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Alias-Free Mamba Neural Operator
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Proposes MambaNO, a linear-complexity neural operator that combines a Mamba state-space "integral" (global scan) with convolution integration (local) within an alias-free, band-limited U-shaped architecture, and reports state-of-the-art PDE benchmark errors.
Problem
Neural operators for PDEs need both global and local function information while staying computationally cheap and respecting alias-free (continuous-discrete equivalence) behavior; existing kernels (GNO, FNO, Transformer NO) are costly or aliasing-prone, and CNO lacks global context.
Approach
Reinterprets the kernel integral over (-inf, x) as a state-space model so that Mamba-style scanning gives a global O(N) integral, combined with a local convolution integral and band-limited activations/up-/down-sampling to keep the operator representation-equivalent.
Key results
- On Poisson in-distribution, MambaNO reaches 0.17% relative median L1 error versus 0.31% for CNO and 7.35% for FNO.
- On Navier-Stokes out-of-distribution, MambaNO achieves 5.95% versus 10.94% for CNO.
- Ablation: pure convolution integration increases test error by 14% and pure Mamba integration by 32% relative to the combined model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| combine global Mamba integration with local convolution integration | MambaNO · Poisson Equation (In) | relative median L1 test error: 0.31% -> 0.17% | Helped |
| combine global Mamba integration with local convolution integration | MambaNO · Navier-Stokes Equation (Out) | relative median L1 test error: 10.94% -> 5.95% | Helped |
| combine global Mamba integration with local convolution integration | MambaNO · Wave Equation (Out) | relative median L1 test error: 1.29% -> 1.22% | Helped |
| combine global Mamba integration with local convolution integration | MambaNO · Smooth Transport (Out) | relative median L1 test error: 0.35% -> 0.34% | Helped |
| combine global Mamba integration with local convolution integration | MambaNO · Discontinuous Transport (Out) | relative median L1 test error: 1.31% -> 1.21% | Helped |
| combine global Mamba integration with local convolution integration | MambaNO · Allen-Cahn Equation (In) | relative median L1 test error: 0.91% -> 0.72% | Helped |
| combine global Mamba integration with local convolution integration | MambaNO · Darcy Flow (Out) | relative median L1 test error: 0.50% -> 0.44% | Helped |
| combine global Mamba integration with local convolution integration | MambaNO · Compressible Euler (Out) | relative median L1 test error: 0.63% -> 0.61% | Helped |
| replace Mamba integration with convolution integration (ablation) | MambaNO · Navier-Stokes Equation | test errors: 1.00 × -> 1.14 × | Hurt |
| remove convolution integration (ablation) | MambaNO · Navier-Stokes Equation | test errors: 1.00 × -> 1.32 × | Hurt |
Surprising findings
- Pure Mamba integration performs worse than the combined model by 32%, indicating local convolution integration is necessary alongside global scanning.
Limitations
- Only evaluated on 2D Cartesian domains; extension to 3D is computationally heavy and increases parameters.
- Global integration increases parameter count in higher dimensions.
Experimental setup
Datasets: Representative PDE Benchmarks (Poisson, Wave, Transport, Allen-Cahn, Navier-Stokes, Darcy, Compressible Euler)
Benchmarks: Poisson Wave Smooth Transport Discontinuous Transport Allen-Cahn Navier-Stokes Darcy Flow Compressible Euler
Metrics: relative median L1 test error inference time parameter count FLOPs
Baselines: GT U-Net ResNet DeepONet FNO CNO
Builds on: Mamba Fourier Neural Operator Convolutional Neural Operator alias-free framework Galerkin Transformer
Closest papers by meaning
- Latent Mamba Operator for Partial Differential Equations (ICML 2025 · similarity 0.91)
- MgNO: Efficient Parameterization of Linear Operators via Multigrid (ICLR 2024 · similarity 0.86)
- Latent Neural Operator for Solving Forward and Inverse PDE Problems (NeurIPS 2024 · similarity 0.85)
- Amortized Fourier Neural Operators (NeurIPS 2024 · similarity 0.85)
- Component Fourier Neural Operator for Singularly Perturbed Differential Equations (AAAI 2024 · similarity 0.85)
- Optimal deep learning of holomorphic operators between Banach spaces (NeurIPS 2024 · similarity 0.85)
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