflow boundaries, the authors introduce a learned ghost-cell padding operator. This shallow network predicts ghost values from boundary-adjacent interior strips, preventing the artifacts associated with fixed extrapolation or zero-padding in local convolutional stencils.

2. Conservation and Training:

   • Conservative Parameterization: The model predicts the cell-centered increment ($\Phi_\theta$) rather than the flux directly. To ensure exact global conservation of the spatial average (under periodic conditions), the learned increment is projected onto the mean-zero subspace before updating the state.
   • Loss Function: The training objective combines a physical-space prediction term (mean absolute error for scalars, relative $L_1$ for multi-component systems) with a high-frequency spectral penalty. This penalty suppresses spurious oscillations near steep fronts by penalizing errors in the high-frequency Fourier modes.

Key Contributions

1. Novel Architecture: The development of a hybrid architecture that integrates a global spectral branch with a local multiresolution convolutional branch, specifically tailored for the coexistence of smooth waves and nonsmooth discontinuities.
2. Performance Improvement: Extensive benchmarks on one- and two-dimensional problems (Linear Advection, Burgers, Shallow Water, and Euler equations) demonstrate that LGNO consistently outperforms parameter-matched FNO baselines. LGNO reduces one-step relative $L_1$ errors by factors of 2 to 5.
3. Long-Time Rollout Superiority: The most significant finding is that LGNO, when trained only on short-time data generated by a high-order WENO-Z scheme, produces long-time autoregressive rollouts that exhibit lower numerical dissipation than the WENO-Z scheme itself, even when WENO-Z is run on a grid twice as fine (e.g., LGNO on $256^2$ vs. WENO-Z on $512^2$). This occurs because the learned one-step map inherits only the low dissipation of the short-time training data, whereas conventional schemes accumulate dissipation at every step of a long integration.
4. Efficiency: The learned operator is orders of magnitude cheaper to evaluate than the conventional high-resolution schemes it outperforms in terms of structural preservation.

Results
The paper evaluates LGNO against FNO baselines and WENO-Z references across several benchmarks:

• One-Dimensional Problems: LGNO accurately preserves shock positions and ramp structures in Burgers and Euler equations, maintaining lower error accumulation over long horizons compared to FNO, which develops unphysical oscillations and smears discontinuities.
• Two-Dimensional Problems: In periodic and outflow Euler cases, LGNO preserves fine vortical and roll-up structures that are smeared by both the FNO baseline and the high-resolution WENO-Z reference.
• Error Metrics: Table 1 reports that LGNO reduces mean errors by factors ranging from 2.18 (Burgers 1D) to 5.09 (Euler 2D periodic) compared to FNO.
• Visual Evidence: Figures 1 and 13 illustrate that LGNO on a coarse grid retains coherent structures and sharp transitions that are lost in FNO predictions and are only partially preserved (with higher dissipation) in finer-grid WENO-Z simulations.

Significance and Claims
The paper claims that with an appropriate architecture and training objective, learned operators can effectively learn discrete flow maps that control long-time numerical dissipation better than the conventional shock-capturing schemes used to generate the training data.

The authors emphasize that this result is counterintuitive: while the training data is produced by a dissipative scheme, the learned operator does not accumulate dissipation in the same manner over long rollouts. Instead, it learns a flow map that preserves the low-dissipation characteristics of the short-time training data. This suggests that data-driven operators have the potential to surpass the long-time accuracy of the very numerical methods that generate their training sets, offering a pathway for accelerated, high-fidelity simulation of hyperbolic conservation laws.

The work concludes by noting that while the current results are promising, future work is needed to analyze the theoretical approximation and stability properties of the local-global architecture and to extend the method to more complex systems and higher dimensions.